Spectral tests for observability and detectability of periodic Markov jump systems with nonhomogeneous Markov chain

This paper addresses observability and detectability of discrete-time periodic systems with nonhomogeneous Markov jump parameter. Popov-Belevitch-Hautus (PBH)-type criteria are proposed for the concerned structural properties. It is shown that, different from stochastic systems with constant coefficients or homogeneous Markov chain, the spectral criteria of considered plants do not rely on a single operator but on a finite sequence of linear evolution operators. By use of the obtained detectability criterion, an extended Lyapunov theorem is established, which relates asymptotic mean square stability to a periodic Lyapunov equation. Further, a difference Riccati equation with periodic coefficients is studied and some sufficient conditions are presented for the existence of stabilizing solution.