Nonlinear System Analysis

All systems are inherently nonlinear in nature. This course deals with the analysis of nonlinear systems. The need for special tools to analyze nonlinear systems arises from the fact that the principle of superposition on which linear analysis is based, fails in the nonlinear case. The course exposes the students to various tools to analyze the behaviour of nonlinear systems, culminating in the stability analysis, which is of paramount importance in control systems.INTENDED AUDIENCE :PhD, MS/M. Tech/ ME, Senior Undergraduate students from EE, ME, AE, PHPRE- REQUISITES : Undergrad control engineering, basic knowledge of differential equations and linear algebra is highly desirable.SUPPORT INDUSTRY : Any robotics, space and defence related industries

Week 1 : Why nonlinear systems? - Non-linear Models of Physical SystemsWeek 2 : Mathematical Preliminaries: Finite dimensional normed spaces, Euclidean space and its topologyWeek 3 : Infinite dimensional Banach spaces - Contraction mapping theoremWeek 4 : Existence and Uniqueness results for solutions to non linear ODEsWeek 5 : ODEs as vector fields - One dimensional systems - Phase portrait of second order linear systems - Equilibrium points, linearization and their classificationWeek 6 : Examples: Simple pendulum, Bead on a hoop, Lotka-Volterra models for predation and competition, biological transcriptional system, van der Pol oscillator and conservative systems, non linear circuits - Limit cyclesWeek 7 : Bifurcations of two dimensional flows: Saddle-node, pitchfork, transcritical and Hopf - their normal formsWeek 8 : Notions of stability - Lyapunov and LaSalle’s theoremsWeek 9 : Finding Lyapunov functions: Linear systems, variable gradient method - Center Manifold TheoremWeek 10 : Physical Non-linearities - Interconnections and feedback - Aizermann’s conjecture - PassivityWeek 11 : PR systems - Dissipation equality - Passive filtersWeek 12 : KYP Lemma - Popov and circle criterion