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Nonlinear Systems

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For a first-year graduate-level course on nonlinear systems. It may also be used for self-study or reference by engineers and applied mathematicians. The text 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. *NEW - Updated to include 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. Moreover, bifurcation is introduced in the context of second-order systems. *NEW - New section added in Chapter 1 - Provides an overview of the book and explains connections between the various analysis and control tools covered in the book. *NEW - The proof of the existence and uniqueness theorem has been moved to an appendix. Prevents students from dealing with the contraction mapping principle in such an early chapter. *NEW - Web page ( - Contains information about the book, detailed description of changes from previous editions, hints on how to organize courses around the textbook, corrections, additional exercises with or without solutions. *Self-contained chapters - Starting from Chapter 5, all the chapters are written to be self-contained or to use limited information from previous chapters, allows for greater flexibility.
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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.


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