the control handbook control system applications second edition electrical engineering handbook_

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MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does
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gogical approach or particular use of the MATLAB® and Simulink® software.
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Library of Congress Cataloging-in-Publication Data
Control system applications / edited by William S. Levine. 2nd ed.
p. cm. (The electrical engineering handbook series)

Includes bibliographical references and index.
ISBN 978-1-4200-7360-7
1. Automatic control. 2. Control theory. I. Levine, W. S. II. Title.
TJ225.C66 2011
629.8 dc22 2010026364
Visit the Taylor & Francis Web site at

and the CRC Press Web site at

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Contents
Preface to the Second Edition xi
Acknowledgments xiii
Editorial Board xv
Editor xvii
Contributors xix
SECTION I Automotive
1 Linear Parameter-Varying Control of Nonlinear Systems with Applications to
Automotive and Aerospace Controls
1-1
Hans P. Geering
2 Powertrain Control 2-1
Davor Hrovat, Mrdjan Jankovic, Ilya Kolmanovsky, Stephen Magner, and Diana Yanakiev
3 Vehicle Controls 3-1
Davor Hrovat, Hongtei E. Tseng, Jianbo Lu, Josko Deur, Francis Assadian, Francesco Borrelli,
and Paolo Falcone

4 Model-Based Supervisory Control for Energy Optimization of Hybrid-Electric
Vehicles
4-1
Lino Guzzella and Antonio Sciarretta
5 Purge Scheduling for Dead-Ended Anode Operation of PEM Fuel Cells 5-1
Jason B. Siegel, Anna G. Stefanopoulou, Giulio Ripaccioli, and Stefano Di Cairano
SECTION II Aerospace
6 Aerospace Real-Time Control System and Software 6-1
Rongsheng (Ken) Li and Michael Santina
7 Stochastic Decision Making and Aerial Surveillance Control Strategies
for Teams of Unmanned Aerial Vehicles
7-1
Raymond W. Holsapple, John J. Baker, and Amir J. Matlock
8 Control Allocation 8-1
Michael W. Oppenheimer, David B. Doman, and Michael A. Bolender
9 Swarm Stability 9-1
Veysel Gazi and Kevin M. Passino
vii
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viii Contents
SECTION III Industrial
10 Control of Machine Tools and Machining Processes 10-1
Jaspreet S. Dhupia and A. Galip Ulsoy
11 Process Control in Semiconductor Manufacturing 11-1
Thomas F. Edgar
12 Control of Polymerization Processes 12-1

Babatunde Ogunnaike, Grégory François, Masoud Soroush, and Dominique Bonvin
13 Multiscale Modeling and Control of Porous Thin Film Growth 13-1
Gangshi Hu, Xinyu Zhang, Gerassimos Orkoulas, and Panagiotis D. Christofides
14 Control of Particulate Processes 14-1
Mingheng Li and Panagiotis D. Christofides
15 Nonlinear Model Predictive Control for Batch Processes 15-1
Zoltan K. Nagy and Richard D. Braatz
16 The Use of Multivariate Statistics in Process Control 16-1
Michael J. Piovoso and Karlene A. Hoo
17 Plantwide Control 17-1
Karlene A. Hoo
18 Automation and Control Solutions for Flat Strip Metal Processing 18-1
Francesco Alessandro Cuzzola and Thomas Parisini
SECTION IV Biological and Medical
19 Model-Based Control of Biochemical Reactors 19-1
Michael A. Henson
20 Robotic Surgery 20-1
Rajesh Kumar
21 Stochastic Gene Expression: Modeling, Analysis, and Identification 21-1
Mustafa Khammash and Brian Munsky
22 Modeling the Human Body as a Dynamical System: Applications
to Drug Discovery and Development
22-1
M. Vidyasagar
SECTION V Electronics
23 Control of Brushless DC Motors 23-1
Farhad Aghili
24 Hybrid Model Predictive Control of the Boost Converter 24-1
Raymond A. DeCarlo, Jason C. Neely, and Steven D. Pekarek
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Contents ix
SECTION VI Networks
25 The SNR Approach to Networked Control 25-1
Eduardo I. Silva, Juan C. Agüero, Graham C. Goodwin, Katrina Lau, and Meng Wang
26 Optimization and Control of Communication Networks 26-1
Srinivas Shakkottai and Atilla Eryilmaz
SECTION VII Special Applications
27 Advanced Motion Control Design 27-1
Maarten Steinbuch, Roel J. E. Merry, Matthijs L. G. Boerlage, Michael J. C. Ronde, and
Marinus J. G. van de Molengraft
28 Color Controls: An Advanced Feedback System 28-1
Lalit K. Mestha and Alvaro E. Gil
29 The Construction of Portfolios of Financial Assets: An Application
of Optimal Stochastic Control
29-1
Charles E. Rohrs and Melanie B. Rudoy
30 Earthquake Response Control for Civil Structures 30-1
Jeff T. Scruggs and Henri P. Gavin
31 Quantum Estimation and Control 31-1
Matthew R. James and Robert L. Kosut
32 Motion Control of Marine Craft 32-1
Tristan Perez and Thor I. Fossen
33 Control of Unstable Oscillations in Flows 33-1
Anuradha M. Annaswamy and Seunghyuck Hong
34 Modeling and Control of Air Conditioning and Refrigeration Systems 34-1
Andrew Alleyne, Vikas Chandan, Neera Jain, Bin Li, and Rich Otten

Index Index-1
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Preface to the
Second Edition
As you may know, the first edition of The Control Handbook was very well received. Many copies were
sold and a gratifying number of people took the time to tell me that they found it useful. To the publisher,
these are all reasons to do a second edition. To the editor of the first edition, these same facts are a modest
disincentive. The risk that a second edition will not be as good as the first one is real and worrisome. I
have tried very hard to insure that the second edition is at least as good as the first one was. I hope you
agree that I have succeeded.
I have made two major changes in the second edition. The first is that all the Applications chapters
are new. It is simply a fact of life in engineering that once a problem is solved, people are no longer as
interested in it as they were when it was unsolved. I have tried to find especially inspiring and exciting
applications for this second edition.
Secondly, it has become clear to me that organizing the Applications book by academic discipline is
no longer sensible. Most control applications are interdisciplinary. For example, an automotive control
system that involves sensors to convert mechanical signals into electrical ones, actuators that convert
electrical signals into mechanical ones, several computers and a communication network to link sensors
and actuators to the computers does not belong solely to any specific academic area. You will notice that
the applications are now organized broadly by application areas, such as automotive and aerospace.
One aspect of this new organization has created a minor and, I think, amusing problem. Several
wonderful applications did not fit into my new taxonomy. I originally grouped them under the title
Miscellaneous. Several authors objected to the slightly pejorative nature of the term “miscellaneous.”
I agreed with them and, after some thinking, consulting with literate friends and with some of the
library resources, I have renamed that section “Special Applications.” Regardless of the name, they are
all interesting and important and I hope you will read those articles as well as the ones that did fit my

organizational scheme.
There has also been considerable progress in the areas covered in the Advanced Methods book. This
is reflected in the roughly two dozen articles in this second edition that are completely new. Some of
these are in two new sections, “Analysis and Design of Hybrid Systems” and “Networks and Networked
Controls.”
There have even been a few changes in the Fundamentals. Primarily, there is greater emphasis on
sampling and discretization. This is because most control systems are now implemented digitally.
I have enjoyed editing this second edition and learned a great deal while I was doing it. I hope that you
will enjoy reading it and learn a great deal from doing so.
William S. Levine
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xii Preface to the Second Edition
MATLAB
®
and Simulink
®
are registered trademarks of The MathWorks, Inc. For product
information, please contact:
The MathWorks, Inc.
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Fax: 508-647-7001
E-mail:
Web: www.mathworks.com

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Acknowledgments
The people who were most crucial to the second edition were the authors of the articles. It took a great
deal of work to write each of these articles and I doubt that I will ever be able to repay the authors for
their efforts. I do thank them very much.
The members of the advisory/editorial board for the second edition were a very great help in choosing
topics and finding authors. I thank them all. Two of them were especially helpful. Davor Hrovat took
responsibility for the automotive applications and Richard Braatz was crucial in selecting the applications
to industrial process control.
It is a great pleasure to be able to provide some recognition and to thank the people who helped
bring this second edition of The Control Handbook into being. Nora Konopka, publisher of engineering
and environmental sciences for Taylor & Francis/CRC Press, began encouraging me to create a second
edition quite some time ago. Although it was not easy, she finally convinced me. Jessica Vakili and Kari
Budyk, the project coordinators, were an enormous help in keeping track of potential authors as well
as those who had committed to write an article. Syed Mohamad Shajahan, senior project executive at
Techset, very capably handled all phases of production, while Richard Tressider, project editor for Taylor
& Francis/CRC Press, provided direction, oversight, and quality control. Without all of them and their
assistants, the second edition would probably never have appeared and, if it had, it would have been far
inferior to what it is.
Most importantly, I thank my wife Shirley Johannesen Levine for everything she has done for me over
the many years we have been married. It would not be possible to enumerate all the ways in which she
has contributed to each and everything I have done, not just editing this second edition.
William S. Levine
xiii
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Editorial Board
Frank Allgöwer
Institute for Systems Theory and
Automatic Control
University of Stuttgart
Stuttgart, Germany
Tamer Ba¸sar
Department of Electrical and
Computer Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Richard Braatz
Department of Chemical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Christos Cassandras
Department of Manufacturing Engineering
Boston University
Boston, Massachusetts
Davor Hrovat
Research and Advanced Engineering
Ford Motor Company
Dearborn, Michigan
Naomi Leonard
Department of Mechanical and
Aerospace Engineering
Princeton University

Princeton, New Jersey
Masayoshi Tomizuka
Department of Mechanical
Engineering
University of California, Berkeley
Berkeley, California
Mathukumalli Vidyasagar
Department of Bioengineering
The University of Texas at Dallas
Richardson, Texas
xv
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Editor
William S. Levine received B.S., M.S., and Ph.D. degrees from the Massachusetts Institute of Technology.
He then joined the faculty of the University of Maryland, College Park where he is currently a research
professor in the Department of Electrical and Computer Engineering. Throughout his career he has
specialized in the design and analysis of control systems and related problems in estimation, filtering, and
system modeling. Motivated by the desire to understand a collection of interesting controller designs,
he has done a great deal of research on mammalian control of movement in collaboration with several
neurophysiologists.
He isco-author of Using MATLABto Analyze andDesign Control Systems,March 1992. SecondEdition,
March 1995. He is the coeditor of The Handbook of Networked and Embedded Control Systems, published
by Birkhauser in 2005. He is the editor of a series on control engineering for Birkhauser. He has been
president of the IEEE Control Systems Society and the American Control Council. He is presently the
chairman of the SIAM special interest group in control theory and its applications.
He is a fellow of the IEEE, a distinguished member of the IEEE Control Systems Society, and a

recipient of the IEEE Third Millennium Medal. He and his collaborators received the Schroers Award
for outstanding rotorcraft research in 1998. He and another group of collaborators received the award
for outstanding paper in the IEEE Transactions on Automatic Control, entitled “Discrete-Time Point
Processes in Urban Traffic Queue Estimation.”
xvii
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Contributors
Farhad Aghili
Division of Spacecraft Engineering
Canadian Space Agency
Saint-Hubert, Quebec, Canada
Juan C. Agüero
School of Electrical Engineering and
Computer Science
The University of Newcastle
Callaghan, New South Wales, Australia
Andrew Alleyne
Department of Mechanical Science and
Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Anuradha M. Annaswamy
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Francis Assadian

Department of Automotive Engineering
Cranfield University
Cranfield, United Kingdom
John J. Baker
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan
Matthijs L. G. Boerlage
Renewable Energy Systems and
Instrumentation
General Electric Global Research
Munich, Germany
Michael A. Bolender
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio
Dominique Bonvin
Automatic Control Laboratory
Swiss Federal Institute of Technology
in Lausanne
Lausanne, Switzerland
Francesco Borrelli
Department of Mechanical Engineering
University of California, Berkely
Berkeley, California
Richard D. Braatz
Department of Chemical Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Vikas Chandan
Department of Mechanical Science and

Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Panagiotis D. Christofides
Department of Chemical and
Biomolecular Engineering
Department of Electrical Engineering
University of California, Los Angeles
Los Angeles, California
Francesco Alessandro Cuzzola
Danieli Automation
Buttrio, Italy
Raymond A. DeCarlo
Department of Electrical and
Computer Engineering
Purdue University
West Lafayette, Indiana
Josko Deur
Mechanical Engineering and Naval Architecture
University of Zagreb
Zagreb, Croatia
xix
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xx Contributors
Jaspreet S. Dhupia
School of Mechanical and

Aerospace Engineering
Nanyang Technological University
Singapore
Stefano Di Cairano
Ford Motor Company
Dearborn, Michigan
David B. Doman
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio
Thomas F. Edgar
Department of Chemical Engineering
University of Texas at Austin
Austin, Texas
Atilla Eryilmaz
Electrical and Computer
Engineering Department
The Ohio State University
Columbus, Ohio
Paolo Falcone
Signals and Systems Department
Chalmers University of Technology
Goteborg, Sweden
Thor I. Fossen
Department of Engineering Cybernetics
and
Centre for Ships and Ocean Structures
Norwegian University of Science and
Technology
Trondheim, Norway
Grégory François

Automatic Control Laboratory
Swiss Federal Institute of Technology in
Lausanne
Lausanne, Switzerland
Henri P. Gavin
Department of Civil and
Environmental Engineering
Duke University
Durham, North Carolina
Veysel Gazi
Department of Electrical and Electronics
Engineering
TOBB University of Economics and Technology
Ankara, Turkey
Hans P. Geering
Measurement and Control Laboratory
Swiss Federal Institute of Technology
Zurich, Switzerland
Alvaro E. Gil
Xerox Research Center
Webster, New York
Graham C. Goodwin
School of Electrical Engineering
and Computer Science
The University of Newcastle
Callaghan, New South Wales, Australia
Lino Guzzella
Swiss Federal Institute of Technology
Zurich, Switzerland
Michael A. Henson

Department of Chemical Engineering
University of Massachusetts Amherst
Amherst, Massachusetts
Raymond W. Holsapple
Control Science Center of Excellence
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio
Seunghyuck Hong
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Karlene A. Hoo
Department of Chemical Engineering
Texas Tech University
Lubbock, Texas
Davor Hrovat
Research and Advanced Engineering
Ford Motor Company
Dearborn, Michigan
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Contributors xxi
Gangshi Hu
Department of Chemical and
Biomolecular Engineering
University of California, Los Angeles
Los Angeles, California

Neera Jain
Department of Mechanical Science and
Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Matthew R. James
College of Engineering and
Computer Science
Australian National University
Canberra, Australia
Mrdjan Jankovic
Research and Advanced
Engineering
Ford Motor Company
Dearborn, Michigan
Mustafa Khammash
Department of Mechanical
Engineering
University of California, Santa Barbara
Santa Barbara, California
Ilya Kolmanovsky
Research and Advanced
Engineering
Ford Motor Company
Dearborn, Michigan
Robert L. Kosut
SC Solutions
Sunnyvale, California
Rajesh Kumar
Department of Computer Science

Johns Hopkins University
Baltimore, Maryland
Katrina Lau
School of Electrical Engineering and
Computer Science
The University of Newcastle
Callaghan, New South Wales, Australia
Bin Li
Department of Mechanical Science and
Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Mingheng Li
Department of Chemical and Materials
Engineering
California State Polytechnic University
Pomona, California
Rongsheng (Ken) Li
The Boeing Company
El Segundo, California
Jianbo Lu
Research and Advanced Engineering
Ford Motor Company
Dearborn, Michigan
Stephen Magner
Research and Advanced Engineering
Ford Motor Company
Dearborn, Michigan
Amir J. Matlock
Department of Aerospace Engineering

University of Michigan
Ann Arbor, Michigan
Lalit K. Mestha
Xerox Research Center
Webster, New York
Roel J.E. Merry
Department of Mechanical Engineering
Eindhoven University of Technology
Eindhoven, the Netherlands
Marinus J. van de Molengraft
Department of Mechanical Engineering
Eindhoven University of Technology
Eindhoven, the Netherlands
Brian Munsky
CCS-3 and the Center for NonLinear
Studies
Los Alamos National Lab
Los Alamos, New Mexico
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xxii Contributors
Zoltan K. Nagy
Chemical Engineering Department
Loughborough University
Loughborough, United Kingdom
Jason C. Neely
Department of Electrical and Computer

Engineering
Purdue University
West Lafayette, Indiana
Babatunde Ogunnaike
Chemical Engineering
University of Delaware
Newark, Delaware
Michael W. Oppenheimer
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio
Gerassimos Orkoulas
Department of Chemical and Biomolecular
Engineering
University of California, Los Angeles
Los Angeles, California
Rich Otten
Department of Mechanical Science
and Engineering
University of Illinois at Urbana–Champaign
Urbana, Illinois
Thomas Parisini
Department of Electrical and Electronics
Engineering
University of Trieste
Trieste, Italy
Kevin M. Passino
Department of Electrical and Computer
Engineering
The Ohio State University
Columbus, Ohio

Steven D. Pekarek
Department of Electrical and Computer
Engineering
Purdue University
West Lafayette, Indiana
Tristan Perez
School of Engineering
The University of Newcastle
Callaghan, New South Wales, Australia
and
Centre for Ships and
Ocean Structures
Norwegian University of Science and
Technology
Trondheim, Norway
Michael J. Piovoso
School of Graduate Professional Studies
Pennsylvania State University
Malvern, Pennsylvania
Giulio Ripaccioli
Department of Informatic Engineering
University of Siena
Siena, Italy
Charles E Rohrs
Rohrs Consulting
Newton, Massachusetts
Michael J. C. Ronde
Department of Mechanical Engineering
Eindhoven University of Technology
Eindhoven, the Netherlands

Melanie B. Rudoy
Department of Electrical and
Computer Science
Massachusetts Institute of Technology
Cambridge, Massachusetts
Michael Santina
The Boeing Company
Seal Beach, California
Antonio Sciarretta
IFP Energies Nouvelles
Rueil-Malmaison, France
Jeff T. Scruggs
Department of Civil and Environmental
Engineering
Duke University
Durham, North Carolina
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Contributors xxiii
Srinivas Shakkottai
Department of Electrical and Computer
Engineering
TexasA&MUniversity
College Station, Texas
Jason B. Siegel
Department of Mechanical Engineering
University of Michigan

Ann Arbor, Michigan
Eduardo I. Silva
Department of Electronic Engineering
Federico Santa María Technical University
Valparaíso, Chile
Masoud Soroush
Department of Chemical and Biological
Engineering
Drexel University
Philadelphia, Pennsylvania
Anna G. Stefanopoulou
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan
Maarten Steinbuch
Department of Mechanical Engineering
Eindhoven University of Technology
Eindhoven, the Netherlands
Hongtei E. Tseng
Research and Advanced Engineering
Ford Motor Company
A. Galip Ulsoy
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan
M. Vidyasagar
Department of Bioengineering
The University of Texas at Dallas
Richardson, Texas
Meng Wang

School of Electrical Engineering
and Computer Science
The University of Newcastle
Callaghan, New South Wales, Australia
Diana Yanakiev
Research and Advanced Engineering
Ford Motor Company
Dearborn, Michigan
Xinyu Zhang
Department of Chemical and Biomolecular
Engineering
University of California, Los Angeles
Los Angeles, California
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I
Automotive
I-1
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1
Linear
Parameter-Varying
Control of Nonlinear

Systems with
Applications to
Automotive and
Aerospace Controls
∗
1.1 Introduction 1-1
1.2 Statement of the Control Problem 1-2
1.3 LPV H
∞
Control 1-3
1.4 Choosing the LPV Weights W
•
(θ, s) 1-6
1.5 Handling PV Time Delays 1-7
1.6 Applications in Automotive Engine Control 1-8
Feedback Fuel Control • Feedforward Fuel Control
1.7 Application in Aircraft Flight Control 1-10
1.8 Conclusions 1-11
References 1-12
Hans P. Geering
Swiss Federal Institute of Technology
1.1 Introduction
In this chapter, a linear parameter-varying (LPV) plant [A(θ), B(θ), C(θ)] with the parameter vector θ is
considered with continuously differentiable system matrices A(θ), B(θ), and C(θ). As described in Section
1.2, such a LPV plant description is typically obtained by linearizing the model of a nonlinear plant
about a nominal trajectory. The control problem, which is considered in this chapter is finding a LPV
continuous-time controller with the system matrices F(θ), G(θ), and H(θ) of its state–space model.
In Section 1.3, the control problem is formulated as an H
∞
problem using the mixed sensitivity

approach. The shaping weights W
e
(θ, s), W
u
(θ, s), and W
y
(θ, s) are allowed to be parameter-varying. The
most appealing feature of this approach isthatit yields a parameter-varying bandwidth ω
c
(θ) of the robust
control system. Choosing appropriate shaping weights is described in Section 1.4. For more details about
the design methodology, the reader is referred to [1–6].
∗
Parts reprinted from H. P. Geering, Proceedings of the IEEE International Symposium on Industrial Electronics—ISIE 2005,
Dubrovnik, Croatia, June 20–23, 2005, pp. 241–246, © 2005. IEEE. With permission.
1-1
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1-2 Control System Applications
In Section 1.5, it is shown, how parameter-varying time-delays in the plant dynamics can be handled
in the framework proposed in Sections 1.3 and 1.4. For more details, consult [5,7].
In Section 1.6, two applications in the area of automotive engine control are discussed. In the first
application [4,5,8], the design of an LPV feedback controller for the fuel injection is shown, which is
suitable over the whole operating envelope of engine.
In the second application [9,10], the philosophy of designing an LPV feedback controller is carried
over to the problem of designing an additional LPV feedforward controller compensating the parameter-
varying wall-wetting dynamics in the intake manifold of the port-injected gasoline engine.

In Section 1.7, the problem of LPV control of the short-period motion of an aircraft is discussed.
1.2 Statement of the Control Problem
We consider the following nonlinear time-invariant dynamic system (“plant”) with the unconstrained
input vector U(t) ∈R
m
, the state vector X(t) ∈ R
n
, and the output vector Y(t) ∈R
p
:
˙
X(t) =f (X(t), U(t)),
Y(t) = g(X(t)),
where f and g are fairly “smooth” continuously differentiable functions.
Let us assume that we have found a reasonable or even optimal open-loop control strategy U
nom
(t) for
a rather large time interval t ∈[0,T] (perhaps T =∞), which theoretically generates the nominal state
and output trajectories X
nom
(t) and Y
nom
(t), respectively.
In order to ensure that the actual state and output trajectories X(t) and Y(t) stay close to the nominal
ones at all times, we augment the open-loop control U
nom
(t) with a (correcting) feedback part u(t). Thus,
the combined open-closed-loop input vector becomes
U(t) = U
nom

(t) +u(t).
Assuming that the errors
x(t) =X(t) −X
nom
(t) and y(t) = Y(t) −Y
nom
(t)
of the state and output trajectories, respectively, can be kept minimum with small closed-loop corrections
u(t), allows us to design a linear (parameter-varying) output feedback controller based on the linearized
dynamics of the plant:
˙x(t) = A(θ)x(t) +B(θ)u(t),
y(t) =C(θ)x(t),
where A(θ), B(θ), and C(θ) symbolically denote the following Jacobi matrices:
A(θ) =
∂f
∂x

X
nom
(t), U
nom
(t)

,
B(θ) =
∂f
∂u

X
nom

(t), U
nom
(t)

,
C(θ) =
∂g
∂x

X
nom
(t)

.
The symbol θ (or more precisely θ(t)) denotes a parameter vector, by which the Jacobi matrices
are parametrized; it contains the reference values X
nom
(t) and U
nom
(t) of the state and control vector,
respectively, but it may also contain additional “exogenous” signals influencing the parameters of the
✐
✐
✐
✐
✐ ✐
Linear Parameter-Varying Control of Nonlinear Systems 1-3
r
e
K

u
s
G
s
y
s
FIGURE 1.1 Schematic representation of the feedback control system.
nonlinear equations describing the dynamics of the plant (e.g., a temperature, which is not included in
the model as a state variable).
By using the symbol θ rather than θ(t), we indicate that we base the design of the feedback controller
on a time-invariant linearized plant at every instant t (“frozen linearized dynamics”).
This leads us to posing the following problem of designing an LPV controller:
For all of the attainable values of the parameter vector θ, design a robust dynamic controller (with a
suitable order n
c
) with the state–space representation
z(t) ∈ R
n
c
,
˙z(t) =A
c
(θ)z(t) +B
c
(θ)e(t),
u
s
(t) = C
c
(θ)z(t),

such that all of the specified quantitative, parameter-dependent performance, and robustness specifica-
tions are met (see Figure 1.1).
In Section 1.3, this rather general problem statement will be narrowed down to a suitable and trans-
parent setting of H
∞
control and the solution will be presented.
1.3 LPV H
∞
Control
In this section, we consider the LPV time-invariant plant
˙x
s
(t) = A
s
(θ)x
s
(t) +B
s
(θ)u
s
(t),
y
s
(t) = C
s
(θ)x
s
(t)
of order n
s

. For the sake of simplicity, we assume that we have a “square” plant, that is, the number of
output signals equals the number of input signals: p
s
=m
s
.
Furthermore, we assume that the input u
s
, the state x
s
, and the output y
s
are suitably scaled, such that
the singular values of the frequency response matrix G
s
(jω) =C
s
[jωI −A
s
]
−1
B
s
are not spread too wide
apart.
For the design of the LPV time-invariant controller K(θ) depicted in Figure 1.1, we use the H
∞
method [1,2]. As a novel feature, we use parameter-dependent weights W
•
(θ, s). This allows in particular

that we can adapt the bandwidth ω
c
(θ) of the closed-loop control system to the parameter-dependent
properties of the plant!
Figure 1.2 shows the abstract schematic of the generic H
∞
control system. Again, K(θ) is the controller,
which we want to design and G(θ, s) is the so-called augmented plant. The goal of the design is finding a
compensator K(θ, s), such that the H
∞
norm from the auxiliary input w to the auxiliary output z is less
w
u
s
G
z
e
K
FIGURE 1.2 Schematic representation of the H
∞
control system.
✐
✐
✐
✐
✐ ✐
1-4 Control System Applications
w
K
G

s
−
W
e
z
e
W
u
z
u
W
y
z
y
FIGURE 1.3 S/KS/T weighting scheme.
than γ (γ ≤ 1), that is,
T
zw
(θ, s)
∞
< γ ≤1
for all of the attainable values of the constant parameter vector θ.
For the H
∞
design we choose the mixed-sensitivity approach. This allows us to shape the singular
values of the sensitivity matrix S( jω) and of the complementary sensitivity matrix T( jω) of our control
system (Figure 1.1), where
S(θ, s) =[I +G
s
(θ, s)K(θ, s)]

−1
T(θ, s) =G
s
(θ, s)K(θ, s)[I +G
s
(θ, s)K(θ, s)]
−1
=G
s
(θ, s)K(θ, s)S(θ, s).
Thus, we choose the standard S
/KS/T weighting scheme as depicted in Figure 1.3. This yields the
following transfer matrix:
T
zw
(θ, s) =
⎡
⎣
W
e
(θ, s)S(θ, s)
W
u
(θ, s)K(θ, s)S(θ, s)
W
y
(θ, s)T(θ, s)
⎤
⎦
.

The augmented plant G (Figure 1.2) has the two input vectors w and u
s
and the two output vectors z
and e, where z consists of the three subvectors z
e
, z
u
, and z
y
(Figure 1.3). Its schematic representation is
shown in more detail in Figure 1.4.
In general,thefour subsystemsG
s
(θ, s), W
e
(θ, s), W
u
(θ, s), and W
y
(θ, s) are LPVtime-invariant systems.
By concatenating their individual state vectors into one state vector x, we can describe the dynamics of
the augmented plant by the following state–space model:
˙x(t) =A(θ)x(t) +

B
1
(θ) B
2
(θ)



w(t)
u
s
(t)


z(t)
e(t)

=

C
1
(θ)
C
2
(θ)

x(t) +

D
11
(θ) D
12
(θ)
D
21
(θ) D
22

(θ)

w(t)
u
s
(t)

.
z
e
z
u
z
y
u
s
e
–
G
s
w
W
e
W
u
W
y
FIGURE 1.4 Schematic representation of the augmented plant.
✐
✐

✐
✐
✐ ✐
Linear Parameter-Varying Control of Nonlinear Systems 1-5
The following conditions are necessary for the existence of a solution to the H
∞
control design
problem
∗
:
1. The weights W
e
, W
u
, and W
y
are asymptotically stable.
2. The plant [A
s
, B
s
]is stabilizable.
3. The plant [A
s
, C
s
]is detectable.
4. The maximal singular value of D
11
is sufficiently small: σ(D

11
) < γ.
5. Rank(D
12
) = m
s
, that is, there is a full feedthrough from u
s
to z.
6. Rank(D
21
) = p
s
=m
s
, that is, there is a full feedthrough to e from w.
7. The system [A, B
1
]has no uncontrollable poles on the imaginary axis.
8. The system [A, C
1
]has no undetectable poles on the imaginary axis.
Remarks
Condition 6 is automatically satisfied (see Figure 1.4). Condition 7 demands that the plant G
s
has no poles
on the imaginary axis. Condition 5 can be most easily satisfied by choosing W
u
as a static system with a
small, square feedthrough: W

u
(s) ≡εI.
In order to present the solution to the H
∞
problem in a reasonably esthetic way, it is useful to introduce
the following substitutions [3,4]:
B =

B
1
B
2

D
1•
=

D
11
D
12

¯
R =

D
T
11
D
11

−γ
2
ID
T
11
D
12
D
T
12
D
11
D
T
12
D
12

¯
S =B
¯
R
−1
B
T
¯
A = A −B
¯
R
−1

D
T
1•
C
1
¯
Q =C
T
1
C
1
−C
T
1
D
1•
¯
R
−1
D
T
1•
C
1
G =

G
1
G
2


=
¯
R
−1
(B
T
K +D
T
1•
C
1
)

R =I −
1
γ
2
D
T
11
D
11
¯
¯
R =D
21

R(γ)
−1

D
T
21
¯
¯
C =C
2
−D
21
G
1
+
1
γ
2
D
21

R
−1
D
T
11
D
12
G
2
¯
¯
S =

¯
¯
C
T
¯
¯
R
−1
¯
¯
C −
1
γ
2
G
T
2
D
T
12
D
12
G
2
−
1
γ
4
G
T

2
D
T
12
D
11

R
−1
D
T
11
D
12
G
2
¯
¯
A = A −B
1
G
1
+
1
γ
2
B
1

R

−1
D
T
11
D
12
G
2
−B
1

R
−1
D
T
21
¯
¯
R
−1
¯
¯
C
¯
¯
Q =B
1

R
−1

B
T
1
−B
1

R
−1
D
T
21
¯
¯
R
−1
D
21

R
−1
B
T
1
.
Remarks
The matrix K has the same dimension as A. The solution of the first algebraic matrix Riccati equation is
given below. The matrix G has the same partitioning as B
T
. The matrices
¯

Q and
¯
¯
Q will automatically be
positive-semidefinite. Remember: all of these matrices are functions of the frozen parameter vector θ.
∗
In order to prevent cumbersome notation, the dependence on the parameter vector θ is dropped in the development of
the results.