We analyse the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. It is well-known that the existence of a positive Ï for which the corresponding discrete time system with stepsize Ï is stable implies the stability of the LSS. Our main goal is to obtain a converse statement, that is to estimate the discretization stepsize Ï > 0 up to a given accuracy Îµ > 0. This would lead to a method for deciding the stability of a continuous time LSS with a guaranteed accuracy. As a first step towards the solution of this problem, we show that for systems of matrices with real spectrum the parameter Ï can be effectively estimated. We prove that in this special case, the discretized system is stable if and only if the Lyapunov exponent of the LSS is smaller than -C Ï , where C is an effective constant depending on the system. The proofs are based on applying Markov-Bernstein type inequalities for systems of exponents. Â© 2013 IEEE.

### Is switching systems stability harder for continuous time systems?

#### Abstract

We analyse the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. It is well-known that the existence of a positive Ï for which the corresponding discrete time system with stepsize Ï is stable implies the stability of the LSS. Our main goal is to obtain a converse statement, that is to estimate the discretization stepsize Ï > 0 up to a given accuracy Îµ > 0. This would lead to a method for deciding the stability of a continuous time LSS with a guaranteed accuracy. As a first step towards the solution of this problem, we show that for systems of matrices with real spectrum the parameter Ï can be effectively estimated. We prove that in this special case, the discretized system is stable if and only if the Lyapunov exponent of the LSS is smaller than -C Ï , where C is an effective constant depending on the system. The proofs are based on applying Markov-Bernstein type inequalities for systems of exponents. Â© 2013 IEEE.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11697/123624`
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