EE 585 – Linear Systems TheoryMihailo Jovanovic,
University of Southern California, Fall 2023
Course descriptionThe course objective is to equip students with the working knowledge of modern linear systems theory. We will establish a balance between statespace methods for analysis/synthesis of linear dynamical systems and frequency domain methods for studying inputoutput properties of multivariable linear systems. The course content will be motivated by examples from different application domains and it will be presented in such a way to make it of interest to students with background in control and dynamical systems, communications, signal and image processing, computer science and engineering, optimization, robotics, machine learning, artificial intelligence, power systems, systems biology, and financial engineering. TopicCourse mechanics; What is the course about? Basic system properties: linearity, time invariance, memory, causality; Statespace models; Equilibrium points; Linearization; Examples of electrical and mechanical systems; Solution to discrete time (DT) systems; State transition matrix; Z transform; Resolvent; Transfer function; Impulse and frequency responses of DT LTI systems; State transition matrix of continuous time (CT) systems; Variation of constants formula; Numerical computation of the state transition matrix; Matrix exponential; Laplace transform; Impulse response and transfer function of CT LTI systems; A doubleintegrator example; Eigenvalue decomposition; Diagonalization of a matrix; Jordan canonical form; Modal decomposition of LTI systems; Normal vs. nonnormal matrices; Modal conditions for stability of LTI systems; Stability of equilibrium points of nonlinear systems; Stability via linearization; Lyapunov functions for LTI systems; Algebraic Lyapunov Equation; Signal norms; System norms; Singular Value Decomposition; Frequency responses of LTI systems; Reachability of discrete time systems; Kalman rank test; Reachability gramian; Minimum energy state transfer; Reachability ellipsoid; Canonical form of unreachable systems; Modal tests for reachability; Popovâ€“Belevitchâ€“Hautus (PBH) reachability test; Controllability of continuous time systems; Observability; Observability gramian; Observability ellipsoid; Infinite horizon uncertainty ellipsoid; Balanced realization; Balanced truncation; Introduction to system identification; KalmanHo algorithm; Pole placement; State estimation; Kalman filter; Separation principle; Observerbased controller; Introduction to optimal control of linear systems; Linear Quadratic Regulator; Algebraic Riccati Equation.
Class schedule
TuTh, 10:00  11:50am, OHE 100B; Aug 22  Dec 1, 2023 Instructor and Teaching Assistant
Text and software
Grading policy
Homework is intended as a vehicle for learning, not as a test. Moderate collaboration with your classmates is encouraged. However, I urge you to invest enough time alone to understand each homework problem, and independently write the solutions that you turn in. Homework is generally handed out every other Thursday, and it is due at the beginning of the class a week later. Late homework will not be accepted. Start early!
