We consider switched linear systems of odes, $\dot{x}(t) = A\left( u(t) \right) x(t)$ where $A(u(t)) \in \cA$, a compact set of matrices. In this paper we propose a new method for the approximation of the upper Lyapunov exponent and lower Lyapunov exponent of the LSS when the matrices in $\cA$ are Metzler matrices (or the generalization of them for arbitrary cone), arising in many interesting applications (see e.g. \cite{FR}). % The method is based on the iterative construction of invariant positive polytopes for a sequence of discretized systems obtained by forcing the switching instants to be multiple of $\Delta^{(k)} t$ where $\Delta^{(k)} t \to 0$ as $k \to \infty$. These polytopes are then used to generate a monotone piecewise-linear joint Lyapunov function on the positive orthant, which gives tight upper and lower bounds for the Lyapunov exponents. As a byproduct we % are able to detect whether the considered system is stabilizable or uniformly stable. The efficiency of this approach is demonstrated in numerical examples, including some of relatively large dimensions.

Polytope Lyapunov functions for stable and for stabilizable LSS

GUGLIELMI N;
2013-01-01

Abstract

We consider switched linear systems of odes, $\dot{x}(t) = A\left( u(t) \right) x(t)$ where $A(u(t)) \in \cA$, a compact set of matrices. In this paper we propose a new method for the approximation of the upper Lyapunov exponent and lower Lyapunov exponent of the LSS when the matrices in $\cA$ are Metzler matrices (or the generalization of them for arbitrary cone), arising in many interesting applications (see e.g. \cite{FR}). % The method is based on the iterative construction of invariant positive polytopes for a sequence of discretized systems obtained by forcing the switching instants to be multiple of $\Delta^{(k)} t$ where $\Delta^{(k)} t \to 0$ as $k \to \infty$. These polytopes are then used to generate a monotone piecewise-linear joint Lyapunov function on the positive orthant, which gives tight upper and lower bounds for the Lyapunov exponents. As a byproduct we % are able to detect whether the considered system is stabilizable or uniformly stable. The efficiency of this approach is demonstrated in numerical examples, including some of relatively large dimensions.
Linear switching systems; Lyapunov exponent; polytope; iterative method; cones; Metzler matrices; joint spectral radius; lower spectral radius
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/3395
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