**[8] R. Y. Qi, G. Tao and B.
Jiang, Fuzzy System Identification and Adaptive Control,
Springer, 2019.**

**[9] D. Deb, J. O.
Burkholder and G. Tao,
Adaptive Compensation of Nonlinear Actuators for Flight Control Applications,
Springer, 2022.**

**Gang
Tao and Petar Kokotovic
(published by John
Wiley & Sons, 1996; ISBN 0-471-15654-X; TJ217.T36 1996)**

**Imperfections
of system components, especially those of actuators and sensors, are
among the factors that severely limit the performance of feedback
control loops, the vital parts of industrial automation, consumer
electronics, and defense and transportation systems. Most often, a
critical imperfection is a nonlinearity which is poorly known,
increases with wear and tear, and varies from component to component.
Components without such imperfections are costly to manufacture, and
their maintenance usually requires specialized personnel. **

**It is
appealing to think of more intelligent approaches to increase the
accuracy achievable with imperfect, but sturdy and inexpensive
components. Can the control system, after a period of learning or
adaptation, recognize the imperfection and compensate for its harmful
effects? With such adaptive controllers, the component specifications
could be greatly relaxed, their cost reduced, and their reliability
increased. **

**This
book points to a direction in which this goal can be achieved for
some of the most common component imperfections: dead-zone, backlash,
and hysteresis. These ``hard'' nonlinearities are ubiquitous in a
wide variety of components: mechanical, hydraulic, pneumatic,
magnetic, piezoelectric, etc. They often serve as aggregate
representations of more complex microscopic phenomena: friction,
viscosity, elasticity, etc. While the ``hard'' nonlinearities have
all but disappeared from the academic texts, they have become more
common in engineering practice, because feedback controls have
entered many new areas of applications. In particular, control
systems have contributed to recent dramatic increases in fuel
efficiency, drivability, and safety of passenger cars. Such
successful applications show that it is more rational to improve
performance with control algorithms than with more expensive
mechanical components. The adaptive inverse methodology presented in
this book is aimed in this direction. **

**The
nonlinearities in this book are approximated by piecewise linear
characteristics. A difficulty with such characteristics is that they
have break-points, so that they are not differentiable. Existing
adaptive control techniques are not applicable to such
nonlinearities. However, a major advantage of the piecewise linear
characteristics is that they admit linear parametrization with
unknown break-point and slope parameters. This property is crucial
for effective design and implementation of robust adaptive control,
one of the main subjects of this book. The unifying theme of the book
is its adaptive inverse approach. Not only are the nonlinear
characteristics linear in their parameters, but so are their
inverses, which, in the case of dead-zone and backlash, are
discontinuous. While the inverses of the actuator nonlinearities are
explicit, those of the sensors have a more complicated implicit form.
The essence of the adaptive inverse approach is that, upon an
adaptation transient, the inverse cancels the effects of the unknown
nonlinear characteristic. In this way a significant improvement of
accuracy and performance is achieved with inexpensive components. In
other words, the adaptation in the controller has ``removed'' the
imperfection of the component. **

**All
the results in this book are new and have evolved from the recent
journal papers of the authors. The style of presentation is aimed at
an audience of practicing engineers and graduate students in
electrical, mechanical, chemical, aeronautical, and computer
engineering departments, as well as those pursuing interdisciplinary
studies such as biomedical engineering. The assumed background is a
standard course in control theory, while the required knowledge of
model reference adaptive control is concisely presented in Appendix
A. **

**Our
interest in the problem of adaptive compensation of ``hard''
nonlinearities was ignited by Jim Winkelman and Doug Rhode, our
colleagues at Ford Motor Company. Several years ago, they presented
to us and Darrel Recker (then a Ph.D. student, now a researcher at
Ford) a problem with a hydraulic valve dead-zone in an automotive
suspension system. The dead-zone's purpose was to prevent the leakage
and maintain the height when the car was parked and the engine was
turned off. However, when the suspension was active, the effect of
the dead-zone was harmful. In his Ph.D. thesis, Darrel Recker
addressed the problem of using adaptation to remove the harmful
effects of the dead-zone. His successful algorithms and experiments
have encouraged us to pursue a broader investigation in this
direction. We acknowledge with gratitude the pioneering contributions
of Darrel Recker and his cooperation in this project. We also greatly
benefited from the experience of Doug Rhode and Jim Winkelman. For
our understanding of hydraulic components we are indebted to Vladimir
Kokotovic, also at Ford. For many years we have been inspired and
helped by Petros Ioannou, University of Southern California, without
whose vast knowledge of robust adaptive control a project like this
would not have been possible. With their patience and understanding
our wives, Lanlin and Anna, generously contributed to the writing of
this book. **

**Our
research summarized in this book was not only initiated, but also
financially supported by, the Ford Motor Company. It was also
supported by the National Science Foundation grant ECS-9203491 and
RIA ECS-9307545 and by the Air Force Office of Scientific Research
grant F-49620-92-J-0495.
Gang Tao
**

**Chapter
1 shows the evolution of the new adaptive inverse approach.
Chapter
2 explains the importance and relevance of the control problem with
nonsmooth nonlinearities.
The key component of the
proposed approach, the inverse, is introduced in Chapter 3, for an
actuator nonlinearity.
Control designs with a fixed
inverse, exact or detuned, continuous-time or discrete-time or
hybrid, are developed in Chapter 4 for systems with actuator
nonlinearities.
Like neither an exact inverse which needs
the nonlinearity knowledge nor a detuned inverse which results in a
compensation error, an adaptive inverse, introduced in Chapter 5, is
able to adaptively cancel the effects of an unknown nonlinearity.
With such an adaptive inverse, adaptive inverse
controllers are designed in Chapter 6 in continuous time and in
Chapter 7 in discrete time, for systems with actuator nonlinearities.
A sensor nonlinearity is more difficult to deal with, as
indicated in Chapter 8, where a more sophisticated inverse design is
also presented to achieve the desired output matching.
Chapter
9 develops adaptive inverse control designs for systems with sensor
nonlinearities.
With partial system knowledge, the order
of an adaptive control design can be reduced and the performance can
be improved, as shown in Chapter 10.
As a further
development of the adaptive inverse approach, Chapter 11 has the
desired inverse control designs for a class of sandwich nonlinear
systems, those with both actuator and sensor nonlinearities.
Appendix A summarizes the model reference adaptive
control theory in a unified and compact form for both the
continuous-time and discrete-time designs with new proofs of the
desired stability and tracking properties.
The
closed-loop signal boundedness with an adaptive inverse controller is
proved in Appendix B for the continuous-time case, in Appendix C for
the discrete-time case, and in Appendix D for sensor nonlinearity
cases.
Bibliography has the most important references, in
particular, the complete collection of the recent results, in the
related research areas.
Finally, Index helps locating
many new concept items used throughout the book. **

**Gang
Tao and Frank F. Lewis, Eds.
(published by Springer,
2001; ISBN 1-85233-384-7; TJ217.A319 2001)**

**Nonsmooth**
**nonlinearities such as
backlash, dead-zone, component failure, friction, hysteresis,
saturation and time delays are common in industrial control systems.
Such nonlinearities are usually poorly known and may vary with time,
and they often limit system performance. Control of systems with
nonsmooth nonlinearities is an important area of control systems
research. A desirable control design approach for such systems should
be able to accommodate system uncertainties. Adaptive methods for the
control of systems with unknown nonsmooth nonlinearities are
particularly attractive in many applications because adaptive control
designs are able to provide adaptation mechanisms to adjust
controller parameters in the presence of parametric, structural and
environmental uncertainties. Most adaptive or nonlinear control
techniques reported in the literature are for linear systems or for
some classes of systems with smooth nonlinearities, but not for
nonsmooth nonlinearities. The need for effective control methods to
deal with nonsmooth nonlinear systems has motivated growing research
activities in adaptive control of systems with such common practical
nonsmooth nonlinearities. Recently, there have been many encouraging
new results on adaptive control problems with backlash, dead-zone,
failures, friction, hysteresis, saturation, and time delays. This
book, entitled ****Adaptive
Control of Nonsmooth Dynamic Systems ****,**
**is aimed at reflecting the
state of the art in designing, analyzing and implementing adaptive
control methods which are able to accommodate uncertain nonsmooth
nonlinearities in industrial control systems.
Backlash,
dead-zone, component failure, friction, hysteresis, saturation, and
time delays are the most common nonsmooth nonlinearities in
industrial control systems. Backlash, a dynamic (with memory)
characteristic, exists in mechanical couplings such as gear trains,
and always limits the accuracy of servo-mechanisms. Dead-zone is a
static input-output relationship which for a range of input values
gives no output; it also limits system performance. Dead-zone
characteristics are often present in amplifiers, motors, hydraulic
valves and even in biomedical actuation systems. Failures of
different types in actuators, sensors and other components of a
control system can cause major system performance deterioration.
Friction exists wherever there is motion or tendency for motion
between two physical components. Friction can cause a steady-state
error or a limit cycle near the reference position and stick-slip
phenomenon at low speed in the conventional linear control of
positioning systems.
Hysteresis, another dynamic
characteristic, exists in electromagnetic and piezoelectric actuators
which are used for micromotion control and high-accuracy positioning.
Saturation is always a potential problem for actuators of control
systems---all actuators do saturate at some level. Actuator
saturation affects the transient performance and even leads to system
instability. Time delays are also important factors to deal with in
order to improve control system performance such as for
teleoperations and in real-time computer control systems.
Although
backlash, dead-zone, failure, friction, hysteresis, saturation, and
time delay characteristics are different, they are all nonsmooth in
nature. Therefore, most existing adaptive control methods are not
applicable. Unfortunately these nonlinearities can severely limit the
performance of feedback systems if not compensated properly.
Moreover, adaptive control of dynamic systems with each of these
nonsmooth characteristics is a control problem that needs a
systematic treatment. It makes the control problem even more
challenging when there are more than one nonlinear characteristic
present in the control system.
In this book it will be
shown how nonsmooth nonlinear industrial characteristics can be
adaptively compensated and how desired system performance is achieved
in the presence of such nonlinearities. The book has 16 chapters on
issues including system modeling, control design, analysis of
stability and robustness, simulation and implementation:
Chapter
One: **

Chapter Two:

Chapter Three:

Chapter Four:

Chapter Five:

Chapter Six:

Chapter Seven:

Chapter Eight:

Chapter Nine:

Chapter Ten:

Chapter Eleven:

Chapter Twelve:

Chapter Thirteen:

Chapter Fourteen:

Chapter Fifteen:

Chapter Sixteen:

The authors of the chapters in this book are experts in their areas of interest and their chapters present new solutions to important issues in adaptive control of industrial systems with nonsmooth nonlinearities such as backlash, dead-zone, failure, friction, hysteresis, saturation, and time delay. These solutions result from recent work in these areas and are believed to be attractive to people from both academia and industry. Adaptive control of nonsmooth dynamical systems is theoretically challenging and practically important. This book is the first book on adaptive control of such systems, addressing all these nonsmooth nonlinear characteristics: backlash, dead-zone, failure, friction, hysteresis, saturation and time delays. Such a book is also aimed at motivating more research activities in the important field of adaptive control of nonsmooth nonlinear industrial systems.

Recent advances in adaptive control of nonsmooth dynamic systems have shown that those practical nonsmooth nonlinear characteristics such as backlash, dead-zone, component failure, friction, hysteresis, saturation and time delays can be adaptively compensated when their parameters are uncertain, as is common in real-life control systems. Rigorous designs have been given for selecting desirable controller structures to meet the control objectives and for deriving suitable algorithms to tune the controller parameters for control of systems with uncertainties in dynamics and nonsmooth nonlinearities. There have been increasing interest and activities in these areas of research, as evidenced by recent conference invited sessions and journal special issues on related topics. It is clear that this is a promising direction of research and there have been many encouraging results. Given the practical importance and theoretical significance of such research, it is time to summarize, unify, and develop advanced techniques for adaptive control of nonsmooth dynamic systems.

Since this book is about some important and new areas of adaptive control research, its contents are intended for people from both academia and industry, including professors, researchers, graduate students who will use this book for research and advanced study, and engineers who are concerned with the fast and precision control of motion systems with imperfections (such as backlash, dead-zone, component failure, friction, hysteresis, saturation and time delays) in mechanical connections, hydraulic servovalves, piezoelectric translators, and electric servomotors, and biomedical actuators systems. The book can be useful for people from aeronautical, biomedical, civil, chemical, electrical, industrial, mechanical and systems engineering, who are working on aircraft flight control, automobile control, high performance robots, materials growth process control, precision motor control, radar and weapons system pointing platforms, VLSI assembly. The adaptive system theory developed in this book is also of interest to people who work on communication systems, signal processing, real-time computer system modeling and control, biosystem modeling and control.

The first editor would like to gratefully acknowledge the partial support from National Science Foundation under grant ECS-9619363 and National Aeronautics and Space Administration under grant NCC-1342 to this project. He would also like to thank his graduate student Xidong Tang for his editorial assistance on this project. The second editor acknowledges the vital support of the Army Research Office under grant DAAD19-99-1-0137.

Gang Tao

Frank L. Lewis

**Avinash**
**Taware and Gang Tao
(published by Springer,
2003; ISBN 3-540-44115-8; TA 342.T43)**

**The
control problem: control of sandwich nonlinear dynamic systems is
addressed in this monograph. Of interest are sandwiched nonsmooth
nonlinearities such as dead-zone, hysteresis and backlash between
dynamic blocks. Some continuous-time control designs are proposed. A
framework for hybrid control is developed to design control schemes
for different cases of the control problem with required
modifications. Friction compensation is addressed for systems with
sandwiched friction along with sandwiched dynamics. The problem of
control of sandwich nonlinear systems with uncertain actuator
failures is introduced, and an adaptive control solution scheme is
developed for this problem. An optimal and nonlinear control solution
is proposed for control of multi-body, multi-input and multi-output
nonlinear systems with joint backlash, flexibility and damping.
The
proposed hybrid control framework employs an inner-loop discrete-time
feedback design and an outer-loop continuous-time feedback design,
combined with a nonlinearity inverse to cancel the nonlinearity
effect, for improving output tracking. The first control design using
this framework is a nominal one with an exact nonlinearity inverse,
which establishes a basic solution to the stated control problem. The
second design is an adaptive one which employs an adaptive inverse to
cancel the unknown sandwiched nonlinearity effect for improving
output tracking. The third one is also an adaptive one using the
framework with a neural network based inverse compensator. The
adaptive inverse is updated from an adaptive law. The neural network
based nonlinearity compensator consists of two neural networks, one
used as an estimator of the sandwiched nonlinearity function and the
other for the compensation itself. The compensator neural network has
neurons that can approximate jump functions such as a dead-zone
inverse. The weights of the two neural networks are tuned using a
modified gradient algorithm. For an adaptive inverse or neural
network based inverse, a control error equation is derived based on
which a desirable tracking error equation is obtained for an adaptive
update or tuning law design. Stability and tracking performance of
the closed-loop control system are analyzed. Simulations are used to
illustrate the effectiveness of the proposed hybrid control designs.
Friction compensation is addressed for a benchmark
sandwich system having sandwiched friction between linear dynamic
blocks as illustrated by a two-body system with load friction plus
joint flexibility and damping. Several non-adaptive and adaptive
compensation designs are analyzed, based on a Model Reference
Adaptive Control (MRAC) scheme that uses static state feedback for
control and dynamic output feedback for parameter adaptation to
achieve output tracking. When applied to the benchmark system,
necessary and sufficient output matching conditions are derived for
friction compensation. Approximate linear parametrizations of
nonlinear friction are developed for adaptive friction compensator
designs. The control problem for a sandwich nonlinear system with
friction sandwiched in between linear and nonlinear dynamics is also
addressed. Whenever load velocity is nonzero, adaptive linearizing
control is designed for such an unknown system with unknown friction.
This linearizing control has a contributing adaptive term that
compensates for the estimated friction. In the case the load velocity
is zero, a maximum-magnitude controller is employed to overcome
static friction effects. The proposed adaptive friction compensation
control schemes promise to bring considerable improvements to the
system performance.
Adaptive tracking control of sandwich
nonlinear systems with actuator failures is formulated and several
suitable control designs are developed, including an adaptive state
feedback control scheme to achieve state tracking, and an adaptive
output feedback controller for output tracking for linear
time-invariant plants with input actuator nonlinearities and
failures. Conditions and controller structures for achieving
plant-model state or output matching in the presence of actuator
failures and nonlinearities are presented. Adaptive laws are designed
for updating the controller parameters when both the plant
parameters, actuator nonlinearities and actuator failure parameters
are unknown. Adaptive inverse compensation is employed for the
unknown actuator nonlinearities. The effectiveness of the proposed
adaptive designs is illustrated with simulation results.
An
optimal and nonlinear solution scheme is proposed for control of
multi-body, multi-input and multi-output nonlinear systems with joint
backlash, flexibility and damping, represented by a gun turret-barrel
model which consists of two subsystems: two motors driving two loads
(turret and barrel) coupled by nonlinear dynamics. The key feature of
such systems is that the backlash is between two dynamic blocks.
Optimal control schemes are employed for backlash compensation and
nonlinear feedback control laws are used for control of nonlinear
dynamics. When one load is in contact phase and the other load is in
backlash phase, a feedback linearization design decouples the
multivariable nonlinear dynamics so that backlash compensation and
tracking control can be both achieved. Nonlinear zero dynamics
systems caused by joint damping are bounded-input, bounded state
stable so that feedback linearization control designs ensure that all
closed-loop signals are bounded and asymptotic tracking is
achievable.
We wish to gratefully acknowledge the
valuable help rendered by institutions and individuals in our
conducting the research presented in this book.
This
research was supported in part by the National Science Foundation
under grant ECS-9619363, by Techno Sciences Inc. under a US Army
subcontract grant, and by NASA Langley Research Center under grant
NCC-1342. We would like to thank their financial support that made
this research possible. We are also thankful to University of
Virginia for a pleasant and supportive environment to do our
research.
We would like to express our gratitude to
Professor Petar Kokotovic for his encouragement, help and support to
this research. We are grateful to Dr. Carole Teolis at
Techno-Sciences Inc. for her collaboration and help in conducting
this research. We would like to thank Professors Petros Ioannou and
Frank Lewis for their interest and comments to this work. We would
also like to thank Professors Zongli Lin, Steve Wilson and Jim Aylor
for their help to our research. We should mention that the research
results on adaptive actuator failure compensation by Shuhao Chen and
Xidong Tang, with the valuable help of Dr. Suresh Joshi of NASA
Langley Research Center, contributed to the framework used in Chapter
9 of this book for actuator failure compensation schemes for systems
with actuator nonlinearities. We would like to recognize the
contribution of Xiaoli Ma and Yi Ling to the work reported in Chapter
10 on control of nonlinear systems with joint backlash, flexibility
and damping (for which Dr. Kenan Ezal's work also inspired our
results), and the contribution of Nilesh Pradhan to the proposed
friction compensation designs in Chapters 7 and 8. We would also like
to express our appreciation for the helpful comments from anonymous
reviewers on this book and our related journal and conference papers
which laid down the foundation for this manuscript.
Finally,
we would like to thank our families for their love and support
without which this project would have never been possibly completed.
Avinash Taware
**

**Gang
Tao
(published by John
Wiley & Sons, 2003; ISBN 0-471-27452-6; TJ217.T34 2003)**

**Adaptive
control is becoming popular in many fields of engineering and science
as concepts of adaptive systems are becoming more attractive in
developing advanced applications. Adaptive control theory is a mature
branch of control theories, and there is a vast amount of literature
on design and analysis of various adaptive control systems using
rigorous methods based on different performance criteria. Adaptive
control faces many important challenges, especially in nontraditional
applications, such as real-time systems, which do not have precise
classical models admissible to existing control designs, or a
physiological system with an artificial heart, whose unknown
parameters may change at a heart beat rate which is also a controlled
variable. To meet the fast growth of adaptive control applications
and theory development, a systematic and unified understanding of
adaptive control theory is thus needed.
In an effort to
introduce such an adaptive control theory, this book presents and
analyzes some common and effective adaptive control design
approaches, including model reference adaptive control, adaptive pole
placement control, and adaptive backstepping control. The book
addresses both continuous-time and discrete-time adaptive control
designs and their analysis; deals with both single-input,
single-output and multi-input, multi-output systems; and employs both
state feedback and output feedback. Design and analysis of various
adaptive control systems are presented in a systematic and unified
framework. The book is a collection of lectures on system modeling
and stability, adaptive control formulation and design, stability and
robustness analysis, and adaptive system illustration and comparison,
aimed at reflecting the state of the art in adaptive control as well
as at presenting its fundamentals. It is a comprehensive book which
can be used as either an academic textbook or technical reference for
graduate students, researchers, engineers, and interested
undergraduate students in the fields of engineering, computer
science, applied mathematics and others, who have prerequisites in
linear systems and feedback control at the undergraduate level.
In
this self-contained book, basic concepts and fundamental principles
of adaptive control design and analysis are covered in 10 chapters.
As a graduate textbook, it is suitable for a one-semester course:
lectures plus reading may cover most of the book without missing
essential material. To help in understanding the topics, at the end
of each chapter, there are problems related to that chapter's
materials as well as technical discussions beyond the covered topics.
A separate manual containing solutions to most of these problems is
also available. At the end of most chapters, there are also some
advanced topics for further study in adaptive control.
Chapter
1 compares different areas of control theory, introduces some basic
concepts of adaptive control, and presents some simple adaptive
control systems, including direct and indirect adaptive control
systems in both continuous and discrete time, as well as an adaptive
backstepping control design for a nonlinear system in continuous
time.
Chapter 2 presents some fundamentals of dynamic
system theory, including system models, system characterizations,
signal measures, system stability theory (including Lyapunov
stability and input--output operator stability), signal convergence
lemmas, and operator norms. In particular, it gives a thorough study
of the Lyapunov direct method for stability analysis, some
time-varying feedback operator stability properties, several
important inequalities for system analysis, some detailed
input--output L^p stability results, various analytical L^p signal
convergence results, some simplified analytical tools for
discrete-time system stability, and multivariable operator norms.
These results, whose proofs are given in detail and are easy to
understand, clarify several important signal and system properties
for adaptive control.
Chapter 3 addresses adaptive
parameter estimation for a general linear model illustrated by a
parametrized linear time-invariant system in either continuous or
discrete time. Detailed design and analysis of a normalized gradient
algorithm and a normalized least-squares algorithm in either
continuous or discrete time are given, including structure,
stability, robustness, and convergence of the algorithms. A
collection of commonly used robust adaptive laws are presented which
ensure robust stability of the adaptive schemes in the presence of
modeling errors. An L^{1+alpha} (alpha >= 1) theory is developed
for adaptive parameter estimation for a linear model, revealing some
important inherent robustness properties of adaptive parameter
estimation algorithms.
Chapter 4 develops two types of
state feedback adaptive control schemes: for state tracking and for
output tracking (and its discrete-time version). For both continuous-
and discrete-time systems, adaptive state feedback for output
tracking control, based on a simple controller structure under
standard model reference adaptive control assumptions, is used as an
introduction to adaptive control of general linear systems. Adaptive
disturbance rejection under different conditions is addressed in
detail; in particular, adaptive output rejection of unmatched input
disturbance is developed based on a derived property of linear
systems. Another development is a derived parametrization of state
feedback using a full- or reduced-order state observer, leading to
the commonly used parametrized controller structures with output
feedback.
Chapter 5 deals with continuous-time model
reference adaptive control using output feedback for output tracking.
The key components of model reference adaptive control theory---a
priori plant knowledge, controller structure, plant--model matching,
adaptive laws, stability, robustness, and robust adaptation---are
addressed in a comprehensive formulation and, in particular,
stability and robustness analysis is given in a simplified framework.
The plant--model matching equation for a standard model reference
controller structure is studied in a tutorial formula. Design and
analysis of model reference adaptive control schemes are given for
plants with relative degree 1 or larger, using a Lyapunov or gradient
method based on a standard quadratic or nonquadratic cost function.
For the relative degree 1 case, an L^{1+alpha} (0 < alpha < 1)
adaptive control design is proposed for reducing output tracking
errors. An L^{1+alpha} (alpha > = 1) theory is developed for
adaptive control with inherent robustness with respect to certain
modeling errors. Robust adaptive control is formulated and solved in
a compact framework. Assumptions on plant unmodeled dynamics are
clarified, and robust adaptive laws are analyzed. Closed-loop signal
boundedness and mean tracking error properties are proved. To develop
adaptive control schemes without using the sign of the high frequency
gain of the controlled plant, a modified controller parametrization
leads to a framework of adaptive control using a Nussbaum gain for
stable parameter adaptation and closed-loop stability and asymptotic
output tracking.
Chapter 6 develops a model reference
adaptive control theory for discrete-time linear time-invariant
plants. A unique plant--model matching equation is derived, with
unique controller parameters specified to ensure exact output
tracking after a finite number of steps. A stable adaptive control
scheme is designed and analyzed which ensures closed-loop signal
boundedness and asymptotic output tracking. It is shown that the
model reference adaptive control system is robust with respect to L^2
modeling errors and with modification is also robust with respect to
L^{1+alpha} (alpha > 1) modeling errors. Thus an L^{1 + alpha}
(alpha > = 1) robustness theory is developed for discrete-time
adaptive control. Robust adaptive laws are derived for discrete-time
adaptive control in the presence of bounded disturbances.
Chapter
7 presents two typical designs (and their analysis) of indirect
adaptive control schemes: indirect model reference adaptive control
and indirect adaptive pole placement control in both continuous and
discrete time. Examples are used to illustrate the design procedures
and analysis methods. For indirect model reference adaptive control
in continuous or discrete time, a concise closed-loop error model is
derived based on which the proof of signal boundedness and asymptotic
output tracking is formed in a feedback and small-gain setting
similar to that for the direct model reference adaptive control
scheme of Chapters 5 and 6. For indirect adaptive pole placement
control, a singularity problem is addressed, and closed-loop
stability and output tracking are analyzed in a unified framework for
both continuous and discrete time. As a comparison, a direct adaptive
pole placement control scheme is presented and discussed for its
potential to avoid the singularity problem.
Chapter 8
conducts a comparison study of several adaptive control schemes
applied to a benchmark two-body system with joint flexibility and
damping, including direct state feedback, direct output feedback,
indirect output feedback, direct--indirect state feedback, and
backstepping state feedback designs, with detailed design and
analysis for the last two designs. With different complexity, they
all ensure closed-loop signal boundedness and asymptotic output
tracking. The design and analysis of the direct--indirect adaptive
control scheme demonstrate some typical time-varying operations on
signals in time-varying systems.
Chapter 9 first gives
the design and analysis of adaptive state feedback state tracking
control for multi-input systems. A multivariable state feedback
adaptive control scheme is derived using LDU decomposition of a plant
gain matrix. Multivariable adaptive control is applied to system
identification. This chapter then develops a unified theory for
robust model reference adaptive control of linear time-invariant
multi-input, multi-output systems in both continuous and discrete
time. Key issues such as a priori plant knowledge, plant and
controller parametrizations, design of adaptive laws, stability,
robustness, and performance are clarified and solved. In particular,
an error model for a coupled tracking error equation is derived, a
robust adaptive law for unmodeled dynamics is designed, a complete
stability and robustness analysis for a general multivariable case is
given, and a unified multivariable adaptive control theory is
established in a form applicable in both continuous and discrete
time. The chapter presents some recent results in reducing a priori
plant knowledge for multivariable model reference adaptive control
using LDU parametrizations of the high frequency gain matrix of the
controlled plant. Model reference adaptive control designs for
multivariable systems with input or output time delays are also
derived. Different adaptive control schemes, including a variable
structure design, a backstepping design, and a pole placement control
design for multivariable systems, are presented. Finally, robust
adaptive control theory is applied to adaptive control of robot
manipulator systems in the presence of parameter variations and
unmodeled dynamics.
Chapter 10 presents a general
adaptive inverse approach for control of plants with uncertain
nonsmooth actuator nonlinearities such as dead-zone, backlash,
hysteresis, and other piecewise-linear characteristics which are
common in control systems and often limit system performance. An
adaptive inverse is employed for cancelling the effect of an actuator
nonlinearity with unknown parameters, and a linear or nonlinear
feedback control law is used for controlling a linear or smooth
nonlinear dynamics following the actuator nonlinearity. This chapter
gives an overview of various state feedback and output feedback
control designs for linear, nonlinear, single-input and
single-output, and multi-input and multi-output plants as well as
open problems in this area of major theoretical and practical
relevance. A key problem is to develop linearly parametrized error
models suitable for developing adaptive laws to update the inverse
and feedback controller parameters, which is solved for various
considered cases. The chapter shows that control systems with
commonly used linear or nonlinear feedback controllers such as a
model reference, PID, pole placement, feedback linearization, or
backstepping can be combined with an adaptive inverse to handle
actuator nonlinearities.
The book is focused on
adaptive control of deterministic systems with uncertain parameters,
dynamics and disturbances. It can also be useful for understanding
the adaptive control algorithms for stochastic systems (see
references for ``Stochastic Systems'' in Section 1.4 for such
algorithms). The material presented has been used and refined in a
graduate course on adaptive control which I have taught for the past
ten years at the University of Virginia to engineering, computer
science, and applied mathematics students. Comments and modifications
to the book can be found at
http://www.people.virginia.edu/~gt9s/wiley-book.
If
used as a reference, this book can be followed in its chapter
sequence for both continuous- and discrete-time adaptive control
system design and analysis. The discrete-time contents are mainly in
Sections 1.5.3 (adaptive control system examples), 2.7 and 2.8
(systems and signals), 3.6 (adaptive parameter estimation), 3.7.2
(robustness of parameter estimation), 3.8.2 (robust parameter
estimation), 4.5 (state feedback adaptive control), Chapter 6 (model
reference adaptive control), Sections 7.3 (indirect model reference
adaptive control and adaptive pole placement control), 9.2
(multivariable model reference adaptive control), and 10.2--10.5
(adaptive actuator nonlinearity inverse control) (both in a unified
continuous- and discrete-time framework). The rest of the book is for
continuous-time adaptive control design and analysis.
If
used as a textbook for students with knowledge of linear control
systems, as a suggestion based on experience at the graduate level,
the instruction may start with Sections 1.4 and 1.5 as an
introduction to adaptive control (one or two lectures, 75 minutes
each). Some basic knowledge of systems, signals, and stability may be
taken from Sections 2.1--2.6 (system modeling, signal norms, Lyapunov
stability, Gronwall-Bellman lemma, small-gain lemma, strictly
positive realness and Lefschetz-Kalman-Yakubovich lemma, signal
convergence lemmas including Lemmas 2.14, 2.15, and 2.16 (Barbalat
lemma) for four or five lectures). Adaptive parameter estimation can
be taught using Sections 3.1--3.6 in four or five lectures, including
some reading assignments of robustness results from Sections 3.7 and
3.8. The design and analysis of adaptive control schemes with state
feedback are presented in Sections 4.1--4.4 (three lectures), while
the discrete-time results in Section 4.5 can be used as reading
materials. Continuous-time model reference adaptive control in
Chapter 5 can be covered in seven or eight lectures (Sections
5.1--5.5, with Section 5.6 as a reading assignment). Indirect
adaptive control in Chapter 7 may need four lectures. One lecture
plus reading is recommended for Chapter 8. Chapters 9 and 10 are for
advanced study as either extended reading or project assignments.
Further reading can be selected from the included extensive list of
references on adaptive systems and control.
In this
book, for a unified presentation of continuous- and discrete-time
adaptive control designs in either the time or frequency domain, the
notation y(t) = G(D)[u](t) (or y(D) = G(D)u(D)) represents, as the
case may be, the time-domain output at time t (or frequency-domain
output) of a dynamic system characterized by a dynamic operator (or
transfer function) G(D) with input u(tau), tau < = t (or u(D)),
where the symbol D is used, in the continuous-time case, as the
Laplace transform variable or the time differentiation operator
D[x](t) = dot{x}(t), t in [0, infty), or, in the discrete-time case,
as the z-transform variable or the time advance operator D[x](t) =
x(t + 1), t in {0, 1, 2, 3, ...}, with x(t) == x(tT) for a sampling
period T > 0.
Adaptive control as knowledge has
no limit and as theory is rigorous. Adaptive control is a field of
science. The universe is mysterious, diverse, and vigorous. The world
is complicated, uncertain, and unstable. Adaptive control deals with
complexity, uncertainty, and instability of dynamic systems. Taoist
philosophy emphasizes simplicity, balance, and harmony of the
universe. A goal of this book is to give a simplified, balanced, and
harmonious presentation of the fundamentals of adaptive control
theory, aimed at improving the understanding of adaptive control,
which, like other control methodologies, brings more simplicity,
balance, and harmony to the dynamic world.
This
book has benefited from many people's help. First, I am especially
grateful to Professors Petros Ioannou and Petar Kokotovic. I was
introduced to the field of adaptive control by Professor Ioannou, and
his continuous support and vigorous instruction were most helpful to
my study and research in adaptive control. Professor Kokotovic has
been a great mentor, and his persistent enthusiasm and continual
encouragement have been most valuable to me in the writing of this
book. Their robust adaptive control theory has been most influential
to my research in adaptive control.
I would like to
particularly acknowledge Professors Karl Astrom, Graham Goodwin, Bob
Narendra, and Shankar Sastry for their work on adaptive control,
which inspired me in research and in writing this book. I would like
to thank Professors Brian Anderson, Anu Annaswamy, Er-Wei Bai, Bob
Bitmead, Stephen Boyd, Marc Bodson, Carlos Canudas de Wit, Han-Fu
Chen, Aniruddha Datta, Michael Demetriou, Manuel De la Sen, Gang
Feng, Li-Chen Fu, Sam Shu-Zhi Ge, Lei Guo, Lui Hsu, Alberto Isidori,
Zhong-Ping Jiang, Dr. Ioannis Kanellakopoulos, Professor Hassan
Khalil, Dr. Bob Kosut, Professors Gerhard Kreisselmeier, P. R. Kumar,
Yoan Landau, Frank Lewis, Wei Lin, Lennart Ljung, Rogelio Lozano,
Iven Mareels, David Mayne, Rick Middleton, Steve Morse, Romeo Ortega,
Marios Polycapou, Laurent Praly, Drs. Darrel Recker, Doug Rhode,
Professors Gary Rosen, Jack Rugh, Ali Saberi, Mark Spong, Yu Tang, T.
J. Tarn, David Taylor, Chang-Yun Wen, John Ting-Yung Wen, and Erik
Ydstie, whose knowledge of adaptive systems and controls helped my
understanding of the field.
I especially thank Professors
Murat Arcak, Ramon Costa, Dr. Suresh Joshi, Professor Miroslav
Krstic, Dr. Jing Sun, and Professor Kostas Tsakalis for their
knowledge and comments, which helped me in writing this book.
I
am thankful to my graduate students Michael Baloh, Lori Brown, Jason
Burkholder, Shu-Hao Chen, Tinya Coles, Warren Dennis, Emin Faruk
Kececi, Yi Ling, Xiao-Li Ma, Raul Torres Muniz, Nilesh Pradhan, Gray
Roberson, Min-Yan Shi, Xi-Dong Tang, Avinash Taware, Ming Tian,
Timothy Waters, and Xue-Rui Zhang, and to computer scientists
Chen-Yang Lu and Ying Lu, and engineer Yi Wu, for their earnest
study, stimulating discussion, and interesting applications of
adaptive control.
I would also like to express my thanks
to my colleagues at the University of Virginia for their support, in
particular, to Professors Milton Adams, Paul Allaire, Jim Aylor,
Zong-Li Lin, Jack Stankovic, Steve Wilson, and Houston Wood, for
their collaboration and help in my teaching and research.
Finally,
I gratefully acknowledge that my study and research on adaptive
control, which led to many of the results in this book, were
supported by grants from the U.S. National Science Foundation and by
a scholarship from the Chinese Academy of Sciences.
Gang
Tao
**

**Gang
Tao, Shuhao Chen, Xidong Tang, Suresh M. Joshi
(published
by Springer, March 2004;
ISBN 1-85233-788-5)**

**Actuator
failures in control systems may cause severe system performance
deterioration and even lead to catastrophic closed-loop system
instability. For example, many aircraft accidents were caused by
operational failures in the control surfaces, such as rudder and
elevator. For system safety and reliability, such actuator failures
must be appropriately accommodated. Actuator failure compensation is
an important and challenging problem for control systems research
with both theoretical and practical significance.
Despite
substantial progress in the area of actuator failure compensation,
there are still many important open problems, in particular those
involving system uncertainties. The main difficulty is that the
actuator failures are uncertain in nature. Very often it is
impossible to predict in advance **

The first three authors wish to gratefully acknowledge the support by the NASA Langley Research Center to this work.

We are especially grateful to our families for their love and their support to our research work, which made this project possible.

Gang Tao, Shuhao Chen, Xidong Tang

Suresh M. Joshi

**Gang
Tao and Jing Sun (editors)
(published by USTC
Press, 2009)**

**Control
systems theory, as an interdisciplinary science that deals with basic
principles underlying the analysis and synthesis of interconnected
systems, has had an enormous impact on the development of basic
physical science, social economy, and advanced technology. Over the
last 50 years, the advancement in control theory and its applications
have played a crucial and prominent role to enable engineering
activities in improving social infrastructure, life quality, and
environment. Advanced theory for feedback control and other control
mechanisms provides foundation and new insights to other branches of
physical sciences such as communication, biomedical, and micro-nano
systems. New control design tools have helped to streamline the
system design and integration tasks for many industries, such as the
process and automotive industry, thereby leading to more effective
and robust products and processes. Widespread applications of
micro-processors, distributed actuators and sensors, and real-time
computing have further extended the domains of control application
and made feedback even more ubiquitous, covering macro systems such
as aircrafts, automobiles as well as micro entities like biology
cells and nano-devices.
While it is evident that control
theory has enabled many technological breakthroughs in aerospace,
automotive, biomedical and other fields, it is equally convincing
that new developments emerged in other fields have offered new
challenges and opportunities for control engineers and researchers.
It is this healthy cross-fertilization between the control theory and
its application domains that has propelled the immense progresses of
the control systems theory and led to the vast amount of scientific
and technical publications in the literature. The field is developing
and expanding rapidly with the stimulation of emerging challenges and
the encouragement of the promising solutions.
This book
presents a collection of diverse topics on some recent advances in
control systems theory and applications, contributed by the authors
who have enthusiastically and persistently worked in this exciting
field. Moreover, most of the authors are alumni of the University of
Science and Technology of China (USTC), who studied in their Alma
Mater during different time periods of her glorious 50 years. The
publication of this book is also intended to be a celebratory event
for the 50th anniversary of the founding of USTC, a commemoratory
testimony to those authors' Alma Mater for her dedication and
contributions to education and research.
**

**The
book consists of 15 chapters whose topics range from different areas
of control systems theory to various control applications: from
adaptive control, control of bifurcations, digital control, fault
tolerance control, H_infty control, learning control, neural and
fuzzy control, nonlinear control, optimization, parameter estimation,
predictive control, robust control, stochastic control, system
identification, variable structure control, to aircraft flight
control, building vibration control, computer control systems,
medical robots, portfolio management, robot formation and control,
and smart structures. The 15 chapters, with their titles and authors
(and their USTC class numbers), are summarized as follows.
Chapter
1: **

**On
the behalf of the USTC alumni authors of this book, we would like to
express our heartfelt gratitude to the teachers of our Alma Mater,
who, with their enthusiasm and dedication, led us to this fascinating
field and taught us the knowledge and skills that allowed us to
explore the subject in various directions presented in this book. Our
experience at our Alma Mater had been life enriching, and it shaped
our personal and professional life in numerous ways. This book is
specially edited and dedicated to our Alma Mater at her 50th
anniversary in the special year of 2008. We would also like to
express our appreciation to the contributions of other authors to
this book, for joining this effort and making this special edition
possible.
In addition, all the authors of this book would
like to thank our colleagues for their intellectual stimulation and
collaboration in our research, our students for their diligent and
conscientious effort and for being our continuous inspiration, and
our universities and our research sponsors for their support to our
professional duties and research activities.
**