Stability under dwell time constraints: Discretization revisited

Abstract

We decide the stability and compute the Lyapunov exponent of continuous-time linear switching systems with a guaranteed dwell time. The main result asserts that the discretization method with step size~$h$ approximates the Lyapunov exponent with the precision~$C\,h^2$, where~$C$ is a constant. Let us stress that without the dwell time assumption, the approximation rate is known to be linear in~$h$. Moreover, for every system, the constant~$C$ can be explicitly evaluated. In turn, the discretized system can be treated by computing the Markovian joint spectral radius of a certain system on a graph. This gives the value of the Lyapunov exponent with a high accuracy. The method is efficient for dimensions up to, approximately, ten; for positive systems, the dimensions can be much higher, up to several hundreds.