Nonlinear Systems by Hassan K. KhalilNonlinear Systems by Hassan K. Khalil

Nonlinear Systems

byHassan K. Khalil

Hardcover | December 18, 2001

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This book is written is such a way that the level of mathematical sophistication builds up from chapter to chapter. It has been reorganized into four parts: basic analysis, analysis of feedback systems, advanced analysis, and nonlinear feedback control. Updated content includes subjects which have proven useful in nonlinear control design in recent years—new in the 3rd edition are: expanded treatment of passivity and passivity-based control; integral control, high-gain feedback, recursive methods, optimal stabilizing control, control Lyapunov functions, and observers. For use as a self-study or reference guide by engineers and applied mathematicians.

This text is intended for a firstyear graduatelevel course on nonlinear systems or control. It may also be used for self study or reference by engineers and applied mathematicians. It is an outgrowth of my experience teaching the nonlinear systems course at Michigan State University, East Lansing. Students taking this course have had ...
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Title:Nonlinear SystemsFormat:HardcoverPublished:December 18, 2001Publisher:Pearson EducationLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0130673897

ISBN - 13:9780130673893

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From the Author

This text is intended for a firstyear graduatelevel course on nonlinear systems or control. It may also be used for self study or reference by engineers and applied mathematicians. It is an outgrowth of my experience teaching the nonlinear systems course at Michigan State University, East Lansing. Students taking this course have had background in electrical engineering, mechanical engineering, or applied mathematics. The prerequisite for the course is a graduatelevel course in linear systems, taught at the level of the texts by Antsaklis and Michel 9, Chen 35, Kailath 94), or Rugh 158. The linear systems prerequisite allowed me not to worry about introducing the concept of "state" and to refer freely to "transfer functions," "state transition matrices," and other linear system concepts. The mathematical background is the usual level of calculus, differential equations, and matrix theory that any graduate student in engineering or mathematics would have. In the Appendix, I have collected a few mathematical facts that are used throughout the book. I have written the text in such a way that the level of mathematical sophistication increases as we advance from chapter to chapter. This is why the second chapter is written in an elementary context. Actually, this chapter could be taught at senior, or even junior, level courses without difficulty. This is also the reason I have split the treatment of Lyapunov stability into two parts. In Sections 4.1 through 4.3, I introduce the essence of Lyapunov stability for autonomous systems where I do not have to worry about technicalities such as uniformity, class k functions, etc. In Sections 4.4 through 4.6, I present Lyapunov stability in a more general setup that accommodates nonautonomous systems and allows for a deeper look into advanced aspects of the stability theory. The level of mathematical sophistication at the end of Chapter 4 is the level to which I like to bring the students, so that they can comfortably read the rest of the text. There is yet a higher level of mathematical sophistication that is assumed in writing the proofs in the Appendix. These proofs are not intended for classroom use. They are included to make the text on one hand, self contained, and, on the other, to respond to the need or desire of some students to read such proofs, such as students continuing on to conduct Ph.D. research into nonlinear systems or control theory. Those students can continue to read the Appendix in a selfstudy manner. This third edition has been written with the following goals in mind: To make the book (especially the early chapters) more accessible to firstyear graduate students. As an example of the changes made toward that end, note the change in Chapter 3: All the material on mathematical background, the contraction mapping theorem, and the proof of the existence and uniqueness theorem have been moved to the Appendix. Several parts of the books have been rewritten to improve readability. To reorganize the book in such a way that makes it easier to structure nonlinear systems or control courses around it. In the new organization, the book has four parts, as shown in the flow chart. A course on nonlinear systems analysis will cover material from Parts 1, 2, and 3, while a course on nonlinear control will cover material from Parts 1, 2, and 4. To update the material of the book to include topics or results that have proven to be useful in nonlinear control design in recent years. New to the third addition are the: expanded treatment of passivity and passivitybased control, integral control, sliding mode control, and highgain observers. Moreover, bifurcation is introduced in the context of secondorder systems. On the technical side, the reader will find Kurzweil’s converse Lyapunov theorem, nonlocal results in Chapters 10 and 11, and new results on integral; control and gain scheduling. To update the exercises. More than 170 new exercises have been included.

Read from the Book

This text is intended for a first-year graduate-level course on nonlinear systems or control. It may also be used for self study or reference by engineers and applied mathematicians. It is an outgrowth of my experience teaching the nonlinear systems course at Michigan State University, East Lansing. Students taking this course have had background in electrical engineering, mechanical engineering, or applied mathematics. The prerequisite for the course is a graduate-level course in linear systems, taught at the level of the texts by Antsaklis and Michel 9, Chen 35, Kailath 94), or Rugh 158. The linear systems prerequisite allowed me not to worry about introducing the concept of "state" and to refer freely to "transfer functions," "state transition matrices," and other linear system concepts. The mathematical background is the usual level of calculus, differential equations, and matrix theory that any graduate student in engineering or mathematics would have. In the Appendix, I have collected a few mathematical facts that are used throughout the book. I have written the text in such a way that the level of mathematical sophistication increases as we advance from chapter to chapter. This is why the second chapter is written in an elementary context. Actually, this chapter could be taught at senior, or even junior, level courses without difficulty. This is also the reason I have split the treatment of Lyapunov stability into two parts. In Sections 4.1 through 4.3, I introduce the essence of Lyapunov stability for autonomous systems where I do not have to worry about technicalities such as uniformity, class k functions, etc. In Sections 4.4 through 4.6, I present Lyapunov stability in a more general setup that accommodates nonautonomous systems and allows for a deeper look into advanced aspects of the stability theory. The level of mathematical sophistication at the end of Chapter 4 is the level to which I like to bring the students, so that they can comfortably read the rest of the text. There is yet a higher level of mathematical sophistication that is assumed in writing the proofs in the Appendix. These proofs are not intended for classroom use. They are included to make the text on one hand, self contained, and, on the other, to respond to the need or desire of some students to read such proofs, such as students continuing on to conduct Ph.D. research into nonlinear systems or control theory. Those students can continue to read the Appendix in a self-study manner. This third edition has been written with the following goals in mind: To make the book (especially the early chapters) more accessible to first-year graduate students. As an example of the changes made toward that end, note the change in Chapter 3: All the material on mathematical background, the contraction mapping theorem, and the proof of the existence and uniqueness theorem have been moved to the Appendix. Several parts of the books have been rewritten to improve readability. To reorganize the book in such a way that makes it easier to structure nonlinear systems or control courses around it. In the new organization, the book has four parts, as shown in the flow chart. A course on nonlinear systems analysis will cover material from Parts 1, 2, and 3, while a course on nonlinear control will cover material from Parts 1, 2, and 4. To update the material of the book to include topics or results that have proven to be useful in nonlinear control design in recent years. New to the third addition are the: expanded treatment of passivity and passivity-based control, integral control, sliding mode control, and high-gain observers. Moreover, bifurcation is introduced in the context of second-order systems. On the technical side, the reader will find Kurzweil's converse Lyapunov theorem, nonlocal results in Chapters 10 and 11, and new results on integral; control and gain scheduling. To update the exercises. More than 170 new exercises have been included.

Table of Contents

All chapters conclude with Exercises.

1. Introduction.

Nonlinear Models and Nonlinear Phenomena. Examples.



2. Second-Order Systems.

Qualitative Behavior of Linear Systems. Multiple Equilibria. Qualitative Behavior Near Equilibrium Points. Limit Cycles. Numerical Construction of Phase Portraits. Existence of Periodic Orbits. Bifurcation. Systems.



3. Fundamental Properties.

Existence and Uniqueness. Continuos Dependence on Initial Conditions and Parameters. Differentiability of solutions and Sensitivity Equations. Comparison Principle.



4. Lyapunov Stability.

Autonomous Systems. The Invariance Principle. Linear Systems and Linearization. Comparison Functions. Nonautonomous Systems. Linear Time-Varying Systems and Linearization. Converse Theorems. Boundedness and Ultimate Boundedness. Input-to-State Stability.



5. Input-Output Stability.

L Stability. L Stability of State Models. L2 Gain. Feedback Systems: The Small-Gain Theorem.



6. Passivity.

Memoryless Functions. State Models. Positive Real Transfer Functions. L2 and Lyapunov Stability. Feedback Systems: Passivity Theorems.



7. Frequency-Domain Analysis of Feedback Systems.

Absolute Stability. The Describing Function Method.



8. Advanced Stability Analysis.

The Center Manifold Theorem. Region of Attraction. Invariance-like Theorems. Stability of Periodic Solutions.



9. Stability of Perturbed Systems.

Vanishing Pertubation. Nonvanishing Pertubation. Comparison Method. Continuity of Solutions on the Infinite Level. Interconnected Systems. Slowly Varying Systems.



10. Perturbation Theory and Averaging.

The Perturbation Method. Perturbation on the Infinite Level. Periodic Perturbation of Autonomous Systems. Averaging. Weekly Nonlinear Second-Order Oscillators. General Averaging.



11. Singular Perturbations.

The Standard Singular Perturbation Model. Time-Scale Properties of the Standard Model. Singular Perturbation on the Infinite Interval. Slow and Fast Manifolds. Stability Analysis.



12. Feedback Control.

Control Problems. Stabilization via Linearization. Integral Control. Integral Control via Linearization. Gain Scheduling.



13. Feedback Linearization.

Motivation. Input-Output Linearization. Full-State Linearization. State Feedback Control.



14. Nonlinear Design Tools.

Sliding Mode Control. Lyapunov Redesign. Backstepping. Passivity-Based Control. High-Gain Observers.



Appendix A. Mathematical Review.


Appendix B. Contraction Mapping.


Appendix C. Proofs.


Notes and References.


Bibliography.


Symbols.


Index.