## [B1] Adaptive Control of Systems with Actuator and Sensor Nonlinearities

## [B2] Adaptive Control of Nonsmooth Dynamic Systems

## [B3] Control of Sandwich Nonlinear Systems

## [B4] Adaptive Control Design and Analysis

## [B5] Adaptive Control of Systems with Actuator Failures

## [B6] Advances in Control Systems Theory and Applications

## [B7] F. Y. Chen, G. Tao and B. Jiang,

Adaptive Control, Science Press, Beijing, 2015 (in Chinese).

Actuator and Sensor Nonlinearities

**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: New Models and Identification Methods for Backlash and Gear
Play, by M. Nordin, P. Bodin
and P.-O. Gutman
Chapter Two: Adaptive Dead Zone Inverses for Possibly Nonlinear Control
Systems, by E.-W. Bai
Chapter Three: Deadzone Compensation in
Motion Control Systems Using Augmented Multilayer Neural Networks, by R. R.
Selmic and F. L. Lewis
Chapter Four: On-line Fault Detection, Diagnosis, Isolation and
Accommodation of Dynamical Systems with Actuator Failures, by M. A. Demetriou and M. M. Polycarpou
Chapter Five: Adaptive Control of Systems with Actuator Failures, by G.
Tao and S. M. Joshi
Chapter Six: Multi-mode System Identification, by E. I. Verriest
Chapter Seven: On Feedback Control of Processes with ``Hard'' Nonlinearities,
by B. Friedland
Chapter Eight: Adaptive Friction Compensation for Servo Mechanisms, by
J. Wang, S. S. Ge and T. H. Lee
Chapter Nine: Relaxed Controls and a Class of Active Material Actuator
Models, by A. Kurdila
Chapter Ten: Robust Adaptive Control of Nonlinear Systems with Dynamic
Backlash-like Hysteresis, by C.-Y. Su, M. Oya and X.-K. Chen
Chapter Eleven: Adaptive Control of a Class of Time-delay Systems in the
Presence of Saturation, by A. M. Annaswamy, S. Evesque, S.-I. Niculescu and A.
P. Dowling
Chapter Twelve: Adaptive Control for Systems with Input Constraints: A
Survey, by J.-W. John Cheng and Y.-M. Wang
Chapter Thirteen: Robust Adaptive Control of Input Rate Constrained Discrete
Time Systems, by G. Feng
Chapter Fourteen: Adaptive Control of Linear Systems with Poles in the
Closed Left Half Plane with Constrained Inputs, by D. A. Suarez-Cerda and
R. Lozano
Chapter Fifteen: Adaptive Control with Input Saturation Constraints, by
C.-S. Zhang
Chapter Sixteen: Adaptive Control of Linear Systems with Unknown Time Delay,
by C.-Y. Wen, Y.-C. Soh and
Y. Zhang
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
Charlottesville, Virginia
Frank L. Lewis
Fort Worth, Texas **

**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.
**