Comments and Expansion of a Result on Common Diagonal Quadratic Lyapunov Functions for Sets of Positive LTI Systems

Abstract

Our work comments a result on common diagonal Lyapunov functions (CDLFs), reported by a recent article, and proposes a substantial extension. The discussed result refers to sets of continuous-time positive linear time-invariant (LTI) systems and ensures the existence of CDLFs with quadratic form. We prove that exactly the same hypothesis as considered by the commented result can guarantee the existence of CDLFs defined by arbitrary Holder vector p-norrns, 1> p>. The proposed expansion, besides the criteria for the existence of CDLFs, also includes procedures for their concrete construction. A separate section of the paper is reserved for sketching the transfer of our approach from continuous-to discrete-time dynamics (although the commented article focusses exclusively on the continuous-time case). An example inspired from the control engineering literature illustrates the applicability of the theoretical developments to a set of 16 continuous-time positive LTI systems.