Nonlinear and Adaptive Control with Applications

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Volume 24 , Issue The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Read the full text. 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.

He would also like to thank his graduate student Xidong Tang for his editorial assistance on this project. Lewis Fort Worth, Texas. 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.


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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.

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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. 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. We would like to thank Professors Petros Ioannou and Frank Lewis for their interest and comments to this work.

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. T34 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.

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Introduction to model predictive control toolbox

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. These results, whose proofs are given in detail and are easy to understand, clarify several important signal and system properties for adaptive control.

Nonlinear and Adaptive Control with Applications | SpringerLink

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. 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 theorya priori plant knowledge, controller structure, plant--model matching, adaptive laws, stability, robustness, and robust adaptationare 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.

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.

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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. 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.