Connotations on a quadratic stability criterion for arbitrary switching linear systems

The work is placed within the area of qualitative analysis of arbitrary switching linear systems, focusing on the existence of quadratic Lyapunov functions for discrete- or continuous-time dynamics. Our investigations express a series of connotations on the utilization of a criterion (amply cited in the control engineering literature) which characterizes the mode-dependent quadratic Lyapunov functions, in the discrete-time case. These connotations are formulated as two distinct theorems, for discrete-time and continuous-time systems, respectively; both theorems provide algebraic conditions that ensure the existence of time-independent quadratic Lyapunov functions, together with the associated sets invariant relative to system trajectories. In the discrete-time case, our algebraic condition includes (as a particular form) the algebraic condition used by the existing criterion mentioned above. Our theorems are accompanied by remarks that create a deeper insight into the dichotomy time-independence vs. time-dependence for quadratic Lyapunov functions. A numerical example addressed by the help of the SeDuMi solver with the YALMIP interface illustrates the key theoretical points discussed for continuous-time arbitrary switching systems.