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: Mondays, 11:00am

Level: graduate

TA: TBD

OVERVIEW

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

Assignments

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

Homework sets

HW1 – due March 29th

HW2 – due April 19th

Projects

Project1 – due May 3rd

Project2 – due May 19th

Notebooks (how to..)

model a free falling object   

 

Lessons

Introduction to hybrid systems

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

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.

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).

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?

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?

Least squares

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