# Nonlinear Control

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Date: Sunday and Tuesday 7:30 – 9:00 (T+3:30 GMT Tehran Local Time) OR Saturday and Monday 23:00 – 00:30 (-5:00 GMT Canada Eastern Time Zone)

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###### Date: Sunday and Tuesday 9:00 – 10:00 (T+3:30 GMT Tehran Local Time) OR Saturday and Monday 00:30 – 01:30 (-5:00 GMT Canada Eastern Time Zone)

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### Tentative Course Timetable

 Time: Teaching Contents Week 1 Introduction: Nonlinear system representation, state-space equations, equilibrium point, multiple isolated equilibria, nonlinear system characteristics, limit cycle, bifurcation, chaos, nonlinear system examples, pendulum, tunnel-diode, mass-spring-damper, negative resistance, Van der Pol, Common nonlinearities such as a relay, dead-zone, saturation, hysteresis. Week 2 2nd Order Systems:  Phase portrait analysis; definitions, singular point, vector fields, phase portrait construction. Qualitative behaviour near equilibrium point; review of linear systems, nonlinear system multiple equilibria, linearization method at the vicinity of equilibria. Week 3 2nd Order Systems: Periodic orbit and limit cycle; definition, types, energy perspective, existence and non-existence theorems, examples. Bifurcation; definition, saddle-node, transcritical, pitchfork, Hopf, global and holonomic orbit bifurcation. Week 4 Lyapunov Stability of Autonomous Systems: Definitions, the concept of Lyapunov analysis, Lyapunov direct method, Lyapunov function, global stability and instability theorem. Week 5 Lyapunov Stability of Autonomous Systems: Invariant set theorems, Krasovskii-Lasalle’s theorem, local and global stability, the region of attraction, attractive limit cycle, linearization and Lyapunov indirect method, Lyapunov equation, Lyapunov function generation and Lyapunov based controller design. Week 6 Advanced Stability Analysis: Motivation example, definition, the notion of uniformity, Class K and KL functions. Non-autonomous Lyapunov analysis, Lyapunov analysis and Class K and KL functions, Lyapunov extensions. Week 7 Advanced Stability Analysis: LTV systems; frozen time conjecture, TV Lyapunov equation, Linearization theorems. Invariance-like theorems; Barbalat’s Lemma, uniformly continuous functions, extensions of invariance theorem. Week 8 Advanced Stability Analysis: Boundedness and ultimate bound, Lyapunov analysis, UUB stability;  Input-to-state stability; linear versus nonlinear input stability, local and global stability theorems. Week 9 Midterm Exam Week 10 Frequency Domain Analysis of Feedback systems:  Feedback system of a linear system and a nonlinear term, absolute stability, Circle and Popov Criteria. Describing function analysis; illustrating example, assumptions and definitions, computing describing functions, Review of Nyquist criterion, existence and stability of limit cycle, examples. Week 11 Feedback Linearization: Concepts, Differential geometry; gradient and jacobian, Lie algebra, involutivity and diffeomorphism. Input-output feedback linearization, relative degree. Week 12 Feedback Linearization: Input-output linearization; relative degree, normal form zero dynamics, non-minimum phase systems, state feedback control, regulation and tracking, Input-state feedback linearization, feasibility conditions, suitable diffeomorphism, state-feedback control. Week 13 Lyapunov-Based Controllers: Backstepping; design Idea, integrator, recursive and general backstepping, examples. Robust nonlinear control, motivating example, switching control, uncertainty description, stabilizing controller, regulation and tracking, chattering, and continuous control, UUB stability. Week 14 Lyapunov-Based Controllers:  Liapunov redesign, regular form, robustification and Lyapunov analysis, 2-norm robust controllers, infinity-norm robust controllers, examples. Week 15 Nonlinear Observers:  Local Observers; Linear observer, Kalman filters, extended Kalman filter; Global observer; certain nonlinear model, uncertain systems, High gain observers.

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##### Reference Materials:
 1 Applied Nonlinear Control, J.J. Slotine and W. Li, Prentice Hall, 1991. 2 Nonlinear Control Systems, A. Isidori, Springer Verlag, 1995. 3 Nonlinear System Analysis, M. Vidyasagar, Prentice-Hall, 1993. 4 Selected Papers
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