Stability analysis of positive descriptor systems

In this paper, we discuss stability properties of positive descriptor systems in the continuous-time as well as in the discrete-time case. We present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems. We show that if the spectral projector onto the finite deflating subspace of the matrix pair (E, A) is nonnegative, then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case. Furthermore, we provide sufficient conditions that guarantee entry-wise nonnegativity along with positive semidefiniteness of solutions of generalised projected Lyapunov equations. As an application of the framework established throughout this paper, we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case.