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| id | title | type | created | modified | tags |
|---|---|---|---|---|---|
| 20250829114522 | Hybrid Systems | permanent | 2025-08-29T15:45:22Z | 2025-09-11T21:46:53Z |
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:
-
Autonomous Switching - Autonomous switches happen depending on state values of
x(t). -
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^nq(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 Qis 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