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Nonlinear Dynamical Systems

Nonlinear Dynamical Systems. Instructors: Prof. Madhu N. Belur and Prof. Harish K. Pillai, Department of Electrical Engineering, IIT Bombay. This course covers basics of nonlinear differential equations that are encountered when dealing with practical dynamical systems in the context of their control. Topics covered include: introduction to nonlinear dynamical systems' features, existence and uniqueness of solutions, Lipschitz condition, linearization and local analysis, classification of equilibrium points (planar case), periodic orbits, Lyapunov theory, Lure problem, sector nonlinearity, Nyquist criterion, Lp stability, small gain theorem, passivity result, circle/Popov criteria, and describing function method. (from nptel.ac.in)

Lecture 19 - Definition for Positive Realness and Kalman-Yakubovich-Popov Theorem


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Lecture 01 - Introduction
Lecture 02 - First Order Systems
Lecture 03 - Classification of Equilibrium Points
Lecture 04 - Lipschitz Functions
Lecture 05 - Existence and Uniqueness Theorems
Lecture 06 - Existence and Uniqueness of Solutions to Differential Equations
Lecture 07 - Lyapunov Theorem on Stability
Lecture 08 - Extension of Lyapunov Theorem in Different Contexts
Lecture 09 - LaSalle's Invariance Principle, Barbashin and Krasovski Theorems, Periodic Orbits
Lecture 10 - Bendixson Criterion and Poincare-Bendixson Criterion
Lecture 11 - Bendixson and Poincare-Bendixson Criteria, van der Pol Oscillator
Lecture 12 - Scilab Simulation of Lotka-Volterra Predator Prey Model, van der Pol Oscillator
Lecture 13 - Signals and Norms of Operators
Lecture 14 - Norms of Signals, Systems (Operators), Finite Gain L2 Stable
Lecture 15 - Nyquist Plots and Nyquist Criterion for Stability
Lecture 16 - Interconnection between Linear System and Nonlinearity, Passive Filters
Lecture 17 - Passive Filters, Dissipation Equality, Positive Real Lemma
Lecture 18 - Positive Real Lemma Proof
Lecture 19 - Definition for Positive Realness and Kalman-Yakubovich-Popov Theorem
Lecture 20 - Kalman-Yakubovich-Popov Lemma/Theorem and Memoryless Nonlinearities
Lecture 21 - Loop Transformations and Circle Criterion
Lecture 22 - Nonlinearities based on Circle Criterion
Lecture 23 - Limit Cycles
Lecture 24 - Popov Criterion, Frequency Domain Theorem
Lecture 25 - Popov Criterion, Frequency Domain Theorem
Lecture 26 - Describing Function Method
Lecture 27 - Describing Function
Lecture 28 - Describing Functions: Optimal Gain
Lecture 29 - Describing Functions: Optimal Gain (cont.)
Lecture 30 - Describing Functions: Jump Hysteresis
Lecture 31 - Describing Functions: Sufficient Conditions for Existence of Periodic Orbits
Lecture 32 - Describing Functions for Nonlinearities
Lecture 33 - Ideal Relay with Hysteresis and Dead Zone
Lecture 34 - Dynamical Systems on Manifolds 1
Lecture 35 - Dynamical Systems on Manifolds 2