On the Lyapunov theorem for singular systems

Abstract

In this paper we revisit the Lyapunov theory for singular systems. There are basically two well known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of [6], [7]. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clari£ed. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of [14].

Practical stability analysis and synthesis of linear descriptor systems with disturbances

“…Hence, the regularity, impulsive behavior and stability of descriptor systems are usually considered simultaneously [4,6] . The following theorem states a sufficient condition for system (1) to be regular, impulsive-free and practically stable.…”

Strongly absolute stability of Lur'e descriptor systems: Popov‐type criteria

“…Lyapunov function candidates for descriptor systems are usually chosen as functions of E x (see [11,23]). Lemma 5 describes a sufficient condition for GLLF (12) to be positive definite and radially unbounded with respect to E x. Theorem 1 of [5] is a special case of Lemma 5 with E = I .…”

On the stability of linear descriptor systems by applying modified Lyapunov equations

“…Until now, only special inertia theorems are known. The results of [1,2] are restricted to systems of index k = 1, where [1] gives some corrections of [2]. The results of [4][5][6][7] require the calculation of the transformation matrices R, S of (2).…”