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Prof. Chris C. Bissell
Department of Telematics
The Open University
Milton Keynes
Great Britain MK7 6AA U.K.
c.c.bissell@open.ac.uk
Real-Time Optimization by
Extremum-Seeking Control, by
Kartik B. Ariyur and Miroslav Krstic,
Wiley, 2003, US$64.95, ISBN
0471468592. Reviewed by Gang Tao.
This interesting and unique book
deals with the optimization of nonlinear dynamical systems using feedback and adaptation, and develops a
systematic extremum-seeking control
methodology supported by rigorous
theory and illustrative applications.
Extremum seeking is a nonmodelbased adaptive control method that
is applicable to control problems
involving a nonlinear plant or control objective, where the nonlinearity has a local minimum or maximum.
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Extremum-seeking control employs
dynamic feedback, sinusoidal perturbation, and adaptive extremum
searching to find an unknown, optimal
operating condition for the plant or
control law. This book builds a solid
bridge between classical optimization
theory, modern feedback control, and
adaptation techniques; provides a collection of useful tools for related
problems; presents diverse applications of extremum-seeking schemes;
and demonstrates the potential of this
methodology for future applications.
Nonmodel-based adaptive control is
useful when it is difficult to obtain a
reliable first principles system model.
In this sense, the book is a novel and
significant addition to the adaptive
control literature.
With 12 chapters, six for fundamental theory and six for advanced
applications, plus six appendix sections, this book gives a self-contained,
rich, and rigorous coverage of
extremum-seeking theory and applications. The book is well organized, its
theme is logically developed, the presentation is compact and clear, and
the illustrations and examples are
informative.
Chapter 1 gives a historical
overview of extremum seeking, presents the design and analysis of an
extremum-seeking algorithm for a single-parameter problem, and lays
down a conceptual foundation for the
extremum-seeking method.
A basic system of extremum seeking consists of a dynamic plant with a
nonlinear function whose unknown
local minimum or maximum defines
an optimal operating condition or
desired output for the plant as well as
a dynamic feedback law, which, by
means of filters and sinusoidal perturbation signals, generates an online
estimate of the minimum or maximum
parameter as the plant input. The
feedback law is designed so that the
plant output error converges to a
neighborhood of the origin, whose
size depends on the amplitude and
the frequency reciprocal of the per-
IEEE Control Systems Magazine
turbation signals. This online optimization problem involves a nonlinear dynamic system with feedback
and adaptation, the solution of which
is derived and analyzed in terms of
frequency and time-domain conditions. In particular, the optimization
problem is solved by using dynamic,
adaptation-based feedback with sinusoidal perturbation, and the adaptive
scheme is nonmodel based. These
features render the extremum-seeking-based algorithms more effective
than classical optimization and adaptive control algorithms in many
advanced applications as presented
in Chapters 7–12.
Chapter 2 deals with extremum
seeking for a multiple-input single-output dynamic plant with a nonlinear
function for which the minimum or
maximum extremum is an unknown
multiparameter vector. A multiplechannel dynamic and adaptive feedback law with sinusoidal perturbation
signals is designed and analyzed to
ensure the desired extremum-seeking
properties. The complexity of the
extremum-seeking algorithm is essential due to dynamic coupling. A diagonal-dominance-based stability criterion is used to characterize a set of
checkable design conditions to handle
such coupling.
Chapter 3 addresses a more general problem of extremum seeking,
namely, slope seeking, where the
objective is to drive the output of
the plant to a value corresponding to
the slope of the reference-to-output
map. Extremum seeking is a special
case of slope seeking, in particular,
the case of zero slope. Slope seeking
is motivated by mission-critical applications such as aeroengine compressor control, aircraft antiskid braking,
minimum power demand formation
flight, and nuclear fusion. The slopeseeking algorithms are designed with
the additional commanded slope
information for both the single-parameter and multiparameter cases.
Chapter 4 considers extremum
seeking in discrete-time, with detailed
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stability analysis based on two-timescale theory. Chapter 5 solves the
extremum-seeking problem for a
class of general nonlinear dynamic
plants whose extremum nonlinear
functions are not separated from the
plant nonlinear dynamics, unlike the
cases considered in the preceding
chapters, where the nonlinear function appears between two linear
dynamic blocks in the plant. The
extremum-seeking system is formulated based on three time scales and is
analyzed using the methods of averaging and singular perturbation.
These solutions set up the frameworks for the corresponding multiparameter extremum-seeking and
slope-seeking designs. Chapter 6 presents an extremum-seeking-based
technique for minimizing the amplitude of the limit cycle of a nonlinear
control system, illustrated by
detailed design and analysis for the
Van der Pol system.
Chapters 7–12 present a series of
five applications of the extremumseeking technique. The first application involves traction maximization in
antilock braking for a car. Between the
wheel and a slippery road surface, the
friction force coefficient is a function
of the wheel slip, and there is a maximum value of this coefficient at some
unknown value of the wheel slip. With
linear acceleration as the plant output,
extremum-seeking estimates this
unknown coefficient for brakingtorque feedback control. Simulation
results indicate that the maximum friction force is reached by extremum
seeking, thus minimizing the car’s time
and distance to stopping.
Chapter 8 shows how to employ
the extremum-seeking technique to
maximize the productivity (mass outflow rate) of a continuous stirred tank
bioreactor. The steady mass outflow
rate is a nonlinear function of the
dilution rate, and the goal of extremum seeking is to adaptively adjust
the dilution rate to converge to an a
priori unknown optimal value to maximize the mass outflow rate. Simula-
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tion results suggest that this goal is
achievable as verified for two practical models of the biomass concentration growth rate function.
Chapter 9 applies extremum-seeking to aircraft formation flight control, aimed at reducing power
demand by flying the wingman aircraft on the peaks of the wake velocity distribution of the leader aircraft.
There is an optimal configuration of
the flight formation that yields maximal power reduction. Extremum seeking is used to adjust the wingmanleader separation to reach the optimal formation by using a formationhold autopilot for wingman control.
The extremum-seeking design procedure has four steps: aerodynamic
interference modeling, robust-control-based formation-hold autopilot,
extremum-seeking formulation, and
design. First, for a formation of large
transport aircraft with experimental
data, a model of the wake of the leading aircraft is specified, the force and
moment on the wingman in the wake
are described, and the wingman
dynamics and equilibrium in the
wake are derived. Second, a two-loop
robust formation-hold autopilot is
designed by combining a relative
velocity tracking loop, consisting of
an internal model plus an LQR gain,
with a separation tracking loop consisting of a PD compensator with a
reference vector signal generated
from extremum seeking. Third, an
intuitive extremum-seeking framework for formation flight power
demand minimization is transformed
into the standard extremum-seeking
structure for which a rigorous design
is available. Finally, the extremumseeking technique developed in
Chapter 2 is used to specify the
design filters and parameters. The
desired system performance is verified by simulations that invoke experimental data.
Chapter 10 features an industrialscale gas turbine combustor application in which the goal is to suppress
pressure oscillations. The need to
IEEE Control Systems Magazine
determine an optimal phase shift
(from pressure measurement to
either fuel injection or acoustic cancellation actuation) for oscillation
suppression is the motivation for
extremum seeking. This application is
performed in several key steps: identification of an averaged pressure
magnitude dynamics, control phase
tuning via extremum seeking, experimentation on a 4-MW single nozzle
combustion rig, and simulation of
instability suppression during engine
transient. The tuned control phase
converges with the optimal phase
shift, which depends on operating
conditions and several unknown parameters, so that the desired adaptive
oscillation suppression is achieved.
Experimental results show that
extremum-seeking control leads to
significant performance improvement.
Finally, extremum seeking is
applied to an aeroengine compressor
in Chapter 11, where system modeling, control design, and simulation
evaluation are considered, and
Chapter 12, where experimental
results are reported. The objective is
to maximize the compressor pressure
rise in the presence of an unknown
compressor characteristic function.
Compressor stall is an instability
phenomenon that can cause a sudden drop in performance known as
pressure rise. To apply extremum
seeking, a model of an axial-flow compression system is derived, and its
equilibria are calculated. In particular, the stall equilibria are specified,
exhibiting deep hysteresis, which is
the so-called stall characteristic. After
a detailed system analysis, two peakpressure extremum-seeking algorithms are presented, including one
for a reduced-order system model
and one for the full-order model, with
two corresponding stabilizing controllers. Simulation results for the fullorder control system show desired
performance with reduced hysteresis. Experimental set-up and results
of extremum seeking are given in
Chapter 12, which demonstrates the
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system structure and desired performance as well. A simulation study of
near-optimal compressor operation
by means of slope seeking is also conducted, indicating that slope seeking
can recover system performance in
the presence of disturbances.
These applications are clearly
presented and technically convincing. Most of the applications are
sophisticated in both system modeling and control design. Their success
proves that extremum seeking is a
powerful technique for significant
engineering applications. These
examples are also representative of
additional systems for which extremum seeking may be effective, such
as power system and magnetic bearing system applications.
In summary, Real-Time Optimization by Extremum-Seeking Control is
a well-written and authoritative
book on this technically important
subject. It has unique demonstrations of important stability concepts
such as bifurcation, as well as analysis tools such as singular perturbations and averaging. The book is an
excellent technical reference or an
advanced textbook for graduate students, researchers, and engineers.
For those who are interested in
dynamic systems and control, I
strongly recommend this book as an
essential resource for learning about
extremum-seeking control and for
motivating further developments in
this subject area.
Stability and Control of Dynamical
Systems with Applications: A Tribute to Anthony N. Michel, by D. Liu
and P.J. Antsaklis, Birkhäuser, 2003,
430 pp, US$89.95, ISBN 0817632336.
Reviewed by Huaguang Zhang.
This book is an extensive compilation of papers presented at a workshop held at the University of Notre
Dame on 5 April 2003. It presents
recent important research results on
stability and control of dynamical
systems by 41 researchers.
The book is organized into three
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major parts incorporating 21 chapters. The first part of the book contains seven chapters on stability
analysis of dynamical systems. Chapter 1 expands wave digital concepts
and relativity theory through some
modifications to Newton’s laws. Chapter 2 studies the notion of time and
establishes a consistent Lyapunov
methodology for nonlinear systems.
Moreover, the extended concept of
the vector Lyapunov function is introduced. Chapter 3 develops a mathematical model for a multibody
attitude system that exposes the
dynamic coupling between the rotational degrees of freedom of the base
body and the deformation or shape
degrees of freedom of the elastic subsystems. Furthermore, results that
guarantee asymptotic stability of this
multibody attitude system are
obtained. Chapter 4 discusses robust
control of uncertain hybrid systems
affected by both parameter variations
and exterior disturbances, and it provides a method for checking attainability. Chapter 5 overviews stability
properties of swarms, and it analyzes
swarm cohesion under very noisy
measurements using Lyapunov stability theory. Chapter 6 presents a necessary and sufficient asymptotic
stability condition for discrete-time,
time-varying, uncertain delay systems, and it applies the result to con-
IEEE Control Systems Magazine
trol problems of a communication
network. Chapter 7 investigates stability and L2 gain properties for
switched symmetric systems. The key
idea is to establish a common Lyapunov function for all of the subsystems in the switched systems.
Comprising six chapters, the second part of the book is concerned
with neural networks and signal processing. Chapter 8 investigates the
approximation capabilities of Gaussian radial basis functions and the
concept of locally compact metric
spaces. Chapter 9 provides a generalized state-space formulation and
learning algorithms for blind source
recovery based on the theory of multivariable optimization. Chapter 10
discusses the theme of approximate
dynamic programming. Furthermore,
it presents a method of direct neuraldynamic programming and its application to helicopter command
tracking. Chapter 11 studies online
approximator-based aircraft state
estimation. Chapter12 proposes and
analyzes a novel dynamic multiobjective evolutionary algorithm.
Chapter 13 introduces set membership adaptive filtering and its novel
feature of data-dependent selective
update of parameter estimates.
The final part of the book covers
power systems and control systems
(Chapters 14–21). Chapter 14 is concerned with trajectory sensitivity
theory and its practical application
to power systems. Chapter 15 investigates the design of a corrective
control strategy after substantial disturbances in large-scale electric
power systems. An analytical
approach in which the system is separated into smaller islands at a slightly reduced capacity is developed.
Chapter 16 expands control methods
for maintaining the stability of the
electric power generation transmission distribution grid. This chapter
also presents a roadmap for the
development of new controls for
power system stability.
Chapter 17 introduces data fusion
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modeling for groundwater system
identification based on Kalman filtering methods and a Markov random
field representation for spatial variations. Chapter 18 provides an introduction to the nominal design problem
along with results for feedback synthesis in an algebraic framework. Chapter
19 introduces the adaptive dynamic
programming algorithm and gives a
detailed proof. Chapter 20 analyzes the
reliability of supervisory control and
data acquisition systems used in offshore oil and gas platforms. Chapter
21 develops call admission control
algorithms, based on signal-to-interfer-
ence ratio, for power-controlled CDMA
cellular networks. In particular, call
admission control algorithms are
developed based on the necessary and
sufficient conditions under which the
power control algorithm will have a
feasible solution.
One welcome feature of the book
is that each chapter includes an
abstract, a detailed introduction,
and a concise conclusion, thereby
significantly assisting the readers’
comprehension. Each chapter is a
helpful guide for anyone engaged in
the analysis and control of dynamical systems, offering ample opportu-
nity for further exploration of the
approaches covered. Rigid mathematical descriptions and logical
derivations are another feature. The
main ideas presented are original,
and the results stated are advanced
and appropriate. The reviewer
believes that the book is an excellent
reference
source
for
researchers and practitioners in the
areas of dynamical systems
research and applications. The book
is well written and well organized,
and it is clear that the authors have
made important research contributions in this field.
Conference Report
(continued from p. 95)
the United States, starting from a regional system to a global system, which evolved according to a specific set of
established rules of noninterference, which in turn evolved
with time.
Oscar Crisalle questioned the name “robust control,”
and he concluded that this word could be misleading. For
example, a robust controller may become fragile if not
properly implemented in a finite word-length processor.
Tempo agreed that the name robust control could be too
narrow, and the use of the terminology uncertain systems
would be more appropriate for its broadness. This notion
may also reflect the development of methods or algorithms, which are not robust in the classical sense. Khargonekar added that the name “electrical engineering” is a
good example of a field that is dynamic and inclusive since
it started with power systems and evolved into electronics
and now incorporates wireless communication and nanotechnology. Hence, he concluded that control should
remain inclusive to embrace very diverse areas.
Başar said that we should periodically introduce new terminology not only to identify specific research groups or
April 2004
subgroups but also for selling our developments to the outside world. The terminology “robust control” was used after
the H∞ developments, and this name later became important for economists who named a field “robust economics.”
This adoption shows the significance of using the right terminology to market control to researchers working in other
areas. Kokotovic mentioned that, after actively working in
the control field for about 45 years, he has never before seen
so much demand for control expertise required in different
disciplines. Although control is extremely popular in many
areas, the real question is whether the traditional control
community will be able to respond to this demand or
whether the demand will be met by specialists in other areas
who develop, or redevelop, methods to suit their needs.
Başar finally drew a brief conclusion on the discussions
made during the panel, noting that the field is alive and
well, and there are lots of opportunities for those who are
willing to take the challenge to extend the boundaries of
robust control.
—Sergio Bittanti
—Patrizio Colaneri
IEEE Control Systems Magazine
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