Linear Optimal Control

The last ten to fifteen years have seen a resurgence of interest in control systems designed to meet robustness and disturbance rejection specifications. Robust stability and robust performance tests, based on the structured singular value, have been developed. The robustness of the linear-quadratic regulator and the linear-quadratic Gaussian controllers have been analyzed. Loop transfer recovery was developed as a means to improve the robustness properties of the linear-quadratic Gaussian controller. control was developed to provide a means of incorporating frequency domain specifications in control system designs.

control also provides an ad hoc means of incorporating robustness specifications into control system designs. Finally, µ-synthesis was developed as a powerful procedure for designing control systems which satisfy robustness and performance specifications.

This book is being written to accomplish a number of tasks. First and foremost, I wish to make the new material on robustness, control, and µ-synthesis accessible to Master's level graduate students. Second, I wish to combine these new results with previous work on optimal control to form a more complete picture of control system design and analysis. Third, I wish to incorporate recent results on robust stability and robust performance analysis into the presentation of linear quadratic Gaussian optimal control. Lastly, I wish to acquaint the student with the CAD tools available for robust optimal controller design.

Special Features

This book has been written to provide Master's level students with access to relatively recent research results on robustness analysis, optimal control, and µ-synthesis. In addition, this material is integrated with linear quadratic Gaussian (H2) optimal control results. The overall treatment is organized in a logical manner, as opposed to being organized along the lines of historical development. A number of more specific features enhance the value of this book as a teaching text.

The results and derivations are simplified by treating special cases whenever this can be done without compromising the student's understanding of the results and methods. In addition, mathematical developments that provide little insight into key derivations, results and/or applications are relegated to the appendix. This approach allows the student to develop a solid grounding in the basics before tackling the mathematical subtleties required to derive the most general results. The practicing engineer can then augment his understanding using more advanced books and research papers, or use computer aided design software to handle the more general cases. While this approach does simplify the derivations, the mathematical level of this text is still quite challenging for most students.

The solutions of both the H2 (linear quadratic Gaussian) control problem and the control problem are based on a common variational approach. The variational approach adds more insight into the optimization process than completing the square. Using a common variational approach in both of these problems also tends to demystify the theory.

The use of computer aided design tools is integrated into the presentation and problems. Matlab with the Control System Toolbox and the µ-synthesis and Analysis Toolbox is the CAD software employed. Software is included for almost every example in the text. For examples which are done analytically, software is included for numerically checking the result. These software programs, and the documentation of these programs, provides a significant learning resource (and also a significant reference source) for the student since virtually all optimal controller design and analysis is performed with the aid of CAD software.

A general treatment of performance including transient performance, tracking performance, and disturbance rejection is given up front along with a treatment of robustness. This organization provides a solid foundation in control system analysis. A thorough performance and robustness analysis can then be performed on the controllers developed subsequently. This approach provides the student with an understanding of what each design does well and what each design does poorly, as opposed to being satisfied that the design is optimal.

Tracking and disturbance rejection are presented in the linear quadratic Gaussian setting, as opposed to the linear quadratic regulator setting. Presenting this material in the linear quadratic regulator setting leaves many students unsure how to incorporate estimation within a tracking system design or within a design tailored for disturbance rejection. The given organization solves this problem by treating the idiosyncrasies involving estimation in these systems.

The book concludes with a case study that compares a design obtained via linear quadratic Gaussian-loop transfer recovery with a design obtained via µ-synthesis. The insight gained through this comparison yields a better understanding of both design methodologies, and provides guidelines on when to apply which design method.

Computer exercises are included for each chapter. These computer exercises familiarize the student with current CAD software. Further, these exercises allow the student to explore design options and "what if" questions concerning the results. My students have frequently told me that a significant part of their learning comes as a result of performing the computer exercises.

Supplemental material consists of a solution manual available to instructors, and a disk that contains the software used for the examples.

Prerequisites

Prerequisites include an introduction to control systems (classical control), probability, state space linear systems, and a working knowledge of linear algebra. In addition, an introduction to random processes is desirable.

Classical control, probability and some linear algebra are part of the undergraduate education of most incoming engineering graduate students. An introduction to state space linear systems is typically accomplished (along with converting the students linear algebra background into a working knowledge) by an introductory graduate course from a book like Chen or Kailath. In addition, this book begins with a review of the relevant state space linear systems material in a multivariable setting (Chapter 2). I usually only present the sections on the singular value decomposition, the principle gains, and internal stability since this material is new to most of my students. But, I recommend that my students skim the remainder of Chapter 2 as a review. I then ask them to inform me of any topics of which they are not familiar, and provide references, when necessary, to bring the students up to speed.

The book contains a terse, but self contained, introduction to random processes (Chapter 3). This chapter also contains many state-space random process results which are not typically included in introductory random process courses. As such, I usually cover this chapter thoroughly.

Organization

The book consists of three parts. The first part covers the analysis of control systems. This part contains a review of multivariable linear systems (Chapter 2), an introduction to vector random processes (Chapter 3), control system performance analysis (Chapter 4), and robustness analysis (Chapter 5). The performance analysis chapter includes transient performance analysis, tracking system analysis, and disturbance rejection. Cost functions are also presented as a means of quantifying performance analysis.

The robustness analysis chapter begins with a review of the Nyquist stability criterion. The Nyquist plot is used to develop the gain margin, the phase margin, and the downside gain margin. The stability robustness interpretation of these classical control stability margins is clearly illustrated. The small gain theorem is presented as a means of determining stability robustness to unstructured perturbations. The structured singular value is then presented as a means of determining both stability and performance robustness to more general structured perturbations.

The second part of this book is devoted to H2, i. e., linear quadratic Gaussian, optimal control. This part is divided into the linear quadratic regulator (Chapter 6), Kalman filtering (Chapter 7), and linear quadratic Gaussian control (Chapter 8).

The chapter on the linear quadratic regulator (LQR) begins with a brief introduction to optimization using variational theory. The results are then used to derive the LQR, both time-varying and steady-state. Application of the LQR is discussed along with cost function selection. Performance and robustness of the steady-state linear quadratic regulator are evaluated in some detail.

The chapter on Kalman filtering begins with an introduction to minimum mean square estimation theory and the orthogonality principle. The Kalman filter, both time-varying and steady-state, is then developed. Application of the Kalman filter is discussed in some detail. Kalman filter performance and robustness are also discussed.

The chapter on linear quadratic Gaussian (LQG) control begins with the development of the stochastic separation principle leading to the structure of the LQG controller. Performance and robustness of the LQG control system are discussed. Loop transfer recovery is presented as a means of increasing LQG robustness when needed. Tailoring the LQG control system for tracking and disturbance rejection is also discussed in this chapter.

The last part of this book is devoted to control. Chapter 9 begins with an introduction to differential games. Differential game theory is used to derive the solution of the suboptimal full information controller. The output estimator is then derived using duality.

The output feedback controller is presented in Chapter 10. The application of this controller to tracking systems, disturbance rejection, and robustness optimization is discussed in detail. The D-K iteration algorithm for µ-synthesis is then presented. A significant case study is presented as a means of contrasting the µ-synthesis and the LQG-loop transfer recovery design methodologies.

The generation of reduced order controllers is presented in the final chapter. This chapter begins by showing that reduced order controller approximation can often be evaluated using a frequency weighted . The general properties of a desirable reduced order approximation are then gleaned from a few examples. Pole-zero truncation and balance truncation are both presented as methods of generating reduced order controllers.

Usage

This book is recommended for use as a text in a two semester Master's level sequence covering linear optimal control. The entire book can be covered thoroughly in two semesters. A two quarter course can also be formed from this material provided the students are well prepared. The two quarter course also necessitates that the material be covered in less depth.

A one semester course which covers robust optimal control can also be based on the material in this book. This course is composed of a review of Chapter 2 followed by a thorough treatment of Chapters 4, 5, 9, 10 and 11.

An additional one semester course on linear quadratic Gaussian control can be taught using this material. This course is composed of a review of Chapter 2, a thorough treatment of Chapter 3, the cost function norm presentation in Chapter 4, the material on unstructured perturbations and the small gain theorem in Chapter 5, and a thorough treatment of Chapters 6, 7 and 8.

Acknowledgments

I would like to express my appreciation to several people who contributed to this project. I am greatly indebted to Roberto Cristi for providing me with many useful suggestions associated with his use of a preliminary version of this text in classes at the Naval Postgraduate School. I would also like to thank all my students at Michigan Technological University for finding errors (big and small) and for letting me know when an explanation was unsatisfactory. I am especially indebted to two students, Willem Van Marian and John Pakkala, who used special diligence in finding errors in earlier versions of this manuscript. My appreciation is also extended to the reviewers for their valuable suggestions. Last, but not least, I wish to thank the editorial staff at Addison-Wesley, especially Royden Tonomura and Paul Becker, for their help in improving the manuscript and in bringing it to production.

Linear Optimal Control: H2 and Hà Methods is a reader-friendly book that features recent research results on robustness, Hà control, and ...m- synthesis. Linear Optimal Control combines these new results with previous work on optimal control to form a complete picture of control system design and analysis.
A comprehensive book, Linear Optimal Control covers the analysis of control systems, H2 (linear quadratic Gaussian), and Hà to a degree not found in many similar books. Its logical organization and its focus on establishing a solid grounding in the basics be fore tackling mathematical subtleties make Linear Optimal Control an ideal learning tool.

Back Cover

A comprehensive book, Linear Optimal Control covers the analysis of control systems, H2 (linear quadratic Gaussian), and to a degree not found in many texts. Its logical organization and its focus on establishing a solid grounding in the basics before tackling mathematical subtleties make Linear Optimal Control an ideal teaching text. The book's structure also makes it suitable for two-semester, one-semester, and two-quarter courses, as well as for professional use.

Features:

• Provides computer projects in each chapter to challenge readers with "what if" questions concerning the choice of cost functions and specifications.
• Simplifies results and derivations by treating special cases whenever possible without compromising the clarity of the results and methods.
• Familiarizes readers with the use of CAD tools for robust optimal controller design; software use is integrated throughout the book, and is included in almost every example.
• Offers a case study that compares the LQG and the µ-synthesis design methodologies and provides guidelines on when to apply which method.
• Presents LQG and control within a common framework— uses variational methods for both solutions.

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