On the Volatility of Optimal Control Policies 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 regulator's control process in discrete time Linear Quadratic Regulators (LQRs). Our investigation in this paper unearths a surprising connection between the cost functional that 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 extreme temporal price volatility, frequently observed in deregulated electricity markets. We model the suppliers and consumers in the market as strategic, price-anticipating participants who submit bids, in the form of price-quantity graphs, to the market operator. We then show that in any such market, at the Nash-Walras equilibrium, improving social welfare must necessarily be traded off with volatility in prices, i.e., it is impossible to reduce volatility in the price of electricity without sacrificing social welfare. Our investigation in this paper unearths a surprising connection between the cost functional that 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 extreme temporal price volatility, frequently observed in deregulated electricity markets. We model the suppliers and consumers in the market as strategic, price-anticipating participants who submit bids, in the form of price-quantity graphs, to the market operator. We then show that in any such market, at the Nash-Walras equilibrium, improving social welfare must necessarily be traded off with volatility in prices, i.e., it is impossible to reduce volatility in the price of electricity without sacrificing social welfare. We then employ this analysis to explain the phenomenon of extreme temporal price volatility, frequently observed in deregulated electricity markets. We model the suppliers and consumers in the market as strategic, price-anticipating participants who submit bids, in the form of price-quantity graphs, to the market operator. We then show that in any such market, at the Nash-Walras equilibrium, improving social welfare must necessarily be traded off with volatility in prices, i.e., it is impossible to reduce volatility in the price of electricity without sacrificing social welfare.
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