Verifiable conditions on asymptotic stabilisability for a class of planar switched linear systems

In this paper, we propose a computer algebra based approach for analyzing asymptotic stabilisability of a class of planar switched linear systems, where subsystems are assumed to be alternatively active. We start with an algebraizable sufficient condition on the existence of stabilizing switching lines and a multiple Lyapunov function. Then, we apply a real root classification based method to under-approximate this condition such that the under-approximation only involves the parametric coefficients. Afterward, we additionally use quantifier elimination to eliminate parameters in the multiple Lyapunov function, arriving at a quantifier-free formula over parameters in the switching lines. According to our intermediate under-approximation as well as our final quantifier-free formula, we can easily design explicit stabilizing switching laws. Moreover, based on a prototypical implementation, we use an illustrating example to show the applicability of our approach. Finally, the advantages of our approach are demonstrated by the comparisons with some related works in the literature.