An extension of the positive real lemma to descriptor systems

The well-known positive real lemma characterizes positive realness of transfer functions of timeinvariant linear systems via the solvability of certain linear matrix inequalities. In this paper, we propose an extension of the positive real lemma and the underlying linear matrix inequalities to descriptor systems. We show that the solvability of these linear matrix inequalities is sufficient and, under an additional assumption, also necessary for positive realness of the transfer function of time-invariant linear descriptor systems. We also study a second system of linear matrix inequalities based on a generalized Lyapunov equation. We show that the solvability of this second system is sufficient, but in general not necessary for positive realness.