---
id: 20250911165736
title: Switched Systems
type: permanent
created: 2025-09-11T20:57:36Z
modified: 2025-09-11T21:09:55Z
tags: []
---

# Switched Systems

Switched systems are those that mix continuous and discrete
dynamics. They are systems that are 'multimodal'. This means
that they can have different continuous dynamic modes.

I'm borrowing form
[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]],
but here's a short description of how they work.

A prototypical switched system is as follows:
$$\dot{x}(t)=f_i ( x(t)), \quad i \in Q \simeq {1,...,N}$$

with two conditions:
1. Each $f_i$ is globally [[Lipschitz Continuous]]
2. The i's are picked in a way that there are finite
   switches in finite time.


There's also this thing called a *continuous switched
system*. A continuous switched system is one that does not
change continuous states when a switch occurs. That is to
say when switching from $i$ to $i'$:

$$f_i(x(t_i),t_i) = f_{i'}(x(t_{i'}),t_{i'})$$

