We consider switched linear systems of odes, ẋ(t)= A(u(t))x(t) where A(u(t)) ∈ A , 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 A are Metzler matrices (or the generalization of them for arbitrary cone), arising in many interesting applications (see e.g. [9]). 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 Δ(k)t where Δ(k)t → 0 as k → ∞. 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 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. © 2013 IEEE.

Polytope joint Lyapunov functions for positive LSS

Guglielmi, Nicola;
2013

Abstract

We consider switched linear systems of odes, ẋ(t)= A(u(t))x(t) where A(u(t)) ∈ A , 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 A are Metzler matrices (or the generalization of them for arbitrary cone), arising in many interesting applications (see e.g. [9]). 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 Δ(k)t where Δ(k)t → 0 as k → ∞. 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 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. © 2013 IEEE.
9781467357173
Control and Systems Engineering; Modeling and Simulation; Control and Optimization
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/3298
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