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Nonlinear Systems: Analysis, Stability and Control Outline

EECS 222 Spring 2007

Linear vs. Nonlinear Chapter 1 of textbook. 1. Nonlinear Phenomena: Multiple Equilibria, Limit Cycles, Complex Dynamics. 2. Simple Nonlinear Models Planar Dynamical Systems Chapter 2 of textbook. 1. Phase Plane Techniques 2. Limit Cycles Poincare Bendixson Theory 3. Multiple Equilibria Index Theory 4. Bifurcations Fold, Pitch Fork, Hopf, Saddle Connection Mathematical Preliminaries Chapter 3 of textbook. 1. Vector Spaces, Subspaces, Norms. 2. Contraction Mapping Theorem 3. Differential Equations, Vector Fields, 4. Relaxation Techniques for integration of differential equations 5. Degree Theory 6. Introduction to Differential Topology (optional with material from Chapter 7 of the book; may defer till later in the semester) Lyapunov Stability and Instability Chapter 5 and 6 of textbook 1

1. Definitions of Stability and Instability 2. Basic Stability Theorems 3. Basic Instability Theorems 4. Converse Lyapunov Theorems 5. Exponential Stability Theorems 6. Specialization to Linear Systems 7. Stabilization of nonlinear systems 8. Circle criterion, absolute stability, Popov criterion Linearization by State Feedback Chapter 9 and 10 of textbook 1. SISO systems: Input Output Linearization 2. SISO systems: Full State Linearization 3. Zero Dynamics 4. Inversion, tracking, stabilization 5. MIMO systems: linearization by static state feedback 6. Full state linearization of MIMO systems 7. Dynamic Extension 8. Sliding Mode Control and Robust Linearization 9. Nonlinear Observers Input Output Analysis and Stability (If time permits: Chapter 4 of textbook) 1. Definitions of Input - Output Stability 2. Small Gain Theorems 3. Passivity and passivity theorems. 4. Harmonic Balance and Describing Functions 5. Connections between Input - Output and State Space Stability Dynamical Systems and Bifurcations (If time permits: Chapter 7 of textbook). 2

1. Qualitative Theory 2. Linear and Nonlinear Maps 3. Closed Orbits, the Poincare map and forced oscillation 4. Structural Stability 5. The Center Manifold Theorem 6. Bifurcations

Instructor S. S. Sastry, 514 Cory and 284 HMMB. Phone Nos. 642-1857 and 643-2200. Office Hours M W 3-4 pm in 284 HMMB. Grading This is a very packed course: you will get as much from the course as you put into the supplementary readings. Homework sets will be due approximately every other week. It is OK to collaborate on the homework. The midterm is in the form of a take home problem set with no-consultation. The final is also a take home final with no consultation. 30 % of the grade will be based on the homework, 30 % on the midterm and 40 % based on the final. Text Book S. S. Sastry, Nonlinear Systems, Analysis, Stability and Control, Springer Verlag, 1999. Supplementary Reading Matter 1. M. Vidyasagar, Nonlinear Systems Analysis, Prentice Hall, 2nd Edition 1992. 2. A. Isidori, Nonlinear Control Systems, Springer Verlag, 3rd Edition, 1995. 3. H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer Verlag, 1990. 4. H. Khalil, Nonlinear Systems, Macmillan, 1992.

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