Linear Systems Theory

ABOUT THE COURSE :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, design and serve as necessary foundation for further study in the area of systems and control. INTENDED AUDIENCE: Graduate Students from Electrical/ Mechanical/ Aerospace / Chemical Engineering PREREQUISITES: 1) Linear Algebra 2) Differential Equations 3) Control Systems Engineering

Week 1: Introduction to Linear systems with Examples Week 2: Math Preliminaries I - Vector Spaces, Bases, Coordinate Transformation, Invariant Subspaces, Inner product, Norms Week 3: Math Preliminaries II - Rank, Types of Matrices, Eigen values, Eigen vectors, Diagonalization, Matrix Factorization Week 4: State Transition Matrix, Solutions to LTI Systems, Solutions to LTV Systems Week 5: Equilibrium points, Linearization, Types of Linearization with Examples Week 6: Stability, Types of Stability, Lyapunov Equation Week 7: Controllability, Reachability, Stabilizability, Tests, Controllable and Reachable Subspaces, Grammians, Controllable Decomposition Week 8: Observability, Constructibility, Detectability, Tests, Subspaces, Grammians, State Estimation, Observable Decomposition Week 9: Kalman Decomposition, Pole Placement, Controller Design Week 10: Observer Design, Duality, Minimal Realization Week 11: Basics of Optimal Control,​ ​ LQR, Ricatti Equation Week 12: LMIs in Control