State Distributions and Minimum Relative Entropy Noise Sequences in Uncertain Stochastic Systems: The Discrete-Time Case

Abstract

This paper is concerned with dissipativity theory and robust performance analysis and design of discrete-time stochastic systems driven by statistically uncertain random noise. The uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence. We discuss a balance equation, dissipation inequality, and superadditivity property for the corresponding conditional relative entropy supply as a function of time. The problem of minimizing the supply, required to drive the system between given state distributions over a specified time horizon, is considered. Such variational problems, involving entropy and probabilistic boundary conditions, are known in the literature as Schrödinger bridge problems. In application to control systems, the minimum required conditional relative entropy supply characterizes the robustness of the system with respect to a statistically uncertain random noise. We obtain a dynamic programming Bellman equation for the minimum required supply and establish a Markov property of the worst-case noise with respect to the state of the system. For multivariable linear systems with a Gaussian white noise sequence as the nominal noise model and Gaussian initial and terminal state distributions, the minimum required conditional relative entropy supply is obtained using an algebraic Riccati equation which admits a closed-form solution. We propose a computable robustness index for such systems subjected to statistically uncertain random noises whose relative entropy rate does not exceed a given threshold, and provide examples to illustrate this approach. We also consider a minimax problem of robust optimization of systems against the class of noises and demonstrate its solution for a robust filter design example.