ArticlePDF Available

Abstract and Figures

Closed-loop turbulence control is a critical enabler of aerodynamic drag reduction, lift increase, mixing enhancement, and noise reduction. Current and future applications have epic proportion: cars, trucks, trains, airplanes, wind turbines, medical devices, combustion, chemical reactors, just to name a few. Methods to adaptively adjust open-loop parameters are continually improving toward shorter response times. However, control design for in-time response is challenged by strong nonlinearity, high-dimensionality, and time-delays. Recent advances in the field of model identification and system reduction, coupled with advances in control theory (robust, adaptive, and nonlinear) are driving significant progress in adaptive and in-time closed-loop control of fluid turbulence. In this review, we provide an overview of critical theoretical developments, highlighted by compelling experimental success stories. We also point to challenging open problems and propose potentially disruptive technologies of machine learning and compressive sensing.
Content may be subject to copyright.
Steven L. Brunton
Department of Mechanical Engineering and
eScience Institute,
University of Washington,
Seattle, WA 98195
Bernd R. Noack
Institut PPRIME, CNRS - Universit
ede
Poitiers - ENSMA, UPR 3346,
D
epartement Fluides, Thermique, Combustion,
CEAT,
F-86036 Poitiers Cedex, France
Institut f
ur Str
omungsmechanik,
Technische Universit
at Braunschweig,
D-38108 Braunschweig, Germany
Closed-Loop Turbulence Control:
Progress and Challenges
Closed-loop turbulence control is a critical enabler of aerodynamic drag reduction, lift
increase, mixing enhancement, and noise reduction. Current and future applications
have epic proportion: cars, trucks, trains, airplanes, wind turbines, medical devices, com-
bustion, chemical reactors, just to name a few. Methods to adaptively adjust open-loop
parameters are continually improving toward shorter response times. However, control
design for in-time response is challenged by strong nonlinearity, high-dimensionality,
and time-delays. Recent advances in the field of model identification and system reduc-
tion, coupled with advances in control theory (robust, adaptive, and nonlinear) are driv-
ing significant progress in adaptive and in-time closed-loop control of fluid turbulence.
In this review, we provide an overview of critical theoretical developments, highlighted
by compelling experimental success stories. We also point to challenging open problems
and propose potentially disruptive technologies of machine learning and compressive
sensing. [DOI: 10.1115/1.4031175]
1 Introduction
Taming turbulence for engineering goals is one of the oldest
and most fruitful academic and technological challenges. One of
the earliest examples is the feathers at the tail of an arrow
invented several thousand years ago. These feathers stabilize the
orientation of the arrow and make the trajectory more predictable
and increase its range. Meanwhile, modern turbulence control has
applications of epic proportion. Examples include drag reduction
of road vehicles, airborne transport, ships and submarines, drag
reduction in pipes and air-conditioning systems, lift increase of
airfoils, efficiency increase of harvesting wind and water energy,
of heat transfer and of chemical and combustion processes—just
to name a few examples.
Animal motion has inspired numerous technical advances in en-
gineering flows [1]. The shape of sharks, dolphins, and whales, for
instance, yields a low drag per volume [2]. It is not an accident
that zeppelins and airplanes have similar shapes. Dolphins are
speculated to delay boundary layer transition by a compliant skin.
This form of transition delay has been applied to submarines and
is under active investigation. Under magnification, the skin of
sharks exhibits riblets [3]. Riblets have been found to reduce drag
by up to 11% in the laboratory [4]. In-flight tests of riblets on an
Airbus passenger airplane have reduced fuel consumption by
2–3%. Some sharks decrease drag by ejecting lubricants during
high-speed chases of their prey. A similar skin-friction reduction
is used in oil pipelines: one added polymer per 10
6
oil molecules
reduces the drag by about 40%. Eagles and other birds have five
feathers at the tip of their wings. These feathers increase the lift
by reducing the pressure shortcut between the low pressure upper
side and the higher pressure lower side. Most modern passenger
airplanes have winglets for the same reason.
The environmental benefit of turbulence control can be illus-
trated with an everyday example: automotive transport. Today,
the annual global CO
2
emissions from cars exceed 22 10
9
tons
and are expected to increase by 57% by 2030. A large portion of
this emission is due to aerodynamic drag [5,6]. At a speed of
50 km/h, the aerodynamic drag accounts for 50% of the total re-
sistance reaching 80% at 130 km/h. A drag reduction of around
25% is currently achievable by active flow control [7]. At a speed
of 120 km/h, this would reduce consumption by about 1.8 l and
would reduce CO
2
by almost 2 kg per 100 km. In normal traffic,
the corresponding reductions are 0.15 l in fuel and 0.73 kg in CO
2
.
For Europe, this drag reduction would mean a reduction of
23 10
6
tonnes of CO
2
emission in one year. To mitigate pollu-
tion, the European government imposes strict norms to car manu-
facturers. By 2020, the mean CO
2
emission per vehicle must not
exceed 95 g CO
2
/km. By 2025, the limit is 75 g CO
2
/km.
Among the countless technologies that will benefit from turbu-
lence control, we highlight the potential benefits for energy and
transportation. Increased lift and reduced drag due to separation
control and transition delay would result in increased payloads
and decreased runway requirements for aircraft and improved effi-
ciency in nearly all vehicles. Considering that transportation
accounts for approximately 20% of global energy consumption, a
small improvement would have a dramatic effect [810]. Active
separation control would also improve the safety of cargo trucks
and trains in strong cross-winds [1113]. Hypersonic vehicles
stand to benefit from active control to prevent the undesirable
ejection of flames out of the combustion chamber and subsequent
quenching. Finally, reducing the amount of turbulent fluctuations
on rotor blades would reduce vibration and improve the life of
rotor hubs on wind turbines and rotorcraft.
Strategies to control laminar and turbulent flows are classified
into three categories [14]: aerodynamic shape optimization, pas-
sive, and active control. The first approach for increasing flow per-
formance is the optimization of the aerodynamic shape. Potential
flow theory, invented about 150 years ago, provides a simple
mathematical foundation. Meanwhile, adjoint-based shape optimi-
zation can be numerically performed for the full Navier–Stokes
equations. As a second step, passive actuators may improve the
performance. Such a device represents a small change of the origi-
nal configuration. One example is the turbulators on wings of pas-
senger airplanes to delay separation. Such passive devices may
come with penalty of parasitic drag. An alternative is the active
control devices, such as fluidic vortex generators, which may be
turned on and off but require energy for their operation. One
advantage is a large dynamic bandwidth, e.g., the excitation of
particular frequencies. Active control may be performed in a pre-
determined open-loop manner, e.g., periodic blowing and suction,
independent of the flow state. The largest gains are, of course,
realized in a closed-loop manner when the actuation is informed
by the sensors recording the flow state.
Until 1990 s, manufacturers have often seen active control as a
remedy for a flawed aerodynamic design [15]. Hence, industrial
interest has been correspondingly low. Meanwhile, aerodynamic
design and passive devices are considered as maturely developed
and active control is pursued to further increase the performance,
particularly for off-design conditions. Three trends foster closed-
loop control. First, the power and reliability of actuators and sen-
sors have dramatically increased, while the price is decreasing.
Manuscript received November 12, 2014; final manuscript received July 25,
2015; published online August 26, 2015. Assoc. Editor: J
org Schumacher.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-1Copyright V
C2015 by ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Second, a sophisticated control logic can be performed under real-
world conditions with increasing computer power and the
advancement of mathematical theories. Third, the experimental
demonstrations of the benefits of closed-loop over open-loop forc-
ing have become overwhelming (see, e.g., Refs. [16,17]).
Most literature on closed-loop flow control falls in one of three
categories: stabilization of laminar flow, adaptive control of turbu-
lence, and model-free tuning of control laws. For the first cate-
gory, there exists a mature theory for the stabilization of laminar
flows with in-time closed-loop control. Early experimental exam-
ples are described in Refs. [18,19], while most studies are based
on direct Navier–Stokes (DNS) solutions (see, e.g., Refs. [20,21]).
“In-time” means that the actuation responds on a time-scale much
smaller than the natural time-scale. Most corresponding publica-
tions are based on linearization of the evolution equation. The
employed evolution equation may be a white-box model, e.g.,
DNS discretization, resolving all features of the flows, a gray-box
model, e.g., proper orthogonal decomposition (POD) models, just
describing the coherent structures, or a black-box model, e.g.,
transfer functions, representing only the input–output behavior.
The control logic based on white box models is the most accurate.
Gray-box and black-box models are less accurate but allow
online-capable control solutions for experiments.
The second category is adaptive control of turbulence usually
based on a manipulation of periodic forcing. Most experimental
success stories belong to this class. “Adaptive” means that the
change of the actuation parameter, such as amplitude or fre-
quency, is slow compared to the natural time-scale. The third
group consists of in-time control of turbulent flows, for instance,
by tuning simple laws, e.g., opposition or proportional-integral-
derivative (PID) control [22]. The inherent nonlinearities of turbu-
lence pose a challenge for model-based in-time control.
Closed-loop control requires decisions on the hardware, such as
the kind, number, location, and dynamic bandwidth of actuators
and sensors. Such decisions may be guided by modern adjoint-
based techniques for linearized equations, i.e., for laminar flow.
For turbulence, these decisions are largely guided by engineering
wisdom from the flow control processes and past experiments.
The control laws may be guessed based on the flow phenomenol-
ogy. These aspects will be touched in Sec. 2. The focus of this
review is the control logic sketched in Fig. 1. This logic is based
on model complexity, e.g., model-free approaches (bottom right
of this figure), black-box (left), gray (middle), and white-box
models (right). Experimental flow control solutions require a deli-
cate compromise between simplicity and accuracy. Many plants
benefit from system reduction approaches. This spectrum of mod-
els will be outlined in Sec. 3. The most complete model-based
control design techniques are available for linear models. These
will be described in Sec. 4. The detailed review of linear control
theory may be surprising in a review about control of nonlinear
Fig. 1 Turbulence control roadmap. For details, see text and the respective sections.
050801-2 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
turbulence processes. We emphasize linear control for three rea-
sons. First, it is a beautiful theory showing clearly the effects of
model accuracy and design parameters. Second, the prevention of
turbulence, i.e., transition control, can largely be done based on a
linear theory. Third, even some strongly nonlinear processes may
be tamed with linear control methods, if the nonlinearity is well
understood. Examples of such “reducible” nonlinear models are
provided in Sec. 5. Model-free approaches for “nonreducible”
nonlinear dynamics are reviewed in Sec. 6. In Sec. 7, we summa-
rize good practices of flow control and promising directions of
future research are discussed in Sec. 8.
This review focuses on modeling and control methods for non-
linear dynamics associated with turbulent fluid flows. Theoretical
and computational aspects of optimal and robust (linear) control
are elaborated in Ref. [23]. A large spectrum of linear system
approaches is presented in Refs. [20,21,24], and periodic excita-
tion is reviewed in Ref. [25]. We illustrate the performance of the
methods by referring to select successful closed-loop control stud-
ies. Preference is given to academic studies for simple geometries.
There are additional books and reviews in the related fields of
flow control for wall flows [2628], for turbulent mixing [2931],
for turbulent jets [32], for combustion [33], for cavity flows
[21,34], for bluff bodies [35], and for actuators [36].
2 Turbulence Control Problem
Turbulence control comprises decisions on the flow control
plant, on the cost functional, on the actuation and sensing, and,
last but not least, on the control logic. The control logic serves to
minimize the cost functional under given constraints. For linear
dynamics, there exist adjoint-based methods for the choice of the
actuators [21] and the sensors [37]. For turbulent flows, the appli-
cation of these methods is limited and the choice is generally
based on experience and engineering wisdom. In this section, we
provide heuristics for turbulence control decisions. Thereafter, a
mathematical foundation of the control logic is elaborated.
2.1 The Flow Control Plant and Associated Goals. In the
sequel, flow is assumed to be within or around a steady boundary
with small unsteady actuators and sensors. Academic flow control
configurations strive at geometric simplicity for enhanced repro-
ducibility and for “clean” understandable physical mechanisms.
Examples include free shear flows from a bluff body, a mixing
layer or a jet and wall-bounded flows in a channel or over a flat
plate. Cavity noise, suppression of aeroelastic oscillation, and
flame-holder combustion serve as examples for multiphysics
flows.
Configurations of industrial importance tend to be geometri-
cally far more complex, such as the flow around a car, truck, train,
airplane, or wind-turbine. Internal flows in pipes, diffusers, com-
bustors, mixers, air-conditioning systems, and buildings are fur-
ther examples. Each of these configurations could profit from
closed-loop control and the range of potential applications has
epic proportion (see Fig. 2). Yet, many flows with complex geo-
metries can locally be approximated by the above-mentioned aca-
demic configurations. Thus, a working flow control experience on
simple test cases is advantageous in the design of efficient control
for industrial purposes.
The aerodynamic performance of transport vehicles and wind-
turbines is based on a force optimization, such as drag reduction,
lift increase, or reduction of fluctuations to prevent early material
fatigue. Examples are transportation trucks (PACCAR), airplane
wings, wind turbine blades, helicopter rotor hubs, and reduction in
structural loads. Combustors, heat exchangers, and chemical
mixers profit from mixing enhancement. Noise reduction is a com-
mon request for greener transport systems. Many of the above
applications can be idealized to aim at the stabilization of an
unstable fixed point or periodic orbit, which we may refer to as
instability suppression. Some internal flows require an improved
destabilization, such as the mixing enhancement in a combustor.
Most control objectives can be formalized in a rigorous mathe-
matical manner. Drag reduction, for instance, reduces by defini-
tion the necessary propulsion or towing power. The optimal drag
reduction may be defined by the maximum energetic benefit, i.e.,
the saving in towing power subtracting the investment in actuation
power. Note that any well-defined control problem requires a
penalization of the actuation power.
Many experimental studies demonstrate that the control is
effective for a well-defined operating condition in a noise-free
wind-tunnel environment. Engineering applications require addi-
tional effectiveness for the intended operating envelope, including
various oncoming velocities, a range of angles of attack, free-
stream turbulence, just to provide few examples. Ideally, robust-
ness of the control is included upfront in an objective comprising
a range of operating conditions, like in linear H
1
control (see
Sec. 4.4). At minimum, the level of robustness needs to be
assessed after a single point optimization of actuation. An under-
standing of the turbulence control mechanisms provides a first
hint on the expectable level of robustness.
2.2 Linear Dynamics. Turbulence is known as the last
unsolved problem of classical mechanics, largely because the
effects of nonlinearity are next to impossible to predict from first
principles. Hence, turbulence control can be seen as an even more
herculean nonlinear problem as not only the unforced state needs
to be predicted but also the effect of a small actuation. Fortu-
nately, there exist a number of configurations for which a linear
dynamics has been shown to be a good working assumption.
Examples include the following cases:
Transition delay. The transition of a laminar into a turbulent
boundary layer is associated with a dramatic rise of skin fric-
tion. Hence, engineering applications include transition delay
with closed-loop control. The laminar state may still be stabi-
lized based on a linearized model. Evidently, stabilization of
a laminar flow has benefits for numerous other
configurations.
Drag reduction in wall turbulence. At high Reynolds num-
bers, active control at the wall does not have the authority to
stabilize the laminar boundary layer. Yet, up to 11% drag
reduction can be achieved with stationary riblets which miti-
gate sweeps in the viscous sublayer [4]. Over 20% drag
reduction can be obtained with linear active control [40].
Arguably, linear control is applicable because the sweep pre-
vention in the viscous sublayer is an effectively laminar pro-
cess, such as transition control.
The in-time actuation response to large-scale coherent struc-
tures may be described a linear model—extending the exam-
ples of drag reduction in wall turbulence. Physically, such a
locally linear model may be derived under similar conditions
as URANS simulations, i.e., if the effect of the unresolved
stochastic velocity component on the dynamically resolved
coherent structures is roughly represented by a temporally
constant eddy viscosity. An example is the mean-field model
for oscillatory fluctuations of turbulence (see Sec. 5.3).
Adaptive control may be subject to a limited linear control.
For instance, the change of cost function may respond line-
arly to small changes of the amplitude and frequency of peri-
odic forcing. This is an implicit working assumption of
extremum seeking control (see Sec. 6.2). Thus, tracking may
be based on locally linear dynamics.
Another recently discovered example of linear dynamics is
the ensemble-averaged actuation response of a turbulent
shear flow [41]. The practical relevance of this observation
still needs to be explored. Studies of forced nonlinear chaotic
systems indicate that the ensemble-averaged effect of a
Heaviside actuation may be described by linear system while
the amplitude dependency is far from linear [42]. Moreover,
the ensemble-averaged response may constitute a small
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-3
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
portion of the fluctuation energy and may not be very rele-
vant for the control goal.
2.3 Turbulence Control Mechanisms. In the following,
common principles of turbulence control are outlined. A control
principle is understood as a phenomenological rationale based on
a simplified physical actuation mechanism. A simple principle has
a low-dimensional plant mimicking relevant aspects of turbulence
control, including linear and nonlinear dynamics.
We start with stabilizing control. A very simple example is
opposition control. Let
da
dt ¼aþb
be a plant with the unstable fixed point a¼0 and actuation b.
Evidently, the control law b¼2awill “oppose” the natural
evolution and stabilize the fixed point. A number of flow control
configurations mimic this behavior. Let us consider
Tollmien–Schlichting (TS) waves over a wall. The wall shall have
a membrane of which the vertical motion can be controlled. A
positive or negative wall-normal velocity fluctuation of a TS wave
can be counteracted by a membrane which moves in the opposite
direction [18]. Similarly, the transverse centerline motion of a
channel flow may be damped by blowing and sucking at opposite
sides of the channel wall in order to counteract this fluctuation.
Thus, an unstable 2D channel flow may be stabilized. Another
example of this category is skin friction reduction. Skin friction is
known to increase with sweeps and ejections, both associated with
wall-normal velocity fluctuation. A simple opposition control
scheme records this velocity fluctuation 10 plus units away from
the wall and opposes this motion by local suction or blowing [43].
A slightly more complex principle is phasor control based on
an oscillatory process. A simple prototypic dynamical system
reads
da1
dt ¼0:1a1a2;da2
dt ¼0:1a2þa1þb
Evidently, the fixed point a
1
¼a
2
¼0 is unstable with respect to
an oscillatory instability with unit frequency and growth-rate of
0.1. The control law b¼0.4a
2
can be seen to stabilize this fixed
Fig. 2 Applications of closed-loop turbulence control: (a) homogeneous grid turbulence (Reproduced with permission from
T. Corke and H. Nagib.); (b) turbulent jet from Bradshaw et al. [38]; (c) Karman vortex street behind a mountain, photo by Bob
Cahalan, NASA GSFC; (d) coherent structures in a mixing layer from Brown and Roshko [39]; (e) thunderstorm; (f) automobile
in a wind tunnel, photo by Robert G. Bulmahn; (g) high-speed train; (h) cargo ship; (i) passenger jet; (j) Blue Angles fighter
jets; (k) automobile engine; (l) turbo jet engine; (m) aircraft engines; (n) wind turbines; (o) heat exchanger flow; (p) rotating
mixer; (q) air conditioner; (r) chocolate mixing; and (s) total artificial heart. Images (e) and (g)–(n) are from the website.
1
Images (c), (f), and (q)–(s) are from the website.
2
Images (o) and (p) were made using the COMSOL Multiphysics
V
R
software and
are provided courtesy of COMSOL.
1
http://pixabay.com
2
https://commons.wikimedia.org
050801-4 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
point by reducing the fluctuation energy with targeted actuation at
the correct phase. Phasor control can also be considered as an
opposition control with respect to the amplitude r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2
1þa2
2
p.
The actuation reduces this amplitude at phases where it is effec-
tive. The corresponding energy-based control design will be ela-
borated in Sec. 5.2. A large class of control laws for suppression
of oscillation may be written in the form
b¼kr cosð/bÞ(1)
where /is the flow phase, bis the phase lag, and kis the gain. In
the given example, /is defined by the polar coordinates a1þia2
¼rexpði/Þ;b¼90 deg, and k¼4. Phasor control (1) is an
analogy for virtually any successful stabilization of an oscillatory
flow, regardless how the control law is derived. The gains may be
constant for assumed linear dynamics or base-flow/energy depend-
ent for weakly nonlinear dynamics. Examples are the stabilization
of a laminar cylinder wake with zero net mass flux actuators [44],
the stabilization of a turbulent wake in an experiment [45], and the
suppression of cavity noise with local forcing [34,46].
A more complex case involves two or few nonlinearly coupled
“clock-works” based on the principle of constructive or destruc-
tive frequency cross-talk. One example is high-frequency forcing
which mitigates a target instability [47,48]. Lower frequency forc-
ing may also serve the same purpose [45,49]. Section 5.4 offers a
least-order dynamical system for such frequency cross-talk. In the
discussed examples, the actuation has a destructive frequency
cross-talk with the target instability. However, the frequency
interaction may also be constructive. In a mixing layer, the excita-
tion of Kelvin–Helmholtz vortices leads to earlier vortex pairing,
i.e., to the excitation of half the actuation frequency. The
Kelvin–Helmholtz instability—and thus the vortex pairing—may
be mitigated by the excitation of higher frequencies. Without a
dynamic model, a careful observation of the flow with respect to
period forcing may provide insight into effective control strat-
egies. A sufficiently strong periodic forcing at the right frequency
may mitigate the target frequency. The loop may be closed on a
long time scale to tune the actuation amplitude to the minimal
level. Alternatively, in-time control may destabilize the corre-
sponding oscillator which mitigates the target instability.
The present mechanisms are helpful concepts which can
explain many results from the literature and can potentially guide
new experiments. Yet, the binary categorization in stabilizing and
destabilizing control is an over-simplification. For many control
goals, like jet noise reduction, the enabling mechanisms are far
from being understood.
In the case of broadband turbulence, no generic simple recipes for
the control law can be offered. Yet, we present a highly promising
machine-learning strategy in Sec. 6. Figure 3summarizes all dis-
cussed control principles with associated methods. The heuristics for
closed-loop turbulence presents a tour de force through the control
strategies and methods, which are elaborated throughout this review.
2.4 Actuators and Sensors. The choice of the actuators and
sensors, their number, location, frequency range, and amplitude
level has a decisive effect on the maximum performance of the
control logic. Actuators may include zero-net mass flux actuators
[5052], piezo-electric actuators [53,54], Festo-valves (intermit-
tent blowing), synthetic jets [5558], plasma actuators [5965],
microelectromechanical systems (MEMS) [6672], and roughness
elements on the wall. Excellent overviews are presented in Refs.
[36,55]. Similarly, sensors may measure velocity, pressure, skin-
friction, and temperature in various frequency resolutions. These are
described in the textbooks of Experimental Fluid Mechanics.
Until this day, the choice of the actuators and sensors in experi-
ments is based on engineering experience and on the hypothetical
actuation mechanism which shall be exploited. For instance, the
sensors are placed before the actuator if the actuation shall coun-
teract upstream perturbations. The sensors are placed downstream
of the actuator, if the actuation mechanism exploits the excited
structures. For opposition control, sensors and actuators should be
at a similar location. For phasor control, it is important that the
actuators are at a high-receptivity point for the oscillatory instabil-
ity, e.g., a point of geometric separation where the sensors can
measure a clean oscillation. The number of rules could easily be
extended. For linear dynamics, sophisticated mathematical meth-
ods have been developed for actuator and sensor placement,
although optimal placement remains elusive. For nonlinear
dynamics, heuristic methods for the optimization of sensor place-
ment exist [21]. In this survey, we focus on the control logic.
2.5 Achievable Performance. An important, yet challenging
question in turbulence control is the achievable performance. Evi-
dently, any investment in improving a control strategy may be
measured in terms of the achievable additional performance. We
constrain the discussion to aerodynamic problems. The minimum
skin friction in a channel flow is conjectured to be associated with
the steady Poiseuille profile. Similarly, the minimum drag of a
cylinder wake may also be conjectured to be associated with the
steady solution. In fact, most drag reduction strategies are formu-
lated as minimization of the fluctuation energy. Yet, control stud-
ies of the cylinder wake exhibit two other potentially desirable
flow states. One is the potential solution which is approximated in
compliant wall actuation by Wu et al. [73]. Interestingly, Choi et
al. [35] reported a better performing optimal control when the cost
function for drag reduction contains the difference between the
controlled flow and potential solution as opposed to the drag itself.
Another solution is closer to the Kirchhoff solution and is
achieved by high-frequency forcing [74]. In fact, the drag of high-
frequency forcing may even be lower than for the completely
Fig. 3 Heuristics of turbulence control. Here, sare the sensor
signals and bare the actuation signals.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-5
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
frequency, it is often called a worst-case scenario norm; therefore,
minimizing the infinity norm provides robustness to worst-case
exogenous inputs.
If we let Pw!Jdenote the transfer function from wto J; then,
the goal of H1control is to construct a controller to minimize the
infinity norm: minkPw!Jk1. This is typically difficult, and no
analytic closed-form solution exists for the optimal controller in
general. However, there are relatively efficient iterative methods
to find a controller such that kPw!Jk1<c, as described in Ref.
[164]. There are numerous conditions and caveats that describe
when this method can be used. In addition, there are computation-
ally efficient algorithms implemented in both MATLAB and PYTHON,
and these methods require relatively low overhead from the user.
Selecting the cost function Jto meet design specifications is a
critically important part of robust control design. Considerations,
such as disturbance rejection, noise attenuation, controller band-
width, and actuation cost, may be accounted for by a weighted
sum of the transfer functions S;T, and KS. In the mixed sensitivity
control problem, various weighting transfer function are used to
balance the relative importance of these considerations at various
frequency ranges. For instance, we may weight Sby a low-pass
filter and KS by a high-pass filter, so that disturbance rejection at
low frequency is promoted and control response at high-frequency
is discouraged. A general cost function may consist of three
weighting filters Fkmultiplying S,T, and KS
F1S
F2T
F3KS
2
43
5
1
Another possible robust control design is called H1loop-
shaping. This procedure may be more straightforward than mixed
sensitivity synthesis for many problems. The method consists of
two major steps. First, a desired open-loop transfer function is
specified based on performance goals and classical control design.
Input and output compensators are constructed to transform the
open-loop system to the desired loop shape. Second, the shaped
loop is made robust with respect to a large class of model uncer-
tainty. Indeed, the procedure of H1loop shaping allows the user
to design an ideal controller to meet performance specifications,
such as rise-time, bandwidth, and settling-time. Typically, a loop
shape should have large gain at low frequency to guarantee accu-
rate reference tracking and slow disturbance rejection, low gain at
high frequencies to attenuate sensor noise, and a cross-over
frequency that ensures desirable bandwidth. The loop transfer
function is then robustified so that there are improved gain and
phase margins.
H2control has been an extremely popular control paradigm
because of its simple mathematical formulation and its tunability
by user input. The advantages of H1control are increasingly real-
ized in flow control, eminent examples being the collaborative
research centers (Sfb 557 and Sfb 1029) lead by King [16,17,175].
Additionally, there are numerous consumer software solutions
that make implementation relatively straightforward. In MATLAB,
mixed sensitivity is accomplished using the mixsyn command
in the robust control toolbox. Similarly, loop-shaping is accom-
plished using the loopsyn command in the robust control
toolbox.
4.4.3 Fundamental Limitations With Implications for Turbulence
Control. As discussed above, we want to minimize the peaks of S
and T. Some peakedness is inevitable, and there are certain system
characteristics that significantly limit performance and robustness.
Most notably, time-delays and right-half plane zeros of the open-
loop system will limit the effective control bandwidth and will
increase the attainable lower-bound for peaks of Sand T. This
contributes to both degrading performance and decreasing
robustness.
Similarly, a plant will suffer from robust performance limita-
tions if the number of poles exceeds the number of zeros by more
than two. These fundamental limitations are quantified in the
waterbed integrals, which are so named because if you push a
waterbed down in one location, it must rise in an another. Thus,
there are limits to how much one can push down peaks in Swith-
out causing other peaks to pop up.
Time delays are somewhat easier to understand, since a time
delay swill introduce an additional phase lag of sx at the fre-
quency x, limiting how fast the controller can respond effectively
(i.e., bandwidth). Thus, the bandwidth for a controller with ac-
ceptable phase margins is typically x
B
<1/s.
Following the discussion in Ref. [166], these fundamental limi-
tations may be understood in relation to the limitations of open-
loop control based on model inversion from Sec. 4.2. If we con-
sider high-gain feedback b¼KðwrsÞfor a system as in Fig. 9
and Eq. (25), but without disturbances or noise, we have
b¼Ke ¼KSwr(27)
We may write this in terms of the complementary sensitivity T,
by noting that since T¼IS, we have T¼LðIþLÞ1¼PKS
b¼P1Twr(28)
Thus, at frequencies where Tis nearly the identity Iand control is
effective, the actuation is effectively inverting the plant P. Even
with sensor-based feedback, perfect control is unattainable. For
example, if the plant Phas right-half plane zeros; then, the actua-
tion signal will become unbounded if the gain Kis too aggressive.
Similarly, limitations arise with time-delays and when the number
of poles of Pexceeds the number of zeros, as in the case of open-
loop model-based inversion.
As a final illustration of the limitation of right-half plane zeros,
we consider the case of proportional control b¼Ks in a single-
input, single output system with plant P(f)¼N(f)/D(f). Here,
roots of the numerator N(f) are zeros of the plant and roots of the
denominator D(f) are poles. The closed-loop transfer function
from reference w
r
to sensors sis given by
sf
ðÞ
wrf
ðÞ¼PK
1þPK ¼NK=D
1þNK=D¼NK
DþNK (29)
For small control gain K, the term NK in the denominator is small,
and the poles of the closed-loop system are near the poles of P,
given by roots of D. As Kis increased, the NK term in the
Fig. 10 General framework for feedback control. The input to
the controller is the system measurements s, and the controller
outputs an actuation signal b. The exogenous inputs wmay
refer to a reference wr, disturbances wd, or sensor noise wn.
The cost function Jmay measure the cost associated with inac-
curacy of reference tracking, expense of control, etc.
050801-14 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
typically very high dimensional, sometimes exceeding the
capacity of computer memory. For example, a high Reynolds
number three-dimensional unsteady flow will exhibit important
spatial structures that span many orders of magnitude in scale.
The Reynolds number can be estimated from the ratio between
the largest-scale structures to the smallest structures in the flow.
Thus, for a generic geometry, the state dimension will scale with
Re
9=4
, along with the memory cost [7779]. The computational
cost will scale with Re
3
because of the addition of multiple tempo-
ral scales, which generally scale with Re
3=4
. For a channel flow,
the scaling may even be worse with Reynolds number, as Re
3
in
space and Re
4
in space and time [80,81]. If a spatial discretization
is required with 1000 elements in each direction; then, a three-
dimensional simulation will contain 10
9
states for every flow vari-
able (velocity, pressure, etc.).
The highest-order fully resolved simulation to date is a wall-
bounded turbulent channel flow with Re
s
¼5200 (Reynolds num-
ber based on the friction velocity), containing 2.4 10
11
states
[81]. This simulation is about 3.5 times larger than the previous
record holder [82], and it uses slightly over 3/4 of a million pro-
cessors in parallel. Even with Moore’s law, it will take nearly 40
yr for this type of computation to become a lightweight “laptop”
computation [83] and decades longer before being useful for
in-time control, since the parallel code takes 7 real seconds per
simulated time-step, as benchmarked in Ref. [81]. However, im-
pressive and useful for design and optimization, it is unclear that
this level of resolution is even necessary for many control
applications.
3.2.2 Modal Representation (Gray-Box). Instead of resolving
every detail of the flow field at all scales, it is often possible to
represent most of the relevant flow features in terms of a much
lower dimensional state. This state represents the amplitudes of
modes, or coherent structures that are likely to be found in the
flow of interest. Galerkin models based on modal expansions con-
stitute one class of gray-box models, which resolve the coherent
structures of the white-box models while accounting for small-
scale fluctuations with subscale closures.
The POD is one of the earliest and most successful modal rep-
resentations used in fluids [84,85], resulting in dominant spatially
coherent structures. POD benefits from a physical interpretation
where modes are ordered hierarchically in terms of the energy
content that they capture in the flow. There are numerous methods
to compute POD, and the snapshot POD [86] is efficient when a
limited number of well-resolved full-state measurements are
available from simulations or experiments. Snapshot POD is
Fig. 5 Schematic illustrating popular choices at the various levels of kinematic and dynamic
descriptions of the turbulent system Pand choices for designing the controller K.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-7
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
based on the singular value decomposition (SVD) [8790], which
is both numerically stable and efficient. POD is known under
other names: principal components analysis (PCA) [91], the
Hotelling transformation [92], Karhunen–Loe`ve decomposition
[93], and empirical orthogonal functions [94]. POD has been
widely used for flow control, as in the case of using proportional
feedback control to reduce turbulent fluctuations around a turret
for aero-optic applications [95]. Extensions of POD specifically
designed for closed-loop feedback control, known as balanced
(BPOD) [96,97], will be discussed in Sec. 4.5.3.
A recent technique known as dynamic mode decomposition
(DMD) combines features of the POD and the discrete Fourier
transform (DFT). The resulting spatial–temporal coherent struc-
tures oscillate in time at fixed frequencies, possibly with growth
or decay [98101]. Like POD, DMD is a snapshot-based method,
making it appealing for simulations and experiments alike. DMD
requires time-resolved snapshots of the same quality as needed for
a Fourier transform, although recent methods have investigated
sub-Nyquist sampled DMD [102]. Finally, DMD has a strong con-
nection to the Koopman operator, which is an infinite dimensional
linear operator describing the evolution of an observable function
of a nonlinear dynamical system on a manifold [100,103107].
It is also possible to construct a modal representation of grow-
ing and decaying features in the flow based on stability modes of
the linearized Navier–Stokes equation and linearized adjoint equa-
tions [108111]. The least damped part of the spectrum deter-
mines the coherent structures and their transient dynamics.
Although completeness of the stability modes of the linearized
Navier–Stokes equation has only been shown for geometrically
simple configurations [112,113], it is generally assumed and has
also been numerically corroborated. For Stokes eigenmodes, aris-
ing from a linearization around vanishing flow, completeness and
orthogonality can be shown for a large class of boundary condi-
tions [114].
In many fluid systems, large transient energy growth may be
experienced, even in linearly stable systems, because of non-
normality of the evolution operator [115,116]. One feature of non-
normality can be large transient growth due to the destructive
interference of nearly parallel eigenvectors, even with very similar
eigenvalues. This phenomenon may be especially pronounced in
shear flows, such as channel flows, and is important for flow
control.
The modal decompositions discussed above are approximations
based on data snapshots of the full system, most likely from a
white-box model or experiments. We then have many of the same
problems as in the white-box models: since the data may be
exceedingly large, even simple tasks, such as computing the inner
product of velocity fields, are cumbersome. Fortunately, the
expense to compute a modal decomposition is a one-time upfront
cost during the training phase, and resulting reduced-order models
are generally much faster in the execution phase. There is a new
software package in PYTHON, called MODRED from “model reduc-
tion,” that efficiently computes various modal decompositions and
reduced-order models for systems with large data [117].
The oldest class of gray-box models is based on vortex repre-
sentations starting with Helmholtz’s vortex laws in 1869. Von
K
arm
an’s kinematic model [118] of the vortex street is one fa-
mous example. Some of these vortex models have been used for
control applications. The F
oppl (1906) vortex model of the cylin-
der wake has, for instance, been used to design a controller stabi-
lizing the wake [119]. Suh’s vortex model [120] of a recirculation
zone has been used for flatness-based control targeting mixing
enhancement [121]. Most vortex models, however, are of hybrid
nature, i.e., vortices are continually produced, merged, or
removed. This implies that the dimension of the state space as
well as the meaning of the coordinates continually changes. Such
hybrid models are a challenge for almost all control methods. We
shall not pause to elaborate these Lagrangian gray box models as
there exist excellent textbooks on the topic (see, e.g., Refs.
[122124]) and the control applications are sparse.
3.2.3 Input–Output (Black-Box). Models that are built purely
on input–output data are referred to as black-box models because
they are opaque with respect to the underlying structure of the
fluid. However, what black-box models lack in flow resolution,
they make up for in rapid identification and low-overhead imple-
mentation. These models may be based on a state space comprised
of the sensor and actuator signals
a¼½sT;bTT
A time-history of sensor signals using time-delay coordinates
a¼½sðtÞT;sðtsÞT;sðt2sÞT;;bðtÞT;T
or these signals and their derivatives
a¼sT;bT;d
dt sT;d
dt bT;

T
3.3 Dynamics: Classification by System Resolution. Some
of the modal representations above are fundamentally linked to a
dynamic model. For instance, DMD results in spatial–temporal
coherent structures along with a low-order model for how these
modes oscillate and/or grow or decay. Likewise, BPOD results in
a balanced linear model. For other representations, it is necessary
to build a model on top of the data separately. The various meth-
ods to determine fin Eq. (2) are illustrated in Fig. 5under the
“dynamics” column.
3.3.1 White-Box Models. For white-box models, the discre-
tized Navier–Stokes equations are used to resolve all flow physics
and nonlinearities. When appropriate, e.g., near a fixed point or
periodic orbit, the linearized Navier–Stokes equations may be
used. As discussed earlier, there are significant hurdles to real-
time implementation of these models based on the computational
load. In addition, full-state estimation may be required when using
a white-box model to control an experiment where measurements
are limited. Estimation may be statistical, such as linear stochastic
estimation [125132], or dynamic, as in the Kalman filter
[133136]. However, these models are able to accurately capture
important dynamic events, such as bifurcations, which define the
phenomenological landscape.
There are a variety of methods in computational fluid dynamics
(CFD), including the finite-element method, spectral elements, fi-
nite volumes, finite differences, immersed boundaries, point-
vortex methods, and many more. As the Reynolds number
increases, DNS becomes prohibitively expensive, and it is neces-
sary to use turbulence models for smaller scale fluctuations. Find-
ing turbulence closures to these models is an active field of
research, and common methods include Reynolds-averaged
Navier–Stokes (RANS) and large eddy simulations (LES). These
may be thought of as off-white box models, since they still resolve
many orders of magnitude in scale. However, these models must
be used with caution and experience.
3.3.2 Gray-Box Models. Reduced-order models may be
obtained in terms of the dynamic interaction of coherent struc-
tures, such as those obtained through POD, BPOD, DMD, or
global stability analysis. The resulting models enable real-time
control in experiments since the computational burden is limited
and state estimation is often feasible based on limited measure-
ments. However, acceptable accuracy is typically only achieved
near the training parameters, for one or a few dominant frequen-
cies, and for a few known nonlinear mechanisms. More robust
mathematical reduced-order models may be obtained for simple
geometries from Hilbert space considerations without advance
data, but at the price of a significantly increased dimension
[137,138].
050801-8 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Galerkin projection of the Navier–Stokes equations onto a
modal basis is a common technique to obtain nonlinear reduced-
order models. The “traditional” Galerkin method [139] requests
the same orthogonal modes for the expansion and as test functions
for the projection. POD–Galerkin models are the most prominent
corresponding example in fluid mechanics [140]. Although BPOD
has an associated linear model, it is possible to use BPOD modes
for a nonlinear Galerkin projection. Gray-box models will be dis-
cussed in more detail in Secs. 4and 5.
3.3.3 Black-Box Models. Transfer functions and state-space
models identified purely from input–output data (i.e., measure-
ments of the sensor and actuation history) are black-box models.
They have similar dynamic bandwidth as gray-box models and in
fact may have identical input–output characteristics. However,
results and predictions lack the clear interpretation available when
model states represent modal amplitudes, as is the case with gray-
box models. A black-box model may be derived from a gray-box
model, and the opposite may also be true if additional snapshot in-
formation is available. For example, it may be possible to relate
the internal model states with linear combinations of the flow
snapshots, resulting in a gray-box model. A major advantage of
black-box models is easy experimental identification, particularly
for linear and weakly nonlinear dynamics.
There are many techniques available to obtain black-box mod-
els, and a full exploration is outside the scope of this review.
However, the eigensystem realization algorithm (ERA) [141,142]
is a promising algorithm that produces balanced models from
input–output data. Other methods are discussed in Refs.
[143,144].
The ERA, discussed more in Sec. 4.5.4, results in a linear state-
space model based on impulse response data, i.e., sensor measure-
ments in response to an impulsive delta function input in the
actuation. The impulse response of a multiple-input, multiple-
output (MIMO) system will be given by hðtÞ, a function of time
with N
s
rows and N
b
columns. It is possible to predict the sensor
measurements sðtÞin response to an arbitrary input signal bðtÞfor
linear systems by convolution of the actuation signal with the
impulse response
sðtÞ¼sð0Þþðt
0
hðtsÞbðsÞds(3)
The notion of convolution may be extended for generic nonlinear
systems, such as in Eq. (2), by a Volterra series [145147]. To
simplify notation, we consider a single-input, single-output
(SISO) system
sðtÞ¼sð0ÞþX
N
k¼1ðt
0ðt
0
hkðts1;;tskÞ
bðs1ÞbðskÞds1dsk(4)
The functions h
k
are called Volterra kernels, and there are exis-
tence and uniqueness theorems for a large class of nonlinear sys-
tems [148]. There are also uses of Volterra series in geometric
control theory [149]. Notice that the first integral in Eq. (4) for
k¼1 is the linear impulse response from Eq. (3).
As a simple example to demonstrate Volterra series, consider a
SISO system with linear dynamics and a quadratic output
nonlinearity
3
d
dt a¼Aa þBb
s¼Ca
ðÞ
2
The output s(t) is the square of the convolution in Eq. (3)
sðtÞ¼sð0Þþ ðt
0
h1ðtsÞbðsÞds

2
¼sð0Þþðt
0ðt
0
h1ðts1Þh1ðts2Þbðs1Þbðs2Þds1ds2
¼sð0Þþðt
0ðt
0
h2ðts1;ts2Þbðs1Þbðs2Þds1ds2
Here, h2ðts1;ts2Þ¼h1ðts1Þh1ðts2Þ, so that static out-
put nonlinearities have simple higher order kernels.
Volterra series have been used with back-stepping and bound-
ary control of PDEs for fluid flow control [150]. They have also
been used for general fluid modeling [151], to capture aerody-
namic and aeroelastic phenomena [152154], and to model and
control plasma turbulence [155157].
Support vector machines (SVMs) [158160] are a new class of
supervised nonlinear models that have a tremendous amount of
potential for the control of complex dynamical systems. SVMs are
supervised models that take input data into a high-dimensional
nonlinear feature space, which may then be mapped back down to
inputs. They are related to the Volterra series above. Knowledge
of the kernel is a central part of the algorithm, and in practice, a
suite of kernels may be tested to optimize model performance.
SVMs have not penetrated the turbulence control literature signifi-
cantly, but we expect this method to become more prominent in
fluids in the future.
3.3.4 Model-Free Approaches. Model-free approaches do not
rely on any underlying model description relating inputs to out-
puts. Instead, these approaches are often based on qualitative
steady-state maps and are generally restricted to existing one or
few-parameter open-loop control. Steady-state maps typically
assume working periodic control and the maps relate input param-
eters to outcomes in some objective function. However, recent
methods allow the identification of controllers using machine
learning and adaptive control.
4 Linear Model-Based Control
Many results in closed-loop turbulence control are specific to
linear systems. For example, one may stabilize an unstable steady
state and delay the transition to turbulence in the boundary layer
or channel flow. In this case, the steady (laminar) solution, which
becomes unstable for postcritical Reynolds numbers, may be sta-
bilized by feedback control.
Such work has encompassed a significant modeling effort (see
Sec. 3) to describe the relevant low-order flow mechanisms to be
suppressed or utilized. Performance issues, such as bandwidth,
disturbance rejection, and noise attenuation, must be balanced
with robustness to model uncertainty and time-delays in sensing,
actuation, or computation. For this reason, there has been a recent
push to move away from H2optimal control techniques (linear-
quadratic regulator (LQR), Kalman estimation, linear-quadratic-
Gaussian (LQG), etc.) to the robust H1controllers [161164].
These controllers guarantee robust performance by penalizing
worst-case performance in the design process. Progress has also
been made in the design and modeling of sensors and actuators
[36], along with their placement in the flow.
There are many instances when linear control strategies may
have significant and direct impact for nonlinear turbulent flows,
even away from steady fixed-point solutions. For example, in
mean-field models, exciting one oscillatory mode with linear con-
trol may suppress other, more energetic, oscillatory modes
[48,47]. In addition, ensemble averages of turbulent flow
responses may be linear as in Refs. [41,165]. Other examples
include the transient control of separation for the fully turbulent
boundary layer [41].
The main goal of this section is to develop an overview of the
linear control framework and provide context for the related liter-
ature in flow control. Ideas, such as control topology (feedback
3
This example was adapted from the notes by Nicholas R. Gamroth.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-9
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
versus feedforward and open-loop versus closed-loop), controll-
ability and observability, state estimation, pole placement, and
robustness (sensitivity, bandwidth, stability), are introduced with
an emphasis on physical interpretation and engineering implica-
tions. Since turbulent flows often have very fast time-scales, many
of the topics above are considered in light of computational com-
plexity and the latency of the control decision, leading to a discus-
sion on reduced-order models.
This section is not meant to be an exhaustive overview of either
linear control theory or its application in flow control. For more
details on control theory, there are two recommended texts, Ref.
[166](Practical Engineering) and Ref. [167](Mathematical
Theory). There have also been a number of excellent recent
reviews involving various aspects of the subject of linear closed-
loop flow control [20,34,168171].
4.1 Linearized Input–Output Dynamics. Often, we are
interested in linearizing Eq. (2) about a steady fixed point as, cor-
responding to a desirable flow state. This leads to a linear system
of equations
d
dt a¼Aa þBb (5a)
s¼Ca þDb (5b)
where each of the matrices ðA;B;C;DÞdepends on the lineariza-
tion point asand bifurcation parameter l. The state vector arefers
to the difference between the current flow and the fixed point as.
The system is linearly stable if all of the eigenvalues of Aare in
the left-half of the complex plane, having negative real parts. The
linear approximation in Eq. (5) will be approximately valid near
fixed points without purely imaginary eigenvalues and away from
critical values of the bifurcation parameter (i.e., when
det df=dl0). Fortunately, if control is effective, the flow state
should stay close to as, where the linear approximation is valid.
Similar linearization may be applied near a periodic orbit of the
flow.
It is possible to represent the state-space model in Eq. (5) as a
transfer function PðfÞrelating the frequency of sinusoidal input
forcing to the magnitude and phase of the output response; here
f¼ix2Cis a Laplace variable
PðfÞ¼CðfIAÞ1BþD(6)
For linear systems, the output frequency will be the same as the
input frequency x, the magnitude is given by jPðfÞj, and the phase
is given by /PðfÞ. For example, consider a SISO linear system,
where an input sinusoid sinðxtÞwill excite an output measure-
ment Asinðxtþ/Þ, where Aand /are the magnitude and phase
angle of the transfer function evaluated at f¼ix:A¼jPðfÞj
and /¼/ðPðfÞÞ. In the context of controls, the system Pis
known as the plant. The roots of the denominator of Pare referred
to as poles and the roots of the numerator are referred to as zeros.
The plant is unstable if any poles have a positive real part; if Pis
based on Eq. (6), then the poles of Pcorrespond to eigenvalues of
A.
Both the state-space and frequency domain representations are
useful. It is often beneficial to design specifications and assess
controller performance in the frequency domain, although it is of-
ten simpler to achieve these goals by manipulating the state-space
system [166]. Generally, an effective open-loop plant will have
high gain at low frequency for disturbance rejection and reference
tracking, low gain at high frequency for noise attenuation, and a
good phase margin at crossover for stability.
4.2 Model-Based Open-Loop Control. With a model of the
form in Eq. (5) or Eq. (6), it may be possible to design an open-
loop control law to achieve some desired specification without the
use of measurement-based feedback or feedforward control. For
instance, if perfect tracking of the reference input wris desired in
Fig. 6, under certain circumstances it may be possible to design a
controller by inverting the plant dynamics P:KðfÞ¼P1ðfÞ.In
this case, the transfer function from reference wrto output sis
given by PP1¼I, so that the output perfectly matches the refer-
ence. However, perfect control is never possible in real-world sys-
tems, and this strategy should be used with caution, since it
generally relies on a number of significant assumptions on the
plant P. First, effective control based on plant inversion requires
extremely precise knowledge of Pand well-characterized, predict-
able disturbances; there is little room for model errors or uncer-
tainties, as there are no sensor measurements to determine if
performance is as expected and no corrective feedback mecha-
nisms to modify the actuation strategy to compensate.
For open-loop control using plant inversion, the plant Pmust
also be stable. It is impossible to fundamentally change the
dynamics of a linear system through open-loop control, and thus
an unstable plant cannot be stabilized without feedback. Attempt-
ing to stabilize an unstable plant by inverting the dynamics will
typically have disastrous consequences. For instance, consider the
following unstable plant with a pole at f¼5 and a zero at
f¼10: P(f)¼(sþ10)/(s5). Inverting the plant would result
in a controller K¼(s5)/(sþ10); however, if there is even the
slightest uncertainty in the model, so that the true pole is at 5 ;
then, the open-loop system will be
Ptrue f
ðÞ
Kf
ðÞ¼s5
s5þ
This system is still unstable, despite the attempted pole cancela-
tion. Moreover, the unstable mode is now nearly unobservable, a
concept that will be discussed later.
In addition to stability, the plant Pmust not have any time
delays or zeros in the right-half plane, and it must have the same
number of poles as zeros. If Phas any zeros in the right-half
plane, then the inverted controller Kwill be unstable, since it will
have right-half plane poles. These plants are called nonminimum
phase, and there have been generalizations to plant inversion,
which provide bounded inverses to these systems [172]. Similarly,
time-delays are not invertible, and if Phas more poles than zeros,
then the resulting controller will not be realizable and may have
extremely large actuation signals b. There are also generalizations
that provide regularized model inversion, where optimization
schemes are applied with penalty terms added to keep the result-
ing actuation signal bbounded. These regularized open-loop con-
trollers are often significantly more effective, with improved
robustness.
Combined, these restrictions on the plant Pimply that model-
based open-loop control should only be used when the plant is
well-behaved, accurately characterized by a model, when distur-
bances are characterized, and when the additional feedback con-
trol hardware is unnecessarily expensive. Otherwise, performance
goals must be modest. Open-loop model inversion is often used in
manufacturing and robotics, where systems are constrained and
well-characterized in a standard operating environment.
4.3 Dynamic Closed-Loop Feedback Control. Feedback
control addresses many of the aforementioned issues with open-
loop control. Namely, closed-loop feedback, as illustrated in Fig.
7, uses sensor measurements to correct for model uncertainties,
Fig. 6 Open-loop control topology
050801-10 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
5.4 Moderately Nonlinear Dynamics. Some oscillatory
flows may be tamed by direct mitigation with models and methods
described in Sec. 5.3. Not all plants, particularly turbulent flows,
have the actuation authority for such a stabilization. However,
periodic forcing at high frequency has mitigated periodic oscilla-
tions in a number of experiments. Examples are the wake stabili-
zation with a oscillatory cylinder rotation [74], suppression of
Kelvin–Helmholtz vortices in transitional shear layers [263],
reduction of the separation zone in a high-lift configuration [48],
and elongation of the dead-water region behind a backward facing
step [47]. Also a periodic frequency at 60–70% of the dominant
shedding frequency may substantially delay the vortex formation
in wall-bounded shear-layers [49] and D-shaped cylinders [45].
Sections 5.4.1 and 5.4.2 outline a corresponding modeling and
control strategy.
5.4.1 Generalized Mean-Field Model. Here, a generalized
mean-field model for such frequency-cross talk phenomena is
reviewed from Ref. [48]. Let uudenote the natural self-amplified
oscillation represented by two oscillatory modes u1and u2. Analo-
gously, the actuated oscillatory fluctuation uais described by two
modes u3and u4. The base-flow deformation due to the unstable
natural frequency x
u
and stable actuation frequency x
a
is
described by the shift-modes u5and u6, respectively. The result-
ing velocity decomposition reads
uðx;tÞ¼usðxÞþuDðx;tÞ
þuuðx;tÞþuaðx;tÞ(53a)
uuðx;tÞ¼a1ðtÞu1ðxÞþa2ðtÞu2ðxÞ(53b)
uaðx;tÞ¼a3ðtÞu3ðxÞþa4ðtÞu4ðxÞ(53c)
uDðx;tÞ¼a5ðtÞu5ðxÞþa6ðtÞu6ðxÞ(53d)
Generalized mean-field arguments yield the following evolu-
tion equation:
d
dt
a1
a2
a3
a4
2
6
6
43
7
7
5¼Aa5;a6
ðÞ
a1
a2
a3
a4
2
6
6
43
7
7
5þBb(54a)
a5¼aua2
1þa2
2
 (54b)
a6¼aaa2
3þa2
4
 (54c)
where
Aða5;a6Þ¼A0þa5A5þa6A6
A0¼
ruxu00
xuru00
00raxa
00xuru
2
6
6
6
6
4
3
7
7
7
7
5
A5¼
buu cuu 00
cuu buu 00
00bau cau
00cau bau
2
6
6
6
6
4
3
7
7
7
7
5
A6¼
bua cua 00
cua bua 00
00baa caa
00caa baa
2
6
6
6
6
4
3
7
7
7
7
5
B¼
0
0
0
g
2
6
6
6
6
4
3
7
7
7
7
5
It should be noted that a vanishing actuation implies
a
3
¼a
4
¼a
6
¼0 and yields the mean-field model of Eq. (52) mod-
ulo index numbering. In the sequel, we assume that the fixed point
is unstable (r
u
>0) and the limit cycle is stable (b
uu
>0). Simi-
larly, a vanishing natural fluctuation a
1
¼a
2
¼a
5
yields another
mean-field model, again modulo index numbering. In the sequel,
we assume that the actuated structures vanish after the end of
actuation, implying r
aa
<0 and b
aa
>0 in the model.
An interesting aspect of Eq. (54) is the frequency cross-talk.
The effective growth-rate for the natural oscillation reads
A11 ¼rubuuauða2
1þa2
2Þbuaaaða2
3þa2
4Þ(55)
The forcing stabilizes the natural instability if and only if b
ua
>0.
Complete stabilization implies A
11
0. From Eq. (55), such com-
plete stabilization is achieved with a threshold fluctuation level at
the forcing frequency
a2
3þa2
4ru
aabua
Thus, increasing the forcing at higher or lower frequency can
decrease the natural frequency.
The generalized mean-field model has been fitted to numerical
URANS simulation data of a high-lift configuration [48] with
high-frequency forcing. This model also accurately describes the
experimental turbulent wake data with a stabilizing low-frequency
forcing [264], as shown in Fig. 15.
5.4.2 Nonlinear Control Design. The model above may guide
in-time control [265] and adaptive control design providing the
minimum effective actuation energy [266].
Figures 16 and 17 show an unactuated and stabilizing transient
with the parameters of Table 1. For simplicity, all nonlinear
Fig. 14 Phase portrait of weakly nonlinear dynamics. The
dashed trajectory corresponds to the unactuated dynamics
while the solid trajectory corresponds to actuated dynamics.
The chosen parameters of Eq. (52) are r
u
50.1, x
u
51, a
u
51,
b
u
51, c
u
50, and the forced decay rate r
c
520.1. The globally
stable limit cycle lies on the parabolic inertial manifold.
050801-22 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
ðtf
0kbðsÞk2ds¼aTWcðtfÞ1a(17)
Thus, if the controllability Gramian is nearly singular; then, tre-
mendous actuation energy is required to control the system. Con-
versely, if the eigenvalues of the Gramian are large; then, the
system is highly controllable.
It is therefore possible to determine a hierarchy of controllable
states, in order of controllability, by taking the eigenvalue decom-
position of Wc, which is positive semidefinite. The eigenvectors
corresponding to the largest eigenvalues are the most controllable
states. This eigendecomposition is closely related to the SVD of
the controllability matrix from Eq. (11), but defined from a
discrete-time system approximating Eq. (5). This connection to
the SVD will be important in model reduction and will be dis-
cussed in Sec. 4.5.1. A similar hierarchy of observable states may
be determined.
4.3.2 Linear Quadratic Gaussian. Nowthatwehaveestab-
lished conditions enabling arbitrary pole placement of the closed-loop
system, we must decide on where to place them. It is mathematically
possible to make closed-loop eigenvalues arbitrarily stable (i.e.,
arbitrarily far in the left-half complex plane) if the system is con-
trollable. However, this may require expensive control expendi-
ture with unrealistic actuation magnitudes. Moreover, very stable
eigenvalues may over-react to noise and disturbances, causing the
closed-loop system to jitter, much as a new driver over-reacting to
vibrations in the steering wheel.
In a LQR controller, the controller Kris chosen to place the
closed-loop poles to minimize a quadratic cost function J. This
cost function balances the desire to regulate the system state to
a¼0with the added objective of small control expenditure
J¼ð1
0ðaTQa þbTRb Þdt (18)
Qis a symmetric positive semidefinite matrix that is chosen to pe-
nalize deviations of the state afrom the set-point a¼0. Similarly,
Ris a symmetric positive definite matrix that is chosen to penalize
control expenditure. Often, Qand Rare chosen to be diagonal mat-
rices, and the magnitude of the diagonal elements may be adjusted
to tune the control performance by adjusting relative penalty ratios.
For example, to increase the aggressiveness of control, the diagonal
entries of Rmay be decreased with respect to those of Q.
The optimal control law that minimizes Jin Eq. (18) is given
by b¼Kra, with Kr¼R1BTX.Xis the unique solution to the
algebraic Riccati equation
ATXþXA XBR1BTXþQ¼0(19)
In LQR, a balance is struck between the stability of the closed-
loop system and the aggressiveness of control. Taking control ex-
penditure into account is important so that the controller does not
over-react to high-frequency noise and disturbances, does not
exceed maximum actuation amplitudes, and is not prohibitively
expensive.
In practice, to make Eq. (5) more realistic, it must be aug-
mented with the addition of white noise disturbance wdand mea-
surement noise wn
d
dt a¼Aa þBb þBwwd(20a)
s¼Ca þDb þwn(20b)
The matrix Bwdetermines the spatial distribution of how distur-
bances enter the state. Each of these noise inputs has a different
covariance matrix: EðwdðtÞwdðsÞTÞ¼VddðtsÞand
EðwnðtÞwnðsÞTÞ¼VndðtsÞ, where E() is the expectation value
and dis the Dirac delta function. The addition of disturbances and
sensor noise is shown in Fig. 8and also in Fig. 9with wr¼0;wr
is the reference input.
In linear-quadratic estimation (LQE), a dual problem to LQR is
solved, resulting in an optimal full-state estimator, as in Eq. (9),
that balances the relative importance of measurement noise and
process noise. The process noise may be an additive stochastic
term, or structural uncertainty in the model. A dual Riccati equa-
tion is solved for Yin the observer gain Kf¼YCTVn
Fig. 8 Linear-quadratic Gaussian controller. The Kalman filter Kfis a dynamical system that takes sensor
measurements sand the actuation signal bto estimate the full-state ^
a. The LQR gain Kris a matrix that multi-
plies the full-state to produce an actuation signal b52Kr^
athat is optimal with respect to the quadratic cost func-
tion in Eq. (18).
Fig. 9 Feedback control with disturbances and noise
050801-12 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
YATþAY YCTV1
nCY þVd¼0(21)
The so-called Kalman filter Kfmay be used in the observer in Eq.
(9), and it is chosen to minimize Eðða^
aÞTða^
aÞÞ given known
covariance Vdand Vn.
The optimal state-feedback (LQR) and optimal state-estimation
(LQE) may be designed independently and then combined. The
separation principle guarantees that when combined, the state-
feedback and state-estimation will remain stable and optimal. The
resulting controller, combining estimation-based full-state feed-
back, is known as a LQG controller, as shown in Fig. 8.
The combined LQG controller may be written as a more gen-
eral observer dynamical system, given by
d
dt ^
a¼^
A^
aþ^
Bs (22a)
b¼^
C^
aþ^
Ds (22b)
For the LQG controller, we set ^
A¼AKfCBKr
þKfDKr;^
B¼Kf;^
C¼Kr, and ^
D¼0, recovering the form of
Eq. (10) with Kras the LQR gain matrix and Kfas the Kalman fil-
ter gain matrix.
The resulting controller, known more generally as an H2con-
troller, optimally balances the effect of Gaussian measurement
noise with process disturbances. Although LQR controllers may
have decent stability margins, there is no guarantee on stability
margins for LQG controllers, as famously demonstrated in Ref.
[161]. This means that even small uncertainties, such as unmod-
eled dynamics, unexpected disturbances, or time-delays, may
destabilize the closed-loop system.
4.4 Robust Control. The notion of robustness and performance
are central in feedback control. The limitations of LQG control have
motivated significant advances in the development of controllers
with robust performance. Robustness typically refers to the ability to
maintain control performance despite model uncertainty, unmodeled
nonlinear dynamics, and unforeseen disturbances, time-delays, etc.,
which are all important for turbulence control. A complete discussion
of robust control is beyond the scope of this review; instead, our goal
is to build an intuition and provide a glimpse of the powerful robust
control machinery for flow systems [174]. For a more complete over-
view with excellent attention to engineering considerations and prac-
tical control design, see Ref. [166].
4.4.1 Sensitivity, Complementary Sensitivity, and Robustness.
As discussed in Sec. 4.3.2, real systems will always include distur-
bances wdand sensor noise wn, as illustrated in Fig. 9. This dia-
gram also includes a commanded reference input wrthat the
controller should track. When designing a robust controller, it is
important to understand how these exogenous inputs affect the
outputs s, the error signal e, and the actuation signal, b. In general,
systems will be particularly sensitive to disturbances and noise at
certain frequencies.
We may express the output sin terms of transfer functions on
the inputs wr;wd, and wn
s¼PdwdþPKðwrswnÞ
IþPKÞs¼PdwdþPKwrPKwn
Therefore, we have
s¼ðIþPKÞ1PK
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
T
wrþðIþPKÞ1
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
S
Pdwd
ðIþPKÞ1PK
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
T
wn(23)
Sis the sensitivity, and Tis the complementary sensitivity.Ifwe
let L¼PK be the loop transfer function, then we have
S¼ðIþLÞ1(24a)
T¼ðIþLÞ1L(24b)
Since we are interested in minimizing the error e(without
noise), the following expression is more useful:
e¼wrs¼SwrSPdwdþTwn(25)
The disturbance plant Pdand the system plant Pare often closely
related. For example, disturbances and control inputs may both be
amplified by a natural convective instability in the flow.
Typically, we will choose the controller Kso that the open-loop
transfer function L¼PK has desirable properties in the frequency
domain. For example, small gain at high frequencies will attenuate
sensor noise. Similarly, high gain at low frequencies will provide
good reference tracking performance. These are intimately related
to the sensitivity Sand complementary sensitivity T.Inparticular,
from Eq. (25),Sshould be small at low frequencies, and Tshould
be small at large frequencies; note that SþT¼I,fromEq.(24).
For performance and robustness, we want the maximum peak
of S,MS¼kSk1, to be as small as possible. From Eq. (25),itis
clear that in the absence of noise, feedback control improves per-
formance (i.e., reduces error) for all frequencies where jSj<1;
thus, control is effective when jTj1. As explained in Ref.
[166], p. 37, all real systems will have a range of frequencies
where jSj>1, in which case performance is degraded. Minimiz-
ing the peak MSmitigates the amount of degradation experienced
with feedback at these frequencies, improving performance. In
addition, the minimum distance of the loop transfer function Lto
the point 1 in the complex plane is given by M1
S. The larger
this distance, the greater the stability margin of the closed-loop
system, improving robustness. These are the two major reasons to
minimize MS.
The controller bandwidth x
B
is the frequency below which
feedback control is effective. This is a subjective definition. Often,
x
B
is the frequency where jSðjxÞj first crosses 3 dB from below.
We would ideally like the controller bandwidth to be as large as
possible without amplifying sensor noise, which is typically high
frequency. However, there are fundamental bandwidth limitations
that are imposed for systems that have time delays or right half
plane zeros [166].
4.4.2 H1Robust Control Design. As discussed above, LQG
controllers are known to have arbitrarily poor robustness margins.
This is a serious problem in turbulence control, where the flow is
wrought with uncertainty and time-delays. H1robust controllers
are used when robustness is important. There are many connec-
tions between H2and H1control, and we refer the reader to the
excellent reference books expanding on this theory [166,167].
Figure 10 shows the most general schematic for closed-loop
feedback control, encompassing H2and H1optimal control strat-
egies. In the generalized theory of modern control, the goal is to
minimize the transfer function from exogenous inputs w(refer-
ence, disturbances, noise, etc.) to the cost function J(accuracy,
actuation cost, time-domain performance, etc.). Both H2and H1
control design result in controllers that minimize different norms
on this fundamental input–output transfer function. In fact, the
symbol H2refers to a Hardy space with bounded two-norm, con-
sisting of stable and strictly proper transfer functions (meaning
gain rolls off at high frequency). The symbol H1refers to a
Hardy space with bounded infinity-norm, consisting of stable and
proper transfer functions (gain does not grow infinite at high fre-
quencies). The infinity norm is defined as
kPk1¢max
xr1ðPðixÞÞ (26)
Here, r
1
denotes the maximum singular value. Since the kk
1
norm is the maximum value of the transfer function at any
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-13
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
frequency, it is often called a worst-case scenario norm; therefore,
minimizing the infinity norm provides robustness to worst-case
exogenous inputs.
If we let Pw!Jdenote the transfer function from wto J; then,
the goal of H1control is to construct a controller to minimize the
infinity norm: minkPw!Jk1. This is typically difficult, and no
analytic closed-form solution exists for the optimal controller in
general. However, there are relatively efficient iterative methods
to find a controller such that kPw!Jk1<c, as described in Ref.
[164]. There are numerous conditions and caveats that describe
when this method can be used. In addition, there are computation-
ally efficient algorithms implemented in both MATLAB and PYTHON,
and these methods require relatively low overhead from the user.
Selecting the cost function Jto meet design specifications is a
critically important part of robust control design. Considerations,
such as disturbance rejection, noise attenuation, controller band-
width, and actuation cost, may be accounted for by a weighted
sum of the transfer functions S;T, and KS. In the mixed sensitivity
control problem, various weighting transfer function are used to
balance the relative importance of these considerations at various
frequency ranges. For instance, we may weight Sby a low-pass
filter and KS by a high-pass filter, so that disturbance rejection at
low frequency is promoted and control response at high-frequency
is discouraged. A general cost function may consist of three
weighting filters Fkmultiplying S,T, and KS
F1S
F2T
F3KS
2
43
5
1
Another possible robust control design is called H1loop-
shaping. This procedure may be more straightforward than mixed
sensitivity synthesis for many problems. The method consists of
two major steps. First, a desired open-loop transfer function is
specified based on performance goals and classical control design.
Input and output compensators are constructed to transform the
open-loop system to the desired loop shape. Second, the shaped
loop is made robust with respect to a large class of model uncer-
tainty. Indeed, the procedure of H1loop shaping allows the user
to design an ideal controller to meet performance specifications,
such as rise-time, bandwidth, and settling-time. Typically, a loop
shape should have large gain at low frequency to guarantee accu-
rate reference tracking and slow disturbance rejection, low gain at
high frequencies to attenuate sensor noise, and a cross-over
frequency that ensures desirable bandwidth. The loop transfer
function is then robustified so that there are improved gain and
phase margins.
H2control has been an extremely popular control paradigm
because of its simple mathematical formulation and its tunability
by user input. The advantages of H1control are increasingly real-
ized in flow control, eminent examples being the collaborative
research centers (Sfb 557 and Sfb 1029) lead by King [16,17,175].
Additionally, there are numerous consumer software solutions
that make implementation relatively straightforward. In MATLAB,
mixed sensitivity is accomplished using the mixsyn command
in the robust control toolbox. Similarly, loop-shaping is accom-
plished using the loopsyn command in the robust control
toolbox.
4.4.3 Fundamental Limitations With Implications for Turbulence
Control. As discussed above, we want to minimize the peaks of S
and T. Some peakedness is inevitable, and there are certain system
characteristics that significantly limit performance and robustness.
Most notably, time-delays and right-half plane zeros of the open-
loop system will limit the effective control bandwidth and will
increase the attainable lower-bound for peaks of Sand T. This
contributes to both degrading performance and decreasing
robustness.
Similarly, a plant will suffer from robust performance limita-
tions if the number of poles exceeds the number of zeros by more
than two. These fundamental limitations are quantified in the
waterbed integrals, which are so named because if you push a
waterbed down in one location, it must rise in an another. Thus,
there are limits to how much one can push down peaks in Swith-
out causing other peaks to pop up.
Time delays are somewhat easier to understand, since a time
delay swill introduce an additional phase lag of sx at the fre-
quency x, limiting how fast the controller can respond effectively
(i.e., bandwidth). Thus, the bandwidth for a controller with ac-
ceptable phase margins is typically x
B
<1/s.
Following the discussion in Ref. [166], these fundamental limi-
tations may be understood in relation to the limitations of open-
loop control based on model inversion from Sec. 4.2. If we con-
sider high-gain feedback b¼KðwrsÞfor a system as in Fig. 9
and Eq. (25), but without disturbances or noise, we have
b¼Ke ¼KSwr(27)
We may write this in terms of the complementary sensitivity T,
by noting that since T¼IS, we have T¼LðIþLÞ1¼PKS
b¼P1Twr(28)
Thus, at frequencies where Tis nearly the identity Iand control is
effective, the actuation is effectively inverting the plant P. Even
with sensor-based feedback, perfect control is unattainable. For
example, if the plant Phas right-half plane zeros; then, the actua-
tion signal will become unbounded if the gain Kis too aggressive.
Similarly, limitations arise with time-delays and when the number
of poles of Pexceeds the number of zeros, as in the case of open-
loop model-based inversion.
As a final illustration of the limitation of right-half plane zeros,
we consider the case of proportional control b¼Ks in a single-
input, single output system with plant P(f)¼N(f)/D(f). Here,
roots of the numerator N(f) are zeros of the plant and roots of the
denominator D(f) are poles. The closed-loop transfer function
from reference w
r
to sensors sis given by
sf
ðÞ
wrf
ðÞ¼PK
1þPK ¼NK=D
1þNK=D¼NK
DþNK (29)
For small control gain K, the term NK in the denominator is small,
and the poles of the closed-loop system are near the poles of P,
given by roots of D. As Kis increased, the NK term in the
Fig. 10 General framework for feedback control. The input to
the controller is the system measurements s, and the controller
outputs an actuation signal b. The exogenous inputs wmay
refer to a reference wr, disturbances wd, or sensor noise wn.
The cost function Jmay measure the cost associated with inac-
curacy of reference tracking, expense of control, etc.
050801-14 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
denominator begins to dominate, and closed-loop poles are
attracted to the roots of N, which are the open-loop zeros of P.
Thus, if there are right-half plane zeros of the open-loop plant P;
then, high-gain proportional control will drive the system unsta-
ble. These effects are often observed in the root locus plot from
classical control theory. In this way, we see that right-half plane
zeros will directly impose limitations on the gain margin of the
controller.
The limitations imposed by a time-delay have significant impli-
cations for turbulence control. First, time-delays will be inevitable
for many sensor/actuator configurations in convective flows [176],
limiting the frequency of disturbances that may be effectively sup-
pressed with feedback. Second, as turbulence time-scales may
become extremely fast, the time it takes to compute a control
action will introduce a latency that is just as deleterious as a time-
delay in the plant. Time-delays and flow time-scales should be
primary considerations when designing feedback controllers for
turbulent flows.
4.4.4 Two Degrees-of-Freedom Control. With the addition of
reference and disturbance feed-forward control, it is possible to
improve the control performance at frequencies where feedback
control is ineffective due to large sensitivity. This more sophisti-
cated two-degrees-of-freedom control is illustrated in Fig. 11.
Again, we may compute the transfer function from inputs to
output
s¼ðIþPKÞ1
½PðKþKref ÞwrþðPdPKdÞwdPKwn
Then, the error becomes
e¼swr¼SSrefwrþSSdPdwdTwn(30)
where the additional feedforward sensitivity functions are
Sref ¼IPKref (31a)
Sd¼IPKdP1
d(31b)
Two-degrees-of-freedom control may be intuitively understood
as the combination of a fast feedforward controller to get close to
a desired reference value, followed by a slow feedback controller
to mitigate model uncertainty and reject unexpected disturbances.
There are explicit bounds on model uncertainty, which determine
when combined inverse-based feedforward and feedback will out-
perform feedback alone [177].
4.5 Balanced Model Reduction. The high-dimensionality
and short time-scales associated with turbulent flows make it
infeasible to implement the model-based control strategies above
in real-time. Moreover, solving for H2and H1optimal control-
lers may be computationally intractable, as they involve either a
high-dimensional Riccati equation solve, or an expensive iterative
optimization. Instead, reduced-order models provide efficient,
low-dimensional representations of the most relevant flow mecha-
nisms. These models result in efficient controllers that may be
applied in real-time for many systems. An alternative is to develop
controllers based on the full-dimensional model and then apply
model reduction techniques directly to the full controller [178].
Model reduction is an essential data reduction that respects the
fact that the data are generated by a dynamic process. If the
dynamic process is a linear time-invariant (LTI) input–output sys-
tem, then there is a wealth of machinery available for model
reduction, and performance bounds may be quantified. Many of
these methods are based on the SVD [87,88,179], and the minimal
realization theory of Ho and Kalman [180,181]. The general idea
is to determine a hierarchical modal decomposition of the flow
state that may be truncated at some model order, only keeping the
most important coherent structures.
The POD [85,182] orders modes based on energy content, and
it has been widely used for a range of fluid dynamic reduced-
order models, many for control. POD is guaranteed to provide an
optimal low-rank basis to capture the maximal energy or variance
in a data set. In some cases, a large number of POD modes may
be required to represent non-normal energy growth, as in wall-
bounded shear flows, such as in Ref. [183].
Instead of ordering modes based on energy, it is possible to
determine a hierarchy of modes that are most controllable and
observable, therefore capturing the most input–output informa-
tion. Such balanced models give equal weighting to the controll-
ability and observability of a state, providing a coordinate
transformation that makes the controllability and observability
Gramians equal and diagonal. It is observed that for many high-
dimensional systems, the control input may only excite a few con-
trollable modes, with the remaining modes being stable. These
models have been extremely successful in the context of flow con-
trol, especially for systems with non-normal growth. However,
computing a balanced model using traditional methods is
extremely expensive computationally. In this section, we describe
the balancing procedure, as well as modern methods for efficient
computation of balanced models. A computationally efficient
suite of algorithms for model reduction and system identification
may be found in Ref. [117].
4.5.1 Discrete-Time Systems and Gramians. Until now, we
have considered continuous-time dynamical systems, as in Eq.
(5). However, the discussion of model reduction is somewhat sim-
plified using a discrete-time model
akþ1¼AdakþBdbk(32a)
sk¼CdakþDdbk(32b)
The index kmay be thought of as the kth sample of a continuous-
time system in Eq. (5), sampled every Dtby an analog-to-digital
converter. Alternatively, the discrete-time system may be a nu-
merical time-stepper. The discrete-time system may be related to
the continuous-time system by the following: Ad
¼expðADtÞ;Bd¼ÐDt
0exp ðAsÞBds;Cd¼C, and Dd¼D. Then,
ak¼aðkDtÞ, and similar for bkand sk. The assumption that bis
constant during each Dtinterval is called a zero-order-hold in con-
trol theory.
The Gramians may be approximated by full-state measurements
of the direct system in Eq. (32) and adjoint system
Fig. 11 Two degrees-of-freedom control with reference track-
ing and disturbance rejection
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-15
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
direct :akþ1¼AdakþBdbk(33a)
adjoint :akþ1¼A
dakþC
dsk(33b)
It is then possible to determine empirical Gramians without solv-
ing the Lyapunov equations in Eq. (15).
WcWe
c¼CdC
d(34a)
WoWe
o¼OdO
d(34b)
Cdand Odare snapshot matrices constructed from impulse
response simulations of the discrete direct and adjoint systems in
Eq. (32). The method of empirical Gramians is quite efficient and
was used in Refs. [181,184,185]. Note that there are N
s
adjoint
impulse response experiments required. This becomes intractable
when there are a large number of outputs (e.g., full-state
measurements).
4.5.2 Goal of Model Reduction. The goal of model reduction
is to obtain a related system ðAr;Br;Cr;DrÞ
~
akþ1¼Ar~
akþBrbk(35a)
sk¼Cr~
akþDrbk(35b)
in terms of a state ~
ak2RNrwith reduced state dimension, N
r
N
a
. Note that bkand skare the same in Eqs. (32) and (35). A bal-
anced reduced-order model should map inputs to outputs as faith-
fully as possible for a given order.
It is therefore important to introduce an operator norm to quan-
tify how similarly Eqs. (32) and (35) act on a given set of inputs.
Typically, we consider the infinity norm of the transfer functions
PðfÞand PrðfÞobtained from the full system (32) and reduced
system (35), respectively,
kPk1¢max
xr1ðPðixÞÞ (36)
To summarize, we seek a reduced-order model (35) of low order,
N
r
N
a
, so the operator norm kPPrk1is small.
4.5.3 Balanced POD. In balanced truncation (BT) [181], a
coordinate transformation is sought that makes the observability
and controllability Gramians equal and diagonal. The balancing
transformation is given by the matrix Bin the eigendecomposition
WcWoB¼BR2(37)
where Ris a diagonal matrix containing Hankel singular values
(HSVs). It is then possible to obtain a reduced-order basis by
choosing the first N
r
columns of Bcorresponding to the N
r
largest
HSVs. It has been demonstrated that modes with small energy
content may be important for control of a given input–output sys-
tem [97,183]. Therefore, instead of truncating based on energy
content, we consider truncating based on HSVs.
In practice, it may be extremely expensive to compute the Gra-
mians Wcand Woby solving a high-dimensional Lyapunov equa-
tion, and the subsequent eigendecomposition is also expensive.
Instead of solving for Gramians directly, it is possible to construct
empirical Gramians using snapshots of direct and adjoint simula-
tions, as discussed in Eq. (34). Strong connections have been
established between POD and BT [96,181185], and in Ref. [96],
POD is used to obtain low-rank approximations to the Gramians.
However, the early methods combining POD and BT do not scale
well with the number of output measurements. In fact, the number
of adjoint simulations required is equal to the number of output
measurements, N
s
, which may be quite large [186].
In Ref. [97], Rowley introduced the method of balanced
(BPOD) to address the aforementioned issues associated with
snapshot-based BT. There are two major advances introduced in
this method, which make it practical to very large dynamical
systems:
(1) Method of snapshots—Instead of computing the balancing
transformation by solving the eigendecomposition in Eq.
(37), it is possible to construct the transformation by com-
puting the SVD of
OdCd(38)
reminiscent of the method of snapshots from Refs.
[86,187,188].
(2) Output projection—To avoid computing N
s
adjoint simula-
tions, it is possible instead to solve an output-projected
adjoint equation
akþ1¼A
dakþC
dUrs(39)
where Uris a rank-N
r
POD truncation.
First, define the generalized Hankel matrix as the product of the
direct (Cd) and adjoint (O
d) snapshot matrices from Eqs. (11) and
(12), for the discrete-time system
H¼OdCd
¼
CdBdCdAdBd CdAmc1
dBd
CdAdBdCdA2
dBd CdAmc
dBd
⯗⯗
..
.
CdAmo1
dBdCdAmo
dBd CdAmcþmo2
dBd
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
(40)
Next, we factor H
4
using the SVD
H¼URV¼½U1U2R10
00

V
1
V
2

¼U1R1V
1(41)
For a given desired model order N
r
N
a
, only the first N
r
col-
umns of Uand Vare kept, along with the first N
r
N
r
block of R.
This yields a bi-orthogonal set of modes given by
direct modes :Ur¼CdVrR1=2
r(42a)
adjoint modes :Wr¼O
dUrR1=2
r(42b)
These modes are bi-orthogonal, W
rUr¼INrNr, and Rowley
[97] showed that the direct and adjoint modes, Uand W, establish
the change of coordinates that balance the empirical Gramians:
U¢CdVR1=2and B1¼W¢R1=2UOd. Moreover, Ur
and Wrare the first N
r
-columns of the balancing transformation.
Now, these modes allow us to project our original system onto
a (balanced) reduced-order model or order N
r
Ar¼W
rAdUr(43a)
Br¼W
rBd(43b)
Cr¼CdUr(43c)
One of the key benefits of BT is that there is an upper bound on
the error of a given order truncation
Upper bound :kPPrk1<2X
n
j¼Nrþ1
rj
4
The powers m
c
and m
o
in Eq. (40) signify that data must be collected until the
matrices Cdand O
dbecome full rank, after which the controllable/observable
subspaces have been sampled.
050801-16 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
gPC [455], and gPC based on short-time flow-map composition
[456]. There has been a recent explosion of interesting methods for
UQ [457] and stochastic dynamics [458,459].
In addition, the finite-time Lyapunov exponent (FTLE) has
been a rapidly developing data-driven technique in the analysis of
time-varying fluid fields [460467]. Like DMD, this method only
relies on velocity field measurements, either from simulations or
experiments. FTLE analysis identifies time-varying coherent
structures in fluids that are analogous to stable and unstable mani-
folds. Thus, FTLE provides a quantitative technique to visualize
flows and identify regions of separation, recirculation, dispersion,
material attraction, and high sensitivity to perturbations. This
method has been used to analyze aortic blood flows for predictions
involving recirculation regions, separation, and stenosis [468].
FTLE has also been used to study biopropulsion [469471], fluid
mixing in large bodies of water [472474], and to understand tur-
bulent structure [463,464,467,475]. FTLE provides a measure of
sensitivity, which is essential in the quantification and manage-
ment of uncertainty. Sensitivity and coherence are closely related
to the calculation of almost invariant sets [476478], using set-
oriented methods [479,480], and to eigenvectors of the Perron
Frobenius operator.
8.2.4 Design of Experiments. Many of the data-driven techni-
ques discussed above suggest innovations in experimental design.
Understanding what measurements must be acquired to design
models and controllers from experimental data is an important
part of turbulence control. UQ and sensitivity analysis can offer
some guidance about what design factors have the most impact on
measurement quality, and where the flow is most sensitive to
actuation. Compressive sensing may allow for improved band-
width through a principled reduction in the spatial and temporal
resolution of measurements required for signal reconstruction.
ML provides an exploratory protocol for actuating the system into
new and beneficial dynamic regimes.
8.3 Advanced Nonlinear Models, Controllers, and
Closures. As discussed above, there may be serious limitations to
physics-based modeling of flows with strongly nonlinear dynam-
ics. Consequently, model-based control will be challenged by the
accuracy of the model. In addition, control design methods gener-
ally assume either a working linearized model or a well-
understood nonlinearity. At the same time, dramatic advances in
data-driven methods, such as system identification and machine
learning, have produced powerful new tools in turbulence control.
This trend is accelerated by the tremendous global resource
investment in machine learning across all physical sciences. How-
ever promising these new methods are, there will continue to be
many compelling reasons to develop improved nonlinear models,
controllers, and closures. First, physics-based models are inter-
pretable and allow for the inclusion of expert human knowledge.
Second, understanding the fundamental reasons why a given con-
troller works is central in developing this human intuition. Such
intuition is critical when deciding on what control strategy (hard-
ware, logic, etc.) to employ in a new situation. Third, a physically
interpretable model might lead to a simple control law with one or
a few easily tunable parameters. These first-principles models can
be expected to develop more slowly as the mathematical chal-
lenges are enormous. Yet, they will undoubtedly remain a critical
part of engineering turbulence control. Hence, understanding of
the physical mechanisms underlying effective turbulence control
will remain a critical enabler—regardless of the control strategy,
lest we lose mastery of the machinery we employ.
There are a host of advanced modeling techniques, including
powerful generalizations of POD, which are useful for obtaining
efficient parameterized models of complex turbulent flows using
high-performance computation [481484]. The gappy POD
method provides the ability to sparsely sample a system and still
evaluate the POD and terms in the Galerkin projection [485,486].
In addition, there are reduced-basis methods for PDEs [487] and
the associated discrete empirical interpolation method [488490],
which approximates nonlinear terms by evaluating the nonlinear-
ity at a few specially determined points. There have also been
powerful advances in the filtering of turbulent systems
[434,491493]. Finally, robust control has also been used as a
method of understanding underlying nonlinear mechanisms in tur-
bulence [494]. In addition, advanced measurement capabilities
contribute to improving our understanding of high Reynolds num-
ber turbulent flows [495,496].
These advanced models are broken into three critical pieces—
modeling, closure, and control design—although the true division
may be more subtle. First, nonlinear model identification needs to
be advanced comprising both structure and parameter identifica-
tion. Significant progress has been made with 4D VAR methods
[497,498]. Second, turbulence closures have always been a critical
part of reduced-order turbulence models. Yet, eddy-viscosity
based subscale models are too coarse to resolve critical frequency
cross-talk mechanisms. Closure schemes based on a Gaussian
approximation [273], on a maximum entropy principle [499501],
and on finite-time thermodynamics [502504] hold corresponding
promises. Finally, the nonlinear theory of control needs to be sig-
nificantly advanced. It is currently unclear how nonlinear models
and closures will be used by control theorists, motivating the need
for a common framework, like the state-space and frequency do-
main framework in Sec. 4for linear systems.
8.3.1 Graph-Theoretic Flow Control. Advances in network
science have recently been applied with success in fluid systems
[505], providing a new set of mathematical techniques for com-
plex systems. The resulting graph models may be based on snap-
shot clusters [439], or on a sparsified graph model for the
underlying vortex network [311]. The integration of methods
from network science and network control theory in turbulence
control is promising, especially since many network control tech-
niques have been developed to handle nonlinear systems.
The past two decades have marked numerous advances in
graph-theoretic control theory surrounding multi-agent systems,
and network science has experienced significant recent attention
[506510]. Networks are often characterized by a large collection
of individuals (represented by nodes) that each executes their own
set of local protocols in response to external stimulus [511]. This
analogy holds quite well for a number of large graph dynamical
systems, including animals flocking [512,513], multirobotic coop-
erative control systems [514], sensor networks [515,516], biologi-
cal regulatory networks [517,518], and the internet [519,520], to
name a few. Similarly, in a fluid packets of vorticity may be
viewed as nodes in a graph, which interact collectively according
to global rules (i.e., governing equations) based on local rules (dif-
fusion, etc.) as well as their external inputs summed across the
entire network (i.e., convection due to induced velocity from the
Biot–Savart law) [311].
In large multi-agent systems, it is often possible to manipulate
the large-scale behavior with leader nodes that enact a larger su-
pervisory control protocol to create a system-wide minima that is
favorable [512,521523]. The fact that birds and fish often act as
local flows with large-scale coherence, and that leaders can
strongly influence and manipulate the large-scale coherent motion
[512,513], is promising when considering network-based fluid
flow control. Based on the network-control methodology used to
analyze schooling fish and flocking birds [512,513], there is an
appealing goal of schooling turbulence by collecting and harness-
ing distributed multiscale eddies into a collective organization, or
community, with favorable large-scale properties.
Recent work investigating the number of leader nodes required
for structural controllability of a network [522] suggests that
large, sparse networks with heterogenous degree distributions,
6
6
The degree distribution of a network is the probability distribution of the
number of connections each node has to other nodes. All scale-free networks are
inherently sparse, with heterogeneous degree distribution [509].
050801-38 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
measurement was analyzed for a turbulent channel flow in Ref.
[209]. Without the suboptimal approximation, a model-predictive
control (MPC) framework may be used for receding-horizon opti-
mal control, although this is computationally intensive. MPC has
been used in Refs. [210,211] and is discussed in more detail in a
recent review [171].
In Bewley and Liu [212], both H2and H1controllers were
developed and applied to single wavenumber pairs in a plane
channel flow with three-dimensional perturbations. In Ref. [213],
robust control was applied to the extended problem of 2D multi-
wavenumber control with localized actuation and sensing in plane
Poiseuille flow. H
ogberg et al. [214] extended the analysis of
Bewley and Liu [212] to include multiple wavenumbers and state
estimation using a Kalman filter based on wall measurements.
Their three-dimensional DNS used boundary condition forcing,
and it was shown that state-estimation was not as effective as full-
state feedback.
H
ogberg and Henningson [215] demonstrated the effectiveness
of linear H2optimal control in spatially developing boundary
layers, even with moderately strong nonlinearities present. In a
follow-up work [216], they extended this analysis to include full-
state estimation based on a number of wall measurements, similar
to Ref. [214].
4.6.2 Use of Reduced-Order Models. As discussed earlier, the
development of controllers for high-dimensional linear systems
may be extremely computationally intensive since they depend on
the solution to Riccati equations, involving OðN3
aÞoperations, or
expensive iterations. Further, once a high-dimensional controller
has been designed, it is expensive to compute the resulting control
action, introducing unwanted latency in the control loop. This la-
tency significantly limits control bandwidth and may prevent real-
time control of high-dimensional systems with fast time-scales.
These issues are especially pronounced in flow control.
In the timely work of A
˚kervik et al. [217], reduced-order mod-
els were developed by using a few non-normal global eigenmodes
of the linearized Navier–Stokes equations as a basis for Galerkin
projection. Based on the reduced-order model, an LQG controller
is able to attenuate global oscillations in a separated boundary
layer. The use of non-normal global modes has been important to
capture the transient energy growth associated with wall-bounded
shear flows. Yet, these approaches have limited applicability to
unstable advection-dominated flows. Transient growth implies a
sensitivity issue: a small difference in the initial condition or a
small perturbation in the system can give rise to significantly dif-
ferent solutions. Hence, unavoidable truncation errors of reduced-
order models based on global modes may cause similar
differences in the solutions. The sensitivity problem is less pro-
nounced at lower Reynolds numbers or for weakly unstable flows.
After the seminal work of Rowley [97], introducing a computa-
tionally efficient method to compute balanced reduced-order
models, the use of balanced models has become a mainstay in
model-based linear flow control. The resulting models are able to
capture transient energy growth with significantly fewer modes
than POD, as demonstrated in Ref. [183] on the transitional channel
flow. BPOD was also shown to outperform standard POD/Galerkin
methods for the separation control of an airfoil at low Reynolds
number [218], using an immersed-boundary method [219,220].
The paper of Bagheri et al. [221] provides one of the most
complete and clearly presented narratives on closed-loop control
based on balanced models. In this work, they develop models
using snapshot-based BT. Snapshot-based methods are necessary
when the state dimension N
a
is large, since the Amatrix is size
N2
a, which may not be representable in system memory. The
matrix-free methods advocated in Ref. [221] use the CFD time-
stepper to find the next flow state, rather than explicitly computing
a large Amatrix. These methods may be used to solve large eigen-
value problems, for instance, using Arnoldi iteration, thus bypass-
ing the creation and factorization of a large matrix. In this paper,
H2control is developed to suppress two-dimensional boundary
layer perturbations. Reference [222] presents a three-dimensional
generalization of Ref. [221] using LQG to delay transition in a
boundary layer.
As described in Sec. 4.5.4, the ERA produces balanced models
that are equivalent to BPOD, but without the need for adjoint sim-
ulations [189]. ERA has been rapidly adopted in flow control
because of the fact that it is based purely on input–output data,
making it a system identification method, rather than a model
reduction method. ERA was used to study and control cavity flow
oscillations and combustion oscillations [223,224]. In Semeraro
et al. [225], ERA models were used to develop LQG control to
suppress three-dimensional TS waves in a transitional boundary
layer using a volume force actuation. This paper is well-written
and thought provoking, and an illustration of the closed-loop con-
trol topology is shown in Fig. 12. Interestingly, one of the earliest
experimental demonstrations of active feedback suppression of
turbulence was applied to cancel TS waves using a downstream
heating element and a phase-shifted measurement feedback,
resulting in an increase in the transition Reynolds number [18].
4.7 Case Study: Wall Turbulence Control and Skin-
Friction Reduction. There is a tremendous industrial motivation
to reduce turbulent skin-friction drag. This problem has thus
Fig. 12 Schematic of the closed-loop controller for transition delay of a flat-plate boundary
layer (Reproduced with permission from Semeraro et al. [225]. Copyright 2013 by Cambridge
University Press). Here, wcorresponds to sensors sand /corresponds to actuators b.
050801-18 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
received significant attention, and corresponding progress has
been made [10,40,226]. Reduced-order models for control based
on LQG and the robustifying loop-transfer recovery were used to
reduce skin-friction drag in a channel flow [227,228], resulting in
eventual relaminarization of the flow. Reduced-order models were
also used with blowing/suction at the wall to reduce skin friction
in simulations [229], and reduced-order Kalman filters were
developed based on BT models [133].
In addition to the mechanisms and techniques discussed, there
have been many efforts to reduce turbulent skin friction drag
[230234] using various techniques, such as traveling-wavelike
actuation including blowing/suction and wall deformation.
Increasing heat transfer while decreasing drag is also an important
area of research [235]. In addition, system identification and con-
trol were used for the linear feed-forward control of turbulent
boundary layer fluctuations by exploiting coherent structures and
predicting their behavior downstream [236]. The same reduction
in turbulent boundary layer fluctuations was explored using
model-based control in an experiment using an array of synthetic
jet actuators [237].
4.8 Case Study: Cavity Flow Control. The work surround-
ing the control of open cavity flows represents one of the most
complete and compelling success stories in active flow control
[238]. Cavity flow control has many applications, such as the sup-
pression of oscillations in weapons bays in high-speed flight
[239], which is an inherently high Reynolds number phenomena.
A more complete discussion of the cavity flow control problem is
discussed in reviews [34,240,241].
Suppression of cavity oscillations has benefited from a combi-
nation of new physical insights combined with simulations [242],
advances in advanced Galerkin modeling [243,244], and the prac-
tical implementation of control in experiments [245248]. Excit-
ing new methods, such as control based on neural networks
[46,249] and the use of ERA [225], have also been investigated.
In some regimes, oscillations are self-sustained and inherently
nonlinear. To identify models suitable for control, Rowley et al.
[245] stabilized the oscillations using Nyquist plots to tune the
phase of controllers based on experimentally obtained frequency
response data. A model was then obtained for the closed-loop sys-
tem using system identification, and model for the unstable system
was derived.
4.9 Potential Impact and Challenges. All of the models dis-
cussed above, and therefore the corresponding controllers, depend
on the placement of actuators and sensors, as these directly affect
Band Cin Eq. (5). Actuator and sensor placement was recently
investigated for boundary layer transition delay [250]. Impor-
tantly, this study provides a clear comparison of feedforward (sen-
sor upstream of actuator) versus feedback (sensor downstream of
actuator) control. In particular, it is shown that disturbance-
feedforward control is effective sometimes, but is sensitive to
additional disturbances and uncertainty. Feedback, on the other
hand, effectively rejects disturbances and compensates for unmod-
eled dynamics. However, if the sensor is too far downstream, the
time-delay between actuation and sensing dramatically limits ro-
bust performance, which is consistent with the discussion above.
Given the performance of drag reduction, in the neighborhood
of 20%, and transition delay cited in the literature, it may be
somewhat surprising that active closed-loop control is not being
utilized on mainstream aircraft, trains, or automobiles. There are a
number of reasons why the studies above are largely numerical.
First, many model-reduction techniques require extensive and
invasive information about the plant, although ERA is less inva-
sive. Second, the time-scales associated with real experimental
and industrial turbulence are extremely fast, so that reducing time
delays is a significant challenge. Time delays may originate from
latency involved in the computation of a control decision, and
they may also arise from convective time scales from actuators to
sensors. Finally, the development of advanced sensor and actuator
hardware will be a major enabling factor in the practical imple-
mentation of these methods.
5 Prototypes for Linear and Nonlinear Dynamics
Despite the powerful tools for linear model reduction and con-
trol, the assumption of linearity is often overly restrictive for real-
world fluids. Turbulent fluctuations are inherently nonlinear, and
often the goal is not to stabilize an unstable fixed point but rather to
change the nature of a turbulent attractor. Moreover, it may be the
case that the control input is either a bifurcation parameter itself, or
closely related to one, such as the control surfaces on an aircraft.
The degree of nonlinearity is most easily characterized in a
Galerkin modeling framework (Sec. 5.1). The subsequent sections
(Secs. 5.25.5) provide prototypic examples of linear, weakly,
moderately, and strongly nonlinear dynamics as evidenced in
many fluid flows and corresponding control strategies. Section 5.6
concludes with enablers and show-stoppers of nonlinear model-
based control design.
5.1 Galerkin Model. In this section, different degrees of sys-
tem nonlinearity are introduced. For simplicity, we consider an
incompressible velocity field uðx;tÞin a finite steady domain x2X.
Let a¼½a1;a2;;aNaTrepresent an N
a
-dimensional approxima-
tion of the flow state. The coordinates a
i
,i¼1,…, N
a
may, for
instance, represent the coefficients in a Galerkin expansion
uðx;tÞ¼usðxÞþX
Na
i¼1
aiðtÞuiðxÞ(46)
where usrepresents the steady Navier–Stokes solution, and
ui;i¼1;;Naare suitable expansion modes.
Let b¼½b1;b2;;bNbTcharacterize the actuation. One exam-
ple involves N
b
volume forces PNb
ibiðtÞgiðxÞwith the individual
fields gi;i¼1;;Nb. To simplify the discussion, only a single
component is assumed, N
b
¼1, and b¼b
1
denotes its amplitude.
The sensor signals s¼½s1;s2;;sNsTmay represent velocity
components. In this case, they are affinely related to expansion
coefficients via Eq. (46). The sensing plays no role in this section.
Focus is placed on finding a proper linearized system whenever
possible.
For the sake of simplicity, we assume the structure of a Galer-
kin system with a single volume force as actuation. Then, the dy-
namics in Eq. (2) take the form
fiða;bÞ¼X
Na
j¼1
lij ajþX
Na
j;k¼1
qijk ajakþgib(47)
The constant term of the dynamics vanishes since the basic mode
usis assumed to be the steady Navier–Stokes solution.
Near the fixed point a0, the quadratic term may be neglected
yielding the linear dynamics (5) of Sec. 4. Physically meaningful
linearizations may also be effected far away from the origin. The
critical element is the notion of “fast” and “slow” modes. The fast
modes describe the evolution of coherent structures and may best
be considered as a fluctuation. Let us assume i¼1;;Nf
arepre-
sent the indices of the fast modes. The slow modes have signifi-
cantly lower frequencies and may best be attributed to a base-flow
variation. Let the remaining indices i¼Nf
aþ1;;Narepresent
such slow modes. By a trivial operation, the dynamics have the
form
fi¼cB
iþX
Nf
a
j¼1
lB
ij ajþhB
iþgib
where
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-19
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
cB
i¼X
Na
j¼Nf
aþ1
lij ajþX
Na
j;k¼Nf
aþ1
qijk ajak
lB
ij ¼X
Na
k¼Nf
aþ1ðqijk þqikjÞak
hB
i¼X
Nf
a
j;k¼1
qijk ajak
Here, the quasi-constant cB
imeasures the distance to the fixed point
a0, the quasi-linear term with the coefficients lB
ij incorporates
slow base-flow changes, and hB
irepresents a quadratic nonlinearity
in the fluctuating modes. hB
ihas slow and fast components. If the vol-
ume force is in feedback with the fluctuations, cB
iþhB
i0ina
short-term average. In this case, fi¼PNf
a
j¼1lB
ij ajþgib;i¼1;;Nf
a
may be a suitable linear representation of the fast dynamics.
The slow modes are slaved to an approximate manifold in state
space defined by dai=dt ¼fiðaÞ0;i¼Nf
aþ1;;Naand are
driven by the Reynolds stress contributions of the fast modes, or,
equivalently, by the slow component of hB
i.
In Secs. 5.2 through 5.5, four prototypic examples are dis-
cussed: (A) an oscillation around the fixed point; (B) a self-
excited amplitude-limited oscillation; (C) frequency cross-talk
with two different frequencies over the base-flow deformation;
and (D) the remaining irreducible cases.
5.2 Linear Dynamics. First, a small oscillatory fluctuation
around a steady solution is considered. Examples include the flow
over a backward-facing step [251] at subcritical Reynolds number
with noise, transition delay of a boundary layer [252], or stabiliza-
tion of a cylinder wake [19,253]. In Sec. 5.2.1, a control-oriented
oscillator model is presented as a least-order description. Section
5.2.2 exemplifies the powerful method of energy-based control
design for this model.
5.2.1 Oscillator Model. The considered flows can be
described by
uðx;tÞ¼usðxÞþuuðx;tÞ(48a)
uuðx;tÞ¼a1ðtÞu1ðxÞþa2ðtÞu2ðxÞ(48b)
Here, ui,i¼1, 2, correspond to the cosine and sine contribution
of the first harmonic or the real and imaginary part of the unstable
complex eigenmode. Higher harmonics are neglected. By con-
struction, the stable or unstable fixed point is as¼0.
The linearized version of the dynamics (47) reads
d
dt a¼A0aþBb(49)
where
A0¼ruxu
xuru
"#
B¼0
g
"#
Without loss of generality, the modes can be rotated so that the
gain in the first component vanishes. A similar equation holds for
the measurement equation. As discussed in Sec. 4.1, the matrices
A0and Bdepend on the fixed point asand the bifurcation parame-
ters l.
The linear dynamics (49) may also be an acceptable approxima-
tion for turbulent flows with dominant oscillatory behavior. In the
triple decomposition, the velocity field is partitioned into the
mean flow u0, the periodic oscillation uu, and an uncorrelated sto-
chastic fluctuation ur. The periodic fluctuation may be described
by two modes (48b), again. Equation (49) is obtained by substitut-
ing the velocity field u¼
uþuuþurin the Navier–Stokes equa-
tion, projecting onto the modes ui, and filtering out anything but
the dominant frequency. In this case, the growth-rate r
u
¼0 has to
vanish and the amplitude selection mechanism cannot be resolved
by the two-dimensional Galerkin model. In particular, the fixed
point 0of the Galerkin model represents the mean flow which is
not the steady solution of the Navier–Stokes equation. Yet, the
model may be good enough to predict the right actuation for
increasing or decreasing the fluctuation near the limit cycle.
5.2.2 Energy-Based Control Design. Control design of the
linear system (49) can be performed with the methods of Sec. 4.
Here, we illustrate the idea of energy-based control which is par-
ticularly suited for nonlinear dynamics. Moreover, energy-based
control has a kinematic relation to phasor control and an energetic
relation to opposition control.
The growth-rate r
u
is assumed positive and small enough so
that the time-scale of amplitude growth is small compared to the
time-scale of oscillation. In this case, the state can be approxi-
mated by a1¼rcos hand a2¼rsin h, where rand x¼dh/dt are
slowly varying functions of time. The amplitude evolution is
given by
dr2
dt ¼2a1
da1
dt þ2a2
da2
dt ¼2rur2þ2ga
2b
The control goal is an exponential decay of the amplitude with
r
c
<0, i.e.,
dr2
dt ¼
!2rcr2
Eliminating the time-derivative in both equation yields
dr2
dt ¼rcr2¼rur2þga
2b
The control command bincreases (decreases) the energy r
2
/2 if it
has the same (different) sign as a
2
. To prevent wasting actuation
energy with the wrong phase, the linear ansatz b¼ka
2is made.
The gain k>0 is determined by substituting a2¼rcos hin the
energy equation and averaging over one period. The resulting con-
trol law reads
b¼2rcru
ga2(50)
The gain increases with the difference between the natural and
design growth rate r
c
r
u
and decreases with the forcing constant
gin the linear dynamics. The factor 2 arises from the fact that the
actuation is only effective in the [0, 1]
T
direction.
It may be emphasized that the achieved decay of the fluctuation
energy is an average value over one period. When a
2
¼0, no
change of the energy is achievable. Second, the construction of
the control law is based on designing a dissipative term ga
2
bin
the power balance. Hence, the actuation mechanism may be con-
sidered an energetic opposition control. Third, the actuation com-
mand bscales with fluctuation amplitude r. Its phase lags 270 deg
with respect to the first coordinate a
1
. Hence, on a kinematical
level, the actuation describes a phasor control. Figure 13 describes
the corresponding unactuated and actuated solution of Eq. (49).
The described energy-based control design is very simple and
immediately reveals the physical mechanism. It is easy to general-
ize for nonlinear systems, particularly if the fluctuation is com-
posed of clean frequency components. Related approaches are
called Lyapunov control design and harmonic balance.
050801-20 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
5.3 Weakly Nonlinear Dynamics. As a refinement to the lin-
earization, mean-field theory is recapitulated [254,255] providing
an important nonlinear amplitude selection mechanism. The onset
of vortex shedding behind a cylinder is one well investigated
example fitting this description [256]. The following two sections
outline the dynamical model and corresponding control design,
respectively.
5.3.1 Mean-Field Model. Qualitatively, mean-field theory
describes the feedback mechanism between the fluctuations and
the base flow. The fluctuation gives rise to a Reynolds stress
which changes the base flow. The base-flow deformation gener-
ally reduces the production of fluctuation energy with increasing
fluctuation level until an equilibrium is reached. The resulting
evolution equations are also referred to as weakly nonlinear
dynamics, as they describe a mild form of nonlinearity.
The fluctuation has the same representation as in Sec. 5.2. How-
ever, the base flow is allowed to vary by another mode u3, called
the zeroth, base-deformation or shift mode. This mode can be
derived from the (linearized) Reynolds equation and it is assumed
to be slaved to the fluctuation level. The resulting velocity field
ansatz reads
uðx;tÞ¼usðxÞþuuðx;tÞþuDðx;tÞ(51a)
uuðx;tÞ¼a1ðtÞu1ðxÞþa2ðtÞu2ðxÞ(51b)
uDðx;tÞ¼a3ðtÞu3ðxÞ(51c)
The evolution equation is given by
d
dt
a1
a2

¼Aa3
ðÞa1
a2

þBb(52a)
a3¼auða2
1þa2
2Þ(52b)
where
Aða3Þ¼A0þa3A3
A0¼ruxu
xuru
"#
A3¼bucu
cubu
"#
B¼0
g
"#
Without loss of generality, a
u
>0. Otherwise, the sign of the
mode u3must be changed. A nonlinear amplitude saturation
requires the Landau constant to be positive b
u
>0.
For a
3
0, Eq. (52) is equivalent to Eq. (49). However, Eq. (52)
has a globally stable limit cycle with radius r1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ru=aubu
pin
the plane a3¼a1
3¼ru=bu, and with center ½0;0;a1
3.
In the framework of weakly nonlinear stability theory, the
growth-rate is considered a linear function of the Reynolds num-
ber r
u
¼j(Re Re
c
), where Re
c
corresponds to its critical value.
The other parameters are considered to be constant. This yields
the famous Landau equation for the amplitude dr=dt ¼
rurbr3;b¼aubuand a corresponding equation for the fre-
quency. In particular, the Landau equation explains the famous
square-root amplitude law r/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Re Rec
pfor supercritical Reyn-
olds numbers assuming a soft bifurcation. We refer to the litera-
ture for a discussion of the hard subcritical bifurcation with
quintic nonlinearity [255].
Mean-field theory explains the stabilizing feedback mechanism
between the harmonic oscillatory structure and the mean-field de-
formation. A refined weakly nonlinear expansion also takes higher
harmonics into account. The first harmonic, via the quadratic
Navier–Stokes term, generates not only a mean-flow deformation
but also a second harmonic which changes, in turn, the mean flow.
The nonlinear interaction of the first and second harmonics pro-
duces a third harmonic, etc. Dusek et al. [257] derived a corre-
sponding harmonic expansion from the Navier–Stokes equation in
the neighborhood of a supercritical Hopf bifurcation. Let >0be
the small amplitude of the first harmonic; then, the nth harmonics
are shown to scale in geometric progression, i.e.,with
n
. Hence,
higher harmonics may be neglected near the onset of fluctuations.
Even for periodic flow far beyond the onset, the second harmonic
is observed to be 1 order of magnitude smaller than the corre-
sponding harmonic component. A similar observation holds for
turbulent flow with dominant periodicity.
Intriguingly, even turbulent flows with dominant periodic
coherent structures may be described by Eq. (52). One example is
the wake behind a finite cylinder [258]. In this case, the rationale
is the velocity decomposition with an added uncorrelated stochas-
tic fluctuation ur:u¼usþa3u3þuuþur. In this case, Eq. (52)
remains valid if urscales with the oscillatory fluctuation level and
the base flow deformation is slaved to this level as well.
5.3.2 Nonlinear Control Design. The mean-field model and
variants thereof have been successfully used for the model-based
stabilization of the cylinder wake at Re ¼100 [44,259,260]. In
these studies, an energy-based control design as discussed in Sec.
5.2 has been used to prescribe a fixed decay rate of the model. Fig-
ure 14 illustrates an unactuated and actuated transient.
It may be noted that Eq. (52) can be considered a linear parame-
ter varying model with the shift-mode amplitude a
3
, or, equiva-
lently, with the fluctuation level r
2
as the parameter. Thus, the
linear control design of Sec. 4is immediately applicable for the
linear system corresponding to the actual value of this parameter.
King et al. [261] elaborated other nonlinear control design techni-
ques for the mean-field model employing input–output lineariza-
tion, Lyapunov-based synthesis, and backstepping. The mean-field
model has also been the basis for optimal control of the
Navier–Stokes equation [262], using the simulation to improve the
model and the model as a surrogate plant for control design.
Fig. 13 Phase portrait of oscillatory linear dynamics (49). The
dashed trajectory corresponds to the unactuated dynamics
while the solid trajectory corresponds to actuated dynamics
with Eq. (50). The chosen parameters are r
u
50.1, r
c
520.1,
x
u
51, g51 implying k50.4.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-21
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
5.4 Moderately Nonlinear Dynamics. Some oscillatory
flows may be tamed by direct mitigation with models and methods
described in Sec. 5.3. Not all plants, particularly turbulent flows,
have the actuation authority for such a stabilization. However,
periodic forcing at high frequency has mitigated periodic oscilla-
tions in a number of experiments. Examples are the wake stabili-
zation with a oscillatory cylinder rotation [74], suppression of
Kelvin–Helmholtz vortices in transitional shear layers [263],
reduction of the separation zone in a high-lift configuration [48],
and elongation of the dead-water region behind a backward facing
step [47]. Also a periodic frequency at 60–70% of the dominant
shedding frequency may substantially delay the vortex formation
in wall-bounded shear-layers [49] and D-shaped cylinders [45].
Sections 5.4.1 and 5.4.2 outline a corresponding modeling and
control strategy.
5.4.1 Generalized Mean-Field Model. Here, a generalized
mean-field model for such frequency-cross talk phenomena is
reviewed from Ref. [48]. Let uudenote the natural self-amplified
oscillation represented by two oscillatory modes u1and u2. Analo-
gously, the actuated oscillatory fluctuation uais described by two
modes u3and u4. The base-flow deformation due to the unstable
natural frequency x
u
and stable actuation frequency x
a
is
described by the shift-modes u5and u6, respectively. The result-
ing velocity decomposition reads
uðx;tÞ¼usðxÞþuDðx;tÞ
þuuðx;tÞþuaðx;tÞ(53a)
uuðx;tÞ¼a1ðtÞu1ðxÞþa2ðtÞu2ðxÞ(53b)
uaðx;tÞ¼a3ðtÞu3ðxÞþa4ðtÞu4ðxÞ(53c)
uDðx;tÞ¼a5ðtÞu5ðxÞþa6ðtÞu6ðxÞ(53d)
Generalized mean-field arguments yield the following evolu-
tion equation:
d
dt
a1
a2
a3
a4
2
6
6
43
7
7
5¼Aa5;a6
ðÞ
a1
a2
a3
a4
2
6
6
43
7
7
5þBb(54a)
a5¼aua2
1þa2
2
 (54b)
a6¼aaa2
3þa2
4
 (54c)
where
Aða5;a6Þ¼A0þa5A5þa6A6
A0¼
ruxu00
xuru00
00raxa
00xuru
2
6
6
6
6
4
3
7
7
7
7
5
A5¼
buu cuu 00
cuu buu 00
00bau cau
00cau bau
2
6
6
6
6
4
3
7
7
7
7
5
A6¼
bua cua 00
cua bua 00
00baa caa
00caa baa
2
6
6
6
6
4
3
7
7
7
7
5
B¼
0
0
0
g
2
6
6
6
6
4
3
7
7
7
7
5
It should be noted that a vanishing actuation implies
a
3
¼a
4
¼a
6
¼0 and yields the mean-field model of Eq. (52) mod-
ulo index numbering. In the sequel, we assume that the fixed point
is unstable (r
u
>0) and the limit cycle is stable (b
uu
>0). Simi-
larly, a vanishing natural fluctuation a
1
¼a
2
¼a
5
yields another
mean-field model, again modulo index numbering. In the sequel,
we assume that the actuated structures vanish after the end of
actuation, implying r
aa
<0 and b
aa
>0 in the model.
An interesting aspect of Eq. (54) is the frequency cross-talk.
The effective growth-rate for the natural oscillation reads
A11 ¼rubuuauða2
1þa2
2Þbuaaaða2
3þa2
4Þ(55)
The forcing stabilizes the natural instability if and only if b
ua
>0.
Complete stabilization implies A
11
0. From Eq. (55), such com-
plete stabilization is achieved with a threshold fluctuation level at
the forcing frequency
a2
3þa2
4ru
aabua
Thus, increasing the forcing at higher or lower frequency can
decrease the natural frequency.
The generalized mean-field model has been fitted to numerical
URANS simulation data of a high-lift configuration [48] with
high-frequency forcing. This model also accurately describes the
experimental turbulent wake data with a stabilizing low-frequency
forcing [264], as shown in Fig. 15.
5.4.2 Nonlinear Control Design. The model above may guide
in-time control [265] and adaptive control design providing the
minimum effective actuation energy [266].
Figures 16 and 17 show an unactuated and stabilizing transient
with the parameters of Table 1. For simplicity, all nonlinear
Fig. 14 Phase portrait of weakly nonlinear dynamics. The
dashed trajectory corresponds to the unactuated dynamics
while the solid trajectory corresponds to actuated dynamics.
The chosen parameters of Eq. (52) are r
u
50.1, x
u
51, a
u
51,
b
u
51, c
u
50, and the forced decay rate r
c
520.1. The globally
stable limit cycle lies on the parabolic inertial manifold.
050801-22 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
terms of the forced oscillator are assumed to vanish, since the
linear term is already stabilizing. For the same reason, all
nonlinear frequency terms are set to zero as the frequency cross-
talk is not affected by small frequency variations. In principle,
Eq. (54) can be generalized for an arbitrary number of
frequencies.
It should be noted that linear, weakly nonlinear, and moderately
nonlinear systems show different actuation response which may
be tested in experiments. For the linear system, a global superpo-
sition principle for the actuation response holds: Let b1ðtÞand
b2ðtÞbe two open-loop actuation commands and a1ðtÞand a2ðtÞ
be the corresponding solutions. Let the linear combination b¼
kb1þlb2with real coefficients kand lbe a new actuation com-
mand. Then, a¼ka1þla2is a corresponding new solution. The
weakly nonlinear system has a similar local superposition princi-
ple for an infinitesimal perturbation of the stable unforced limit
cycle. In contrast, the frequency cross-talk mechanism of the mod-
erately nonlinear system is not resolved by a local linearization
around the stable unforced limit cycle. The linearization of Eq.
(55) removes the frequency cross-talk.
5.5 Strongly Nonlinear Dynamics. Sections 5.5.1 and 5.5.2
outline examples of strongly nonlinear dynamics not fitting in the
previous categories and examples of corresponding control
design, respectively.
5.5.1 Examples of Strongly Nonlinear Dynamics. In the case
of moderately nonlinear dynamics, different frequencies interact
over the slowly varying base flow. The corresponding solutions
are well described by the local linearization around the short-term
averaged flow. In other words, the solution lives on a manifold in
state space and evolves according to a locally linear dynamical
system. This may be a useful approximation even for turbulent
flows with one or few dominant frequencies. The turbulence cas-
cade is, however, dominated by triadic interactions involving non-
vanishing frequencies. The corresponding energy flow from large
scales (low frequencies) to small scales (high frequencies) is not
well characterized by moderately nonlinear dynamics. No lineari-
zation is able to describe such a cascade. A similar behavior
applies to the inverse cascade from dominant to larger scales, e.g.,
via vortex pairing. We shall call such dynamics “strongly non-
linear” implying that they are irreducible to even locally linear
systems. The decay of 2D turbulence by optimal initial conditions
is a beautiful configuration illustrating the complexity associated
with strong nonlinearity [267].
Mathematically, strong nonlinearity may be cast in a form simi-
lar to the linear dynamics in Eq. (5), but with
Fig. 15 Flow visualization of the experimental wake behind a
D-shaped body without (a) and with symmetric low-frequency
actuation (b) (Reproduced with permission from Mark Pastoor.)
The D-shaped body is shown with five pressure sensors on the
rear face, and the arrows at the corners indicate the employed
zero net mass flux actuators. (a) Natural wake with vortex shed-
ding and (b) actuated partially stabilized wake.
Fig. 16 Phase portrait of moderately nonlinear dynamics (54).
The dashed trajectory corresponds to the unactuated dynamics
while the solid trajectory corresponds to actuated dynamics
with Eq. (50). The chosen parameters are enumerated in
Table 1.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-23
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
A¼A0þX
Na
i¼1
aiAi(56)
The variation of the matrix Acan neither be ignored nor consid-
ered slowly varying—in contrast to linear, weakly nonlinear or
moderately nonlinear behavior.
We arrive at the following classification of system dynamics
(see Fig. 18). The dynamics are either linearizable near the fixed
point, or they are locally linearizable with one or few frequencies,
or they belong to the reducible cases. The boundaries between
two neighboring cases are, of course, a bit blurred depending on
the considered error tolerance of the model.
5.5.2 Nonlinear Control Design. The above classification is
based on a spectrum of periodic processes (clock-works). There
exists an analog classification for event-based control. Model-
based control strategies for strongly nonlinear dynamics are
scarce. One programmatic general approach is suboptimal control
in which the optimal control actuation is chosen just for the next
time step [43]. In Ref. [268], a corresponding Lyapunov-based
control strategy has been found effective to suppress jet noise
events. For some well understood configurations, the physical
mechanism can be exploited. For instance, skin-friction of wall
turbulence is known to increase with sweeps and ejections. Hence,
a wall-normal blowing or suction counteracting the wall-normal
velocity at 10 plus units has been found to be a very effective
closed-loop opposition control [43].
Figure 19 displays examples of the single input (one actuator)
and single output (one sensor) system for the four kinds of system
dynamics. The first row shows the excitation of a frequency for a
stable linear system. The second example is the energization of a
stable limit cycle. The third row shows the suppression of a low
frequency by high-frequency forcing, and the last example indi-
cates how a single excitation frequency can change the whole fre-
quency spectrum.
There is a strong increase of complexity as one moves from
taming few frequency peaks to broadband dynamics. For the lat-
ter, even developing reduced-order models suitable for control
design constitutes an unattainable goal with modern techniques.
5.6 Enablers and Challenges of Nonlinear Model-Based
Control. In this section, enablers and challenges of nonlinear
model-based control are discussed. The beauty of the linear (49),
weakly nonlinear (52), and moderately nonlinear models (54) is
that each mode and each dynamical system coefficient have a
clear physical meaning. Moreover, the nonlinear systems have an
inherent tunable robustness. Evidently, more sophisticated models
with more frequencies can be constructed in a similar spirit. Alter-
natively, a POD model with, say, 100 dimensions may be more
accurate but the modes and the system coefficients have no physi-
cal meaning. Moreover, each mode and each coefficient act as a
noise amplifier for estimation and control design tasks. Thus,
more accurate models for a given operating condition tend to be
less effective for control design [44].
In the sequel, a validated recipe for control-oriented least-order
modeling is provided giving preference to the most simple meth-
ods. Focus is placed on the generalized mean-field model which
incorporates the other models as special cases. In addition, an ex-
perimental plant is assumed, which gives access to less data and
thus makes model identification more difficult.
Modes—The domain of the modes should be large enough to
provide instantaneous phase information and small enough to
ignore pure convection effects. The size of the recirculation
region or one or two wavelengths is a good indicator. The
first two modes u1;2may be a pair of Fourier cosine and sine
modes at the natural frequency x
u
of the unactuated flow.
Similarly, the second pair u3;4may be Fourier cosine and
sine modes at the actuated frequency x
a
under periodic forc-
ing. The dominant POD or DMD modes of unactuated and
forced flow are good candidates [48]. Filtering techniques
may be equally suited [269]. The shift modes u5;6are more
difficult as the unstable steady solution usis generally not ac-
cessible from experiments. However, base-flow changes
from modulations and unforced transients may provide u5,
for instance, via a POD of low-pass filtered flows [258].
Now, usmay be inferred from the fixed point of a calibrated
Galerkin model with the averaged flow u0and the expansion
modes u1;2;5. The second shift mode u6points from usto the
average actuated flow huai. An orthonormalization com-
pletes the construction of the basis.
Coefficients of the dynamical system—The 4D Var method is
a powerful technique that is generally applicable for parame-
ter identification [270]. One significant advantage is that no
time-derivative information from flow snapshots is needed.
Fig. 17 Phase portrait of the shift-mode amplitudes a
5
and a
6
,
i.e., the slow dynamics in Eq. (54). Same transient solutions as
in Fig. 16.
Table 1 Parameters of the generalized mean-field model illus-
trated in Fig. 16
Unstable oscillator Stable oscillator
r
u
¼0.1 x
u
¼1r
a
¼0.1 x
a
¼10
b
uu
¼1c
uu
¼0b
au
¼0c
au
¼0
b
ua
¼1c
ua
¼0b
aa
¼0c
aa
¼0
a
u
¼1a
a
¼1g¼1
Fig. 18 Venn diagram for the classification of nonlinearities.
Prototypic examples are for (A), the subcritical flow over
backward-facing step with noise excitation [251]; for (B), the
supercritical onset of vortex shedding [256]; for (C), the sup-
pression of Kelvin–Helmoltz vortices by high-frequency forcing
[47]; and for (D), the decay of 2D turbulence [267].
050801-24 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
One actuation off–on–off transient (from unforced to forced
to unforced state) can provide information about unforced
and periodically forced solutions as well as growth-rates and
frequencies. Generally, the parameter identification problem
tends to be ill-posed, i.e., significantly different dynamical
system coefficients can yield similar Galerkin solutions.
Hence, a critical enabler is a regularization, i.e., a weak
penalization of the difference from a reference dynamical
system. This reference system may be obtained for a
Navier–Stokes-based Galerkin projection (ignoring the third
flow dimension for 2D PIV data) or from a simple propagator
calibration technique.
Sensor-based estimation—Good sensor placement can easily
be inferred from the actuation mechanism. A destabilizing
sensor-based control, for instance, requires knowledge of the
phase of the forced structures and the amplitude of the natu-
ral shedding for gain scheduling. Developing more general
techniques for sensor optimization is a recent area of active
research. No generally applicable robust optimization strat-
egy has emerged yet (see Sec. 8).
Control design—A control law for the model can easily be
designed following conventional wisdom of control theory.
However, any control design should respect the narrow limi-
tations of the model. For instance, the generalized mean-field
model can be stabilized in a tiny fraction of the shedding pe-
riod. The shed vortices in the experiment, however, need a
minimum time to leave the observation domain. In particular,
the model is only applicable for slow transients with the
design frequency content. For most shear flows, the resolv-
able frequencies are hard-coded by the wavelength of the
vorticity in the expansion modes and the convection of the
mean flow.
These recipes for model-based control design follow increas-
ingly powerful data-driven methods. The danger of overfitting
needs to be mitigated by a cross-validation with data not used for
the parameter identification. The modeling may also be based on
first principles, i.e., a Navier–Stokes-based Galerkin projection,
subgrid turbulence closures, actuation modes, deformable expan-
sion modes, and other auxiliary methods. This path provides addi-
tional physical insight at the price of larger effort and the
necessity of a more extensive experience [271].
Reduced and least-order Galerkin models can provide a crisp
analytical description of the actuation mechanism and thus guide
the design of control laws. As such, they can serve as light houses
for terra incognita. However, the construction of robust control-
oriented models is more often an art rather than a fool-proof meth-
odology. By construction, Galerkin models are elliptic, i.e., a local
change of the flow is immediately communicated via the expan-
sion modes in the whole domain. Moreover, Galerkin models can
build up unbounded fluctuation energy. Shear flows, however, are
hyperbolic, i.e., dominated by convection. Excited structures con-
vect out the observation domain. Thus, the fluctuation level is nat-
urally stabilized. Galerkin models lack this convective
stabilization mechanism. This can be considered the root-cause of
the narrow dynamic bandwidth of Galerkin models. Another
related challenge is the change of flow structures under natural or
actuated transients. In simple cases, such as structures with domi-
nant frequencies, these changes may be tracked in deformable
base-flow dependent modes [272,273]. In broadband turbulence,
as in turbulent jets, there is no rationale for such cure. Physically,
an evolving vortex configuration convects downstream and it con-
stitutes a significant challenge to robustly embed a relevant en-
semble of such vortex dynamics in a modal framework.
Some challenges of Galerkin models can be avoided by choos-
ing a suitable sensor-based state space (see, e.g., Sec. 3). The
structure of the linear, weakly nonlinear, and moderately nonlin-
ear models has been derived purely from the linear-quadratic
structure of the Navier–Stokes equation and from frequency filter-
ing arguments. Hence, similar models may be constructed from
the sensor history. These sensor-based models keep all relevant
informations which are accessible in the control experiment and
bypass Galerkin modeling problems with the observation domain,
the unknown steady solution, base-flow dependent expansion
modes, etc. ERA/OKID is a powerful realization of this path for
linear dynamics (see Sec. 4.5.4).
6 Model-Free Control
Often, developing a detailed dynamical system model for a
given set of high-dimensional nonlinear phenomena may not be
the best use of time and resources. After an attractor has been
identified and dynamics painstakingly determined, applying con-
trol strategies will usually shift the attractor significantly, render-
ing models inaccurate. The obvious exception is linear
stabilization of a fixed point, whereby effective control makes the
model more accurate. Alternatively, one may apply adaptive or
model-free approaches to control a complex high-dimensional
system.
There are a wide range of model-free control options, and we
explore a number of methods that have been widely applied in tur-
bulence control. First, open-loop forcing is perhaps the most per-
vasive model-free control strategy, based on its simplicity. Next,
adaptive control may be used as a slow parameter tuning feedback
Fig. 19 Input/output characteristics of different dynamics.
Left: actuation command; right: sensor signal without forcing
(dark), and sensor signal under periodic forcing (light). From
top to bottom: a stable fixed point with periodic excitation (lin-
ear dynamics); a stable limit cycle with locking periodic forcing
(weakly nonlinear dynamics); a stable limit cycle with high-
frequency forcing (moderately nonlinear dynamics); and broad-
band turbulence under periodic forcing (strongly nonlinear
dynamics).
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-25
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
wrapped around a working open-loop control to modify controller
behavior in response to changing environmental conditions.
Extremum-seeking control (ESC) is among the most widely used
adaptive controllers for flow control. Third, in-time control may
be achieved by first specifying a given parameterized control
structure, such as PID control, and then employing tuning method-
ologies to improve performance. Finally, machine-learning con-
trol constitutes a growing collection of data-driven techniques for
structure identification and parameter identification of controllers.
Machine learning (ML) is a rapidly developing field of com-
puter science whereby a complex system may be learned from
observational data, rather than first-principles modeling
[274277]. There is a tremendous potential to incorporate data-
driven modeling techniques, especially for the control of high-
dimensional complex systems, such as turbulence. ML control,
the use of machine-learning techniques to determine effective
output–input maps (i.e., the controller), is a relatively new innova-
tion [278]. The specific machine learning methods discussed here
include adaptive neural networks, genetic algorithms (GA), and
genetic programing (GP) control.
Figure 20 provides an organization of these model-free control
methods. In each method, the structure of the control law must be
determined and the parameters are identified to optimize the con-
trol law. Many of the methods (open-loop forcing, in-time control,
and neural networks) specify the structure of the control law,
while other methods (adaptive control, gradient search, and GA)
identify optimal parameters once the structure has been identified.
Thus, once a given structure has been assumed, there is a choice
of parameter identification algorithm. Among the methods dis-
cussed here, GP control is the only method where both the struc-
ture and the parameters of the control law are identified. Other
important considerations include whether or not the method adap-
tation or learning occurs predominantly online or offline, and
whether the parameter optimization finds local extrema or has the
possibility of exploring the global parameter space.
It is important to note that model-free control methodologies
may be applied to numerical or experimental systems with little
modification. All of these model-free methods have some sort of
macroscopic objective function, typically based on sensor meas-
urements (past and present). The objective may be drag reduction,
mixing enhancement, or noise reduction, among others.
6.1 Open-Loop Forcing. Periodic forcing is widely used in
turbulence control, largely because of its ease of implementation
and the lack of the need for a model of the flow [25]. Periodic
forcing may be used to modify the dominant frequency of a flow,
resulting in a lock-on with the forcing frequency. Alternatively,
the forcing may modify the broadband frequency content, exploit-
ing nonlinear frequency cross-talk.
In either case, open-loop forcing does not take advantage of
sensor measurements, including reference or disturbance measure-
ments for feedforward control and downstream sensors for feed-
back control. This limits the ability of open-loop forcing to reject
disturbances, adapt to slow variations in flow parameters, or com-
pensate for unmodeled dynamics. Most importantly, open-loop
strategies are unable to stabilize unstable flows, regardless of the
forcing strategy.
6.2 Adaptive Control
6.2.1 ESC Methodology. There are numerous adaptive control
techniques, although ESC [279,280] has gained the most traction
in fluid dynamics. ESC is attractive for complex systems because
it does not rely on an underlying model and it has guaranteed con-
vergence and stability under certain well-defined conditions
[279,280]. ESC may be applied to track local maxima of an objec-
tive function, such as mixing enhancement or drag reduction, de-
spite disturbances, and varying system parameters. Although
adaptive control may generally be implemented in-time, it is over-
whelmingly used in flow control as a slower feedback tuning the
parameters of a working open-loop controller. There are a number
of reasons for this, including sensor and actuator bandwidth, the
fact that turbulent fluctuations may be viewed as a fast disturb-
ance, and the fact that many objective functions, such as mixing,
require integration over a slow time-scale. However, this slow
feedback has many benefits, such as maintaining performance de-
spite slow changes to environmental conditions.
ESC is an advanced method of perturb-and-observe, whereby a
sinusoidal input perturbation is used to estimate the gradient of an
objective function Jto be maximized (or minimized). The objec-
tive function is typically based on sensor measurements, s,
although it ultimately depends on the choice of input signal b, via
the plant dynamics relating bto s. Often, for ESC, the variable b
may be a parameter that describes the actuation signal, such as the
frequency of periodic forcing.
A schematic of the extremum seeking control architecture is
shown in Fig. 21 for a single scalar input b, although the methods
readily generalize to vector-valued inputs b. A schematic objec-
tive function J(b), for static plant dynamics, is shown in Fig. 22.
In ESC, a sinusoidal perturbation is added to ^
b, the best approx-
imation of the input that maximizes the objective function
Fig. 20 Overview of model-free control methods discussed in Sec. 6. Model-free control involves the choice of
control law structure as well as the optimization of controller parameters.
050801-26 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
b¼^
bþMsinðxtÞ(57)
This input perturbation passes through the system dynamics,
resulting in an output signal Jthat varies sinusoidally about some
mean value. To remove the mean, the output is high-pass filtered,
resulting in J
HP
. A simple high-pass filter is represented in the fre-
quency domain as
HP f
ðÞ¼f
fþxHP
(58)
where fis the Laplace variable, and x
HP
is the filter frequency,
chosen to be slow compared with the perturbation frequency x.It
is possible to multiply the high-pass filtered output signal by the
input sinusoid, resulting in a demodulated signal n,whichismostly
positive if the input bis to the left of the optimal value b
*
and
which is mostly negative if bis to the right of the optimal value b
*
n¼MsinðxtÞJHP (59)
Finally, the demodulated signal nis integrated into ^
b, the best esti-
mate of the optimizing value
d
dt
^
b¼kn(60)
so that the system estimate ^
bis steered toward the optimal input
b
*
. Here, kis the integral gain, which determines how aggres-
sively the actuation responds to a nonoptimal input.
The demodulated signal nmeasures the gradient of the objec-
tive function, so that the algorithm converges more rapidly when
the gradient is larger. To see this, first assume constant plant dy-
namics, so that JðsÞis simply a function of the input
JðbÞ¼Jð^
bþMsinðxtÞÞ. Expanding J(b) in the perturbation am-
plitude M, which is assumed to be small, yields
Jb
ðÞ
¼J^
bþMsin xt
ðÞ

¼J^
b
ðÞ
þ@J
@bb¼^
bMsin xt
ðÞ
þOM2
ðÞ
The leading-order term in the high-pass filtered signal is
JHP @J=@bjb¼^
bMsinðxtÞ. Averaging nover one period yields
navg ¼x
2pð2p=x
0
Msin xt
ðÞ
JHP dt
¼x
2pð2p=2
0
@J
@bb¼^
b
M2sin2xt
ðÞ
dt
¼M2
2
@J
@bb¼^
b
Thus, for the case of trivial plant dynamics, the average signal
n
avg
is proportional to the gradient of the objective function Jwith
respect to the input b.
In general, Jmay be time-varying and the plant relating bto s
may have nonlinear dynamics that operate on a faster timescale
than the perturbation x, complicating the simplistic averaging
analysis above. This general case of ESC applied to a nonlinear
dynamical system was analyzed by Krstic´ and Wang [279], where
they developed powerful stability guarantees based on a separa-
tion of timescales and a singular perturbation analysis. It is also
possible to modify the basic algorithm outlined above by adding a
phase /to the sinusoidal input perturbation in Eq. (59). In Ref.
[279], there was an additional low pass filter x
LP
/(fþx
LP
) placed
before the integrator to extract the DC component of the demodu-
lated signal n. Finally, there is an extension to extremum-seeking
called slope-seeking, where instead of a zero slope, a specific
slope is sought [280]. Slope-seeking is preferred when there is not
an extremum, as in the case when control inputs saturate. Often
extremum-seeking is used for frequency selection and slope-
seeking is used for amplitude selection.
6.2.2 Examples of Adaptive Control in Turbulence.
Extremum-seeking has been widely applied in turbulence control,
largely because of its ease of use and equation-free implementation.
ESC was used in Refs. [281,282] to reduce the drag over a bluff-
body in an experiment at moderate Reynolds number (Re ¼20,000).
The objective function weighted drag reduction against energy
expended by the actuation, a rotating cylinder on the upper trailing
edge of the backward facing step, to obtain efficient drag reduction.
In Ref. [283], ESC was used for separation control on a high-
lift configuration. The experiment consisted of spanwise pressure
Fig. 21 Schematic illustrating the components of an ESC. A
sinusoidal perturbation is added to the best guess of the input b,
passing through the plant, and resulting in a sinusoidal output
perturbation. The high-pass filter removes the DC gain and
results in a zero-mean output perturbation, which is then multi-
plied (demodulated) by the same input perturbation. This
demodulated signal is finally integrated into the best guess ^
bfor
the optimizing input b.
Fig. 22 Schematic illustrating ESC on for a static objective
function J(b). The output perturbation (light) is in phase when
the input is left of the peak value (i.e., b<b*) and out of phase
when the input is to the right of the peak (i.e., b>b*). Thus, inte-
grating the product of input and output sinusoids moves ^
b
toward b*.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-27
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
sensors and pulsed jets on the leading edge of the single-slotted
flap for actuators. This work demonstrated enhanced lift over
open-loop forcing, both at large angles of attack where the flow
was separated and at moderate angles of attack where flow
remained attached. They use the slope-seeking extension of ESC.
This work also developed both SISO and MIMO controllers.
ESC has also been used to control thermoacoustic modes across
a range of frequencies in an industrial scale, 4 MW gas turbine
combustor [284,285]. It has also been utilized for separation con-
trol in a planar diffusor that is fully turbulent and stalled [286],
and to control jet noise [287].
The ESC relies on a separation of timescales where the system
dynamics are the fastest, the periodic perturbation is moderate, and
the high-pass filter cut-off is slow. Changes to the plant dynamics, ei-
ther varying parameters or disturbances, are assumed to the slowest
timescale. It is possible to improve the performance of ESC in some
instances by adding additional filters and phase delays. For example,
extended Kalman filters were used as the filters in Ref. [288]tocon-
trol thermoacoustic instabilities in a combustor experiment. The dra-
matic performance improvement of the modified ESC algorithm
from Ref. [288] is reproduced in Fig. 23. Kalman filters were also
used in Ref. [289] to reduce the flow separation and increase the
pressure ratio in a high-pressure axial fan using an injected pulsed air
stream. The use of a Kalman filter improved the controller speed by
a factor of 10 over traditional ESC. The implication is that the con-
troller may compensate for disturbances and changes in environmen-
tal conditions that are ten times faster than before.
The external perturbation used in ESC may be infeasible for
some applications, such as optimizing aircraft control surfaces for ef-
ficient flight. In Ref. [290], atmospheric turbulent fluctuations were
used as the perturbation for optimization of aircraft control based on
ESC. The strategy of using natural system perturbations for ESC is
promising. ESC has also been used for parameter tuning in PID con-
trollers [291] as well as to tune PI controllers to stabilize a model of
nonlinear acoustic oscillation in a combustion chamber [292].
6.3 In-Time Control
6.3.1 Control Law Parameterizations and Tuning Methodolo-
gies. Often, the structure of a control law may be decided by an
expert in the loop with engineering intuition based on previous
experience. The control structure is often chosen for a combina-
tion of flexibility and simplicity, as in the ubiquitous PID control
bt
ðÞ¼kPst
ðÞþkIðt
t0
ss
ðÞ
dsþkD
d
dt st
ðÞ
The PID control above is parameterized by three constants, k
P
,k
I
,
and k
D
, the proportional, integral, and derivative gains. When the
controller is parameterized, as in PID control, it is possible to tune
the controller to improve performance and meet specifications
(rise-time, bandwidth, overshoot, etc.) by identifying locally or
globally optimal parameters. Without an underlying system
model, there is tremendous freedom in choosing these parameters.
This parameter tuning may be based on intuition, trial-and-error,
or on a physical mechanism, as in the case of opposition control
in Sec. 6.3.2.
6.3.2 Case Study: Opposition Control in Wall Turbulence. In
1994, Choi et al. introduced a method of active feedback control
to reduce the drag in a fully developed turbulent boundary layer
flow [43]. In DNS, a 20–30% reduction in drag was achieved by
imposing a surface boundary condition (blowing/suction) to
oppose the effect of vortices in the near wall region. This opposi-
tion control was an early model-free control approach based on
intuition about flow physics and drag mechanisms, rather than a
mathematical model. References [10,27,40,208,293] provide
reviews of opposition control. One of the earliest experimental
demonstrations of active feedback suppression was Liepmann and
Nosenchuck [18], where they canceled T–S waves using a down-
stream heating element and a phase-shifted measurement feed-
back. This may be seen as a predecessor of the popular opposition
control, and the experiments were quite successful, resulting in a
significant increase in the transitional Reynolds number for a flat
plate experiment.
There are many extensions to opposition control. The best per-
formance of skin-friction reduction [43] was achieved when sen-
sors were located at y
þ
¼10, which is not nearly as practical as
measurements at the wall. This led to the development of a neural
network architecture to optimize the mapping from surface meas-
urements to opposition control [22], as shown in Fig. 24. A drag
reduction of about 20% was achieved in a low Reynolds number
turbulent channel flow. Some studies suggest that opposition con-
trol does not scale favorably with increasing Reynolds number
[294], although there are studies suggesting significant potential
drag reduction, even at high Reynolds numbers, assuming perfect
damping of near-wall fluctuations [295]. The method of opposi-
tion control is also strongly dependent on the amplitude and phase
of the actuation response [296].
Opposition control has also been generalized to utilize piston
actuation [297] and has been used to reduce drag in a DNS of tur-
bulent channel flow using wall deformation [298]. Drag reduction
has also been achieved in DNS of pipe flow using opposition con-
trol [299,300] and suboptimal control [301]. Opposition control
has also been investigated in the context of stochastically forced
non-normal dynamical systems [302].
Recently, opposition control has been explained in the context
of resolvent analysis, whereby the Fourier transformed
Navier–Stokes equations are viewed as an input–output system
[303]. In this approach, the convective terms are viewed as the
input and the turbulent flow field is the output, and a SVD of the
resolvent operator indicates the forcing that results in the largest
gain in the response.
6.4 Neural Network Based Control. Many model-based
open-loop controllers in fluid dynamics are based on the inversion
of a model, as discussed in Sec. 4.2. When an input–output func-
tion is not well-approximated in a simple linear or quadratic
framework, it is often necessary to employ more sophisticated
methods, such as artificial neural networks (ANNs). ANNs are a
Fig. 23 Acoustic pressure reduction in combustor experiment
with modified ESC algorithm. The main peak is reduced by
about a factor of 60 when control is applied (Reproduced with
permission from Gelbert et al. [288]. Copyright 2012 by
Elsevier).
050801-28 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
construct in machine learning that attempts to mimic the computa-
tional flexibility observed in the brains of animals. In particular, a
number of individual computational components, or neurons, may
be connected in a graph structure with inputs and outputs. By
exposing this network to example stimulus, it is possible to train
the network to perform complex tasks, though either supervised or
unsupervised reinforcement learning. There is a tendency to use a
gradient search to determine network weights, although there are
many variations in the literature [304,305].
In turbulence, there are examples where neural networks have
been used for both modeling and control [249,306,307]. As men-
tioned earlier, neural networks have also been used to optimize
opposition control [22]. Interestingly, the POD, also known as
PCA, may be trained in a neural network [308]. The neural net-
work framing of PCA also allows for powerful nonlinear general-
izations [309,310]. Recent work has demonstrated the application
of network-theoretic tools more generally to fluid modeling [311],
resulting in a graph theoretic model of the vortex dynamics in a
flow.
In general, neural networks are adaptable and may approximate
any input–output function to arbitrary precision with enough
layers and enough training. However, they are often susceptible to
local minima and may result in overfit models when trained on
too much data. In recent years, SVMs [158160] have begun to
replace neural networks for a number of reasons. First, SVMs
result in global solutions that have simple geometric interpreta-
tion. SVMs also scale favorably for systems with very large input
spaces. However, multilayer neural networks have seen a recent
resurgence in activity with the associated field of deep learning
[312314]. The fact that these algorithms have been developed to
scale to extremely large data sets (i.e., by Google, etc.) is promis-
ing for the mining of high-Reynolds number turbulence data.
6.5 GA-Based Control. An important class of machine learn-
ing algorithms is based on evolutionary algorithms that mimic the
process of optimization by natural selection, whereby a population
of individuals compete in a given task and rules exist to propagate
successful strategies to future generations. Evolutionary algo-
rithms are typically employed to find near globally optimal solu-
tions when there are multiple extrema and gradient searches
would not work. They may also provide an alternative to the
extremely expensive Monte Carlo search algorithm, which does
not scale well with high-dimensional parameter spaces. In this
section, evolutionary algorithms are employed for parameter iden-
tification of controllers in the GA [315317]. In Sec. 6.6, evolu-
tionary algorithms are employed for both parameter and structure
identification of controllers in GP [318,319]. The implementation
of evolutionary algorithms for engineering control is relatively
recent [278].
In both GAs and GP, an initial generation of candidate parame-
ters or controllers, called individuals, is randomly populated and
the performance of each individual is quantified by some cost
function for that particular simulation or experiment. The cost
function balances various design goals and constraints, and it
should be minimized by an effective individual. In the case of
GAs, the individuals correspond to parameter values to be identi-
fied in a parameterized model, as shown in Fig. 25. In GP, the
individual corresponds to both the structure of the control law and
the specific parameters, as shown in Fig. 26 (see Sec. 6.6).
After an initial generation is populated with individuals, the
performance of each individual is assessed based on their per-
formance on the relevant cost function. Individuals resulting in a
lower cost function have a higher probability of being selected for
Fig. 24 Illustration of the benefits of opposition control (bot-
tom) in contrast to the unforced system (top). Contours of
streamwise vorticity are plotted in a cross-flow plane. Negative
contours are indicated with dashed lines (Reproduced with per-
mission from Lee et al. [22]. Copyright 1997 by AIP Publishing
LLC).
Fig. 25 Illustration of a possible binary representation of
parameters used in GAs. This example has two parameters,
each represented with a 3-bit binary number.
Fig. 26 Illustration of function tree representation used in GP
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-29
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 27 Genetic operations are used to advance generations of individuals in GAs. Operations are elitism (E), replication
(R), crossover (C), and mutation (M). For each individual of generation k11, after the elitism step, a genetic operation is
chosen randomly according to a predetermined probability distribution. The individuals participating in this operation are
selected from generation kwith probability related to their fitness (e.g., inversely proportional to the cost function).
Fig. 28 Genetic operations are used to advance generations of functions in GP
050801-30 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
the next generation. Successful individuals advance to the next
generation according to a handful of rules or genetic operations:
Elitism (optional)—a given number of top-performing indi-
viduals are copied directly to the next generation. Elitism
guarantees that the best individuals of the next generation
will not perform worse in a noise-free environment.
Replication—a statistically selected individual is copied
directly to the next generation. Replication, also called (asex-
ual) reproduction in GP, has a positive memory function.
Crossover—two statistically selected individuals exchange
randomly selected values or structures and then advance to
the next generation. Crossover has an exploitation purpose
and tends to breed better individuals.
Mutation—some portion of a statistically selected individual
is modified with new values or structures. Mutation has an
explorative function as it can discover better minima of the
cost function.
The top-performing individuals from each generation are
advanced to the next generation using these four genetic opera-
tions, and a handful of new random individuals are added for vari-
ety. This is illustrated conceptually for the GA in Fig. 27. These
generations are evolved until the algorithm converges or perform-
ance is within a desired range.
There are no guarantees that the evolutionary algorithms will
converge, although they have been successful in a wide range of
applications and may converge to a nearly globally optimal solu-
tion. There are a number of choices that can improve the perform-
ance and convergence time of these algorithms. For example, the
number of individuals in a generation, the number of generations,
the rate of each genetic operation, and the schedule for advancing
top-performers all determine the quality of solution and speed of
convergence.
GAs generally involve a large-scale parameter identification in
a possibly high-dimensional space. Thus, these methods are typi-
cally applied to tune control laws with predetermined structure.
Early efforts in using machine learning for flow control involved
the application of GAs for parameter optimization in open-loop
control [306]. Applications included jet mixing [320], optimiza-
tion of noisy combustion processes [321], wake control and drag
reduction [322,323], and drag reduction of linked bodies [324].
These early GAs optimized cost functions by specifying input
forcing parameters without taking into account sensor feedback.
However, the method was also applied to tune the parameters of
H1controllers in a combustion experiment [325].
6.6 GP Control. GP [318,319] achieves both structure and
parameter identification of input–output maps using evolutionary
algorithms. In genetic programing control (GPC), GP is used to
iteratively learn and refine a nonlinear mapping from the sensors
to the actuators to achieve some control objective. The resulting
control law is determined by sequential mathematical operations
on combinations of sensors and constants, which may be repre-
sented in a recursive tree structure where each branch is a signal
and the merging points denote a mathematical operation, as illus-
trated in Fig. 26. The sensors and constants are the “leaves,” and
each subsequent merging of branches results in a more finely
tuned mapping. The “root,” where all branches eventually merge,
is the signal that is fed into the actuation.
The same evolutionary operations of elitism, replication, cross-
over, and mutation, described in Sec. 6.5, are used to advance
individuals across generations in GP. These operations are shown
in Fig. 28 for GP function trees. The probability of each operation
is chosen to balance exploration with exploitation.
Recently, GPC has been applied on a set of benchmark turbu-
lence control experiments that exhibit various levels of complex-
ity [263,265,326329]. The interaction of the GPC paradigm with
a dynamical system is illustrated in Fig. 29. It is important to reit-
erate that the methods are based on GP so that both control struc-
ture and parameters are identified, as opposed to GAs, which are
useful for parameter optimization only. These examples demon-
strate the ability of machine-learning control using GP to produce
desired macroscopic behavior (e.g., drag reduction, mixing
enhancement, etc.) for a variety of flow configurations. These
flow configurations include a generalized mean-field model [329],
the mixing layer with pulsed actuation jets on the splitter plate
[263,265,327,329], the backward facing step controlled by a slot-
ted jet [265,328], and a turbulent separated boundary layer [265].
The first experimental demonstration of machine-learning con-
trol employing GP was performed in a mixing-layer with a veloc-
ity ratio of approximately 1:3—both for laminar and turbulent
boundary layers [263]. The flow was actuated with 96 equidis-
tantly spaced streamwise facing jets and sensed with a rake of 24
equidistantly spaced hot-wire sensors downstream. Machine
learning control (MLC) yields a control law which increases the
mixing-layer width by 67%. Thus, MLC performs 20% better as
compared to the best periodic forcing while simultaneously reduc-
ing the volume flux by over 46% (see Fig. 30)[265]. In addition,
the MLC performance has been shown to be robust against large
changes of oncoming velocities implying a transition from lami-
nar to turbulent boundary layers. Also other experimental studies
Fig. 29 Schematic of closed-loop feedback control using GP for optimization. Various controllers in a population compete to
minimize a cost function J, and the best performing individual controllers may advance to the next generation according to
the optimization procedure on the right (Reproduced with permission from Fig. 4of Duriez et al. [265]. Copyright 2014 by T.
Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N.
Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-31
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
with laminar and turbulent wall-bounded shear flows have shown
that MLC can significantly outperform the best open-loop control
and increase robustness against varying operating conditions
[265,328,330].
The performance of GP for turbulence control is striking, pro-
viding robust performance in extremely nonlinear regimes that are
not amenable to traditional model-based or model-free control
design. Yet, there are a number of unexplored extensions of GP
for the control of complex dynamical systems. First, in addition to
sensor measurements and constants, a modified GP algorithm
could create nonlinear mappings on time-delayed sensor measure-
ments or measurements that have been passed through LTI filters.
Second, the inputs of a control law may consist of suitable time-
periodic functions. Thus, open-loop actuation may be optimized.
Third, the control law arguments may comprise sensors and periodic
functions so that GPC can choose between open- and closed-loop
control or combine them. Finally, it will also be interesting to aug-
ment the machine-learning control to include reference tracking and
disturbance rejection, among other classical control methods.
7 Conclusions
Understanding turbulence has been a central engineering chal-
lenge of the past century, with the clear goal of passively manipu-
lating and eventually actively controlling turbulence. Many
advances in improved turbulence models have resulted in devices
that enable passive manipulation. The focus has rapidly shifted
from passive control of turbulence to actively controlling it, and
this will surely be a central focus of engineering efforts during the
present century. Put simply, our ability to control and manipulate
turbulence will be a deciding factor in our ability to advance key
technologies, addressing challenges in energy, security, transpor-
tation, medicine, and many other endeavors.
Our goal throughout this review has been to explore the possi-
bilities associated with turbulence control in various contexts and
for different problems. We have emphasized aspects of model
complexity and resolution as well as control logic and design
objectives. In this section, we summarize this review from three
perspectives: historical, practical, and industrial. In Sec. 8,we
point toward exciting future directions that we believe will be par-
ticularly impactful for closed-loop turbulence control.
As discussed throughout this review, there is not one single
method of turbulence control, just as there is not a single type of
turbulence. Fortunately, as difficult as it has been to develop
improved models of turbulence, control strategies may be quite
robust to model imperfections and uncertainties. It was pointed
out in Ref. [20] that models suitable for control may not be suita-
ble for accurate prediction. Hopefully, controlling turbulence will
prove to be a more manageable task than understanding it.
7.1 Historical Perspective. Progress and challenges in turbu-
lence control may be understood more clearly in a historical con-
text. The story of turbulence control is one of the oldest and
richest chapters in humans’ engineering history, marking great
technological and theoretical strides along with ongoing struggles.
Fig. 30 “Pseudovisualizations of the TUCOROM experimental mixing layer demonstrator for three cases: (I) unforced base-
line (width W5100%), (II) the best open-loop benchmark (width W5155%), and (III) MLC closed-loop control (width
W5167%). The velocity fluctuations recorded by 24 hot-wires probes are shown as contour-plot over the time t(abscissa)
and the sensor position y(ordinate). The black stripes above the controlled cases indicate when the actuator is active (taking
into account the convective time). The average actuation frequency achieved by the MLC control is comparable to the open-
loop benchmark.” The relative mixing cost function of the natural flow, open-loop forcing, and machine-learning control is
shown in (b), and the mixing layer is shown in (a) (Reproduced with permission from Parezanovic et al. [263]. Copyright 2015
by Springer). A lower cost function Jindicates improved mixing (Reproduced with permission from Duriez et al. [265]. Copy-
right 2014 by T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond,
M. Abel, N. Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).
050801-32 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
An in-depth treatment of this fascinating history is beyond the
scope of this review; there are other excellent sources [331,332].
The first chapter of quantitative research in flow control can be
considered to be shape optimization. While Lilienthal optimized
the wings of his gliders purely based on careful experiments in the
1880s, airfoil design has profited immensely from potential flow
theory around 1900. Joukowski formulated his famous lift theo-
rem for two-dimensional airfoils in 1906 after visiting Lilienthal
and observing his experiments. Prandtl generalized the lift theory
for finite wings with his famous lifting line theory in 1923. Further
progress in quantitative shape optimization using potential theory
is associated with the names of Betz, von K
arm
an, and Theo-
dorsen, to name a few. More recently, these achievements have
been generalized for use in adjoint-based shape optimization using
the full Navier–Stokes equations.
The discovery of boundary-layer theory by Prandtl in 1904 has
stimulated active control, as suction and blowing were rapidly
realized to change important boundary-layer properties. The first
experimental demonstrations included stabilization of boundary
layers with steady suction. These efforts have enjoyed a thorough
mathematical treatment entering Schlichting’s famous book on
boundary-layer theory [333]. In the past century, many other
forms of actuators have been invented with continually increasing
authority and dynamic bandwidth [36]. Destabilizing control
using unsteady periodic forcing was considered later for mixing
enhancements of shear flows [334].
In the 1940s, the foundations of modern turbulence theory were
laid by Kolmogorov, Landau, Millionshtchikov, Monin, von Neu-
mann, Obhukov, and Yaglom, to name only a few (see, e.g., Refs.
[335,336]). Neither the energy cascade nor the mathematical closure
approaches have entered main-stream flow control methodologies
until now. Yet, the need for closures in nonlinear control design is
increasingly realized as an important grand challenge problem.
Around 1950, passive actuators were explored and optimized.
The vortex generator [337] for separation mitigation is one promi-
nent example. These developments are continued—hydrophobe
surfaces for ships, riblets for airplanes, and spoilers on cars serv-
ing as prominent examples.
In the 1980s, chaos theory has stimulated fluid mechanics to-
ward the search for universal transition scenarios, for low-
dimensional strange attractors [338], and for chaos control
[339,340]. The success of quantitative nonlinear dynamics meth-
ods for turbulent flows has been limited. Yet, nonlinear dynamics
has entered fluid dynamics thinking with the qualitative notions of
strange attractors, domains of attraction, edge states, and bifurca-
tions, just to name a few.
In the same period, the first closed-loop control experiments
were performed for the suppression of boundary layer instabilities
[18], for wake stabilization [19], for skin friction reduction of tur-
bulent boundary layers [43], etc. The first control designs were
based on heuristic considerations, like superposition of traveling
waves and opposition control. Rapidly, control theory has entered
fluid mechanics [27,212] and is providing a solid foundation for
model identification and control design. Even the nomenclature,
like “plant” for actuated flow and “order” for the model dimen-
sion, is largely borrowed from control theory.
In the late 1980s, the foundation of many current reduced-order
models for flow control has been laid by the pioneering POD
model of wall turbulence [341], the snapshot method for POD
[86,188,187], and other early work on coherent structures
[342344]. This development of control-oriented reduced-order
models is attracting increasingly many researchers with numerical
and experimental control demonstrations. At the same time, the
challenges of POD models are becoming more obvious. A mini-
mum requirement and challenge for model-based control design is
that the model describes unactuated and actuated states as well as
the transients between both. A more comprehensive presentation
is provided in Ref. [271].
Starting around 2000, the technological development of MEMS
with actuators and sensors of increasing performance [69] has
stimulated the development of more complex hierarchical control
laws, based on globally coupled local sensing and control units.
This area is undergoing rapid development. The complexity of
turbulent dynamics has been partially addressed by a novel form
of model-free control design. The control laws have been cast in
the form of ANNs with the pioneering computational demonstra-
tions in wall turbulence [22]. Currently, many other machine-
learning methods are entering fluid mechanics for an increasing
number of analysis, modeling, and control tasks.
There are two common and recurring themes in this historical
perspective on turbulence control. First, advances in theory drive
progress in control design. These theoretical advances may
involve first-principles understanding of instabilities and turbu-
lence itself. Additionally, theoretical advances in external fields,
such as linear systems, control theory, and dynamical systems,
have all had a transformative impact on the direction and progress
of flow control efforts.
The next general rule is that whatever is possible on the existing
hardware will be fully exploited by the software. Physical demon-
strations of flow control have been necessarily tied to advances in
sensor and actuator hardware and to growth in computational
capabilities. With recent developments in micromanufacturing
techniques, including MEMS, many industrial applications of
closed-loop flow control are becoming increasingly feasible.
7.2 Current Practices. Here, we attempt to summarize some
of the current “best practices” gathered from success stories in the
literature (Fig. 31). Because of the mercurial nature of the field,
this discussion may be viewed as a work in progress. The specific
control strategy will vary depending on whether the goal is to
minimize or maximize a cost function, track a reference value of
some quantity, or stabilize an unstable steady state and reject
disturbances.
It is reasonable to start by assessing whether or not superposi-
tion holds. If so, then it is possible to develop balanced linear
input–output models, either gray-box or black-box, and then
design optimal linear feedback controllers. These controllers may
be optimal in the sense of minimizing the control expenditure,
attenuating sensor noise, and rejecting disturbances while mini-
mizing the error of the state to some reference value. Alterna-
tively, if model uncertainty is large and stability margins are
critical; then, it is possible to optimize the controller for robust
performance. Linear control has been especially successful in
delaying the transition to turbulence, which involves stabilizing
an unstable steady state.
If superposition does not hold, then the flow may be dominated
by oscillatory components. If there are a few dominant frequen-
cies in the flow, then it is possible to develop a mean-field model
and resulting nonlinear control law. This strategy has been proven
effective in suppressing shedding behind bluff bodies.
If there are not strong, isolated oscillatory components, but
rather there is broadband frequency cross-talk, then open-loop
control may be able to modify the spectrum toward desirable
specifications. It may be necessary to explore the input parameter
space to identify the most effective combination of inputs for
open-loop actuation. Once these input directions are identified, an
adaptive control algorithm, such as extremum-seeking, may be
applied to find a locally optimal open-loop forcing. Effective
open-loop strategies may then be emulated in closed-loop with
added robustness.
A very general approach, including open- and closed-loop con-
trol, is enabled by machine-learning methods. Machine-learning
control can be configured to yield either open-loop or closed-loop
control, depending on the choice of inputs. Although the resulting
controllers may approach globally minimizing values of the cost
function, there is no guaranteed convergence, and depending on
the dimension of the input space, convergence may be slow. More
work is needed to extend MLC to include two-degrees-of-freedom
reference tracking control to suppress disturbances via feedback.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-33
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
The progress in turbulence control, both passive and active, has
been significant, as evidenced by the ability to relaminarize flows
and reduce drag, to control separation, and to modulate mixing.
However, each of these three canonical flow control problems
presents unique challenges to large-scale industrial implementa-
tion, which in turn motivate new opportunities for research. Note
that the association of the following three flows with correspond-
ing control methodologies is not inclusive, but instead covers a
large portion of the cases in the literature.
7.2.1 Linear Control and Transition. Linear feedback flow
control benefits from the most well-developed theoretical founda-
tions and a set of clear goals. For instance, it is often used to stabi-
lize unstable laminar steady states and delay the transition to
turbulence, reducing drag. These techniques amount to a system-
atic approach to identify and suppress unstable structures and dis-
turbances in the flow utilizing sensors and actuators. Many
feedforward techniques in the optimal linear control of turbulent
boundary layers closely resemble opposition control, except that
future model predictions of disturbances are used to cancel flow
structures after they have been deformed and convected with the
flow. Feedback control offers better robustness to unmodeled dy-
namics and disturbances, although it is sensitive to time-delays
which are inevitable in convective flows, as it takes time for the
effect of actuation to reach downstream sensors. The limitations
imposed by time delays motivate the need for a denser array of
sensors and actuators for more complex convective flows, which
in turn motivate bio-inspired sensing and actuation along with
local computations.
7.2.2 Nonlinear Control and Separation. Nonlinear control
presents a significant opportunity to manipulate flows that are not
close enough to a fixed point for linear flow control. For example,
massively separated flows are in a fully nonlinear regime, and
manipulation of coherent structures requires an understanding of
their nonlinear coupling. In contrast to linear control, which
involved suppressing structures, nonlinear control involves redi-
recting some flow structures, possibly to suppress others. The big-
gest issue for nonlinear flow regimes is the lack of theoretical
understanding of nonlinear turbulence models and closures. More-
over, many tools from control theory are developed for linear sys-
tems and do not generalize to nonlinear problems. The need for
improved nonlinear reduced-order models and turbulence closures
will be an important area of turbulence research, facilitating
advances in turbulence control.
7.2.3 Model-Free Control and Mixing. Effective linear con-
trol will suppress disturbances and bring the flow state closer to
the fixed point where a model was developed. However, applying
control to a nonlinear flow that is on an unsteady attractor, rather
than a fixed point, may significantly distort the attractor, rendering
models invalid. It may be prohibitively difficult to develop models
that are sufficiently general to predict control responses. Instead
of developing detailed models that will then be rendered invalid
by control, model-free approaches are promising when it is desira-
ble to change the nature of an attractor, for example, to increase
mixing. The most challenging nonlinear problems are excellent
candidates for advanced model-free techniques. Data-driven
methods, including compressive sensing, machine learning, and
uncertainty quantification (UQ), may be increasingly effective in
the development of control strategies for highly complex, fully
turbulent problems.
7.3 Industrial Applications. There are many industrial
research programs centered around turbulence control, but it is not
always clear how control results will scale to high Reynolds num-
ber flows. For instance, how does the actuation velocity, energy
input, etc., scale with the Reynolds number of the problem? In
cases where we do understand the scaling, such as the spatial and
temporal scales of flow perturbations for disturbance rejection
through linear feedback control, the scaling is not favorable.
Results suggest that at industrial Reynolds numbers, a finer mesh
of sensors and actuators with improved bandwidth and more
powerful computational capabilities will be required for similar
transition delay and drag reduction.
As a consequence of the large Reynolds numbers, industrial
flows are often exceedingly complex, with broadband frequency
cross-talk and many orders of magnitude scale separation in space
and time. It is entirely possible that data-driven techniques, such
as machine-learning control methodologies, will enable turbu-
lence control in these situations before we are fully able to under-
stand the mechanisms. However complex the flow, the multitude
of industrial and defense applications will continue to drive
research developments in turbulence control for decades to come.
8 Future Developments Beyond Control Theory
In the past, control theory has shaped flow control and signifi-
cantly influenced the path from open-loop to closed-loop control.
Flow control and control theory were both mature disciplines with
little synergy before the 1990s. After techniques from mathemati-
cal control theory were embedded in flow control, dramatic pro-
gress has been made. The luminary words of N. Wiener are apt in
this situation: “The most fruitful areas for the growth of the scien-
ces were those which had been neglected as a no-man’s land
between the various established fields”[76]. In the following, we
sketch three mature disciplines which have not fully integrated
into mainstream fluid mechanics, but which are likely to dramati-
cally improve the complexity and performance of flow control in
the future.
The first area includes advances in sensor and actuator hard-
ware, as well as principled or heuristic placement of sensors and
actuators in the flow. Recent developments in biologically
inspired sensing and actuation are motivated by the extreme per-
formance observed in biological turbulence control, and they
remain promising for engineering flow control. Next, advances in
data-driven modeling and control are poised to leverage the grow-
ing movement in data science. These new techniques include
machine learning, compressive sensing, and UQ, all of which
have direct relevance for in-time closed-loop flow control. Finally,
developments in first-principles modeling and control of turbu-
lence will remain a critical backbone of efforts in flow control.
These developments may include a common control-theoretic
framework generalizing linear systems theory to handle various
classes of flow nonlinearity, as well as developments in turbulence
closures. Combining bottom-up (theoretical) advances in turbu-
lence models and control theory with top-down (data-driven)
approaches will enable hybrid controllers with greater flexibility
and robustness (see Fig. 32). These directions are necessarily a
product of the authors’ experiences and biases, and there are cer-
tainly many more fruitful directions that remain unlisted in this
review.
8.1 Bio-Inspired Sensing and Actuation. Sensors and actua-
tors are the workhorses of active flow control. Advances in hard-
ware will be a critical determining factor in the ultimate success
and adoption of turbulence control in industry. These advances
include smaller, higher-bandwidth, cheaper, and more reliable
devices that may be integrated directly into existing hardware,
such as wings. There have been many advances in actuator hard-
ware, as reviewed in Ref. [36] and discussed in Sec. 2.4. These
have included plasma actuators, MEMS, fluidic oscillators
[345,346], synthetic jets, and zero-net-mass-flux actuators, such as
piezoelectric actuators. Advances in sensors, such as the nano-
scale thermal anemometry probe [347349], are facilitating ever
finer measurements of turbulent systems. We expect that contin-
ued developments in sensing and actuation hardware will continue
to drive advances in turbulence control. In addition, we predict
significant progress in the effective placement of sensors and
actuators for a given control objective. This progress should be
marked by theoretical breakthroughs in the first-principles optimal
050801-34 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
placement as well as improved heuristics for key engineering
flows.
8.1.1 Cheap Hardware and Local Computations. Active sep-
aration control has not been adopted in the automobile industry, in
part because of low fuel prices and the perception among auto-
makers that drivers are unwilling to pay the upfront cost for hard-
ware upgrades. Reducing the cost of control hardware, either
through manufacturing advances or economies of scale, will be
important to gain traction in the consumer automobile market.
Other markets, such as air travel or commercial shipping, may
be willing to bear upgrade costs for significant improvements to
fuel economy or improved range of operability. However, modify-
ing the design and manufacture of an airplane wing is extremely
costly and will likely require advances in the manufacture of em-
bedded or surface-laminated sensors and actuators. Because the
smallest eddies become smaller with increased flow velocity,
transition delay on a modern aircraft may involve a fine web of
integrated sensors and actuators. Moreover, time-scales also
shrink with faster flows, and it may be necessary to perform local
computations in a neighborhood of the sensors and actuators. This
will have a twofold benefit: first, local computations will greatly
reduce the transfer of data associated with a fine mesh of sensors,
and second, local computations will reduce the latency in a con-
trol decision.
These ideas are already being explored in the context of bio-
inspired engineering [350369] and biomanipulation [370376].
It is observed that birds, bats, insects, fish, and swimming mam-
mals routinely harness unsteady and turbulent fluid phenomena to
improve their propulsive efficiency, maximize lift and thrust, and
enhance maneuverability [364,377388]. They achieve this per-
formance with robustness to external factors and disturbances,
rapid changes in flight conditions, such as gusts, and even signifi-
cant changes to body geometry.
Fig. 31 Flow chart illustrating the hierarchy of active control approaches. This diagram is
conservative in giving preference to the most established techniques. If the task is optimiza-
tion or minimization of measurement time, machine-learning control may be an earlier branch.
Top panel depicts a turbulent jet from Bradshaw et al. [38].
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-35
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Insect flight is particularly remarkable, since aerodynamic time-
scales are faster than the time it takes for signal transduction from
the visual system to the central nervous system [389]. In addition
to centralized computations, local computations are being per-
formed in the shoulder muscles and these decisions are fed
directly to motor neuron outputs. Thus, the insect’s neural flight
control system is largely centered around the mechanosensory
system, comprised of sparsely distributed sensors that act on the
fastest time scales and provide the lowest latency [390,391].
Nearly, all flying insects contain microscopic-embedded strain
sensors in their wings, called campaniform sensilla, along with
other mechanosensors, such as antennas and halters, which are
small passive wings that serve as gyroscopes [392]. Moths, for
example, have up to hundreds of these campaniform sensors,
more densely spaced at the wing base. Similarly, birds sense flow
disturbances with their feathers [393], and bats with tiny hairs on
the surfaces of their wings [394396].
Biological systems suggest a strategy for sensing turbulence
based on a large number of relatively inaccurate or noisy sensors,
as opposed to a few accurate, reliable sensors. In the correct esti-
mation framework, utilizing the law of large numbers, it is possi-
ble that many low-fidelity sensors may be more accurate and
robust in the aggregate than a few accurate sensors. This strategy
has already been employed in a number of diverse settings, rang-
ing from finance to optical detection with cameras [397] to distrib-
uted sensor networks.
8.1.2 Sensor and Actuator Placement. As discussed above,
sensor and actuator placement is of central importance in turbu-
lent flow control. This is especially clear in the context of linear
control, where the matrix Cis determined entirely by the sensor
placement and type, and the matrix Bis similarly determined by
the actuators. These matrices, in conjunction with the dynamics
A, determine to what extent flow states are observable and
controllable.
The optimal locations of a small number of actuators and sen-
sors were recently determined for the H2optimal control of the
complex Ginzburg–Landau equation in Ref. [37]. In Ref. [398],
sensor/actuator placement was investigated in the context of the
cylinder wake, and both direct and adjoint modes were used to
determine regions of high sensitivity to disturbances. Optimal
actuation is also explored in the form of small upstream jets to
delay the transition to turbulence in a pipe flow [399401]. As
mentioned previously, sensor and actuator placement was also
investigated for a transitional boundary layer [250].
Determining optimal sensor and actuator placement in general,
even for linear feedback control, is an important unsolved prob-
lem. Currently, optimal placement can only be determined using a
brute-force combinatorial search, at least with existing mathemati-
cal machinery. This combinatorial search does not scale well to
larger problems, and even with Moore’s law, exponentially
increasing computer power does not grow quickly enough to help.
Advances in compressive sensing, however, may provide convex
algorithms to almost certainly find optimal sensor/actuator place-
ment, under certain conditions. Ideas from compressive sensing
have recently been used to determine the optimal sensor locations
for categorical decision making based on high-dimensional data
[402] with applications in dynamic processes [403].
8.2 Data-Driven Modeling and Control. As researchers
strive to make more complete measurements and simulations of
increasingly complex flows, we as a community find ourselves in
a deluge of data. This burden of data will only become more acute
as sensor arrays are manufactured with finer spatial and temporal
resolution. Some of the foundational techniques used in the analy-
sis of big data [404], such as dimensionally reduction and parallel
computation, were developed extensively in the fluids community.
However, many of the most promising recent techniques have
been largely applied to static data problems in artificial intelli-
gence and computer vision. There is a tremendous opportunity
ahead to embrace and innovate new techniques in compressive
sensing, machine learning, and other data-driven methods applied
to the rich dynamical system of turbulence; data-driven methods
are also being applied more broadly in the aerospace industry
[405]. For an overview of data techniques applied to dynamical
systems, see Refs. [403,406].
8.2.1 Compressive Sensing. Compressive sensing [407411]
has the potential to be a disruptive technology in turbulence con-
trol, marking one of the most important breakthrough in computa-
tional mathematics since the FFT. Compressive sensing allows
complex, high-dimensional signals and states to be reconstructed
from surprisingly few measurements, as long as the high-
dimensional signal is sparse in some basis; most natural signals
are. Sparsity that means the vector, written in a transformed basis,
contains mostly zeros. For instance, image and audio signals are
sparse in Fourier or wavelet bases, as evidenced by their high
degree of compressibility (e.g., JPEG and MP3 compression).
In fluid dynamics, signals of interest may represent flow field
snapshots with many degrees-of-freedom from PIV [412] or the
time-history of hot-wire measurements in the wake of a turbulent
flow. The first example is analogous to an image, and the second
example is analogous to a multichannel audio signal.
The conventional wisdom in signal processing and data acquisi-
tion is that a signal must be sampled at twice the rate of the high-
est frequency present, the so-called Shannon–Nyquist sampling
rate [413,414], for perfect signal reconstruction. However, when a
signal is sparse in some basis, meaning that its vector representa-
tion in those coordinates contains mostly zeros, it is possible to
relax this sampling rate restriction. A reduction in the sampling
rate may have a dramatic effect in bandwidth limited control
applications.
A vector a2RNais K-sparse in a transformation basis U2
RNaNaif a¼Unand n2RNahas exactly Knonzero elements;
this means that amay be represented in a low-dimensional sub-
space of U. Compressive sensing provides a framework to deter-
mine the Knonzero coefficients in n, and therefore to determine a,
from the measurements s2RNs
Fig. 32 Schematic illustrating a roadmap for future develop-
ment. We envision the synthesis of classical control theory with
data-driven methods for the development of hybrid controllers.
Both top-down and bottom-up approaches will contribute to a
better understanding of nonlinearities, which will in turn con-
tribute to the development of more effective controllers.
050801-36 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
s¼Ca ¼CUn(61)
The measurements (i.e., rows of C) must be incoherent with
respect to the columns of U, meaning that they are not strongly
correlated, and there must be NsOðKlogðNa=KÞÞ measure-
ments. Note that this is a significant reduction in measurements
when KN
a
. For a signal athat is sparse in the Fourier domain,
Uis an inverse discrete Fourier transform, and spatial or temporal
point measurements are incoherent, since they are broadband in
frequency. This is very convenient, since many signals of interest
are sparse in the frequency domain and point measurements are
realistic in many engineering applications.
Compressive sensing has recently been applied to reduce the
data acquisition required in PIV systems [102,415418]. Time-
resolved PIV may be prohibitively expensive for high-speed,
high-Reynolds number flows because of the multiscale nature of
turbulence. PIV systems are often limited by the bandwidth of
data transfer. Reducing the spatial or temporal sampling rate using
compressive sensing may open up orders of magnitude more com-
plex flows to PIV analysis. Other advances in time-resolved PIV
involve using a Kalman smoother to combine a time-resolved
point measurement with nontime-resolved PIV data [135].
Before the advent of compressive sensing, finding the K-sparse
solution nto Eq. (61) involved a combinatorial brute force search
among all possible vectors. This is a nonpolynomial (NP) hard
problem, meaning that the computational complexity does not
scale as a polynomial of the size of the problem N
a
. With com-
pressive sensing, it is possible to find the K-sparse vector in poly-
nomial time with high probability using techniques in convex
minimization [419422]. The ability to solve these problems in
polynomial time means that we will be able to solve proportion-
ally more challenging problems with the continued progress of
Moore’s law of exponential growth in computing power.
Recently, compressive sensing techniques have been applied to
the (previously) NP hard problem of sensor placement for categor-
ical discrimination in high-dimensional data [402].
Although compressive sensing has been extremely successful in
image processing, it has not been widely applied to dynamical
systems, with some notable exceptions [423427]. In fact, fluid
dynamics is one of the first fields in dynamical systems to adopt
compressive sensing techniques. Compressive sensing and other
sparsity-based ideas have been used recently in the computation
of the DMD [102,428430]. In addition, POD modes have been
used as a data-driven sparsifying basis [102,417,431,432]. Indeed,
the application of dimensionality reduction and compressive sens-
ing to dynamical systems is synergistic, since low-rank attractors
facilitate sparse measurements. Other applications of compressive
sensing in fluids are provided in Refs. [433435].
8.2.2 Machine Learning. ML comprises a set of tools that
extend classic dimensionality reduction techniques to automati-
cally generate models that both learn from and improve with more
data [275277,406]. In fluid dynamics, dimensionality reduction
techniques, such as POD or DMD, may be thought of as library
building. In ML, these low-rank libraries are stored for each
dynamic regime, and they may be used to rapidly characterize a
system. These models are used for categorical decision, pattern
recognition, high-dimensional regression, occlusion inference in
data with incorrect or missing values, and outlier rejection. More-
over, these methods improve as more data are collected, and they
may either be trained by expert supervision or used in an unsuper-
vised context to elucidate underlying patterns that may not have
been readily apparent to a human investigator. ML offers a para-
digm shift in modeling and control based on engineering data, lev-
eraging both human expertise and the statistical power of a large
sample size for quantifiably improved decisions and diagnostics
based on features mined from high-dimensional data.
Turbulence control may be characterized by many different
attractors in a small region of parameter space. With classification
protocols, it is possible to characterize underlying bifurcation
parameters from relatively few measurements of a complex sys-
tem [431437]. Once the system is characterized, the high-level
control system may jump to a previously determined control strat-
egy, which may be good but suboptimal for the particular instanta-
neous parameters. Then, feedback may be applied to reduce
errors. If a new region is found that is not amenable to previous
controllers, it is characterized and incorporated into the library.
System identification may be thought of as a form of machine
learning, where training data are used to generate a model based
on observed patterns. It is hoped that the model is valid on new
inputs that were not used for training, providing a so-called cross-
validation. Decreasing the amount of data required for the training
and execution of the model is often important when a prediction
or decision is required quickly, as in turbulence control. Compres-
sive sensing and machine learning have already begun to be com-
bined for sparse decision making [402,403,406,432,438], which
may dramatically reduce the latency in a control decision. Many
of these methods involve clustering techniques, which are a cor-
nerstone of machine learning. Cluster-based reduced-order models
are especially promising and have recently been developed in flu-
ids [439], building on cluster analysis [440], and transition matrix
models [441].
Turbulence control based on GP has had a number of recent
successes in canonically challenging flow control problems
[263,265,326329]. These advances motivate renewed effort to
integrate aspects of machine learning and control theory. For
instance, adding reference tracking and robust performance to
machine-learning control may provide the best of both strategies.
8.2.3 UQ and Equation-Free Methods. Although many engi-
neering phenomena satisfy governing equations, it is often the
case that the high-level questions of interest are far removed from
first-principles analysis. Moreover, initial conditions, parameters,
and even the equations of motion may only be known with some
certainty. As an alternative to classical equation-based
approaches, model-free methods have emerged in the past decade
and have the potential to transform the control of complex sys-
tems. Equation-free analysis [442445] and UQ [446448] are
two crucial fields that have direct relevance for turbulence model-
ing and control.
For instance, in fluid dynamics, DMD has been useful for iden-
tifying important spatial–temporal coherent structures that are
spatially correlated and share the same time dynamics (i.e.,
growth, decay, oscillation, etc.) [98101]. This method is purely
data-driven, making it equally useful for data from simulations
and experiments. Moreover, a low-order dynamical system is
identified, which may be used for short-time predictions and
closed-loop feedback control to shape the system dynamics to-
ward a desired outcome. Recently, DMD has been extended to
disambiguate the internal dynamics from externally applied con-
trol [195]. Other extensions to DMD include video analysis [449],
streaming data sets [450], and data with noise [451].
Two of the defining characteristics of turbulent flows are that
they are chaotic and that it is difficult or impossible to measure
every scale simultaneously with arbitrary precision. The inherent
uncertainty in the measured flow state may dramatically affect
future predictions as the uncertainty propagates through the dy-
namical system. This is especially challenging when the dynamics
are chaotic, as probabilistic descriptions of the uncertainty will
become stretched and folded by the nonlinearity [452]. The field
of UQ has arisen to characterize such systems. Generalized poly-
nomial chaos (gPC) provides a modern perspective on an old tech-
nique [453], approximating the evolution of stochastic dynamical
systems through a Galerkin projection using a set of orthogonal
polynomials. These polynomials are chosen to describe the proba-
bility distribution of uncertain quantities [446448]. gPC can
become quite expensive for long-time evaluation of uncertain
quantities, because of the significant distortion of trajectories and
probability densities. To address this, various extensions have
been developed, such as multi-element gPC [454], time-dependent
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-37
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
gPC [455], and gPC based on short-time flow-map composition
[456]. There has been a recent explosion of interesting methods for
UQ [457] and stochastic dynamics [458,459].
In addition, the finite-time Lyapunov exponent (FTLE) has
been a rapidly developing data-driven technique in the analysis of
time-varying fluid fields [460467]. Like DMD, this method only
relies on velocity field measurements, either from simulations or
experiments. FTLE analysis identifies time-varying coherent
structures in fluids that are analogous to stable and unstable mani-
folds. Thus, FTLE provides a quantitative technique to visualize
flows and identify regions of separation, recirculation, dispersion,
material attraction, and high sensitivity to perturbations. This
method has been used to analyze aortic blood flows for predictions
involving recirculation regions, separation, and stenosis [468].
FTLE has also been used to study biopropulsion [469471], fluid
mixing in large bodies of water [472474], and to understand tur-
bulent structure [463,464,467,475]. FTLE provides a measure of
sensitivity, which is essential in the quantification and manage-
ment of uncertainty. Sensitivity and coherence are closely related
to the calculation of almost invariant sets [476478], using set-
oriented methods [479,480], and to eigenvectors of the Perron
Frobenius operator.
8.2.4 Design of Experiments. Many of the data-driven techni-
ques discussed above suggest innovations in experimental design.
Understanding what measurements must be acquired to design
models and controllers from experimental data is an important
part of turbulence control. UQ and sensitivity analysis can offer
some guidance about what design factors have the most impact on
measurement quality, and where the flow is most sensitive to
actuation. Compressive sensing may allow for improved band-
width through a principled reduction in the spatial and temporal
resolution of measurements required for signal reconstruction.
ML provides an exploratory protocol for actuating the system into
new and beneficial dynamic regimes.
8.3 Advanced Nonlinear Models, Controllers, and
Closures. As discussed above, there may be serious limitations to
physics-based modeling of flows with strongly nonlinear dynam-
ics. Consequently, model-based control will be challenged by the
accuracy of the model. In addition, control design methods gener-
ally assume either a working linearized model or a well-
understood nonlinearity. At the same time, dramatic advances in
data-driven methods, such as system identification and machine
learning, have produced powerful new tools in turbulence control.
This trend is accelerated by the tremendous global resource
investment in machine learning across all physical sciences. How-
ever promising these new methods are, there will continue to be
many compelling reasons to develop improved nonlinear models,
controllers, and closures. First, physics-based models are inter-
pretable and allow for the inclusion of expert human knowledge.
Second, understanding the fundamental reasons why a given con-
troller works is central in developing this human intuition. Such
intuition is critical when deciding on what control strategy (hard-
ware, logic, etc.) to employ in a new situation. Third, a physically
interpretable model might lead to a simple control law with one or
a few easily tunable parameters. These first-principles models can
be expected to develop more slowly as the mathematical chal-
lenges are enormous. Yet, they will undoubtedly remain a critical
part of engineering turbulence control. Hence, understanding of
the physical mechanisms underlying effective turbulence control
will remain a critical enabler—regardless of the control strategy,
lest we lose mastery of the machinery we employ.
There are a host of advanced modeling techniques, including
powerful generalizations of POD, which are useful for obtaining
efficient parameterized models of complex turbulent flows using
high-performance computation [481484]. The gappy POD
method provides the ability to sparsely sample a system and still
evaluate the POD and terms in the Galerkin projection [485,486].
In addition, there are reduced-basis methods for PDEs [487] and
the associated discrete empirical interpolation method [488490],
which approximates nonlinear terms by evaluating the nonlinear-
ity at a few specially determined points. There have also been
powerful advances in the filtering of turbulent systems
[434,491493]. Finally, robust control has also been used as a
method of understanding underlying nonlinear mechanisms in tur-
bulence [494]. In addition, advanced measurement capabilities
contribute to improving our understanding of high Reynolds num-
ber turbulent flows [495,496].
These advanced models are broken into three critical pieces—
modeling, closure, and control design—although the true division
may be more subtle. First, nonlinear model identification needs to
be advanced comprising both structure and parameter identifica-
tion. Significant progress has been made with 4D VAR methods
[497,498]. Second, turbulence closures have always been a critical
part of reduced-order turbulence models. Yet, eddy-viscosity
based subscale models are too coarse to resolve critical frequency
cross-talk mechanisms. Closure schemes based on a Gaussian
approximation [273], on a maximum entropy principle [499501],
and on finite-time thermodynamics [502504] hold corresponding
promises. Finally, the nonlinear theory of control needs to be sig-
nificantly advanced. It is currently unclear how nonlinear models
and closures will be used by control theorists, motivating the need
for a common framework, like the state-space and frequency do-
main framework in Sec. 4for linear systems.
8.3.1 Graph-Theoretic Flow Control. Advances in network
science have recently been applied with success in fluid systems
[505], providing a new set of mathematical techniques for com-
plex systems. The resulting graph models may be based on snap-
shot clusters [439], or on a sparsified graph model for the
underlying vortex network [311]. The integration of methods
from network science and network control theory in turbulence
control is promising, especially since many network control tech-
niques have been developed to handle nonlinear systems.
The past two decades have marked numerous advances in
graph-theoretic control theory surrounding multi-agent systems,
and network science has experienced significant recent attention
[506510]. Networks are often characterized by a large collection
of individuals (represented by nodes) that each executes their own
set of local protocols in response to external stimulus [511]. This
analogy holds quite well for a number of large graph dynamical
systems, including animals flocking [512,513], multirobotic coop-
erative control systems [514], sensor networks [515,516], biologi-
cal regulatory networks [517,518], and the internet [519,520], to
name a few. Similarly, in a fluid packets of vorticity may be
viewed as nodes in a graph, which interact collectively according
to global rules (i.e., governing equations) based on local rules (dif-
fusion, etc.) as well as their external inputs summed across the
entire network (i.e., convection due to induced velocity from the
Biot–Savart law) [311].
In large multi-agent systems, it is often possible to manipulate
the large-scale behavior with leader nodes that enact a larger su-
pervisory control protocol to create a system-wide minima that is
favorable [512,521523]. The fact that birds and fish often act as
local flows with large-scale coherence, and that leaders can
strongly influence and manipulate the large-scale coherent motion
[512,513], is promising when considering network-based fluid
flow control. Based on the network-control methodology used to
analyze schooling fish and flocking birds [512,513], there is an
appealing goal of schooling turbulence by collecting and harness-
ing distributed multiscale eddies into a collective organization, or
community, with favorable large-scale properties.
Recent work investigating the number of leader nodes required
for structural controllability of a network [522] suggests that
large, sparse networks with heterogenous degree distributions,
6
6
The degree distribution of a network is the probability distribution of the
number of connections each node has to other nodes. All scale-free networks are
inherently sparse, with heterogeneous degree distribution [509].
050801-38 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
such as scale-free turbulence networks, are especially difficult to
control. In particular, the number of driver nodes may be quite
large for these systems, as compared with a regular or random
graph with more homogeneous degree distribution. However, full
structural controllability of a turbulent vortex network is not nec-
essarily a reasonable or desirable goal. In fact, these results are
consistent with physical intuition that controlling every single tur-
bulent eddy would require immense control authority with distrib-
uted actuation over a number of spatial locations.
In flow control, especially for turbulent fluids, the goal is not
necessarily to control every minor eddy in the flow, but rather to
manipulate the base flow and large-scale coherent structures that
mediate quantities of interest (lift, drag, etc.). In addition, practi-
cally, we are more interested in the degree of controllability
[524], rather than structural controllability, of large-scale vortical
structures in order to manipulate the base flow. Even with the goal
of increasing mixing, which involves increasing the level of turbu-
lence, individual eddies need not be controlled, but rather control
should manipulate the statistical distribution of eddies that deter-
mine dominant energy balances in the fluid.
8.3.2 Markov Model-Based Control. One elegant strategy
comprising modeling, closures, and control design starts with the
Liouville equation for the probability distribution [525]. The prob-
ability density pða;tÞof Eq. (2a)evolves according to the Liou-
ville equation
@tpða;tÞþr
a½fða;bÞpða;tÞ ¼ 0
where r
a
is the Nabla operator in the state space. The elegance
lies in the fact that this evolution equation is a linear conservation
equation for the probability distribution and thus allows to take
advantage of the sophisticated linear control design methods for
nonlinear dynamics. In addition, the probability distribution gives
immediate access to engineering goals, like average drag, average
lift, or level of fluctuation. Such averages are not provided by line-
ar(ized) models. However, the challenge is that the Liouville
equation of the Navier–Stokes equation [526] is a functional equa-
tion and thus computationally far more demanding than the origi-
nal equation. Yet, a suitable cluster-based phase-space
discretization may reduce this Liouville equation down to a low-
dimensional linear Markov model [439,440] providing easy
access to linear control design. Intriguingly, the adjoint of the
Liouville equation is related to the Koopman operator
[439,527,528]. This deep relation still waits to be exploited in
probability control. In summary, there are plenty of promising
opportunities advancing nonlinear model-based control.
Acknowledgment
We have benefited tremendously from many fruitful discussions
on machine learning, compressive sensing, and control of com-
plex dynamical systems with Nathan Kutz, Josh Proctor, and Bing
Brunton. We acknowledge highly stimulating discussions with the
local TUCOROM team (Jean-Paul Bonnet, Laurent Cordier, Jo
el
Delville, Thomas Duriez, Eurika Kaiser, Kai von Krbek, Jean-
Charles Laurentie, Vladimir Parezanovic´, and Andreas Spohn),
with the OpenLab PPRIME/PSA team (Diogo Barros, Cecile Li,
Jacques Bor
ee, and Tony Ruiz), with the SepaCoDe team man-
aged by Azeddine Kourta, Andreas Spohn, and Michel Stanislas,
with Ambrosys (Markus W. Abel and Marc Segond), with the
SFB 880, in particular Rolf Radespiel, Peter Scholz, and Richard
Semaan, with the ARC team (Robert Niven and Steven Waldrip),
and with our close visitors and collaborators Jean-Luc Aider,
Shervin Bagheri, Maciej Balajewicz, Zacharias Berger, Jason
Bourgeois, Helmut Eckelmann, Nicolas Gautiers, Mark Glauser,
Hans-Christian Hege, Jens Kasten, Sini
sa Krajnovic´ , Jacques
Lewalle, Kervin Low, Robert Martinuzzi, Marek Morzy
nski,
Christian Nayeri, Jan
Osth, Oliver Paschereit, Brian Polagye,
Bartosz Protas, Michael Schlegel, Tamir Shaqarin, Sam Taira,
Jonathan Tu, and Dave Williams.
S.L.B. would also like to gratefully acknowledge and thank
Clancy Rowley, Richard Murray, Rob Stengel, and Naomi Leon-
ard, who each found unique ways to make control theory come to
life. S.L.B. has also had the pleasure of learning about bio-
inspired fluids from Tom Daniel and turbulence from Lex Smits,
Jim Riley, and Gigi Martinelli. S.L.B. was generously supported
by the Department of Mechanical Engineering, and as a Data Sci-
ence Fellow in the eScience Institute (NSF, Moore-Sloan Founda-
tion, and Washington Research Foundation) at the University of
Washington, as well as by the Department of Energy and Boeing.
B.R.N. is deeply indebted to his turbulence control mentors
Andrzej Banaszuk, Andreas Dillmann, Helmut Eckelmann, Rudi-
bert King, and William K. George who shared and fueled the pas-
sion for the field. B.R.N. acknowledges generous funding from
the French Agence Nationale de la Recherche (ANR) via the Sen-
ior Chair of Excellence TUCOROM and the SepaCoDe project,
from the OpenLab PPRIME/PSA by Peugeot-Citro
en, from the
region Poitou-Charentes, from the German Science Foundation
via the Collaborative Research Center SFB 880, from the Rector-
Funded Visiting Professorship of the UNSW, Canberra, Australia,
from the US National Research Foundatiation via the PIRE Grant
No. OISE-0968313 from Ambrosys GmbH, Potsdam, Germany
and from Bernd Noack Cybernetics Foundation.
We are deeply indebted to both anonymous referees for provid-
ing valuable suggestions regarding many technical aspects of the
manuscript.
Nomenclature
a¼model state
A¼matrix function of the dynamics
^
a¼estimated model state
Ai¼matrix of the quadratic nonlinearity
A0¼Jacobian of dynamics at as
as¼steady fixed point
ðA;B;C;DÞ¼linearized state-space system
ð^
A;^
B;^
C;^
DÞ¼controller state-space system
ðAd;Bd;Cd;DdÞ¼discrete-time system
ðAr;Br;Cr;DrÞ¼reduced-order model
b¼actuation input
balancing transformation
b¼optimal actuation
controllability matrix
Cd¼discrete-time controllability matrix
dissipation in TKE
D¼discretization operator
e¼error signal
E¼expectation value
f¼dynamics
F
k
¼prefilter for mixed sensitivity synthesis
g¼volume force
forcing in TKE
h¼impulse response function
H¼Hankel matrix
h
k
¼Volterra kernels
H2¼hardy space with bounded two norm
H1¼hardy space with bounded infinity norm
I¼identity matrix
J¼cost function
K¼controller
turbulent kinetic energy (TKE)
k
D
¼derivative control gain
k
i
¼gain of the Galerkin system
k
I
¼integral control gain
k
P
¼proportional control gain
Kd¼disturbance feed-forward control
Kf¼Kalman filter
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-39
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Kr¼regulator
Kref ¼reference feed-forward control
L¼loop transfer function (L¼PK)
¼Laplace transform
l
ij
¼coefficients of the linear term in the Galerkin
system
m¼output function
M¼amplitude of ESC input
M
S
¼maximum peak of sensitivity function
N
a
¼number of states
N
b
¼number of actuators
N
s
¼number of sensors
N
r
¼number of reduced-order states
Nf
a¼number of fast states
observability matrix
Od¼discrete-time observability matrix
P¼plant model
production in TKE
Pd¼disturbance model
Pr¼reduced-order plant model
Pdes ¼desired plant
Q¼LQR state weight
q
ijk
¼coefficients of the quadratic term in the
Galerkin system
R¼LQR actuation weight
Re ¼Reynolds number
s¼sensor output
S¼sensitivity function
^
s¼estimated sensor output
Sd¼sensitivity function to disturbance
Sref ¼sensitivity function to reference
t¼time
T¼complementary sensitivity
U¼left singular vectors from SVD
Ur¼first N
r
columns of U
uðx;tÞ¼fluid velocity field
u0ðx;tÞ¼velocity fluctuations
uaðx;tÞ¼actuation component
usðxÞ¼steady Navier–Stokes solution
uuðx;tÞ¼unsteady component
uDðx;tÞ¼mean-flow deformation
V¼right singular vectors from SVD
Vd¼disturbance covariance
Vn¼noise covariance
Vr¼first N
r
columns of V
w¼exogenous inputs
wd¼disturbance
wn¼sensor noise
wr¼reference input
Wc¼controllability Gramian
Wo¼observability Gramian
x¼position vector
X¼unknown variable in Riccati equation
Y¼unknown variable in dual Riccati equation
a
a
¼manifold parameter associated with ua
a
u
¼manifold parameter associated with uu
b
u
¼nonlinear damping parameter
b
aa
¼nonlinear damping parameter of uaon ua
b
au
¼nonlinear damping parameter of uuon ua
b
ua
¼nonlinear damping parameter of uaon uu
b
uu
¼nonlinear damping parameter of uuon uu
c
u
¼nonlinear frequency parameter
c
**
¼nonlinear frequency parameter (superscripts
have analogous meaning as in b)
Dt¼time step
d(t)¼Dirac delta function
f¼Laplace variable
l¼bifurcation parameters
n¼sparse vector for compressive sensing
R¼singular values from SVD
Rr¼first N
r
N
r
singular values
r
a
¼growth rate associated with ua
r
u
¼growth rate associated with uu
r
c
¼designed growth rate with control
s¼time delay
/¼phase variable
U¼direct BPOD modes
Ur¼first N
r
direct BPOD modes
W¼adjoint BPOD modes
Wr¼first N
r
adjoint BPOD modes
x¼oscillation frequency
x
a
¼frequency associated with ua
x
u
¼frequency associated with uu
References
[1] Fish, F. E., and Lauder, G. V., 2006, “Passive and Active Flow Control by
Swimming Fishes and Mammals,” Annu. Rev. Fluid Mech.,38, pp. 193–224.
[2] Ahlb orn, B. K., 2004, Zoological Physics, Springer-Verlag, Berlin.
[3] Dean, B., and Bhushan, B., 2010, “Shark-Skin Surfaces for Fluid-Drag Reduc-
tion in Turbulent Flow: A Review,” Philos. Trans. R. Soc. A,368(1929), pp.
4775–4806.
[4] Bechert, D. W., Bruse, M., Hage, W., van der Hoeven, J. G. T., and Hoppe,
G., 1997, “Experiments on Drag-Reducing Surfaces and Their Optimization
With an Adjustable Geometry,” J. Fluid Mech.,338, pp. 59–87.
[5] Gilli
eron, P., and Kourta, A., 2010, “Aerodynamic Drag Reduction by Vertical
Splitter Plates,” Exp. Fluids,48(1), pp. 1–16.
[6] Grandemange, M., Ricot, D., Vartanian, C., Ruiz, T., and Cadot, O., 2014,
“Characterisation of the Flow Past Real Road Vehicles With Blunt After-
bodies,” Int. J. Aerodyn.,4(1), pp. 24–42.
[7] Pfeiffe r, J., and King, R., 2012, “Multivariable Closed-Loop Flow Control of
Drag and Yaw Moment for a 3D Bluff Body,” AIAA Paper No. 2012-2802.
[8] Gad- el Hak, M., 1989, “Flow Control,” ASME Appl. Mech. Rev.,42(10), pp.
261–293.
[9] Gad-el Hak, M., and Tsai, H. M., 2006, Transition and Turbulence Con trol,
Vol. 8, World Scientific, Singapore.
[10] Kim, J., 2011, “Physics and Control of Wall Turbulence for Drag Reduction,”
Philos. Trans. R. Soc. A,369(1940), pp. 1396–1411.
[11] Baker, C., Jones, J., Lopez-Calleja, F., and Munday, J., 2004, “Measurements
of the Cross Wind Forces on Trains,” J. Wind Eng. Ind. Aerodyn.,92(7), pp.
547–563.
[12] Baker, C., 2010, “The Flow Around High Speed Trains,” J. Wind Eng. Ind.
Aerodyn.,98(6), pp. 277–298.
[13] Schetz, J. A., 2001, “Aerodynamics of High-Speed Trains,” Annu. Rev. Fluid
Mech.,33(1), pp. 371–414.
[14] Gad-el Hak, M., 1996, “Modern Developments in Flow Control,” ASME
Appl. Mech. Rev.,49(7), pp. 365–379.
[15] Barber, T. J., 1999, private communication.
[16] King, R., 2007, “Active Flow Control,” Notes on Numerical Fluid Mechanics
and Interdisciplinary Design, Vol. 95, Springer, Berlin.
[17] King, R., 2010, “Active Flow Control II,” Notes on Numerical Fluid Mechan-
ics and Interdisciplinary Design, Vol. 108, Springer, Berlin.
[18] Liepmann, H., and Nosenchuck , D., 1982, “Active Control of Laminar-
Turbulent Transition,” J. Fluid Mech.,118, pp. 201–204.
[19] Roussopoulos, K., 1993, “Feedback Control of Vortex Shedding at Low
Reynolds Numbers,” J. Fluid Mech.,248, pp. 267–296.
[20] Kim, J., and Bewley, T., 2007, “A Linear Systems Approach to Flow Control,”
Annu. Rev. Fluid Mech.,39, pp. 383–417.
[21] Sipp, D., Marquet, O., Meliga, P., and Barbagallo, A., 2010, “Dynamics and
Control of Global Instabilities in Open-Flows—A Linearized Approach,”
ASME Appl. Mech. Rev.,63(3), p. 030801.
[22] Lee, C., Kim, J., Babcock, D., and Goodman, R., 1997, “Application of Neural
Networks to Turbulence Control for Drag Reduction,” Phys. Fluids,9(6), pp.
1740–1747.
[23] Medjo, T. T., Temam, R., and Ziane, M., 2008, “Optimal and Robust Control
of Fluid Flows: Some Theoretical and Computational Aspects,” ASME Appl.
Mech. Rev.,61(1), p. 010802.
[24] Bewley, T. R., 2001, “Flow Control: New Challenges for a New Renaissance,”
Prog. Aerosp. Sci.,37(1), pp. 21–58.
[25] Greenblatt, D., and Wygnanski, I. J., 2000, “The Control of Flow Separation
by Periodic Excitation,” Prog. Aerosp. Sci.,36(7), pp. 487–545.
[26] Bushnell, D. M., and McGinley, C. B., 1989, “Turbulence Control in Wall
Flows,” Annu. Rev. Fluid Mech.,21, pp. 1–20.
[27] Moin, P., and Bewley, T., 1994, “Feedback Control of Turbulence,” ASME
Appl. Mech. Rev.,47(6S), pp. S3–S13.
[28] Lumley, J., and Blossey, P., 1998, “Control of Turbulence,” Annu. Rev. Fluid
Mech.,30, pp. 311–327.
[29] Gutmark, E. J., Schadow, K. C., and Yu, K. H., 1994, “Methods for Enhanced
Turbulence Mixing in Supersonic Shear Flows,” ASME Appl. Mech. Rev.,
47(6S), pp. S188–S192.
[30] Aamo, O. M., and Krstic´, M., 2002, Flow Control by Feedback: Stabilization
and Mixing, Springer-Verlag, London.
050801-40 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[31] Dimotakis, P. E., 2005, “Turbulent Mixing,” Annu. Rev. Fluid Mech.,37,pp.
329–35 6.
[32] Mankbadi, R. R., 1992, “Dynamics and Control of Coherent Struct ures in
Turbulent Jets,” ASME Appl. Mech. Rev.,45(6), pp. 219–248.
[33] Dowling, A. P., and Morgans, A. S., 2005, “Feedback Control of Combustion
Oscillations,” Annu. Rev. Fluid Mech.,37, pp. 151–182.
[34] Rowley, C., and Williams, D., 2006, “Dynamics and Control of High-
Reynolds Number Flows Over Open Cavities,” Annu. Rev. Fluid Mech.,38,
pp. 251–276.
[35] Choi, H., Jeon, W.-P., and Kim, J., 2008, “Control of Flow Over a Bluff
Body,” Annu. Rev. Fluid Mech.,40, pp. 113–139.
[36] Cattafesta, L., 2011, “Actu ators for Active Flow Control,” Annu. Rev. Fluid
Mech.,43, pp. 247–272.
[37] Chen, K. K., and Rowley, C. W., 2011, “H
2
Optimal Actuator and Sensor
Placement in the Linearised Complex Ginzburg-Landau System,” J. Fluid
Mech.,681, pp. 241–260.
[38] Bradshaw, P., Ferriss, D. H., and Johnson, R., 1964, “Turbulence in the
Noise-Producing Region of a Circular Jet,” J. Fluid Mech.,19(4), pp.
591–624.
[39] Brown, G. L., and Roshko, A., 1974, “On Density Effects and Large Structure
in Turbulent Mixing Layers,” J. Fluid Mech.,64(4), pp. 775–816.
[40] Kim, J., 2003, “Control of Turbulent Boundary Layers,” Phys. Fluids,15(5),
pp. 1093–1105.
[41] Siauw, W., Bonnet, J.-P., Tensi, J., Cordier, L., Noack, B. R., and Cattafesta,
L. I., 2010, “Transient Dynamics of the Flow Around a NACA0015 Airfoil
Using Fluid Vortex Generators,” Int. J. Heat Fluid Flow,31(3), pp. 450–459.
[42] Shaqarin, T., 2014, private communication.
[43] Choi, H., Moin, P., and Kim, J., 1994, “Active Turbulence Control for Drag
Reduction in Wall-Bounded Flows,” J. Fluid Mech.,262, pp. 75–110.
[44] Gerhard, J., Pastoor, M., King, R., Noack, B. R., Dillmann, A., Morzy
nski,
M., and Tadmor, G., 2003, “Model-Based Control of Vortex Shedding Using
Low-Dimensional Galerkin Models,” AIAA Paper No. 2003-4262.
[45] Pastoor, M., Henning, L., Noack, B. R., King, R., and Tadmor, G., 2008,
“Feedback Shear Layer Control for Bluff Body Drag Reduction,” J. Fluid
Mech.,608, pp. 161–196.
[46] Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J., and
Myatt, J., 2007, “Feedback Control of Subsonic Cavity Flows Usin g Reduced-
Order Models,” J. Fluid Mech.,579, pp. 315–346.
[47] Vukasinovic, B., Rusak, Z., and Glezer, A., 2010, “Dissipative, Small-Scale
Actuation of a Turbulent Shear Layer,” J. Fluid Mech.,656, pp. 51–81.
[48] Luc htenburg, D. M., G
unter, B., Noack, B. R., King, R., and Tadmor, G. A.,
2009, “Generalized Mean-Field Model of the Natural and Actuated Flows
Around a High-Lift Configuration,” J. Fluid Mech.,623, pp. 283–316.
[49] Aider, J.-L., 2014, private communication.
[50] Gordon, M., and Soria, J., 2002, “PIV Measurements of a Zero-Net-Mass-Flux
Jet in Cross Flow,” Exp. Fluids,33(6), pp. 863–872.
[51] Cater, J. E., and Soria, J., 2002, “The Evolution of Round Zero-Net-Mass-Flux
Jets,” J. Fluid Mech.,472, pp. 167–200.
[52] Zhang, P., Wang, J., and Feng, L., 2008, “Review of Zero-Net-Mass-Flux Jet
and Its Application in Separation Flow Control,” Sci. China Ser. E, Technol.
Sci.,51(9), pp. 1315–1344.
[53] Cattafesta, L. N., Garg, S., and Shukla, D., 2001, “Development of Piezoelec-
tric Actuators for Active Flow Control,” AIAA J.,39(8), pp. 1562–1568.
[54] Gallas, Q., Holman, R., Nishida, T., Carroll, B., Sheplak, M., and Cattafesta,
L., 2003, “Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet
Actuators,” AIAA J.,41(2), pp. 240–247.
[55] Glezer, A., and Amitay, M., 2002, “Synthetic Jets,” Annu. Rev. Fluid Mech.,
34, pp. 503–529.
[56] Smith, B. L., and Glezer, A., 1998, “The Formation and Evolution of Synthetic
Jets,” Phys. Fluids,10(9), pp. 2281–2297.
[57] Holman, R., Utturkar, Y., Mittal, R., Smith, B. L., and Cattafesta, L., 2005,
“Formation Criterion for Synthetic Jets,” AIAA J.,43(10), pp. 2110–2116.
[58] You, D., and Moin, P., 2008, “Active Control of Flow Separation Over an Air-
foil Using Synthetic Jets,” J. Fluids Struct.,24(8), pp. 1349–1357.
[59] Moreau, E., 2007, “Airflow Control by Non-Thermal Plasma Actuators,” J.
Phys. D: Appl. Phys.,40(3), p. 605.
[60] Hanson, R. E., Lavoie, P., and Naguib, A. M., 2010, “Effect of Plasma Actua-
tor Excitation for Controlling Bypass Transition in Boundary Layers,” AIAA
Paper No. 2010-1091.
[61] Hanson, R. E., Bade, K. M., Belson, B. A., Lavoie, P., Naguib, A. M., and
Rowley, C. W., 2014, “Feedback Control of Slowly-Varying Transient Growth
by an Array of Plasma Actuators,” Phys. Fluids,26(2), p. 024102.
[62] Huang, J., Corke, T. C., and Thomas, F. O., 2006, “Plasma Actuators for
Separation Control of Low-Pressure Turbine Blades,” AIAA J.,44(1), pp.
51–57.
[63] Roth, J. R., Sherman, D. M., and Wilkinson, S. P., 2000, “Electrohydrodynamic
Flow Control With a Glow-Discharge Surface Plasma,” AIAA J.,38(7), pp.
1166–1172.
[64] Post, M. L., and Corke, T. C., 2004, “Separation Control on High Angle of
Attack Airfoil Using Plasma Actuators,” AIAA J.,42(11), pp. 2177–2184.
[65] Hanson, R. E., Lavoie, P., Naguib, A. M., and Morrison, J. F., 2010,
“Transient Growth Instability Cancelation by a Plasma Actuator Array,” Exp.
Fluids,49(6), pp. 1339–1348.
[66] Ho, C.-M., and Tai, Y.-C., 1996, “Review: MEMS and Its Applications for
Flow Control,” ASME J. Fluids Eng.,118(3), pp. 437–447.
[67] Ho, C.-M., and Tai, Y.-C., 1998, “Micro-Electro-Mechanical Systems
(MEMS) and Fluid Flows,” Annu. Rev. Fluid Mech.,30, pp. 579–612.
[68] Naguib, A., Christophorou, C., Alnajjar, E., Nagib, H., Huang, C., and Najafi,
K., 1997, “Arrays of MEMS-Based Actuators for Control of Supersonic Jet
Screech,” AIAA Paper No. 1997-1963.
[69] L
ofdahl, L., and Gad-el-Hak, M., 1999, “MEMS Applications in Turbulence
and Flow Control,” Prog. Aeronaut. Sci.,35(2), pp. 101–203.
[70] Huang, C., Christophorou, C., Najafi, K., Naguib, A., and Nagib, H. M., 2002,
“An Electrostatic Microactuator System for Application in High-Speed Jets,”
Microelectromech. Syst., J.,11(3), pp. 222–235.
[71] Suzuki, H., Kasagi, N., and Suzuki, Y., 2004, “Active Control of an Axisym-
metric Jet With Distributed Electromagnetic Flap Actuators,” Exp. Fluids,
36(3), pp. 498–509.
[72] Kasagi, N., Suzuki, Y., and Fukagata, K., 2009, “Microelectromechanical
Systems-Based Feedback Control of Turbulence for Skin Friction Reduction,”
Annu. Rev. Fluid Mech.,41, pp. 231–251.
[73] Wu, J., Wang, L., and Tadmor, J., 2007, “Suppression of the Von Karman
Vortex Street Behind a Circular Cylinder by a Traveling Wave Generated by a
Flexible Surface,” J. Fluid Mech.,574, pp. 365–391.
[74] Thiria, B., Goujon-Durand, S., and Wesfreid, J. E., 2006, “The Wake of a Cyl-
inder Performing Rotary Oscillations,” J. Fluid Mech.,560, pp. 123–147.
[75] Bergm ann, M., Cordier, L., and Brancher, J.-P., 2005, “Optimal Rotary Con-
trol of the Cylinder Wake Using Proper Orthogonal Decomposition Reduced
Order Model,” Phys. Fluids,17(9), p. 097101.
[76] Wiener, N., 1948, Cybernetics or Control and Communication in the Animal
and the Machine, 1st ed., MIT Press, Boston.
[77] Kolmogorov, A., 1941, “The Local Structure of Turbulence in Incompressible
Viscous Fluid for Very Large Reynolds Number,” Dokl. Akad. Nauk. SSSR,
30, pp. 9–13.
[78] Kolmogorov, A., 1941, “On Degeneration (Decay) of Isotropic Turbulence,”
Dokl. Akad. Nauk SSSR,31, pp. 538–540.
[79] Landau, L. D., and Lifshitz, E. M., 1987, “Fluid Mechanics,” Course of
Theoretical Physics, 2nd ed., Vol. 6, Pergamon Press, Oxford, UK.
[80] Pope, S., 2000, Turbulent Flows, 1st ed., Cambridge University Press,
Cambridge, UK.
[81] Lee, M., Malaya, N., and Moser, R. D., 2013, “Petascale Direct Numerical
Simulation of Turbulent Channel Flow on Up to 786 k Cores,” International
Conference on High Performance Computing, Networking, Storage and Anal-
ysis (SC’13), Denver, CO, Nov. 17–21, p. 61.
[82] Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K., and Uno, A., 2003,
“Energy Dissipation Rate and Energy Spectrum in High Resolution Direct
Numerical Simulations of Turbulence in a Periodic Box,” Phys. Fluids,15(2),
pp. L21–L24.
[83] Moore, G. E., 1965, “Cramming More Components Onto Integrated Circuits,”
Electronics,38(8), pp. 114–117.
[84] Lumley, J., 1970, Stochastic Tools in Turbulence, Academic Press, New York.
[85] Holmes, P., Lumley, J. L., Berkooz, G., and Rowley, C. W., 2012, Turbulence,
Coherent Structures, Dynamical Systems and Symmetry, 2nd ed., Cambridge
University Press, Cambridge, UK.
[86] Sirovich, L., 1987, “Turbulence and the Dynamics of Coherent Structures,
Part I—Coherent Structures,” Q. Appl. Math.,XLV(3), pp. 561–571.
[87] Golub, G. H., and Reinsch, C., 1970, “Singular Value Decomposition and
Least Squares Solutions,” Numer. Math.,14(5), pp. 403–420.
[88] Golub, G., and Kahan, W., 1965, “Calculating the Singular Values and
Pseudo-Inverse of a Matrix,” J. Soc. Ind. Appl. Math., Ser. B: Numer. Anal.,
2(2), pp. 205–224.
[89] Trefethen, L. N., and Bau, D., III., 1997, Numerical Linear Algebra, Vol. 50,
SIAM, Philadelphia.
[90] Antoulas, A. C., 2005, Approximation of Large-Scale Dynamical Systems,
SIAM, Philadelphia.
[91] Pearson, K., 1901, “On Lines and Planes of Closest Fit to Systems of Points in
Space,” Philos. Mag.,2(7–12), pp. 559–572.
[92] Hotelling, H., 1933, “Analysis of a Complex of Statistical Variables Into
Principal Components,” J. Educ. Psychol.,24(6), pp. 417–441.
[93] Karhunen, K., 1946, “Zur Spektraltheorie Stochastischer Prozesse,” Ann.
Acad. Sci., Fennicae, Ser. A. I., Math.-Phys.,37, pp. 1–79.
[94] Lorenz, E., 1956, “Empirical Orthogonal Functions and Statistical Weather
Prediction,” Department of Meteorology, Statistical Forecasting Project, MIT,
Cambridge, MA, Scientific Report No. 1.
[95] Andino, M. Y., Wallace, R. D., Glauser, M. N., Camphouse, R. C., Schmit, R.
F., and Myatt, J. H., 2011, “Boundary Feedback Flow Control: Proportional
Control With Potential Application to Aero-Optics,” AIAA J.,49(1), pp.
32–40.
[96] Willcox, K., and Peraire, J., 2002, “Balanced Model Reduction Via the Proper
Orthogonal Decomposition,” AIAA J.,40(11), pp. 2323–2330.
[97] Rowley, C., 2005, “Model Reduction for Fluids Using Balanced Proper Or-
thogonal Decomposition,” Int. J. Bifurcation Chaos,15(3), pp. 997–1013.
[98] Schmid, P. J., and Sesterhenn, J., 2008, “Dynamic Mode Decomposition of
Numerical and Experimental Data,” 61st Annual Meeting of the APS Division
of Fluid Dynamics, San Antonio, TX, Nov. 23–25, American Physical Society,
College Park, MD, pp. 208.
[99] Schmid, P. J., 2010, “Dynamic Mode Decomposition for Numerical and
Experimental Data,” J. Fluid Mech.,656, pp. 5–28.
[100] Rowley, C. W., Mezic´ , I., Bagheri, S., Schlatter, P., and Henningson, D.,
2009, “Spectral Analysis of Nonlinear Flows,” J. Fluid Mech.,645, pp.
115–127.
[101] Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L., and Kutz, J. N.,
2014, “On Dynamic Mode Decomposition: Theory and Applications,” J. Com-
put. Dyn.,1(2), pp. 391–421.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-41
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[102] Tu, J. H., Rowley, C. W., Kutz, J. N., and Shang, J. K., 2014, “Spectral Analy-
sis of Fluid Flows Using Sub-Nyquist-Rate PIV Data,” Exp. Fluids,55(9), p.
1805.
[103] Koopman, B. O., 1931, “Hamiltonian Systems and Transformation in Hilbert
Space,” Proc. Natl. Acad. Sci.,17(5), pp. 315–318.
[104] Mezic´, I., and Banaszuk, A., 2004, “Comparison of Systems With Complex
Behavior,” Phys. D: Nonlinear Phenom.,197(1), pp. 101–133.
[105] Mezic´, I., 2005, “Spectral Properties of Dynamical Systems, Model Reduction
and Decompositions,” Nonlinear Dyn.,41(1–3), pp. 309–325.
[106] Budi
sic´ , M., Mohr, R., and Mezic´, I., 2012, “Applied Koopmanism,” Chaos:
Interdiscip. J. Nonlinear Sci.,22(4), p. 047510.
[107] Mezic´, I., 2013, “Analysis of Fluid Flows Via Spectral Properties of the Koop-
man Operator,” Annu. Rev. Fluid Mech.,45, pp. 357–378.
[108] Schmid, P. J., and Hennigson, D. S., 2001, Stability and Transition in Shear
Flows, Springer, New York.
[109] Schmid, P. J., 2007, “Nonmodal Stability Theory,” Annu. Rev. Fluid Mech.,
39, pp. 129–162.
[110] The ofilis, V., 2011, “Global Linear Instability,” Annu. Rev. Fluid Mech.,43,
pp. 319–352.
[111] Schmid, P. J., and Brandt, L., 2014, “Analysis of Fluid Systems: Stability,
Receptivity, Sensitivity,” ASME Appl. Mech. Rev.,66(2), p. 024803.
[112] Grosch, C. E., and Salwen, H., 1978, “The Continuous Spectrum of the
Orr-Sommerfeld Equation Part I—The Spectrum and the Eigenfunctions,” J.
Fluid Mech.,87, pp. 33–54.
[113] Salwen, H., and Grosch, C. E., 1981, “The Continuous Spectrum of the Orr-
Sommerfeld Equation. Part 2—Eigenfunction Expansions,” J. Fluid Mech.,
104, pp. 445–465.
[114] Joseph, D. D., 1976, “Stability of Fluid Motions I & II,” Springer Tracts in
Natural Philosophy, Vols. 26 and 27, Springer, New York.
[115] Boberg, L., and Brosa, U., 1988, “Onset of Turbulence in a Pipe,” Z. Natur-
forsch., 43a, pp. 697–726.
[116] Trefethen, L. N., Trefethen, A. E., Reddy, S. C., and Driscol l, T. A., 1993,
“Hydrodynamic Stability Without Eigenvalues,” Science,261(5121), pp.
578–584.
[117] Belson, B. A., Tu, J. H., and Rowley, C. W., 2014, “Algorithm 945: MODRED
A Parallelized Model Reduction Library,” ACM Trans. Math. Software,40(4),
p. 30.
[118] von Karman, T., 1912, “
Uber Den Mechanismus des Widerstands, den Ein
Bewegter Korper in Einer Fl
ussigkeit Erf
ahrt,” G
ottinger Nachrichten, Math.
Phys. Kl., 1912, pp. 547–556.
[119] F
oppl, L., 1913, “Wirbelbewegung hinter einem Kreiszylinder,” Sitzb. d. k.
Bayer. Akad. d. Wiss., 1, pp. 1.
[120] Suh, Y., 1993, “Periodic Motion of a Point Vortex in a Corner Subject to a
Potential Flow,” J. Phys. Soc. Jpn.,62, pp. 3441–3445.
[121] Noack, B. R., Mezic´, I., Tadmor, G., and Banaszuk, A., 2004, “Optimal Mix-
ing in Recirculation Zones,” Phys. Fluids,16(4), pp. 867–888.
[122] Lugt, H., 1996, Introduction to Vortex Theory, Vortex Flow Press, Potomac, MA.
[123] Cottet, G. H., and Koumoutsakos, P., 2000, Vortex Methods—Theory and
Practice, Cambridge University Press, Cambridge, UK.
[124] Wu, J.-Z., Ma, H.-Y., and Zhou, M.-D., 2006, Vorticity and Vortex Dynamics,
1st ed., Springer, Berlin.
[125] Adrian, R., and Moin, P., 1988, “Stochastic Estimation of Organized Turbu-
lent Structure: Homogeneous Shear Flow,” J. Fluid Mech.,190, pp. 531–559.
[126] Nicoud, F., Baggett, J., Moin, P., and Cabot, W., 2001, “Large Eddy Simula-
tion Wall-Modeling Based on Suboptimal Control Theory and Linear Stochas-
tic Estimation,” Phys. Fluids,13(10), pp. 2968–2984.
[127] Bonnet, J.-P., Cole, D., Delville, J., Glauser, M. N., and Ukeiley, L. S., 1998,
“Stochastic Estimation and Proper Orthogonal Decomposition—
Complementary Techniques for Identfying Structure,” Exp. Fluids,17(5), pp.
307–314.
[128] Glauser, M. N., Higuchi, H., Ausseur, J., and Pinier, J., 2004, “Feedback Con-
trol of Separated Flows,” AIAA Paper No. 2004-2521.
[129] Ausseur, J. M., Pinier, J. T., Glauser, M. N., Higuchi, H., and Carlson, H.,
2006, “Experimental Development of a Reduced-Order Model for Flow Sepa-
ration Control,” AIAA Paper No. 2006-1251.
[130] Tinney, C., Coiffet, F., Delville, J., Hall, A., Jordan, P., and Glauser, M., 2006,
“On Spectral Linear Stochastic Estimation,” Exp. Fluids,41(5), pp. 763–775 .
[131] Hudy, L. M., Naguib, A., and Humphreys, W. M., 2007, “Stochastic Estima-
tion of a Separated-Flow Field Using Wall-Pressure-Array Measurements,”
Phys. Fluids,19(2), p. 024103.
[132] Pinier, J. T., Ausseur, J. M., Glauser, M. N., and Higuchi, H., 2007,
“Proportional Closed-Loop Feedback Control of Flow Separation,” AIAA J.,
45(1), pp. 181–190.
[133] Farrell, B. F., and Ioannou, P. J., 2001, “State Estimation Using a Reduced-
Order Kalman Filter,” J. Atmos. Sci.,58(23), pp. 3666–3680.
[134] King, R., and Gilles, E., 1985, “Multiple Kalman Filters for Early Detection of
Hazardous States,” International Conference Industrial Process Modelling and
Control, Hangzhou, China, June 6–9, pp. 130–138.
[135] Tu, J. H., Griffin, J., Hart, A., Rowley, C. W., III, L. N. C., and Ukeiley, L. S.,
2013, “Integration of Non-Time-Resolved PIV and Time-Resolved Velocity
Point Sensors for Dynamic Estimation of Velocity Fields,” Exp. Fluids,54(2),
p. 1429.
[136] Welch, G., and Bishop, G., 1995, “An Introduction to the Kalman Filter,” Uni-
versity of North Carolina, Chapel Hill, NC, Technical Report 95-041.
[137] Busse, F. H., 1991, “Numerical Analysis of Secondary and Tertiary States of
Fluid Flow and Their Stability Properties,” Appl. Sci. Res.,48(3–4), pp.
341–351.
[138] Noack, B. R., and Eckelmann, H., 1994, “A Global Stability Analysis of the
Steady and Periodic Cylinder Wake,” J. Fluid Mech.,270, pp. 297–330.
[139] Fletcher, C. A. J., 1984, Computational Galerkin Methods, 1st ed., Springer,
New York.
[140] Holmes, P., Lumley, J. L., and Berkooz, G., 1998, Turbulence, Coherent
Structures, Dynamical Systems and Symmetry, 1st ed., Cambridge University
Press, Cambridge, UK.
[141] Juang, J. N., and Pappa, R. S., 1985, “An Eigensystem Realiza tion Algorithm
for Modal Parameter Identification and Model Reduction,” J. Guid., Control,
Dyn.,8(5), pp. 620–627.
[142] Juang, J. N., 1994, Applied System Identification, Prentice-Hall, Upper Saddle
River, NJ.
[143] Ljung, L., 2001, “Black-Box Models From Input–Output Measurements,”
18th IEEE Instrumentation and Measurement Technology Conference (IMTC
2001), Budapest, May 21–23, pp. 138–146.
[144] Ljung, L., 1999, System Identificat ion: Theory for the User, Prentice-Hall,
Upper Saddle River, NJ.
[145] Crouch, P., 1981, “Dynamical Realizations of Finite Volterra Series,” SIAM J.
Control Optim.,19(2), pp. 177–202.
[146] Boyd, S., Chua, L. O., and Desoer, C. A., 1984, “Analytical Foundations of
Volterra Series,” IMA J. Math. Control Inf.,1(3), pp. 243–282.
[147] Boyd, S., and Chua, L. O., 1985, “Fading Memory and the Problem of
Approximating Nonlinear Operators With Volterra Series,” IEEE Trans.
Circuits Syst.,32(11), pp. 1150–1161.
[148] Lesiak, C., andKrener, A. J., 1978, “The Existence and Uniqueness of Volterra Se-
ries for Nonlinear Systems,” IEEE Trans. Autom. Control,23(6), pp. 1090–1095.
[149] Brockett, R. W., 1976, “Volterra Series and Geometric Control Theory,”
Automatica,12(2), pp. 167–176.
[150] Krstic´, M., Smyshlyaev, A., and Vazquez, R., 2006, “Boundary Control of
PDEs and Applications to Turbulent Flows and Flexible Structures,” IEEE
Chinese Control Conference (CCC 2006), Harbin, China, Aug. 7–11, pp.
PL–4–PL–16.
[151] Floriani, E., de Wit, T. D., and Le Gal, P., 2000, “Nonlinear Interactions in a
Rotating Disk Flow: From a Volterra Model to the Ginzburg–Landau Equa-
tion,” Chaos: Interdiscip. J. Nonlinear Sci.,10(4), pp. 834–847.
[152] Tromp, J. C., and Jenkins, J. E. A., 1990, “Volterra Kernel Identification
Scheme Applied to Aerodynamic Reactions,” AIAA Paper No. 90-2803.
[153] Prazenica, R. J., Reisenthel, P. H., Kurdila, A. J., and Brenner, M. J., 2007,
“Volterra Kernel Extrapolation for Modeling Nonlinear Aeroelastic Systems
at Novel Flight Conditions,” J. Aircr.,44(1), pp. 149–162.
[154] Balajewicz, M., and Dowell, E., 2012, “Reduced-Order Modeling of Flutter
and Limit-Cycle Oscillations Using the Sparse Volterra Series,” J. Aircr.,
49(6), pp. 1803–1812.
[155] Balikhin, M., Bates, I., and Walker, S., 2001, “Identification of Linear and
Nonlinear Processes in Space Plasma Turbulence Data,” Adv. Space Res.,
28(5), pp. 787–800.
[156] Vazquez, R.,and Krstic´, M., 2007, Control of Turbulent and Magnetohydrodynamic
Channel Flows: Boundary Stabilization and State Estimation, Springer, New York.
[157] Estrada, T., Happel, T., Hidalgo, C., Ascasibar, E., and Blanco, E., 2010,
“Experimental Observation of Coupling Between Turbulence and Sheared Flows
During LH Transitions in a Toroidal Plasma,” Europhys. Lett.,92(3), p. 35001.
[158] Smola, A. J., and Sch
olkopf, B., 2004, “A Tutorial on Support Vector
Regression,” Stat. Comput.,14(3), pp. 199–222.
[159] Sch
olkopf, B., and Smola, A. J., 2002, Learning With Kernels: Support Vector
Machines, Regularization, Optimization, and Beyond, MIT Press, Cambridge, MA.
[160] Suykens, J. A., and Vandewalle, J., 1999, “Least Squares Support Vector
Machine Classifiers,” Neural Process. Lett.,9(3), pp. 293–300.
[161] Doyle, J. C., 1978, “Guaranteed Margins for LQG Regulators,” IEEE Trans.
Autom. Control,23(4), pp. 756–757.
[162] Doyle, J. C., and Stein, G., 1981, “Multivariable Feedback Design: Concepts for
a Classical/Modern Synthesis,” IEEE Trans. Autom. Control,26(1), pp. 4–16.
[163] Glover, K., and Doyle, J. C., 1988, “State-Space Formulae for All Stabilizing
Controllers That Satisfy an H
1
-Norm Bound and Relations to Risk
Sensitivity,” Syst. Control Lett.,11(3), pp. 167–172.
[164] Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., 1989, “State-
Space Solutions to Standard H
2
and H
1
Control Problems,” IEEE Trans.
Autom. Control,34(8), pp. 831–847.
[165] Schlinker, R., Simonich, J., Shannon, D., Reba, R., Colonius, T., Gudmunds-
son, K., and Ladeinde, F., 2009, “Supersonic Jet Noise From Round and
Chevron Nozzles: Experimental Studies,” AIAA Paper No. 2009-3257.
[166] Skogestad, S., and Postlethwaite, I., 1996, Multivariable Feedback Control,
Wiley, Chichester, UK.
[167] Dullerud, G. E., and Paganini, F., 2000, “A Course in Robust Control Theory:
A Convex Approach,” Texts in Applied Mathematics, Springer, Berlin.
[168] Scott Collis, S., Joslin, R. D., Seifert, A., and Theofilis, V., 2004, “Issues in
Active Flow Control: Theory, Control, Simulation, and Experiment,” Prog.
Aerosp. Sci.,40(4), pp. 237–289.
[169] Rowley, C. W., and Batten, B. A., 2008, “Dynamic and Closed-Loo p Con-
trol,” Fundamentals and Applications of Modern Flow Control (Progress in
Astronautics and Aeronautics, Vol. 231), American Institute of Aeronautics
and Astronautics, Reston, VA, pp. 115–148.
[170] Bagheri, S., Hoepffner, J., Schmid, P. J., and Henningson, D. S., 2009,
“Input–Output Analysis and Control Design Applied to a Linear Mod el of
Spatially Developing Flows,” ASME Appl. Mech. Rev.,62(2), p. 020803.
[171] Fabbiane, N., Semeraro, O., Bagheri, S., and Henningson, D. S., 2014,
“Adaptive and Model-Based Control Theory Applied to Convectively Unsta-
ble Flows,” ASME Appl. Mech. Rev.,66(6), p. 060801.
050801-42 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[172] Devasia, S., Chen, D., and Paden, B., 1996, “Nonlinear Inversion-Based Out-
put Tracking,” IEEE Trans. Autom. Control,41(7), pp. 930–942.
[173] Krstic´, M., and Banaszuk, A., 2006, “Multivariable Adaptive Control of Insta-
bilities Arising in Jet Engines,” Control Eng. Pract.,14(7), pp. 833–842.
[174] Bewley, T. R., Temam, R., and Ziane, M., 2000, “A General Framework for
Robust Control in Fluid Mechanics,” Phys. D,138(3–4), pp. 360–392.
[175] King, R., Active Flow and Combustion Control, Vol. 127, Springer Interna-
tional Publishing, Cham, Switzerland.
[176] Kerstens, W., Pfeiffer, J., Williams, D., King, R., and Colonius, T., 2011,
“Closed-Loop Control of Lift for Longitudinal Gust Suppression at Low Reyn-
olds Numbers,” AIAA J.,49(8), pp. 1721–1728.
[177] Devasia, S., 2002, “Should Model-Based Inverse Inputs be Used as Feedfor-
ward Under Plant Uncertainty?” IEEE Trans., Autom. Control,47(11), pp.
1865–1871.
[178] Chen, K. K., and Rowley, C. W., 2013, “Normalized Coprime Robust Stability
and Performance Guarantees for Reduced-Order Controllers,” IEEE Trans.
Autom. Control,58(4), pp. 1068–1073.
[179] Businger, P. A., and Golub, G. H., 1969, “Algorithm 358: Singular Value
Decomposition of a Complex Matrix [F1, 4, 5],” Commun. ACM,12(10), pp.
564–565.
[180] Ho, B. L., and Kalman, R. E., 1965, “Effective Construction of Linear State-
Variable Models From Input/Output Data,” 3rd Annual Allerton Conference
on Circuit and System Theory, Monticello, IL, Oct. 20–22, pp. 449–459.
[181] Moore, B. C., 1981, “Principal Component Analysis in Linear Systems:
Controllability, Observability, and Model Reduction,” IEEE Trans. Autom.
Control,26(1), pp. 17–32.
[182] Berkooz, G., Holmes, P., and Lumley, J., 1993, “The Proper Orthogonal
Decomposition in the Analysis of Turbulent Flows,” Annu. Rev. Fluid Mech.,
25, pp. 539–575.
[183] Ilak, M., and Rowley, C. W., 2008, “Modeling of Transitional Channel Flow
Using Balanced Proper Orthogonal Decomposition,” Phys. Fluids,20(3), p.
034103.
[184] Lall, S., Marsden, J. E., and Glava
ski, S., 1999, “Empirical Model Reduction
of Controlled Nonlinear Systems,” International Federation of Automatic
Control (IFAC) World Congress, Beijing, July 5–9, pp. 473–478.
[185] Lall, S., Marsden, J. E., and Glava
ski, S., 2002, “A Subspace Approach to Bal-
anced Truncation for Model Reduction of Nonlinear Control Systems,” Int. J.
Rob. Nonlinear Control,12(6), pp. 519–535.
[186] Laub, A. J., Heath, M. T., Paige, C., and Ward, R., 1987, “Computation of
System Balancing Transformations and Other Applications of Simultaneous
Diagonalization Algorithms,” IEEE Trans. Autom. Control,32(2), pp.
115–122.
[187] Sirovich, L., 1987, “Turbulence and the Dynamics of Coherent Structures,
Part III—Dynamics and Scaling,” Q. Appl. Math., XLV, pp. 583–590.
[188] Sirovich, L., 1987, “Turbulence and the Dynamics of Coherent Structures,
Part II—Symmetries and Transformations,” Q. Appl. Math., XLV, pp.
573–582.
[189] Ma, Z., Ahuja, S., and Rowley, C. W., 2011, “Reduced Order Models for Con-
trol of Fluids Using the Eigensystem Realization Algorithm,” Theor. Comput.
Fluid Dyn.,25(1), pp. 233–247.
[190] Luchtenburg, D. M., and Rowley, C. W., 2011, “Model Reduction Using
Snapshot-Based Realizations,” Bull. Am. Phys. Soc., 56,p.
BAPS.2011.DFD.H19.4.
[191] Tu, J. H., and Rowley, C. W., 2012, “An Improved Algorithm for Balanced
POD Through an Analytic Treatment of Impulse Response Tails,” J. Comput.
Phys.,231(16), pp. 5317–5333.
[192] Juang, J. N., Phan, M., Horta, L. G., and Longman, R. W., 1991,
“Identification of Observer/Kalman Filter Markov Parameters: Theory and
Experiments,” NASA Langley Research Center, Hampton, VA, NASA Tech-
nical Memorandum No. 104069.
[193] Phan, M., Juang, J. N., and Longman, R. W., 1992, “Identification of Linear-
Multivariable Systems by Identification of Observers With Assigned Real
Eigenvalues,” J. Astronaut. Sci., 40(2), pp. 261–279.
[194] Phan, M., Horta, L. G., Juang, J. N., and Longman, R. W., 1993, “Linear Sys-
tem Identification Via an Asymptotically Stable Observer,” J. Optim. Theory
Appl.,79(1), pp. 59–86.
[195] Proctor, J. L., Brunton, S. L., and Kutz, J. N., 2014, “Dynamic Mode Decom-
position With Control: Using State and Input Snapshots to Discover Dynami-
cs,” arXiv:1409.6358.
[196] Barkley, D., and Tuckerman, L. S., 1999, “Stability Analysis of Perturbed
Plane Couette Flow,” Phys. Fluids,11(5), pp. 1187–1195.
[197] Bayly, B. J., Orszag, S. A., and Herbert, T., 1988, “Instability Mechanisms in
Shear-Flow Transition,” Annu. Rev. Fluid Mech.,20(1), pp. 359–391.
[198] Orszag, S. A., and Patera, A. T., 1983, “Secondary Instability of Wall-
Bounded Shear Flows,” J. Fluid Mech.,128, pp. 347–385.
[199] Ruelle, D., and Takens, F., 1971, “On the Nature of Turbulence,” Commun.
Math. Phys.,20(3), pp. 167–192.
[200] Aamo, O. M., Krstic´, M., and Bewley, T. R., 2003, “Control of Mixing
by Boundary Feedback in 2D Channel Flow,” Automatica,39(9), pp.
1597–1606.
[201] Bagheri, S., and Henningson, D. S., 2011, “Transition Delay Using Control
Theory,” Philos. Trans. R. Soc. A,369(1940), pp. 1365–1381.
[202] Abergel, F., and Temam, R., 1990, “On Some Control Problems in Fluid
Mechanics,” Theor. Comput. Fluid Dyn.,1(6), pp. 303–325.
[203] Jameson, A., 2003, “Aerodynamic Shape Optimization Using the Adjoint
Method” (VKI Lecture Series on Aerodynamic Drag Prediction and Reduc-
tion), von Karman Institute of Fluid Dynamics, Rhode-St-Genese, Belgium.
[204] Reuther, J. J., Jameson, A., Alonso, J. J., Rimlinger, M. J., and Saunders, D., 1999,
“Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint For-
mulation and Parallel Computers, Part 1,” J. Aircr.,36(1), pp. 51–60.
[205] Jameson, A., Martinelli, L., and Pierce, N., 1998, “Optimum Aerodynamic
Design Using the Navier–Stokes Equations,” Theor. Comput. Fluid Dyn.,
10(1–4), pp. 213–237.
[206] Reuther, J., Jameson, A., Farmer, J., Martinelli, L., and Saunders, D., 1996,
“Aerodynamic Shape Optimization of Complex Aircraft Configurations Via
an Adjoint Formulation,” Research Institute for Advanced Computer Science ,
NASA Ames Research Center, Mountain View, CA, Report No. NASA-CR-
203275.
[207] Choi, H., Temam, R., Moin, P., and Kim, J., 1993, “Feedback Control for
Unsteady Flow and Its Application to the Stochastic Burgers Equation,” J.
Fluid Mech.,253, pp. 509–543.
[208] Bewley, T., and Moin, P., 1994, “Optimal Control of Turbulent Channel
Flows,” Act. Control Vib. Noise, ASME DE-Vol. 75, pp. 221–227.
[209] Lee, C., Kim, J., and Choi, H., 1998, “Suboptimal Control of Turbulent Chan-
nel Flow for Drag Reduction,” J. Fluid Mech.,358, pp. 245–258.
[210] Bewley, T. R., Moin, P., and Temam, R., 2001, “DNS-Based Predictive Con-
trol of Turbulence: An Optimal Benchmark for Feedback Algorithms,” J.
Fluid Mech.,447, pp. 179–225.
[211] Collis, S. S., Chang, Y., Kellogg, S., and Prabhu, R., 2000, “Large Eddy Simu-
lation and Turbulence Control,” AIAA Paper No. 2000-2564.
[212] Bewley, T., and Liu, S., 1998, “Optimal and Robust Control and Estimation of
Linear Paths to Transition,” J. Fluid Mech.,365, pp. 305–349.
[213] Baramov, L., Tutty, O. R., and Rogers, E., 2000, “Robust Control of Plane
Poiseuille Flow,” AIAA Paper No. 2000-2684.
[214] H
ogberg, M., Bewley, T. R., and Henningson, D. S., 2003, “Linear Feedback
Control and Estimation of Transition in Plane Channel Flow,” J. Fluid Mech. ,
481, pp. 149–175.
[215] H
ogberg, M., and Henningson, D. S., 2002, “Linear Optimal Control Applied
to Instabilities in Spatially Developing Boundary Layers,” J. Fluid Mech.,
470, pp. 151–179.
[216] Chevali er, M., Hœpffner, J., A
˚kervik, E., and Henningson, D., 2007, “Linear
Feedback Control and Estimation Applied to Instabilities in Spatially Devel-
oping Boundary Layers,” J. Fluid Mech.,588, pp. 163–187.
[217] A
˚kervik, E., Hœpffner, J., Ehrenstein, U., and Henningson, D. S., 2007,
“Optimal Growth, Model Reduction and Control in Separated Boundary-
Layer Flow Using Global Eigenmodes,” J. Fluid Mech.,579, pp. 305–314.
[218] Ahuja, S., Rowley, C. W., Kevrekidis, I. G., Wei, M., Colonius, T., and Tad-
mor, G., 2007, “Low-Dimensional Models for Control of Leading-Edge Vorti-
ces: Equilibria and Linearized Models,” AIAA Paper No. 2007-709.
[219] Colonius, T., and Taira, K., 2008, “A Fast Immersed Boundary Method Using
a Nullspace Approach and Multi-Domain Far-Field Boundary Conditions,”
Comput. Methods Appl. Mech. Eng.,197(25–28), pp. 2131–2146.
[220] Taira, K., and Colonius, T., 2007, “The Immersed Boundary Method: A Pro-
jection Approach,” J. Comput. Phys.,225(2), pp. 2118–2137.
[221] Bagheri, S., Brandt, L., and Henningson, D., 2009, “Input–Output Analysis,
Model Reduction and Control of the Flat-Plate Boundary Layer,” J. Fluid
Mech.,620, pp. 263–298.
[222] Semeraro, O., Bagheri, S., Brandt, L., and Henningson, D. S., 2011, “Feedback
Control of Three-Dimensional Optimal Disturbances Using Reduced-Order
Models,” J. Fluid Mech.,677, pp. 63–102.
[223] Illingworth, S. J., Morgans, A. S., and Rowley, C. W., 2010, “Feedback
Control of Flow Resonances Using Balanced Reduced-Order Models,” J.
Sound Vib.,330(8), pp. 1567–1581.
[224] Illingworth, S. J., Morgans, A. S., and Rowley, C. W., 2012, “Feedback Con-
trol of Cavity Flow Oscillations Using Simple Linear Models,” J. Fluid Mech.,
709, pp. 223–248.
[225] Semeraro, O., Bagheri, S., Brandt, L., and Henningson, D. S., 2013,
“Transition Delay in a Boundary Layer Flow Using Active Control,” J. Fluid
Mech.,731, pp. 288–311.
[226] Moarref, R., and Jovanovic´, M. R., 2012, “Model-Based Design of Transverse Wall
Oscillations for Turbulent Drag Reduction,” J. Fluid Mech.,707, pp. 205–240.
[227] Cortelezzi, L., Lee, K., Kim, J., and Speyer, J., 1998, “Skin-Friction Drag
Reduction Via Robust Reduced-Order Linear Feedback Control,” Int. J. Com-
put. Fluid Dyn.,11(1–2), pp. 79–92.
[228] Cortelezzi, L., and Speyer, J., 1998, “Robust Reduced-Order Controller of
Laminar Boundary Layer Transitions,” Phys. Rev. E,58(2), pp. 1906–1910.
[229] Lee, K. H., Cortelezzi, L., Kim, J., and Speyer, J., 2001, “Application of
Reduced-Order Controller to Turbulent Flows for Drag Reduction,” Phys.
Fluids,13(5), pp. 1321–1330.
[230] Kasagi, N., Hasegawa, Y., and Fukagata, K., 2009, “Toward Cost-Effective
Control of Wall Turbulence for Skin Friction Drag Reduction,” Advances in
Turbulence XII, Springer, Berlin, pp. 189–200.
[231] Fukagata, K., Kobayashi, M., and Kasagi, N., 2010, “On the Friction Drag
Reduction Effect by a Control of Large-Scale Turbulent Structures,” J. Fluid
Sci. Technol.,5(3), pp. 574–584.
[232] Mamori, H., Fukagata, K., and Hoepffner, J., 2010, “Phase Relationship in
Laminar Channel Flow Controlled by Traveling-Wave-Like Blowing or
Suction,” Phys. Rev. E,81(4), p. 046304.
[233] Kametani, Y., and Fukagata, K., 2011, “Direct Numerical Simulation of Spa-
tially Developing Turbulent Boundary Layers With Uniform Blowing or
Suction,” J. Fluid Mech.,681, pp. 154–172.
[234] Nakanishi, R., Mamori, H., and Fukagata, K., 2012, “Relaminarization of Tur-
bulent Channel Flow Using Traveling Wave-Like Wall Deformation,” Int. J.
Heat Fluid Flow,35, pp. 152–159.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-43
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[235] Kasa gi, N., Hasegawa, Y., Fukagata, K., and Iwamoto, K., 2012, “Control of
Turbulent Transport: Less Friction and More Heat Transfer,” ASME J. Heat
Transfer,134(3), p. 031009.
[236] Rathnasingham, R., and Breuer, K. S., 1997, “System Identification and
Control of a Turbulent Boundary Layer,” Phys. Fluids,9(7), pp. 1867–1869.
[237] Rathnasingham, R., and Breuer, K. S., 2003, “Active Control of Turbulent
Boundary Layers,” J. Fluid Mech.,495, pp. 209–233.
[238] Rowley, C. W., 2002, “Modeling, Simulation, and Control of Cavity Flow
Oscillations,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
[239] Cattafesta, L., Shukla, D., Garg, S., and Ross, J., 1999, “Development of an
Adaptive Weapons-Bay Suppression System,” AIAA Paper No. 1999-1901.
[240] Cattafesta, L., Williams, D., Rowley, C., and Alvi, F., 2003, “Review of
Active Control of Flow-Induced Cavity Resonance,” AIAA Paper No. 2003-
3567.
[241] Cattafesta, L. N., III, Song, Q., Williams, D. R., Rowley, C. W., and Alvi, F.
S., 2008, “Active Control of Flow-Induced Cavity Oscillations,” Prog. Aerosp.
Sci.,44(7), pp. 479–502.
[242] Rowley, C. W., Colonius, T., and Basu, A. J., 2002, “On Self-Sustained Oscil-
lations in Two-Dimensional Compressible Flow Over Rectangular Cavities,”
J. Fluid Mech.,455, pp. 315–346.
[243] Rowley, C. W., Colonius, T., and Murray, R. M., 2000, “POD Based Models
of Self-Sustained Oscillations in the Flow Past an Open Cavity,” AIAA Paper
No. 2000-1969.
[244] Rowley, C., Colonius, T., and Murray, R., 2004, “Model Reduction for
Compressible Flows Using POD and Galerkin Projection,” Physica D,
189(1–2), pp. 115–129.
[245] Rowley, C. W., Williams, D. R., Colonius, T., Murray, R. M., MacMartin, D.
G., and Fabris, D., 2002, “Model-Based Control of Cavity Oscillations. Part
II: System Identification and Analysis,” AIAA Paper No. 2002-0972.
[246] Samimy, M., Debiasi, M., Caraballo, E., Malone, J., Little, J.,
Ozbay, H., Efe,
M., Yan, X., Yuan, X., DeBonis, J., Myatt, J., and Camphouse, R., 2004,
“Exploring Strategies for Closed-Loop Cavity Flow Control,” AIAA Paper
No. 2004-0576.
[247] Rowley, C. W., Williams, D. R., Colonius, T., Murray, R. M., and Macmy-
nowski, D. G., 2006, “Linear Models for Control of Cavity Flow Oscillations,”
J. Fluid Mech.,547, pp. 317–330.
[248] Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., and Little, J.,
2007, “Reduced-Order Model-Based Feedback Control of Subsonic Cavit y
Flows—An Experimental Approach,” Notes on Numerical Fluid Mechanics
and Multidisciplinary Design (NNFM), Vol. 25, Springer, Berlin, pp.
211–230.
[249] Efe, M., Debiasi, M., Yan, P.,
Ozbay, H., and Samimy, M., 2005, “Control of
Subsonic Cavity Flows by Neural Networks—Analytical Models and Experi-
mental Validation,” AIAA Paper No. 2005-294.
[250] Belson, B. A., Semeraro, O., Rowley, C. W., and Henningson, D. S., 2013,
“Feedback Control of Instabilities in the Two-Dimensional Blasius Boundary
Layer: The Role of Sensors and Actuators,” Phys. Fluids,25(5), p. 054106.
[251] Herv
e, A., Sipp, D., Schmid, P. J., and Samuelides, M., 2012, “A Physics-
Based Approach to Flow Control Using System Identification,” J. Fluid
Mech.,702, pp. 26–58.
[252] Semeraro, O., Pralits, J. O., Rowley, C. W., and Henningson, D. S., 2013,
“Riccati-Less Approach for Optimal Control and Estimation: An Application
to Two-Dimensional Boundary Layers,” J. Fluid Mech.,731, pp. 394–417.
[253] Weller, J., Camarri, S., and Iollo, A., 2009, “Feedback Control by Low-Order
Modelling of the Laminar Flow Past a Bluff Body,” J. Fluid Mech.,634, pp.
405–418.
[254] Stuart, J., 1958, “On the Non-Linear Mechanics of Hydrodynamic Stability,”
J. Fluid Mech.,4(1), pp. 1–21.
[255] Stuart, J., 1971, “Nonlinear Stability Theory,” Annu. Rev. Fluid Mech.,3, pp.
347–370.
[256] Schumm, M., Berger, E., and Monkewitz, P., 1994, “Self-Excited Oscillations
in the Wake of Two-Dimensional Bluff Bodies and Their Control,” J. Fluid
Mech.,271, pp. 17–53.
[257] Dusek, J., Le Gal, P., and Frauni
e, P., 1994, “A Numerical and Theoretical
Study of the First Hopf Bifurcation in a Cylinder Wake,” J. Fluid Mech.,264,
pp. 59–80.
[258] Bourgeois, J. A., Martinuzzi, R. J., and Noack, B. R., 2013, “Generalised
Phase Average With Applications to Sensor-Based Flow Estimation of the
Wall-Mounted Square Cylinder Wake,” J. Fluid Mech.,736, pp. 316–350.
[259] Luchtenburg, M., Tadmor, G., Lehmann, O., Noack, B. R., King, R., and Mor-
zy
nski, M., 2006, “Tuned POD Galerkin Models for Transient Feedback Reg-
ulation of the Cylinder Wake,” 44th AIAA Aerospace Sciences Meeting,
Reno, NV, Jan. 9–12, AIAA Paper 2006-1407.
[260] Tadmor, G., Lehmann, O., Noack, B. R., Cordier, L., Delville, J., Bonnet,
J.-P., and Morzy
nski, M., 2011, “Reduced Order Models for Closed-Loop
Wake Control,” Philos. Trans. R. Soc. A,369(1940), pp. 1513–1523.
[261] King, R., Seibold, M., Lehmann, O., Noack, B. R., Morzy
nski, M., and Tad-
mor, G., 2005, “Nonlinear Flow Control Based on a Low Dimensional Model
of Fluid Flow,” Control and Observer Design for Nonlinear Finite and Infinite
Dimensional Systems (Lecture Notes in Control and Information Sciences,
Vol. 322), T. Meurer, K. Graichen, and E. Gilles, eds., Springer, Berlin, pp.
369–386.
[262] Bergmann, M., and Cordier, L., 2008, “Optimal Control of the Cylinder Wake
in the Laminar Regime by Trust-Region Methods and POD Reduced Order
Models,” J. Comput. Phys.,227(16), pp. 7813–7840.
[263] Parezanovic, V., Laurentie, J.-C., Duriez, T., Fourment, C., Delville, J.,
Bonnet, J.-P., Cordier, L., Noack, B. R., Segond, M., Abel, M., Shaqarin, T.,
and Brunton, S. L., 2015, “Mixing Layer Manipulation Experiment—From
Periodic Forcing to Machine Learning Closed-Loop Control,” J. Flow Turbul.
Combust.,94(1), pp. 155–173.
[264] Aleksic, K., Luchtenburg, D. M., King, R., Noack, B. R., and Pfeiffer, J.,
2010, “Robust Nonlinear Control Versus Linear Model Predictive Control of a
Bluff Body Wake,” AIAA Paper No. 2010-4833.
[265] Duriez, T., Parezanovic, V., Laurentie, J.-C., Fourment, C., Delville , J., Bon-
net, J.-P., Cordier, L., Noack, B. R., Segond, M., Abel, M., Gautier, N., Aider,
J.-L., Raibaudo, C., Cuvier, C., Stanislas, M., and Brunton, S. L., 2014,
“Closed-Loop Control of Experimental Shear Flows Using Machine
Learning,” AIAA Paper No. 2014-2219.
[266] Luchtenburg, D. M., Schlegel, M., Noack, B. R., Aleksic´, K., King, R., Tad-
mor, G., and G
unther, B., 2010, “Turbulence Control Based on Reduced-
Order Models and Nonlinear Control Design,” Active Flow Control II (Notes
on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 108), R.
King, ed., Springer-Verlag, Berlin, pp. 341–356.
[267] Farazmand, M. M., Kevlahan, N. K.-R., and Protas, B., 2011, “Controlling the
Dual Cascade of Two-Dimensional Turbulence,” J. Fluid Mech.,668, pp.
202–222.
[268] Schlegel, M., Noack, B. R., Comte, P., Kolomenskiy, D., Schneider, K., Farge,
M., Scouten, J., Luchtenburg, D. M., and Tadmor, G., 2009, “Reduced-Order
Modelling of Turbulent Jets for Noise Control,” Numerical Simulation of Tur-
bulent Flows and Noise Generation: Results of the DFG/CNRS Research
Groups FOR 507 and FOR 508 (Notes on Numerical Fluid Mechanics and
Multidisciplinary Design (NNFM)), Springer-Verlag, Berlin, pp. 3–27.
[269] John, C., Noack, B. R., Schlegel, M., Tr
oltzsch, F., and Wachsmuth, D., 2010,
“Optimal Boundary Control Problems Related to High-Lift Configurations,”
Active Flow Control II (Notes on Numerical Fluid Mechanics and Multidisci-
plinary Design), R. King, ed., Springer-Verlag, Berlin.
[270] Cordier, L., Noack, B. R., Daviller, G., Delvile, J., Lehnasch, G., Tissot, G.,
Balajewicz, M., and Niven, R., 2013, “Control-Oriented Model Identific ation
Strategy,” Exp. Fluids,54, p. 1580.
[271] Noack, B. R., Morzy
nski, M., and Tadmor, G. E., 2011, Reduced-Order
Modelling for Flow Control (CISM Courses and Lectures, Vol. 528),
Springer-Verlag, Berlin.
[272] Morzy
nski, M., Stankiewicz, W., Noack, B. R., Thiele, F., and Tadmor, G.,
2006, “Generalized Mean-Field Model for Flow Control Using Continuous
Mode Interpolation,” AIAA Paper No. 2006-3488.
[273] Sapsis, T. P., and Majda, A., 2013, “Statistically Accurate Low-Order Models
for Uncertainty Quantification in Turbulent Dynamical Systems,” Proc. Natl.
Acad. Sci.,110(34), pp. 13705–13710.
[274] Mitchell, T. M., 1997, Machine Learning, McGraw-Hill, Maidenhead, UK.
[275] Duda, R. O., Hart, P. E., and Stork, D. G., 2000, Pattern Classification, Wiley-
Interscience, New York.
[276] Bishop, C. M., 2006, Pattern Recognition and Machine Learning, Vol. 1,
Springer, New York.
[277] Murphy, K. P., 2012, Machine Learning: A Probabilistic Perspective, MIT
Press, Cambridge, MA.
[278] Fleming, P. J., and Purshouse, R. C., 2002, “Evolutionary Algorithms in Control
Systems Engineering: A Survey,” Control Eng. Pract.,10(11), pp. 1223–1241.
[279] Krstic´, M., and Wang, H., 2000, “Stability of Extremum Seeking Feedback for
General Nonlinear Dynamic Systems,” Automatica,36(4), pp. 595–601.
[280] Ariyur, K. B., and Krstic´, M., 2003, Real-Time Optimization by Extremum-
Seeking Control, Wiley, Hoboken, NJ.
[281] Beaudoin, J., Cadot, O., Aider, J., and Wesfreid, J. E., 2006, “Bluff- Body
Drag Reduction by Extremum-Seeking Control,” J. Fluids Struct.,22(6), pp.
973–978.
[282] Beaudoin, J.-F., Cadot, O., Aider, J.-L., and Wesfreid, J.-E., 2006, “Drag
Reduction of a Bluff Body Using Adaptive Control Methods,” Phys. Fluids,
18(8), p. 085107.
[283] Becker, R., King, R., Petz, R., and Nitsche, W., 2007, “Adaptive Closed-Loop
Control on a High-Lift Configuration Using Extremum Seeking,” AIAA J.,
45(6), pp. 1382–1392.
[284] Banaszuk, A., Zhang, Y., and Jacobson, C. A., 2000, “Adaptive Control of
Combustion Instability Using Extremum-Seeking,” American Control Confer-
ence (ACC), Chicago, June 28–30, Vol. 1, pp. 416–422.
[285] Banaszuk, A., Ariyur, K. B., Krstic´ , M., and Jacobson, C. A., 2004, “An
Adaptive Algorithm for Control of Combustion Instability,” Automatica,
40(11), pp. 1965–1972.
[286] Banaszuk, A., Narayanan, S., and Zhang, Y., 2003, “Adaptive Control of Flow
Separation in a Planar Diffuser,” AIAA Paper No. 2003-617.
[287] Maury, R., Keonig, M., Cattafesta, L., Jordan, P., and Delville, J., 2012,
“Extremum-Seeking Control of Jet Noise,” Aeroacoustics,11(3–4), pp.
459–474.
[288] Gelbert, G., Moeck, J. P., Paschereit, C. O., and King, R., 2012, “Advanced
Algorithms for Gradient Estimation in One- and Two-Parameter Extremum
Seeking Controllers,” J. Process Control,22(4), pp. 700–709.
[289] Wiederhold, O., King, R., Noack, B. R., Neuhaus, L., Neise, W., Enghard, L.,
and Swoboda, M., 2009, “Extensions of Extremum-Seeking Control to
Improve the Aerodynamic Performance of Axial Turbomachines,” AIAA
Paper No. 092407.
[290] Krieger, J. P., and Krstic´, M., 2011, “Extremum Seeking Based on Atmos-
pheric Turbulence for Aircraft Endurance,” J. Guid. Control Dyn.,34(6), pp.
1876–1885.
[291] Killingsworth, N. J., and Krstic´, M., 2006, “PID Tuning Using Extremum
Seeking: Online, Model-Free Performance Optimization,” IEEE Control Syst.
Mag.,26(1), pp. 70–79.
050801-44 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[292] Krstic´, M., Krupadanam, A., and Jacobson, C., 1999, “Self-Tuning Control of
a Nonlinear Model of Combustion Instabilities,” IEEE Trans. Control Syst.
Technol.,7(4), pp. 424–436.
[293] Koumoutsakos, P., 1997, “Active Control of Turbulent Channel Flow,” Center
for Turbulence Research, Stanford University, Stanford, CA, Annual Research
Briefs, C, pp. 289–297.
[294] Pamie`s, M., Garnier, E., Merlen, A., and Sagaut, P., 2007, “Response of a Spa-
tially Developing Turbulent Boundary Layer to Active Control Strategies in
the Framework of Opposition Control,” Phys. Fluids,19(10), p. 108102.
[295] Iwamoto , K., Fukagata, K., Kasagi, N., and Suzuki, Y., 2005, “Friction Drag
Reduction Achievable by Near-Wall Turbulence Manipulation at High Reyn-
olds Numbers,” Phys. Fluids,17(1), p. 011702.
[296] Chung, Y. M., and Talha, T., 2011, “Effectiveness of Active Flow Control for
Turbulent Skin Friction Drag Reduction,” Phys. Fluids,23(2), p. 025102.
[297] Rebbeck, H., and Choi, K.-S., 2001, “Opposition Control of Near-W all Turbu-
lence With a Piston-Type Actuator,” Phys. Fluids,13(8), pp. 2142–2145.
[298] Endo, T., Kasagi, N., and Suzuki, Y., 2000, “Feedback Control of Wall Turbu-
lence With Wall Deformation,” Int. J. Heat Fluid Flow,21(5), pp. 568–575.
[299] Fukagata, K., and Kasagi, N., 2002, “Active Control for Drag Reduction in
Turbulent Pipe Flow,” Engineering Turbulence Modelling and Experiments 5,
W. Rodi and N. Fueyo, eds., Elsevier Science, Oxford, UK, pp. 607–616.
[300] Fukagata, K., and Kasagi, N., 2003, “Drag Reduction in Turbulent Pipe Flow
With Feedback Control Applied Partially to Wall,” Int. J. Heat Fluid Flow,
24(4), pp. 480–490.
[301] Fukagata, K., and Kasagi, N., 2004, “Suboptimal Control for Drag Reduction
Via Suppression of Near-Wall Reynolds Shear Stress,” Int. J. Heat Fluid Flow,
25(3), pp. 341–350.
[302] Farrell, B. F., and Ioannou, P. J., 1996, “Turbulence Suppression by Active
Control,” Phys. Fluids,8(5), pp. 1257–1268.
[303] Luhar, M., Sharma, A. S., and McKeon, B. J., 2014, “Opposition Control
Within the Resolvent Analysis Framework,” J. Fluid Mech.,749, pp. 597–626.
[304] Cheng, B., and Titterington, D. M., 1994, “Neural Networks: A Review From
a Statistical Perspective,” Statistical Science,9(1), pp. 2–30.
[305] Haykin, S., 2004, Neural Networks: A Comprehensive Foundatio n, Prentice
Hall, Upper Saddle River, NJ.
[306] M
uller, S., Milano, M., and Koumoutsakos, P., 1999, “Application of Machine
Learning Algorithms to Flow Modeling and Optimization,” Center for Turbu-
lence Research Annual Research Briefs, Stanford University, Stanford, CA,
pp. 169–178.
[307] Milano, M., and Koumoutsakos, P., 2002, “Neural Network Modeling for
Near Wall Turbulent Flow,” J. Comput. Phys.,182(1), pp. 1–26.
[308] Oja, E., 1992, “Principal Components, Minor Components, and Linear Neural
Networks,” Neural Networks,5(6), pp. 927–935.
[309] Oja, E., 1997, “The Nonlinear PCA Learning Rule in Independent Component
Analysis,” Neurocomputing,17(1), pp. 25–45.
[310] Karhunen, J., and Joutsensalo, J., 1994, “Representation and Separation of Signals
Using Nonlinear PCA Type Learning,” Neural Networks,7(1), pp. 113–127.
[311] Nair, A. G., and Taira, K., 2015, “Network-Theoretic Approach to Sparsified
Discrete Vortex Dynamics,” J. Fluid Mech.,768, pp. 549–571.
[312] Ciresan, D., Meier, U., and Schmidhuber, J., 2012, “Multi-Column Deep Neu-
ral Networks for Image Classification,” IEEE Conference on Computer Vision
and Pattern Recognition (CVPR), Providence, RI, June 16–21, pp. 3642–3649.
[313] Dean, J., Corrado, G., Monga, R., Chen, K., Devin, M., Mao, M., Senior, A.,
Tucker, P., Yang, K., Le, Q. V., and Ng, A. Y., 2012, “Large Scale Distributed
Deep Networks,” Advances in Neural Information Processing Systems 25,F.
Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger, eds., Curran Asso-
ciates, Inc., Red Hook, NY. pp. 1223–1231.
[314] Hinton, G., Deng, L., Yu, D., Dahl, G. E., Mohamed, A.-R., Jaitly, N., Senior,
A., Vanhoucke, V., Nguyen, P., Sainath, T. N., and Kingsbury, B., 2012, “Deep
Neural Networks for Acoustic Modeling in Speech Recognition: The Shared
Views of Four Research Groups,” Signal Process. Mag.,29(6), pp. 82–97.
[315] Holland, J. H., 1975, Adapt ation in Natural and Artificial Systems: An
Introductory Analysis With Applications to Biology, Control, and Artificial
Intelligence, University of Michigan Press, Ann Arbor, MI.
[316] Davis, L., 1991, Handbook of Genetic Algorithms, Vol. 115, Van Nostrand
Reinhold, New York.
[317] Goldberg, D. E., 2006, Genetic Algorithms, Pearson Education India, Delhi,
India.
[318] Koza, J. R., 1992, Genetic Programming: On the Programming of Computers
by Means of Natural Selection, Vol. 1, MIT Press, Cambridge, MA.
[319] Koza, J. R., Bennet, F. H., III, and Stiffelman, O., 1999, “Genetic Program-
ming as a Darwinian Invention Machine,” Genetic Programming, Springer,
Berlin, pp. 93–108.
[320] Koumoutsakos, P., Freund, J., and Parekh, D., 2001, “Evolution Strategies for
Automatic Optimization of Jet Mixing,” AIAA J.,39(5), pp. 967–969.
[321] Buche, D., Stoll, P., Dornberger, R., and Koumoutsakos, P., 2002,
“Multiobjective Evolutionary Algorithm for the Optimization of Noisy Com-
bustion Processes,” IEEE Trans. Systems, Man, and Cybernet., Part C,32(4),
pp. 460–473.
[322] Poncet, P., Cottet, G.-H., and Koumoutsakos, P., 2005, “Control of Three-
Dimensional Wakes Using Evolution Strategies,” C. R. Mec.,333(1), pp. 65–77.
[323] Fukagata, K., Kern, S., Chatelain, P., Koumoutsakos, P., and Kasagi, N., 2008,
“Evolutionary Optimization of an Anisotropic Compliant Surface for Turbu-
lent Friction Drag Reduction,” J. Turbul.,9(35), pp. 1–17.
[324] Gazzola, M., Vasilyev, O. V., and Koumoutsakos, P., 2011, “Shape Optimiza-
tion for Drag Reduction in Linked Bodies Using Evolution Strategies,” Com-
put. Struct.,89(11), pp. 1224–1231.
[325] Hansen, N., Niederberger, A. S., Guzzella, L., and Koumoutsakos, P., 2009,
“A Method for Handling Uncertainty in Evolutionary Optimization With an
Application to Feedback Control of Combustion,” IEEE Trans. Evol. Comput.,
13(1), pp. 180–197.
[326] Noack, B. R., Duriez, T., Cordier, L., Segond, M., Abel, M., Brunton, S. L.,
Morzy
nski, M., Laurentie, J.-C., Parezanovic, V., and Bonnet, J.-P., 2013,
“Closed-Loop Turbulence Control With Machine Learning Methods,” Bull.
Am. Phys. Soc., 58(18), p. 418.
[327] Parezanovic´, V., Duriez, T., Cordier, L., Noack, B. R., Delville, J., Bonnet,
J.-P., Segond, M., Abel, M., and Brunton, S. L., 2014, “Closed-Loop Control
of an Experimental Mixing Layer Using Machine Learning Control,” pr eprint
arXiv:1408.3259.
[328] Gautier, N., Aider, J.-L., Duriez, T., Noack, B. R., Segond, M., and Abel, M.,
2015, “Closed-Loop Separation Control Using Machine Learning,” J. Fluid
Mech.,770, pp. 442–457.
[329] Duriez, T., Parezanovic´, V., Cordier, L., Noack, B. R., Delville, J., Bonnet,
J.-P., Segond, M., and Abel, M., 2014, “Closed-Loop Turbulence Control
Using Machine Learning,” preprint arXiv:1404.4589.
[330] Gautier, N., 2014, “Flow Control Using Optical Sensors,” Ph.D. thesis, Ecole
Doctorale: Sciences M
ecaniques, Acoustique,
Electronique & Robotique
(UPMC), ESPCI, Laboratoire PMMH, Paris.
[331] Gunzburger, M. D., 2003, Perspectives in Flow Control and Optimization,
Vol. 5, SIAM, Philadelphia.
[332] Williams, D., and MacMynowski, D., “Brief History of Flow Control,” Funda-
mentals and Applications of Modern Flow Control, Vol. 231, R. Joslin and D.
Miller, eds., American Institute of Aeronautics and Astronautics, Reston, VA,
pp. 1–20.
[333] Schlichting, H., 1979, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York.
[334] Fiedler, H., and Fernholz, H.-H., 1990, “On the Management and Control of
Turbulent Shear Flows,” Prog. Aeronaut. Sci.,27(4), pp. 305–387.
[335] McComb, D., 1991, The Physics of Fluid Turbulence, 1st ed., Clarendon Press,
Oxford, UK.
[336] Frisch, U., 1995, Turbulence, 1st ed., Cambridge University Press, Cambridge,
UK.
[337] Taylor, H., 1947, “The Elimination of Diffuser Separation by Vortex Gener-
ators,” United Aircraft Corporation, East Hartford, CT, Technical Report No.
R.4012-3.
[338] Lorenz, E. N., 1963, “Deterministic Nonperiodic Flow,” J. Atmos. Sci.,20(2),
pp. 130–141.
[339] Ott, E., Grebogi, C., and Yorke, J. A., 1990, “Controlling Chaos,” Phys. Rev.
Lett.,64(23), p. 2837.
[340] Sch
oll, E., and Schuster, H. G., 2007, Handbook of Chaos Control, Wiley-
VCH, Weinheim, Germany.
[341] Aubry, N., Holmes, P., Lumley, J. L., and Stone, E., 1988, “The Dynamics of
Coherent Structures in the Wall Region of a Turbulent Boundary Layer,” J.
Fluid Mech.,192, pp. 115–173.
[342] Glauser, M. N., Leib, S. J., and George, W. K., 1987, Coherent Structures in
the Axisymmetric Turbulent Jet Mixing Layer, Springer, Berlin.
[343] George, W. K., 1988, “Insight Into the Dynamics of Coherent Structures From
a Proper Orthogonal Decomposition,” Symposium on Near Wall Turbulence,
Dubrovnik, Yugoslavia, May 16–20.
[344] Glauser, M. N., and George, W. K., 1992, “Application of Multipoint
Measurements for Flow Characterization,” Exp. Therm. Fluid Sci.,5(5), pp.
617–632.
[345] Guyot, D., Paschereit, C. O., and Raghu, S., 2009, “Active Combustion Con-
trol Using a Fluidic Oscillator for Asymmetric Fuel Flow Modulation,” Int. J.
Flow Control,1(2), pp. 155–166.
[346] Bobusch, B. C., Woszidlo, R., Bergada, J., Nayeri, C. N., and Paschereit, C.
O., 2013, “Experimental Study of the Internal Flow Structures Inside a Fluidic
Oscillator,” Exp. Fluids,54(6), p. 1559.
[347] Vallikivi, M., Hultmark, M., Bailey, S., and Smits, A., 2011, “Turbulence
Measurements in Pipe Flow Using a Nano-Scale Thermal Anemometry
Probe,” Exp. Fluids,51(6), pp. 1521–1527.
[348] Bailey, S. C., Kunkel, G. J., Hultmark, M., Vallikivi, M., Hill, J. P., Meyer,
K. A., Tsay, C., Arnold, C. B., and Smits, A. J., 2010, “Turbulence Measure-
ments Using a Nanoscale Thermal Anemometry Probe,” J. Fluid Mech.,663,
pp. 160–179.
[349] Hultmark, M., Vallikivi, M., Bailey, S., and Smits, A., 2012, “Turbulent Pipe
Flow at Extreme Reynolds Numbers,” Phys. Rev. Lett.,108(9), p. 094501.
[350] Daniel, T. L., 1988, “Forward Flapping Flight From Flexible Fins,” Can. J.
Zool.,66(3), pp. 630–638.
[351] Anderson, J. M., Streitlien, K., Barrett, D. S., and Triantafyllou, M. S., 1998,
“Oscillating Foils of High Propulsive Efficiency,” J. Fluid Mech.,360, pp. 41–72.
[352] Triantafyllou, M. S., and Triantafyllou, G. S., 1995, “An Efficient Swimming
Machine,” Sci. Am.,272(3), pp. 64–71.
[353] Allen, J. J., and Smits, A. J., “Energy Harvesting Eel,” J. Fluids Struct.,
15(3–4), pp. 629–640.
[354] Combes, S. A., and Daniel, T. L., 2001, “Shape, Flapping and Flexion: Wing
and Fin Design for Forward Flight,” J. Exp. Biol.,204(12), pp. 2073–2085.
[355] Clark, R. P., and Smits, A. J., 2006, “Thrust Production and Wake Structure of
a Batoid-Inspired Oscillating Fin,” J. Fluid Mech.,562, pp. 415–429.
[356] Buchholz, J. H., and Smits, A. J., 2008, “The Wake Structure and Thrust Per-
formance of a Rigid Low-Aspect-Ratio Pitching Panel,” J. Fluid Mech.,
603(May), pp. 331–365.
[357] Song, A., Tian, X., Israeli, E., Galvao, R., Bishop, K., Swartz, S., and Breuer,
K., 2008, “Aeromechanics of Membrane Wings With Implications for Animal
Flight,” AIAA J.,46(8), pp. 2096–2106.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-45
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[358] Taira, K., and Colonius, T., 2008, “Effect of Tip Vortices in Low-Reynolds-
Number Poststall Flow Control,” AIAA J.,47(3), pp. 749–756.
[359] Taira, K., and Colonius, T., 2009, “Three-Dimensional Flows Around Low-
Aspect-Ratio Flat-Plate Wings at Low Reynolds Numbers,” J. Fluid Mech.,
623, pp. 187–207.
[360] Whittlesey, R. W., Liska, S. C., and Dabiri, J. O., 2010, “Fish Schooling as a
Basis for Vertical-Axis Wind Turbine Farm Design,” Bioinspiration Biomi-
metics,5(3), p. 035005.
[361] Faruque, I., and Humbert, J. S., 2010, “Dipteran Insect Flight Dynamics. Part 1:
Longitudinal Motion About Hover,” J. Theor. Biol.,264(2), pp. 538–552.
[362] Faruque, I., and Humbert, J. S., 2010, “Dipteran Insect Flight Dynamics. Part
2: Lateral–Directional Motion About Hover,” J. Theor. Biol.,265(3), pp.
306–313.
[363] Humbert, J. S., and Hyslop, A. M., 2010, “Bioinspired Visuomotor Con-
vergence,” IEEE Trans. Rob.,26(1), pp. 121–130.
[364] Shelley, M. J., and Zhang, J., 2011, “Flapping and Bending Bodies Interacting
With Fluid Flows,” Annu. Rev. Fluid Mech.,43, pp. 449–465.
[365] Leftwich, M. C., Tytell, E. D., Cohen, A. H., and Smits, A. J., 2012, “Wake
Structures Behind a Swimming Robotic Lamprey With a Passively Flexible
Tail,” J. Exp. Biol.,215(3), pp. 416–425.
[366] Dewey, P. A., Carriou, A., and Smits, A. J., 2012, “On the Relationship
Between Efficiency and Wake Structure of a Batoid-Inspired Oscillating Fin,”
J. Fluid Mech.,691, pp. 245–266.
[367] Nawroth, J. C., Lee, H., Feinberg, A. W., Ripplinger, C. M., McCain, M. L.,
Grosberg, A., Dabiri, J. O., and Parker, K. K., 2012, “A Tissue-Engineered
Jellyfish With Biomimetic Propulsion,” Nat. Biotechnol.,30, pp. 792–797.
[368] Roth, E., Sponberg, S., and Cowan, N., 2014, “A Comparative Approach to
Closed-Loop Computation,” Curr. Opin. Neurobiol.,25, pp. 54–62.
[369] Cowan, N. J., Ankarali, M. M., Dyhr, J. P., Madhav, M. S., Roth, E., Sefati, S.,
Sponberg, S., Stamper, S. A., Fortune, E. S., and Daniel, T. L., 2014,
“Feedback Control as a Framework for Understanding Tradeoffs in Biology,”
Integr. Comp. Biol.,54(2), pp. 223–237.
[370] Dickinson, M. H., and Gotz, K. G., 1996, “The Wake Dynamics and Flight
Forces of the Fruit Fly Drosophila melanogaster,” J. Exp. Biol.,199(9), pp.
2085–2104.
[371] Sane, S. P., and Dickinson, M. H., 2001, “The Control of Flight Force by
a Flapping Wing: Lift and Drag Production,” J. Exp. Biol.,204(15), pp.
2607–2626.
[372] Frye, M. A., and Dickinson, M. H., 2001, “Fly Flight: A Model for the Neural
Control of Complex Behavior,” Neuron,32(3), pp. 385–388.
[373] Ghose, K., Horiuchi, T. K., Krishnaprasad, P. S., and Moss, C. F., 2006,
“Echolocating Bats Use a Nearly Time-Optimal Strategy to Intercept Prey,”
PLoS Biol.,4(5), p. e108.
[374] Hedenstr
om, A., Johansson, L., Wolf, M., Von Busse, R., Winter, Y., and
Spedding, G., 2007, “Bat Flight Generates Complex Aerodynamic Tracks,”
Science,316(5826), pp. 894–897.
[375] Riskin, D. K., Willis, D. J., Iriarte-D
ıaz, J., Hedrick, T. L., Kostandov, M.,
Chen, J., Laidlaw, D. H., Breuer, K. S., and Swartz, S. M., 2008, “Quantifying
the Complexity of Bat Wing Kinematics,” J. Theor. Biol.,254(3), pp.
604–615.
[376] Hubel, T. Y., Hristov, N. I., Swartz, S. M., and Breuer, K. S., 2009, “Time-
Resolved Wake Structure and Kinematics of Bat Flight,” Exp. Fluids,46(5),
pp. 933–943.
[377] Fish, F. E., and Hui, C. A., 1991, “Dolphin Swimming—A Review,” Mamm.
Rev.,21(4), pp. 181–195.
[378] Fish, F. E., 1996, “Transitions From Drag-Based to Lift-Based Propulsion in
Mammalian Swimming,” Am. Zool.,36(6), pp. 628–641.
[379] Dickinson, M. H., Lehmann, F. O., and Sane, S. P., 1999, “Wing Rotation
and the Aerodynamic Basis of Insect Flight,” Science,284(5422), pp.
1954–1960.
[380] Birch, J., and Dickinson, M., 2001, “Spanwise Flow and the Attachment of the
Leading-Edge Vortex on Insect Wings,” Nature,412(6848), pp. 729–733.
[381] Sane, S. P., 2003, “The Aerodynamics of Insect Flight,” J. Exp. Biol.,206(23),
pp. 4191–4208.
[382] Liao, J. C., Beal, D. N., Lauder, G. V., and Triantafyllou, M. S., 2003, “Fish
Exploiting Vortices Decrease Muscle Activity,” Science,302(5650), pp.
1566–1569.
[383] Tytell, E. D., and Lauder, G. V., 2004, “The Hydrodynamics of Eel Swim-
ming. I. Wake Structure,” J. Exp. Biol.,207(11), pp. 1825–1841.
[384] Lauder, G. V., and Tytell, E. D., 2005, “Hydrodynamics of Undulatory
Propulsion,” Fish Physiol.,23, pp. 425–468.
[385] Videler, J. J., Samhuis, E. J., and Povel, G. D. E., 2004, “Leading-Edge Vortex
Lifts Swifts,” Science,306(5703), pp. 1960–1962.
[386] Wang, Z. J., 2005, “Dissecting Insect Flight,” Annu. Rev. Fluid Mech.,37,
pp. 183–210.
[387] Dabiri, J. O., 2009, “Optimal Vortex Formation as a Unifying Principle in Bio-
logical Propulsion,” Annu. Rev. Fluid Mech.,41, pp. 17–33.
[388] Wu, T. Y., 2011, “Fish Swimming and Bird/Insect Flight,” Annu. Rev. Fluid
Mech.,43, pp. 25–58.
[389] Collett, T. S., and Land, M. F., 1975, “Visual Control of Flight Behaviour in
the Hoverfly Syritta pipiens L.,” J. Comp. Physiol. A,99(1), pp. 1–66.
[390] Fayyazuddin, A., and Dickinson, M. H., 1996, “Haltere Afferents Provide
Direct, Electronic Input to a Steering Motor Neuron in the Blowfly,
Calliphora,” J. Neurosci.,16(16), pp. 5225–5232.
[391] Fox, J. L., and Daniel, T. L., 2008, “A Neural Basis for Gyroscopic Force
Measurement in the Halteres of Holorusia,” J. Comp. Physiol. A,194(10), pp.
887–897.
[392] Sane, S. P., Dieudonne, A., Willis, M. A., and Daniel, T. L., 2007, “Antennal
Mechanosensors Mediate Flight Control in Moths,” Science,315(5813), pp.
863–866.
[393] Brown, R. E., and Fedde, M. R., 1993, “Airflow Sensors in the Avian Wing,”
J. Exp. Biol.,179(1), pp. 13–30.
[394] Sterbing-D’Angelo, S. J., and Moss, C. F., 2014, “Air Flow Sensing in Bats,”
Flow Sensing in Air and Water, Springer, Berlin, pp. 197–213.
[395] Sterbing-D’Angelo, S., Chadha, M., Chiu, C., Falk, B., Xian, W., Barcelo, J.,
Zook, J. M., and Moss, C. F., 2011, “Bat Wing Sensors Support Flight Con-
trol,” Proc. Natl. Acad. Sci.,108(27), pp. 11291–11296.
[396] Dickinson, B., 2010, “Hair Receptor Sensitivity to Changes in Laminar
Boundary Layer Shape,” Bioinspiration Biomimetics,5(1), p. 016002.
[397] Massey, T., Kapur, R., Dabiri, F., Vu, L. N., and Sarrafzadeh, M., 2007,
“Localization Using Low-Resolution Optical Sensors,” IEEE International
Conference on Mobile Adhoc and Sensor Systems (MASS 2007), Pisa, Italy,
Oct. 8–11.
[398] Giannetti, F., and Luchini, P., 2007, “Structural Sensitivity of the First Insta-
bility of the Cylinder Wake,” J. Fluid Mech.,581, pp. 167–197.
[399] Hof, B., de Lozar, A., Avila, M., Tu, X., and Schneider, T. M., 2010,
“Eliminating Turbulence in Spatially Intermittent Flows,” Science,327(5972),
pp. 1491–1494.
[400] McKeon, B. J., 2010, “Controlling Turbulence,” Science,327(5927), pp.
1462–1463.
[401] Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D., and Hof, B., 2011,
“The Onset of Turbulence in Pipe Flow,” Science,333(6039), pp. 192–196.
[402] Brunton, B. W., Brunton, S. L., Proctor, J. L., and Kutz, J. N., 2013, “Optimal
Sensor Placement and Enhanced Sparsity for Classification,” preprint
arXiv:1310.4217.
[403] Proctor, J. L., Brunton, S. L., Brunton, B. W., and Kutz, J. N., 2014,
“Exploiting Sparsity and Equation-Free Architectures in Complex Systems
(Invited Review),” Eur. Phys. J. Spec. Top.,223(13), pp. 2665–2684.
[404] Hey, A. J., Tansley, S., and Tolle, K. M., 2009, The Fourth Paradigm: Data-
Intensive Scientific Discovery, Microsoft Research, Redmond, WA.
[405] Allaire, D., Biros, G., Chambers, J., Ghattas, O., Kordonowy, D., and Willcox,
K., 2012, “Dynamic Data Driven Methods for Self-Aware Aerospace Vehi-
cles,” Proc. Comput. Sci.,9, pp. 1206–1210.
[406] Kutz, J. N., 2013, Data-Driven Modeling & Scientific Computation: Methods
for Complex Systems & Big Data, Oxford University Press, Oxford, UK.
[407] Cande`s, E. J., 2006, “Compressive Sampling,” International Congress of
Mathematics, Madrid, Aug. 22–30, Vol. 3, pp. 1433–1452.
[408] Donoho, D. L., 2006, “Compressed Sensing,” IEEE Trans. Inf. Theory,52(4),
pp. 1289–1306.
[409] Baraniuk, R. G., 2007, “Compressive Sensing,” IEEE Signal Process. Mag.,
24(4), pp. 118–120.
[410] Tropp, J. A., and Gilbert, A. C., 2007, “Signal Recovery From Random Meas-
urements Via Orthogonal Matching Pursuit,” IEEE Trans. Inf. Theory,53(12),
pp. 4655–4666.
[411] Cande`s, E. J., and Wakin, M. B., 2008, “An Introduction to Compressive
Sampling,” IEEE Signal Process. Mag.,25(2), pp. 21–30.
[412] Willert, C. E., and Gharib, M., 1991, “Digital Particle Image Velocimetry,”
Exp. Fluids,10(4), pp. 181–193.
[413] Nyquist, H., 1928, “Certain Topics in Telegraph Transmission Theory,” Trans.
AIEE,47(2), pp. 617–644.
[414] Shannon, C. E., 1948, “A Mathematical Theory of Communication,” Bell
Syst. Tech. J.,27(3), pp. 379–423.
[415] Petra, S., and Schn
orr, C., 2009, “TomoPIV Meets Compressed Sensing,”
Pure Math. Appl.,20(1–2), pp. 49–76.
[416] Becker, F., Wieneke, B., Petra, S., Schr
oder, A., and Schn
orr, C., 2012,
“Variational Adaptive Correlation Method for Flow Estimation,” IEEE Trans.
Image Process.,21(6), pp. 3053–3065.
[417] Bai, Z., Wimalajeewa, T., Berger, Z., Wang, G., Glauser, M., and Varshney,
P. K., 2013, “Physics Based Compressive Sensing Approach Applied to Air-
foil Data Collection and Analysis,” AIAA Paper No. 2013-0772.
[418] Bai, Z., Wimalajeewa, T., Berger, Z., Wang, G., Glauser, M., and Varshney,
P. K., 2014, “Low-Dimensional Approach for Reconstruction of Airfoil Data
Via Compressive Sensing,” AIAA J.,53(4), pp. 920–933.
[419] Cande`s, E. J., Romberg, J., and Tao, T., 2006, “Robust Uncertainty Principles:
Exact Signal Reconstruction From Highly Incomplete Frequency
Information,” IEEE Trans. Inf. Theory,52(2), pp. 489–509.
[420] Cande`s, E. J., Romberg, J., and Tao, T., 2006, “Stable Signal Recovery From
Incomplete and Inaccurate Measurements,” Commun. Pure Appl. Math.,
59(8), pp. 1207–1223.
[421] Cande`s, E. J., and Tao, T., 2006, “Near Optimal Signal Recovery From Ran-
dom Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory,
52(12), pp. 5406–5425.
[422] Boyd, S., and Vandenberghe, L., 2009, Convex Optimization, Cambridge Uni-
versity Press, Cambridge, UK.
[423] Mathelin, L., and Gallivan, K. A., 2012, “A Compress ed Sensing Approach
for Partial Differential Equations With Random Input Data,” Commun.
Comput. Phys.,12(4), pp. 1–36.
[424] Schaeffer, H., Caflisch, R., Hauck, C. D., and Osher, S., 2013, “Sparse
Dynamics for Partial Differential Equations,” Proc. Natl. Acad. Sci. U.S.A.,
110(17), pp. 6634–6639.
[425] Mackey, A., Schaeffer, H., and Osher, S., “On the Compressive Spectral
Method,” Multiscale Model. & Simul.,12(4), pp. 1800–1827.
[426] Tran, G., Schaeffer, H., Feldman, W. M., and Osher, S. J., 2014, “An L1 Pen-
alty Method for General Obstacle Problems,” preprint arXiv:1404.1370.
050801-46 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[427] Shi, J. V., Yin, W., Sankaranarayanan, A. C., and Baraniuk, R. G., “Video Com-
pressive Sensing for Dynamic MRI,” BMC Neurosci.,13(Suppl 1), p. 183.
[428] Jovanovic´, M. R., Schmid, P. J., and Nichols, J. W., 2014, “Sparsity-
Promoting Dynamic Mode Decomposition,” Phys. Fluids,26(2), p. 024103.
[429] Brunton, S. L., Proctor, J. L., and Kutz, J. N., “Compressive Sampling and
Dynamic Mode Decomposition,” arXiv:1312.5186.
[430] Gueniat, F., Mathelin, L., and Pastur, L., 2015, “A Dynamic Mode Decompo-
sition Approach for Large and Arbitrarily Sampled Systems,” Phys. Fluids,
27(2), p. 025113.
[431] Bright, I., Lin, G., and Kutz, J. N., 2013, “Compressive Sensing and Machine
Learning Strategies for Characterizing the Flow Around a Cylinder With Lim-
ited Pressure Measurements,” Phys. Fluids,25(12), p. 127102.
[432] Brunton, S. L., Tu, J. H., Bright, I., and Kutz, J. N., 2014, “Compressive
Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in
Nonlinear Dynamical Systems,” SIAM J. Appl. Dyn. Syst.,13(4), pp.
1716–1732.
[433] Tayler, A. B., Holland, D. J., Sederman, A. J., and Gladden, L. F., 2012,
“Exploring the Origins of Turbulence in Multiphase Flow Using Compressed
Sensing MRI,” Phys. Rev. Lett.,108(26), p. 264505.
[434] Branicki, M., and Majda, A. J., 2014, “Quantifying Bayesian Filter Perform-
ance for Turbulent Dynamical Systems Through Information Theory,” Com-
mun. Math. Sci.,12(5), pp. 901–978.
[435] Bourguignon, J.-L., Tropp, J., Sharma, A., and McKeon, B., 2014, “Compact
Representation of Wall-Bounded Turbulence Using Compressive Sampling,”
Phys. Fluids,26(1), p. 015109.
[436] Fu, X., Brunton, S. L., and Kutz, J. N., 2014, “Classification of Birefringence
in Mode-Locked Fiber Lasers Using Machine Learning and Sparse Repre-
sentation,” Opt. Express,22(7), pp. 8585–8597.
[437] Brunton, S. L., Fu, X., and Kutz, J. N., 2014, “Self-Tuning Fiber Lasers,”
IEEE J. Sel. Top. Quantum Electron.,20(5), p. 1101408.
[438] Wright, J., Yang, A., Ganesh, A., Sastry, S., and Ma, Y., 2009, “Robust Face
Recognition Via Sparse Representation,” IEEE Trans. Pattern Anal. Mach.
Intell. (PAMI),31(2), pp. 210–227.
[439] Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Davil-
ler, G., and Niven, R. K., 2014, “Cluster-Based Reduced-Order Modelling of a
Mixing Layer,” J. Fluid Mech.,754, pp. 365–414.
[440] Burkardt, J., Gunzburger, M., and Lee, H.-C., 2004, “Centroidal Voronoi
Tessellation-Based Reduced-Order Modeling of Complex Systems,” SIAM J.
Sci. Comput.,28(2), pp. 459–484.
[441] Schneider, T. M., Eckhardt, B., and Vollmer, J., 2007, “Statistical Analysis of
Coherent Structures in Transitional Pipe Flow,” Phys. Rev. E,75(6), pp.
66–313.
[442] Gear, C. W., Kevrekidis, I. G., and Theodoropoulos, C., “‘Coarse’ Integration/
Bifurcation Analysis Via Microscopic Simulators: Micro-Galerkin Methods,”
Comput. Chem. Eng.,26(7–8), pp. 941–963.
[443] Gorban, A., Kazantzis, N. K., Kevrekidis, I. G.,
Ottinger, H., and Theodoro-
poulos, C., eds., 2006, Model Reduction and Coarse-Graining Approaches for
Multiscale Phenomena, Springer-Verlag, Berlin.
[444] Kevrekidis, I. G., Gear, C. W., Hyman, J. M., Kevrekidis, P. G., Runborg, O.,
and Theodoropoulos, C., 2003, “Equation-Free, Coarse-Grained Multiscale
Computation: Enabling Microscopic Simulators to Perform System-Level
Analysis,” Commun. Math. Sci.,1(4), pp. 715–762.
[445] Sirisup, S., Karniadakis, G. E., Xiu, D., and Kevrekidis, I. G., 2005,
“Equation-Free/Galerkin-Free POD-Assisted Computation of Incompressible
Flows,” J. Comput. Phys.,207(2), pp. 568–587.
[446] Xiu, D., and Karniadakis, G. E., 2002, “The Wiener–Askey Polynomial Chaos
for Stochastic Differential Equations,” SIAM J. Sci. Comput.,24(2), pp.
619–644.
[447] Xiu, D., and Karniadakis, G. E., 2003, “Modeling Uncertainty in Flow Simula-
tions Via Generalized Polynomial Chaos,” J. Comput. Phys.,187(1), pp.
137–167.
[448] Xiu, D., 2010, Numerical Methods for Stochastic Computations: A Spectral
Method Approach, Princeton University Press, Princeton, NJ.
[449] Grosek, J., and Kutz, J. N., 2014, “Dynamic Mode Decomposition for
Real-Time Background/Foreground Separation in Video,” preprint
arXiv:1404.7592.
[450] Hemati, M. S., Williams, M. O., and Rowley, C. W., 2014, “Dynamic
Mode Decomposition for Large and Streaming Datasets,” preprint
arXiv:1406.7187.
[451] Dawson, S., Hemati, M., Williams, M., and Rowley, C., 2014, “Characterizing
and Correcting for the Effect of Sensor Noise in the Dynamic Mode Decom-
position,” Bull. Am. Phys. Soc., 59(20), p. 428.
[452] Aref, H., 1984, “Stirring by Chaotic Advection,” J. Fluid Mech. ,143, pp.
1–21.
[453] Wiener, N., 1938, “The Homogeneous Chaos,” Am. J. Math.,60(4), pp.
897–936.
[454] Wan, X., and Karniadakis, G. E., 2005, “An Adaptive Multi-Element General-
ized Polynomial Chaos Method for Stochastic Differential Equations,” J.
Comput. Phys.,209(2), pp. 617–642.
[455] Gerritsma, M., van der Steen, J.-B., Vos, P. E. J., and Karniadakis, G. E.,
2010, “Time-Dependent Generalized Polynomial Chaos,” J. Comput. Phys.,
229(22), pp. 8333–8363.
[456] Luchtenburg, D. M., Brunton, S. L., and Rowley, C. W., 2014, “Long-Time
Uncertainty Propagation Using Generalized Polynomial Chaos and Flow Map
Composition,” J. Comput. Phys.,274, pp. 783–802.
[457] Le Ma^
ıtre, O. P., and Knio, O. M., 2010, Spectral Methods for Uncertainty
Quantification, Springer, Dordrecht.
[458] Sapsis, T. P., and Lermusiaux, P. F., 2009, “Dynamically Orthogonal Field
Equations for Continuous Stochastic Dynamical Systems,” Physica D,
238(23–24), pp. 2347–2360.
[459] Sapsis, T. P., and Lermusiaux, P. F., 2012, “Dynamical Criteria for the Evolu-
tion of the Stochastic Dimensionality in Flows With Uncertainty,” Physica D,
241(1), pp. 60–76.
[460] Haller, G., 2001, “Distinguished Material Surfaces and Coherent Structures in
Three-Dimensional Fluid Flows,” Physica D,149(4), pp. 248–277.
[461] Haller, G., 2002, “Lagrangian Coherent Structures From Approximate Veloc-
ity Data,” Phys. Fluids,14(6), pp. 1851–1861.
[462] Shadden, S. C., Lekien, F., and Marsden, J. E., 2005, “Definition and
Properties of Lagrangian Coherent Structures From Finite-Time Lyapunov
Exponents in Two-Dimensional Aperiodic Flows,” Physica D,212(3–4), pp.
271–304.
[463] Green, M. A., Rowley, C. W., and Haller, G., 2007, “Detection of Lagrangian
Coherent Structures in 3D Turbulence,” J. Fluid Mech.,572, pp. 111–120.
[464] Mathur, M., Haller, G., Peacock, T., Ruppert-Felsot, J. E., and Swinney, H. L.,
2007, “Uncovering the Lagrangian Skeleton of Turbulence,” Phys. Rev. Lett.,
98(14), p. 144502.
[465] Brunton, S. L., and Rowley, C. W., 2010, “Fast Computation of FTLE Fields
for Unsteady Flows: A Comparison of Methods,” Chaos,20(1), p. 017503.
[466] Farazmand, M., and Haller, G., 2012, “Computing Lagrangian Coherent Struc-
tures From Their Variational Theory,” Chaos,22(1), p. 013128.
[467] Kafiabad, H. A., Chan, P. W., and Haller, G., 2012, “Lagrangian Detection of
Aerial Turbulence for Landing Aircraft,” J. Appl. Meteorol. Climatol.,30(12),
pp. 2808–2819.
[468] Shadden, S. C., Astorino, M., and Gerbeau, J. F., 2010, “Computational Anal-
ysis of an Aortic Valve Jet With Lagrangian Coherent Structures,” Chaos,
20(1), p. 017512.
[469] Wilson, M. M., Peng, J., Dabiri, J. O., and Eldredge, J. D., 2009, “Lagrangian
Coherent Structures in Low Reynolds Number Swimming,” J. Phys.: Condens.
Matter,21(20), p. 204105.
[470] Green, M. A., Rowley, C. W., and Smits, A. J., 2011, “The Unsteady Three-
Dimensional Wake Produced by a Trapezoidal Pitching Panel,” J. Fluid
Mech.,685, pp. 117–145.
[471] Peng, J., and Dabiri, J. O., 2008, “The ‘Upstream Wake’ of Swimming and
Flying Animals and Its Correlation With Propulsive Efficiency,” J. Exp. Biol.,
211(16), pp. 2669–2677.
[472] Bollt, E. M., Luttman, A., Kramer, S., and Basnayake, R., 2012, “Me asurable
Dynamics Analysis of Transport in the Gulf of Mexico During the Oil Spill,”
Int. J. Bifurcation Chaos,22(3), p. 1230012.
[473] Lekien, F., Coulliette, C., Mariano, A. J., Ryan, E. H., Shay, L. K., Haller, G.,
and Marsden, J. E., 2005, “Pollution Release Tied to Invariant Manifolds: A
Case Study for the Coast of Florida,” Physica D,210(1–2), pp. 1–20.
[474] Mezic´, I., Loire, S., Fonoberov, V. A., and Hogan, P., 2010, “A New Mixing
Diagnostic and Gulf Oil Spill Movement,” Science,330(6003), pp. 486–489.
[475] Padberg, K., Hauff, T., Jenko, F., and Junge, O., 2007, “Lagrangian Structures
and Transport in Turbulent Magnetized Plasmas,” New J. Phys.,9(11), p.
400.
[476] Froyland, G., and Padberg, K., 2009, “Almost-Invariant Sets and Invariant
Manifolds—Connecting Probabilistic and Geometric Descriptions of Coherent
Structures in Flows,” Physica D,238(16), pp. 1507–1523.
[477] Froyland, G., Santitissadeekorn, N., and Monahan, A., 2010, “Transport in
Time-Dependent Dynamical Systems: Finite-Time Coherent Sets,” Chaos,
20(4), p. 043116.
[478] Tallapragada, P., and Ross, S. D., 2013, “A Set Oriented Definition of Finite-
Time Lyapunov Exponents and Coherent Sets,” Commun. Nonlinear Sci.
Numer. Simul.,18(5), pp. 1106–1126.
[479] Dellnitz, M., Froyland, G., and Junge, O., 2001, “The Algorithms Behind
Gaio—Set Oriented Numerical Methods for Dynamical Systems,” Ergodic
Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fieldler,
ed., Springer, Dordrecht, pp. 145–174.
[480] Dellnitz, M., and Junge, O., 2002, “Set Oriented Numerical Metho ds for Dy-
namical Systems,” Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed.,
Elsevier, Amsterdam, pp. 221–264.
[481] Carlberg, K., Bou-Mosleh, C., and Farhat, C., 2011, “Efficient Non-Linear
Model Reduction Via a Least-Squares Petrov–Galerkin Projection and Com-
pressive Tensor Approximations,” Int. J. Numer. Methods Eng.,86(2), pp.
155–181.
[482] Avellaneda, M., and Majda, A. J., 1990, “Mathematical Models With Exact
Renormalization for Turbulent Transport,” Commun. Math. Phys.,131(2), pp.
381–429.
[483] Amsallem, D., Zahr, M. J., and Farhat, C., 2012, “Nonlinear Model Order
Reduction Based on Local Reduced-Order Bases,” Int. J. Numer. Methods
Eng.,92(10), pp. 891–916.
[484] Carlberg, K., Farhat, C., Cortial, J., and Amsallem, D., 2013, “The GNAT
Method for Nonlinear Model Reduction: Effective Implementation and Appli-
cation to Computational Fluid Dynamics and Turbulent Flows,” J. Comput.
Phys.,242, pp. 623–647.
[485] Everson, R., and Sirovich, L., 1995, “Karhunen–Loeve Procedure for Gappy
Data,” JOSA A,12(8), pp. 1657–1664.
[486] Willcox, K., 2006, “Unsteady Flow Sensing and Estimation Via the
Gappy Proper Orthogonal Decom position,” Comput. Fluids,35(2), pp.
208–226.
[487] Barrault, M., Maday, Y., Nguyen, N. C., and Patera, A. T., 2004, “An ‘Empiri-
cal Interpolation’ Method: Application to Efficient Reduced-Basis Discretiza-
tion of Partial Differential Equations,” C. R. Math.,339(9), pp. 667–672.
Applied Mechanics Reviews SEPTEMBER 2015, Vol. 67 / 050801-47
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[488] Chaturantabut, S., and Sorensen, D. C., 2010, “Nonlinear Model Reduction
Via Discrete Empirical Interpolation,” SIAM J. Sci. Comput.,32(5), pp.
2737–2764.
[489] Chaturantabut, S., and Sorensen, D. C., 2012, “A State Space Error Estimate
for POD-DEIM Nonlinear Model Reduction,” SIAM J. Numer. Anal.,50(1),
pp. 46–63.
[490] Peherstorfer, B., Butnaru, D., Willcox, K., and Bungartz, H.-J., 2014,
“Localized Discrete Empirical Interpolation Method,” SIAM J. Sci. Comput.,
36(1), pp. A168–A192.
[491] Majda, A. J., and Kramer, P. R., 1999, “Simplified Models for Turbulent Dif-
fusion: Theory, Numerical Modelling, and Physical Phenomena,” Phys. Rep.,
314(4), pp. 237–574.
[492] Majda, A. J., Harlim, J., and Gershgorin, B., 2010, “Mathematical Strategies
for Filtering Turbulent Dynamical Systems,” Discrete Contin. Dyn. Syst.,
27(2), pp. 441–486.
[493] Majda, A. J., and Harlim, J., 2012, Filtering Complex Turbulent Systems,
Cambridge University Press, Cambridge, UK.
[494] Maynard Gayme, D., 2010, “A Robust Control Approach to Understanding
Nonlinear Mechanisms in Shear Flow Turbulence,” Ph.D. thesis, California
Institute of Technology, Pasadena, CA.
[495] Marusic, I., and Hutchins, N., 2005, “Experimental Study of Wall Turbulence:
Implications for Control,” Transition and Turbulence Control, World Scien-
tific, Singapore, pp. 207–246.
[496] Smits, A. J., McKeon, B. J., and Marusic, I., 2011, “High-Reynolds Number
Wall Turbulence,” Annu. Rev. Fluid Mech.,43(1), pp. 353–375.
[497] Cacuci, D. G., Navon, I. M., and Ionescu-Bujor, M., 2013, Computational
Methods for Data Evaluation and Assimilation, Chapman & Hall, Oxford,
UK.
[498] Cordier, L., Abou El Majd, B., and Favier, J., 2010, “Calibration of POD
Reduced-Order Models Using Tikhonov Regularization,” Int. J. Numer. Meth-
ods Fluids,63(2), pp. 269–296.
[499] Kapur, J. N., and Kevasan, H. K., 1992, Entropy Optimization Principles With
Applications, 1st ed., Academic Press, Boston.
[500] Noack, B. R., and Niven, R. K., 2012, “Maximum-Entropy Closure for a
Galerkin System of Incompressible Shear Flow,” J. Fluid Mech.,700, pp.
187–213.
[501] Noack, B. R., and Niven, R. K., 2013, “A Hierarchy of Maximum-Entropy
Closures for Galerkin Systems of Incompressible Flows,” Comput. Math.
Appl.,65(10), pp. 1558–1574.
[502] Andresen, B., 1983, “Finite-Time Thermody namics,” Physics Laboratory II,
1st ed., University of Copenhagen, Copenhagen, Denmark.
[503] Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzy
nski, M.,
Comte, P., and Tadmor, G., 2008, “A Finite-Time Thermodynamics of
Unsteady Fluid Flows,” J. Non-Equilibr. Thermodyn.,33(2), pp. 103–148.
[504] Noack, B. R., Schlegel, M., Morzy
nski, M., and Tadmor, G., 2010, “System
Reduction Strategy for Galerkin Models of Fluid Flows,” Int. J. Numer. Meth-
ods Fluids,63(2), pp. 231–248.
[505] Taira, K., 2015, private communication.
[506] Watts, D. J., and Strogatz, S. H., 1998, “Collective Dynamics of ‘Small-
World’ Networks,” Nature,393(6684), pp. 440–442.
[507] Barab
asi, A.-L., and Albert, R., 1999, “Emergence of Scaling in Random
Networks,” Science,286(5439), pp. 509–512.
[508] Barab
asi, A.-L., 2009, “Scale-Free Networks: A Decade and Beyond,” Sci-
ence,325(5939), pp. 412–413.
[509] Del Genio, C. I., Gross, T., and Bassler, K. E., 2011, “All Scale-Free Networks
are Sparse,” Phys. Rev. Lett.,107(17), p. 178701.
[510] Barzel, B., and Barab
asi, A.-L., 2013, “Universality in Network Dynamics,”
Nat. Phys.,9(10), pp. 673–681.
[511] Newman, M. E., 2003, “The Stru cture and Function of Complex Networks,”
SIAM Rev.,45(2), pp. 167–256.
[512] Leonard, N. E., and Fiorelli, E., 2001, “Virtual Leaders, Artificial Potentials
and Coordinated Control of Groups,” 40th IEEE Conference on Decision and
Control, Orlando, FL, Dec. 4–7, Vol. 3, pp. 2968–2973.
[513] Olfati-Saber, R., 2006, “Flocking for Multi-Agent Dynamic Systems: Algo-
rithms and Theory,” IEEE Trans. Autom. Control,51(3), pp. 401–420.
[514] Balch, T., and Arkin, R. C., 1998, “Behavior-Based Formation Control for
Multirobot Teams,” IEEE Trans. Rob. Autom.,14(6), pp. 926–939.
[515] Cortes, J., Martinez, S., Karatas, T., and Bullo, F., 2002, “Coverage Control
for Mobile Sensing Networks,” IEEE International Conference on Robotics
and Automation (ICRA ’02), Washington, DC, May 11–15, Vol. 2, pp.
1327–1332.
[516] Leonard, N. E., Paley, D. A., Lekien, F., Sepulchre, R., Fratantoni, D. M., and
Davis, R. E., 2007, “Collective Motion, Sensor Networks, and Ocean
Sampling,” Proc. IEEE,95(1), pp. 48–74.
[517] Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., and Alon,
U., 2002, “Network Motifs: Simple Building Blocks of Complex Networks,”
Science,298(5594), pp. 824–827.
[518] Luscombe, N. M., Babu, M. M., Yu, H., Snyder, M., Teichmann, S. A., and
Gerstein, M., 2004, “Genomic Analysis of Regulatory Network Dynam ics
Reveals Large Topological Changes,” Nature,431(7006), pp. 308–312.
[519] Low, S. H., Paganini, F., and Doyl e, J. C., 2002, “Internet Congestion Con-
trol,” Control Syst.,22(1), pp. 28–43.
[520] Doyle, J. C., Alderson, D. L., Li, L., Low, S., Roughan, M., Shalunov, S.,
Tanaka, R., and Willinger, W., 2005, “The “Robust Yet Fragile” Nature of the
Internet,” Proc. Natl. Acad. Sci. U.S.A.,102(41), pp. 14497–14502.
[521] Rahmani, A., Ji, M., Mesbahi, M., and Egerstedt, M., 2009, “Controllability of
Multi-Agent Systems From a Graph-Theoretic Perspective,” SIAM J. Control
Optim.,48(1), pp. 162–186.
[522] Liu, Y.-Y., Slotine, J.-J., and Barabasi, A.-L., 2011, “Controllability of Com-
plex Networks,” Nature,473(7346), pp. 167–173.
[523] Lin, F., Fardad, M., and Jovanovic´ , M. R., 2014, “Algorithms for Leader
Selection in Stochastically Forced Consensus Networks,” IEEE Trans. Auto-
mat. Control,59(7), pp. 1789–1802.
[524] Cowan, N. J., Chastain, E. J., Vilhena, D. A., Freudenberg, J. S., and Berg-
strom, C. T., 2012, “Nodal Dynamics, Not Degree Distributions, Determine
the Structural Controllability of Complex Networks,” PloS One,7(6), p.
e38398.
[525] Brockett, R., 2012, “Notes on the Control of the Liouville Equation,” Control
of Partial Differential Equations (Lecture Notes in Mathematics, Vol. 2048),
F. Alabau-Boussouira, R. Brockett, O. Glass, J. Le Rousseau, and E. Zuazua,
eds., Springer, Berlin, pp. 101–129.
[526] Hopf, E., 1951, “Statistical Hydromechanics and Functional Analysis,” J.
Ration. Mech. Anal., 1, pp. 87–123.
[527] Bagheri, S., 2013, “Koopman-Mode Decomposition of the Cylinder Wake,” J.
Fluid Mech.,726, pp. 596–623.
[528] Bagheri, S., 2014, “Effects of Weak Noise on Oscillating Flows: Linking
Quality Factor, Floquet Modes and Koopman Spectrum,” Phys. Fluids,26(9),
p. 094104.
050801-48 / Vol. 67, SEPTEMBER 2015 Transactions of the ASME
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09/10/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Supplementary resource (1)

... Projecting DNS data on the spatial modes yields a time series of the three modal coefficients. A dynamical system is fit to the modal coefficient time series using Sparse Identification of Nonlinear Dynamics (SINDy) [6]. We apply constraints to ensure the dynamical system respects the energy-preserving nature of the nonlinearity in the Navier-Stokes equation [9]. ...
... The selected skew-symmetric nonlinear operator is identical in form to the Galerkin model first employed for the cylinder wake [28]. This well-known model has previously been discussed in detail [6,29,44]. There are two primary nonlinear effects. ...
... A 3-state dynamical system with up to quadratic nonlinearity was calibrated to describe the spatial mode interactions by projecting simulation data onto each mode. The dynamical system was found as the solution of a regression problem [6] with constraints to ensure the nonlinearity be energy-preserving [9]. Our modeling approach does not require knowledge of the full-order governing equations, making it wholly non-intrusive [30]. ...
Article
Full-text available
We apply data-driven techniques to construct a nonlinear 3-mode model of a Kolmogorov-like flow transitioning from steady to periodic. Data from direct numerical simulation that include features of experimental realizations of Kolmogorov-like flow are used to build the model. Our low-order modeling methodology does not require knowledge of the underlying governing equations. The 3-mode basis for the model is determined solely from data and the sparse identification of nonlinear dynamics framework (SINDy) is used to fit a dynamical system describing modal interactions. We impose constraints within the SINDy framework to ensure the resulting model will possess energy-preserving nonlinear terms that are consistent with the underlying flow physics. We use the low-order model to determine an appropriate equilibrium solution to stabilize, thereby avoiding searching for equilibrium solutions in the full-order system. The model is linearized about the identified equilibrium solution and subsequently used to design feedback controllers that successfully suppress an oscillatory instability when applied in direct numerical simulations—a testament to the model’s ability to capture the underlying dynamics that are most relevant for flow control. Our results confirm that low-order models obtained in a purely data-driven framework can be implemented for flow control in experimentally-realizable Kolmogorov-like flow.
... The rapid development of artificial intelligence brings new vitality to the field of flow control, with reinforcement learning (RL) emerging as a key machine learning algorithm that effectively addresses complex decision-making and control problems [6,5]. RL agents possess the ability for long-term planning and decision-making, continuously adjusting their behavior strategies in response to environmental feedback, making it a powerful tool for autonomous learning and decision-making [23,3]. ...
... Comparing jet positions, 1 shows the slowest growth and lowest final reward, highlighting convergence challenges. 5 achieves moderate convergence, while 2 , 3 , and 4 demonstrate rapid convergence within 1,000 episodes. Between 1,000 and 3,000 episodes, the reward curves plateau, confirming stable and effective flow control strategies that reliably reduce and control . ...
Preprint
Full-text available
This study utilizes deep reinforcement learning (DRL) to develop flow control strategies for circular and square cylinders, enhancing energy efficiency and minimizing energy consumption while addressing the limitations of traditional methods. We find that the optimal jet placement for both square and circular cylinders is at the main flow separation point, achieving the best balance between energy efficiency and control effectiveness. For the circular cylinder, positioning the jet at approximately 105°from the stagnation point requires only 1% of the inlet flow rate and achieves an 8% reduction in drag, with energy consumption one-third of that at other positions. For the square cylinder, placing the jet near the rear corner requires only 2% of the inlet flow rate, achieving a maximum drag reduction of 14.4%, whereas energy consumption near the front corner is 27 times higher, resulting in only 12% drag reduction. In multi-action control, the convergence speed and stability are lower compared to single-action control, but activating multiple jets significantly reduces initial energy consumption and improves energy efficiency. Physically, the interaction of the synthetic jet with the flow generates new vortices that modify the local flow structure, significantly enhancing the cylinder's aerodynamic performance. Our control strategy achieves a superior balance between energy efficiency and control performance compared to previous studies, underscoring its significant potential to advance sustainable and effective flow control.
... Many typical turbulence scenarios have been involved with machine learning control, such as airfoil flow, 2,3 cylinder flow, 4-6 mixing layer, 7,8 turbulent jet, [9][10][11] flow separation, 12 cavity flow, 13 blunt body flow, 14,15 and boundary layer. 16 The vast majority of active turbulence control studies are performed in a model-free manner, 17 as control-oriented modeling of the actuation response from broadband frequency dynamics is still a challenge. Steady or periodic operation of a single actuator may be optimized by gradient-based approaches for one or few parameters, like extremum seeking. ...
Article
Full-text available
We propose an automated analysis of the flow control behavior from an ensemble of control laws and associated time-resolved flow snapshots. The input may be the rich database of machine learning control optimizing a feedback law for a cost function in the plant. The proposed methodology provides (1) insights into control landscape which maps control laws to performance including extrema and ridge lines, (2) a catalogue of representative flow states and their contribution to cost function for investigated control laws, and (3) a visualization of the dynamics. Key enablers are classification and feature extraction methods of machine learning. The analysis is successfully applied to the stabilization of a mixing layer with sensor-based feedback driving an upstream actuator. The fluctuation energy is reduced by 26%. The control replaces unforced Kelvin–Helmholtz vortices with subsequent vortex pairing by higher frequency Kelvin–Helmholtz structures of lower energy. The algorithm picks up the most effective sensors s7′ and s22′ from 25 sensors. The best control law exhibits a net upward force with high frequency. The learning curve shows the difficulty to stabilize the mixing layer with only a few individuals distributed below Jsn=1. This is also verified in the control landscape exhibiting a pretty small distance with the unforced case. The fluctuation contribution of each centroid tends to be lower with the increasing performance of the control law. These efforts target a human interpretable, fully automated analysis of MLC identifying qualitatively different actuation regimes, distilling corresponding coherent structures, and developing a digital twin of the plant.
... Sirovich [55,56] introduced the method of snapshots for computing POD modes via a simple data-driven procedure involving singular value decomposition. Since POD modes represent coherent structures in the flow field, they can be used for developing ROM of unsteady flows [4,14,16,20,22,30] and efficient control design algorithm via neural networks [15,18,21,41,60]. ...
Preprint
Proper-orthogonal decomposition (POD) based reduced-order models (ROM) of structurally dominant fluid flow can support a wide range of engineering applications. Yet, although they perform well for unsteady laminar flows, their straightforward extension to turbulent flows fails to capture the effects of small scale eddies and often leads to divergent solutions. Several approaches to mimic nonlinear closure terms modeling techniques within ROM frameworks have been employed to include the effect of higher modes that are often neglected. Recent success of neural network based models show promising results in modeling the effects of turbulence. In this study, we augment POD-ROM with a recurrent neural network (RNN) to develop ROM for turbulent flows. We simulate a three dimensional flow past a circular cylinder at Reynolds number of 1000. We first compute the POD modes and project the Navier-Stokes equations onto the limited number of modes in a Galerkin approach to develop a conventional ROM and LES-inspired ROM for comparison. We then develop a hybrid model by integrating the output of Galerkin projection ROM and long short-term memory (LSTM) RNN and term it as a physics-guided machine learning (PGML) model. The novelty of this study is to introduce a hybrid model that integrates LES inspired ROM and RNN to achieve more accurate and reliable predictions of turbulent flows. The results demonstrate that PGML for higher temporal coefficients outperforms the conventional and LES-inspired ROM.
... Turbulent flows lead to significantly greater energy losses compared to laminar flows, presenting a major challenge in various engineering applications (Brunton & Noack 2015). For instance, wall friction contributes to approximately 50 % of total resistance in aircraft, up to 90 % in submarines, and nearly all resistance in pipeline flows (Gad-el-Hak & Blackwelder 1989). ...
Article
Full-text available
Deep reinforcement learning (DRL) is employed to develop control strategies for drag reduction in direct numerical simulations of turbulent channel flows at high Reynolds numbers. The DRL agent uses near-wall streamwise velocity fluctuations as input to modulate wall blowing and suction velocities. These DRL-based strategies achieve significant drag reduction, with maximum rates 35.6%35.6\,\% at Reτ180Re_{\tau }\thickapprox 180 , 30.4%30.4\,\% at Reτ550Re_{\tau }\thickapprox 550 , and 27.7%27.7\,\% at Reτ1000Re_{\tau }\thickapprox 1000 , outperforming traditional opposition control methods. An expanded range of wall actions further enhances drag reduction, although effectiveness decreases at higher Reynolds numbers. The DRL models elevate the virtual wall through blowing and suction, aiding in drag reduction. However, at higher Reynolds numbers, the amplitude modulation of large-scale structures significantly increases the residual Reynolds stress on the virtual wall, diminishing the drag reduction. Analysis of budget equations provides a systematic understanding of the underlying drag reduction dynamics. The DRL models reduce skin friction by inhibiting the redistribution of wall-normal turbulent kinetic energy. This further suppresses the wall-normal velocity fluctuations, reducing the production of Reynolds stress, thereby decreasing skin friction. This study showcases the successful application of DRL in turbulence control at high Reynolds numbers, and elucidates the nonlinear control mechanisms underlying the observed drag reduction.
... Through episodes of consecutive actions, neural-network weights are updated, optimizing policies to maximize expected rewards. For recent advances in flow control using MARL, interested readers are directed to Belus et al. (2019), Brunton et al. (2015), , where significant progress and insights have been reported. ...
Article
Full-text available
This study presents novel drag reduction active-flow-control (AFC) strategies for a three-dimensional cylinder immersed in a flow at a Reynolds number based on freestream velocity and cylinder diameter of ReD=3900Re_D=3900. The cylinder in this subcritical flow regime has been extensively studied in the literature and is considered a classic case of turbulent flow arising from a bluff body. The strategies presented are explored through the use of deep reinforcement learning. The cylinder is equipped with 10 independent zero-net-mass-flux jet pairs, distributed on the top and bottom surfaces, which define the AFC setup. The method is based on the coupling between a computational-fluid-dynamics solver and a multi-agent reinforcement-learning (MARL) framework using the proximal-policy-optimization algorithm. This work introduces a multi-stage training approach to expand the exploration space and enhance drag reduction stabilization. By accelerating training through the exploitation of local invariants with MARL, a drag reduction of approximately 9%9\% is achieved. The cooperative closed-loop strategy developed by the agents is sophisticated, as it utilizes a wide bandwidth of mass-flow-rate frequencies, which classical control methods are unable to match. Notably, the mass cost efficiency is demonstrated to be two orders of magnitude lower than that of classical control methods reported in the literature. These developments represent a significant advancement in active flow control in turbulent regimes, critical for industrial applications.
Preprint
We propose a novel method for model-based time super-sampling of turbulent flow fields. The key enabler is the identification of an empirical Galerkin model from the projection of the Navier-Stokes equations on a data-tailored basis. The basis is obtained from a Proper Orthogonal Decomposition (POD) of the measured fields. Time super-sampling is thus achieved by a time-marching integration of the identified dynamical system, taking the original snapshots as initial conditions. Temporal continuity of the reconstructed velocity fields is achieved through a forward-backwards integration between consecutive measured Particle Image Velocimetry measurements of a turbulent jet flow. The results are compared with the interpolation of the POD temporal coefficients and the low-order reconstruction of data measured at a higher sampling rate. In both cases, the results obtained show the ability of the method to reconstruct the dynamics of the flow with small errors during several flow characteristic times.
Conference Paper
Full-text available
A novel framework for closed-loop control of turbulent flows is tested in an experimental mixing layer flow. This framework, called Machine Learning Control (MLC), provides a model-free method of searching for the best control law (see talk of B.~R.\ Noack). Here, MLC is benchmarked against classical open-loop actuation of the mixing layer. Results show that this method is capable of producing sensor-based control laws which can rival or surpass the best open-loop forcing, and be robust to changing flow conditions. Additionally, MLC can detect non-linear mechanisms present in the controlled plant, and exploit them to find a better type of actuation than the best periodic forcing. Other experimental shear-flow control studies with MLC will be presented in a talk by T.\ Duriez.
Book
Data evaluation and data combination require the use of a wide range of probability theory concepts and tools, from deductive statistics mainly concerning frequencies and sample tallies to inductive inference for assimilating non-frequency data and a priori knowledge. Computational Methods for Data Evaluation and Assimilation presents interdisciplinary methods for integrating experimental and computational information. This self-contained book shows how the methods can be applied in many scientific and engineering areas. After presenting the fundamentals underlying the evaluation of experimental data, the book explains how to estimate covariances and confidence intervals from experimental data. It then describes algorithms for both unconstrained and constrained minimization of large-scale systems, such as time-dependent variational data assimilation in weather prediction and similar applications in the geophysical sciences. The book also discusses several basic principles of four-dimensional variational assimilation (4D VAR) and highlights specific difficulties in applying 4D VAR to large-scale operational numerical weather prediction models.