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M  "Zettelkasten/Permanent Notes/20250829114522-hybrid-systems.md"

A  "Zettelkasten/Permanent Notes/20250911165736-switched-systems.md"

A  "Zettelkasten/Permanent Notes/20250911170650-lipschitz-continuous.md"

M  "Zettelkasten/Permanent Notes/Literature Notes/LIT-20250911143337-multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems.md"

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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^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

1.4 KiB Raw Blame History

Hybrid Systems

1.4 KiB

Raw Blame History