Dissipativity of Uncontrollable Systems, Storage Functions, 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. In this paper 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. 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 link Lyapunov functions for autonomous systems to storage functions of an uncontrollable system. The set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature. 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 most of the results in this paper, we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis; we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity.

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