New results in dissipativity of uncontrollable systems and Lyapunov functions

Dissipative systems have played an important role in the analysis and synthesis of dynamical systems. The commonly used definition of dissipativity often requires an assumption on the controllability of the system. However, it is very natural to think of Lyapunov functions as storage functions for autonomous systems with power supplied to the system equal to zero. We use a definition of dissipativity that is slightly different (and less often used in the literature) to study a linear, time-invariant, possibly uncontrollable dynamical system. This paper contains various results in the context of uncontrollable dissipative systems that smoothly bridge the gap between storage functions for controllable dissipative systems and Lyapunov functions for autonomous systems. We provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not ¿mixed¿; i.e., no pair of uncontrollable poles is symmetric about the imaginary axis: this condition is known to be related to the solvability of a Lyapunov equation. We show that for an uncontrollable system the set of storage functions is unbounded, and that the unboundedness arises precisely due to the set of Lyapunov functions for an autonomous linear system being unbounded. Further, we show that stabilizability of a system results in this unbounded set becoming bounded from below. Positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive. In this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions. In the context of autonomous systems, we prove that the Lyapunov operator is onto if and only if its image has observable symmetric rank one matrices.