# Hybrid Systems

Hybrid systems comprise the collection of dynamical systems that exhibit both continuous and discrete behaviors. An example of such a system can be found in a common switched electrical circuit, voltages and currents governed by continuously changing classical electrical network laws can also change discontinuously due to the opening and closing of switches. Another quintessential dynamical system with hybrid behavior is that of the bouncing ball, in free fall the equations of motion observe classical Newtonian mechanics but at each bounce a discontinuity is observed in the instantaneous velocity of the ball. To model such systems, a variety of methods have been proposed however, for the purposes of this article, we will present the modeling framework proposed by Goebel et al.[1]

A model of a hybrid system ${\displaystyle {\mathcal {H}}}$ is given by a system of differential and difference equations or inclusions represented by four elements of data

{\displaystyle {\begin{aligned}\\\qquad \qquad {\mathcal {H}}\ {\begin{cases}{\dot {x}}\ \ =f(x)&&x\in C\\x^{+}\!=g(x)&&x\in D\end{cases}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}\\\qquad \qquad {\mathcal {H}}\ {\begin{cases}{\dot {x}}\ \ \in F(x)&&x\in C\\x^{+}\!\in G(x)&&x\in D\end{cases}}\\\\\end{aligned}}}

where ${\displaystyle x}$ is the hybrid system state, ${\displaystyle C}$ is the flow set that governs the differential equation ${\displaystyle f(x)}$ or inclusion ${\displaystyle F(x)}$, and ${\displaystyle D}$ is the jump set that governs the difference equation ${\displaystyle g(x)}$ or inclusion ${\displaystyle G(x)}$.

## References

1. R. Goebel, R. G. Sanfelice, and A. R. Teel Hybrid Dynamical Systems: Modeling, Stability, and Robustness , New Jersey, Princeton University Press, 2012.