Analysis of Hybrid Systems - 13

Question 1

What is the issue with stability for a hybrid system?

Answer 1

Even if a hybrid system is Lyapunov stable in all discrete states, the hybrid
system is not necessarily stable! Switching can destabilize a system.

Question 2

What are the necessary conditions for the common Lyapunov function?

Answer 2

A common Lyapunov function is necessary and sufficient.

If the continuous systems of all discrete states share a common Lyapunov
function, the hybrid dynamics is stable. If a hybrid system is stable for arbitrary sequences of discrete states, all
discrete states share a common Lyapunov function.

Question 3

What is a possible solution to deal with a system which is not arbitrarily switched?

Answer 3

One often requires
multiple Lyapunov functions to prove stability.