# EE 585 – Linear Systems Theory

Mihailo Jovanovic, University of Southern California, Fall 2023

## Course description

The course objective is to equip students with the working knowledge of modern linear systems theory. We will establish a balance between state-space methods for analysis/synthesis of linear dynamical systems and frequency domain methods for studying input-output 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.

## Topic

Course mechanics; What is the course about? Basic system properties: linearity, time invariance, memory, causality; State-space 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 double-integrator example; Eigenvalue decomposition; Diagonalization of a matrix; Jordan canonical form; Modal decomposition of LTI systems; Normal vs. non-normal 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; Kalman-Ho algorithm; Pole placement; State estimation; Kalman filter; Separation principle; Observer-based 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

• Instructor
Mihailo Jovanovic
Office: EEB 344
Office hours: Tu noon - 1pm (or by appointment)

• Teaching Assistant
TBA
Office: TBA
Office hours: TBA

## Text and software

• Supplementary text
Joao P. Hespanha
Linear Systems Theory
Princeton University Press, First Edition, ISBN-10: 0-691-14021-9

• Software
Homework sets will make a use of Matlab