Dual switched positive systems – a less conservative condition for diagonal quadratic stability

The key result of the paper exploits the duality of arbitrary switching positive linear systems, in order to derive a sufficient condition for the existence and construction of diagonal quadratic copositive Lyapunov functions. This condition is less conservative than a result also relying on duality, recently reported in the literature; our condition operates with milder inequalities than those required by the existing result. We prove that the algebraic relaxation of these inequalities is related to a relaxation of the primal and dual systems’ properties, referring to the trajectory growth rates, or, equivalently, to the spectra of the associated column representatives. The existing result is incorporated into the new one as a particular case, and the advantages offered by the latter are illustrated by a relevant numerical example. Our developments focus on the continuous-time case, so as to permit direct comparisons with the existing result; some brief discussion suggests how the proposed approach can be applied, mutatis mutandis, to discrete-time dynamics.