A new procedure to solve generalized Lyapunov equations

For stability analysis of discrete-time descriptor systems, various generalized Lyapunov equations have been proposed in the literature. However, positiveness of the solutions for these well known Lyapunov equations are not biunivocaly related to causal state trajectories that go to zero as the time goes to infinity, even under observability assumptions. We propose in this paper two new Lyapunov equations to deal with this problem. As these Lyapunov equations depend on two unknown matrices, P and R, solutions are a prime concern in this approach. In this paper we propose an algorithm to solve these new Lyapunov equations for stability test of descriptor systems.