---
id: 20250829114522
title: Hybrid Systems
type: permanent
created: 2025-08-29T15:45:22Z
modified: 2025-09-11T21:46:53Z
tags: []
---

# Hybrid Systems

I'm borrowing a lot from
[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]].

Hybrid systems are those that combine continuous and
discrete dynamics together. This is usually some sort of
finite automata + differential equations.

Hybrid systems can be written like:

$$\dot{x}(t) = \xi(t), \quad t\geq 0$$

where $x(t)$ is the continuous component of the state.
$\xi(t)$ is a vector field that depends on $x(t)$ and the
hybrid dynamics.

Switching between modes (aka discontinuities in $\xi(\cdot)$)
can happen in one of two ways:

1. **Autonomous Switching** - Autonomous switches happen
   depending on state values of $x(t)$.

2. **Controlled Switching** - $\xi(\cdot)$ changes abruptly
   in response to a control command.

One may write a continuous time autonomous hybrid system
like this:

$$\dot{x}(t) = f(x(t), q(t))$$
$$q(t) = \nu(x(t), q(t^-))$$

where:
- $x(t) \in R^n$
- $q(t) \in Q \simeq {1,...,N}$
- $f(\cdot,q): R^n \rightarrow R^n,q \in Q$, with each
[[lipschitz-continuous]]. These are the *continuous
dynamics*.
- $\nu: R^n \times Q \rightarrow Q$ is the *finite dynamics*

A controlled system might be written as:
$$\dot{x}(t) = f(x(t), q(t), u(t))$$
$$q(t) = \nu(x(t), q(t^-), u(t))$$

where:
- $u(t) \in R^m$