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58 lines
1.4 KiB
Markdown
58 lines
1.4 KiB
Markdown
---
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id: 20250829114522
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title: Hybrid Systems
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type: permanent
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created: 2025-08-29T15:45:22Z
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modified: 2025-09-11T21:46:53Z
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tags: []
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---
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# Hybrid Systems
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I'm borrowing a lot from
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[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]].
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Hybrid systems are those that combine continuous and
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discrete dynamics together. This is usually some sort of
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finite automata + differential equations.
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Hybrid systems can be written like:
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$$\dot{x}(t) = \xi(t), \quad t\geq 0$$
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where $x(t)$ is the continuous component of the state.
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$\xi(t)$ is a vector field that depends on $x(t)$ and the
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hybrid dynamics.
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Switching between modes (aka discontinuities in $\xi(\cdot)$)
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can happen in one of two ways:
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1. **Autonomous Switching** - Autonomous switches happen
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depending on state values of $x(t)$.
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2. **Controlled Switching** - $\xi(\cdot)$ changes abruptly
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in response to a control command.
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One may write a continuous time autonomous hybrid system
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like this:
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$$\dot{x}(t) = f(x(t), q(t))$$
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$$q(t) = \nu(x(t), q(t^-))$$
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where:
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- $x(t) \in R^n$
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- $q(t) \in Q \simeq {1,...,N}$
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- $f(\cdot,q): R^n \rightarrow R^n,q \in Q$, with each
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[[lipschitz-continuous]]. These are the *continuous
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dynamics*.
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- $\nu: R^n \times Q \rightarrow Q$ is the *finite dynamics*
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A controlled system might be written as:
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$$\dot{x}(t) = f(x(t), q(t), u(t))$$
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$$q(t) = \nu(x(t), q(t^-), u(t))$$
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where:
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- $u(t) \in R^m$
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