Storage functions for dissipative linear systems are quadratic state functions

This paper deals with dissipative dynamical systems which can be used as models for physical phenomena in which the energy exchange with their environment plays a role. In a dissipative dynamical system, the control of energy is done via the supply rate and a storage function. The supply rate is the rate at which energy flows into the system, a storage function is a function that measures the amount of energy that is stored inside the system. In this paper, we argue that for linear dynamical systems with quadratic supply rates, any storage function can be represented as a quadratic function of any state variable of a linear dynamical system whose dynamics is obtained by combining the dynamics of the original system and the dynamics of the supply rate.