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# Dynamical System and Control

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## Overview

This course ‘Dynamical systems and control’ is a basic course offered to PG students and final year UG students of Engineering/Science background. The objective of this course is to enhance the understanding of the theory, properties and applications of various dynamical and control systems. After completing the course one may be able to understand some of the important aspects of dynamical systems such as mathematical modeling, well posedness (existence, uniqueness and stability) of the considered problem. The participants will also be conversant with the controllability, stabilizability and optimal control aspects of a dynamical system.
Most Dynamical systems-physical, social, biological, engineering are often conveniently expressed (modeled) in the form of differential equations with or without control. Such mathematical models can provide an insight into the behavior of real life system if appropriate mathematical theory and techniques are applied. In this context this course has tremendous applications in diverse fields of engineering and technology.

PREREQUISITES: Basic concepts from Linear Algebra and Ordinary Differential Equations

## Syllabus

Week 1  :  Formulation of physical systems-I, Formulation of physical systems-II, Existence and uniqueness theorems-I, Existence and uniqueness theorems-II, Linear systems-I
Week 2  :   Linear Systems-II, Solution of linear systems-I, Solution of linear systems–II, Solution of linear systems-III, Fundamental Matrix-I
Week 3 :  Fundamental Matrix-II, Fundamental matrices for non- autonomous systems, Solution of non-homogeneous systems , Stability of systems: Equilibrium points, Stability of linear autonomous systems-I
Week 4  :  Stability of linear autonomous systems-II, Stability of linear autonomous systems-III, Stability of weakly non-  linear systems-I, Stability of weakly non-  linear systems-II, Stability of non-  linear systems using linearization
Week 5  :   Properties  of phase portrait, Properties  of orbits, Phase portrait : Types of critical points, Phase portrait of linear differential equations-I, Phase portrait of linear differential equations-II
Week 6  :   Phase portrait of linear differential equations-III, Poincare Bendixson Theorem, Limit cycle , Lyapunov stability-I, Lyapunov stability–II
Week 7 :   Introduction to Control Systems-I, Introduction to Control Systems-II, Controllability of Autonomous Systems, Controllability of Non-autonomous Systems, Observability-I Week 8  :  Observability-II,  Results on Controllability and Observability,  Companion Form,  Feedback Control-I, Feedback Control-II Week 9  :   Feedback Control-III, Feedback Control-IV,  State Observer, Stabilizability, Introduction to Discrete Systems-I
Week 10  :  Introduction to Discrete Systems-II,  Lyapunov Stability Theory-I,  Lyapunov Stability Theory-II,  Lyapunov Stability Theory-III, Optimal Control- I
Week 11  :   Optimal Control-II, Optimal Control-III, Optimal Control- IV, Optimal Control for Discrete Systems-I,  Optimal Control for Discrete Systems-II Week 12  :  Controllability of Discrete Systems, Observability of Discrete Systems, Stability for Discrete Systems, Relation between Continuous and Discrete Systems-I,  Relation between Continuous and Discrete Systems-II

#### Taught by

Prof. N. Sukavanam Prof. D.N. Pandey

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