Linear System Theory

COURSE OUTLINE: This course will provide a thorough introduction to the theory of Linear Systems with emphasis on Control related concepts. First, mathematical models describing the fundamental properties that govern the behavior of systems will be developed. We will cover time invariant, time varying, continuous and discrete time systems. This course will cover concepts of stability, controllability, observability, and design and serve as a necessary foundation for further study in the area of systems and control.

System Models - Part 01.
Linear Algebra - Span, Basis and Subspaces.
Introduction to Linear Systems.
System Models - Part 02.
System Models - Part 03.
General Representation.
Sets, Functions and Fields.
Linear Algebra - Vector Spaces and Metric Spaces.
Linear Algebra - Linear Maps and Matrices.
Tutorial 1 on Linear Algebra.
Linear Algebra - Diagonalization and Jordan Forms.
Linear Algebra - Eigen Decomposition and Singular Value Decomposition.
Tutorial 2 on Linear Algebra.
Linear Algebra - Change of Basis and Similarity Transformation.
Linear Algebra - Invariant Subspaces, Eigen Values & Eigen Vectors.
Linear Algebra - Fundamental Subspaces and Rank-Nullity.
Solutions to LTI Systems.
State Transition Matrix for LTI systems.
Forced Reponse of Continuous and Discrete LTI system.
State Transition Matrix and Solutions to LTV systems.
mod05lec21-Equilibrium Points.
mod05lec22-Limit Cycles and Linearization.
Supplementary Lecture: Comparison Lemma and Lyapunov Stabilty.
Stability of Discrete Time Systems.
Lyapunov Stability.
Stability Analysis & Types of Stability.
Stabilizability.
Controllable Decomposition.
Controllability of Discrete Time Systems.
Controllability Tests.
Controllability Matrix and Controllable Systems.
Controllability and Reachability.
Observability for Discrete Time Systems and Observability Tests.
Gramians and Duality.
Observability.
Observable Decompositon and Detectability.
Tutorial for Modules 9 and 10.
Control Design using Pole Placement.
Canonical Forms and State Feedback Control.
Kalman Decomposition and Minimal Realisation.
mod10lec42-Design of Observer and Observer based Controller.
mod10lec41-State Estimation and Output Feedback.
mod11lec45-Tutorial for Module 11.
mod11lec44-Feedback Invariant and Algebraic Ricatti Equation.
mod11lec43-Optimal Control and Linear Quadratic Regulator (LQR).
mod12lec46-Linear Matrix Inequalities.
mod12lec47-Properties of LMIs and Delay LMIs.