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

# 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.

## Lessons

**List of lectures**

### 1. Introduction to hybrid systems

### 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).

### 4. Existence of solutions

Local existence; uniqueness of solutions; fundamental lemmas.

### 5. Stability of continuos-time systems (review)

Stability for continuous systems; stability in the sense of Lyapunov; asymptotic stability; Lyapunov’s second method; stability of linear systems.

### 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?

### 8. Midterm review

Midterm review.

### 9. Identification of LTI systems (review)

Identification of LTI systems; definition; two approaches: time domain + frequency domain; review: transfer functions + Bode plots.

### 10. Least squares

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

### 11. Discrete Kalman filter

State estimation; discrete Kalman filter; state-space observer (no-noise case); simulations.

### 12. Identification of hybrid systems

Identification of hybrid systems; definition; PWA systems; a clustering technique for PWA systems.

### 13. Hybrid control design

### 14. Feedback stabilization