On testing stability and flow-invariance of arbitrary switching positive systems via linear copositive Lyapunov functions

The paper proposes an algebraic setting for the concrete construction of linear copositive Lyapunov functions associated with arbitrary switching positive systems; this approach complements a series of already reported results that are limited to the existence problem. Our development encompasses both discrete- and continuous-time dynamics, in a unifying manner, based on sets of quasi-linear inequalities and their solvability. The construction procedure can provide the linear copositive Lyapunov function exhibiting the optimal or ε-suboptimal decreasing rate. The procedure exploits the Perron-Frobenius eigenstructure of the representative matrix (built for columns), which possesses the greatest eigenvalue; the role of (ir)reducibility of this matrix is analyzed for some of mostly encountered practical cases. To illustrate the applicability of our developments, a numerical example from literature is considered.