Hybrid systems

Hybrid systems are often necessary when the simplifications introduced by continuous models mask effects that are worthy of attention. This lecture introduces fundamental concepts and tools for the study of dynamical systems that combine both continuous and discrete entities (differential equations and discrete events).

Instructor: Jorge Finke

Office hours: TBD

Level: graduate

TA: TBD

OVERVIEW

Week Lectures
1 Lec 1 - Intro to hybrid systems
2 Lec 2 - Hybrid automata
3 Lec 3 - Classification of hybrid time sets
4 Lec 4 - Existence of hybrid solutions
5 Lec 5 - Stability of continuos-time systems (review)
6 Lec 6 - Stability of switched systems
7 Lec 7 - Stability of hybrid systems
8 Lec 8 - Midterm review
9 Lec 9 - Identification of LTI systems (review)
10 Lec 10 - Least squares
11 Lec 11 - Discrete Kalman filter
12 Lec 12 - Identification of hybrid systems
13 Lec 13 - Hybrid control design
14 Lec 14 - feedback stabilization
15 Project presentations

Assignments

To access the problem sets and lab assignments please use the password provided in class. 

 

Homework sets

HW1 – due September 9th

HW2 – due October 3rd 

Project

Project 1 – due November 4th

Project 2 – due November 25th

Notebooks (how to..)

 

 

Lessons

List of lectures

1. Introduction to hybrid systems

Dynamics of love! Stable? No! Why? Continuos + discrete dynamics; other examples of hybrid systems; notation; classification of dynamical systems; examples.

2. Hybrid automata

Review of continuous systems; State space form
; existence and uniqueness of solutions for continuous systems; hybrid automata (deterministic case); definitions; time sets + executions; classification of solutions.

3. Classification of hybrid time sets

Modeling issues
; existence of executions
; time sets and executions; two fundamental concepts: reachable states + transition states; Lemma 1 (non-blocking); Lemma 2 (deterministic).

6. Stability of switched systems

Switched systems (a subclass of hybrid systems); stability properties of switched systems; multiple Lyapunov functions; common Lyapunov function; commuting system matrices; Do solutions converge to an equilibrium? How to choose a stabilizing sequence?

7. Stability of hybrid systems

Switched systems (a subclass of hybrid systems); stability properties of switched systems; multiple Lyapunov functions; common Lyapunov function; commuting system matrices; Do solutions converge to an equilibrium? How to choose a stabilizing sequence?

10. Least squares

Identification of system parameters; Least Squares; Recursive Least Squares; Weighted Least Squares.