On the Volatility of Optimal Control Policies and the Capacity of a Class of Linear Quadratic Regulators

It is well known that highly volatile control laws, while theoretically optimal for certain systems, are undesirable from an engineering perspective, being generally deleterious to the controlled system. In this article we are concerned with the temporal volatility of the control process of the regulator in discrete time Linear Quadratic Regulators (LQRs). Our investigation in this paper unearths a surprising connection between the cost functional which an LQR is tasked with minimizing and the temporal variations of its control laws. We first show that optimally controlling the system always implies high levels of control volatility, i.e., it is impossible to reduce volatility in the optimal control process without sacrificing cost. We also show that, akin to communication systems, every LQR has a $Capacity~Region$ associated with it, that dictates and quantifies how much cost is achievable at a given level of control volatility. This additionally establishes the fact that no admissible control policy can simultaneously achieve low volatility and low cost. We then employ this analysis to explain the phenomenon of temporal price volatility frequently observed in deregulated electricity markets.