New Risk-Averse Control Paradigm for Stochastic Two-Time-Scale Systems and Performance Robustness

Abstract

This work is concerned with the optimal control of stochastic two-time-scale linear systems with performance measure in a finite-horizon integral-quadratic form. Nature, modeled by stationary Wiener processes whose mean and covariance statistics are known, malevolently affects the state dynamics and output observations of the control problem class. With particular focus on the system performance robustness, the use of higher-order statistics or cumulants associated with the performance measure of chi-squared random variable type makes it possible to restate the stochastic control problem as the solution of a deterministic one, which subsequently allows disregarding all sample-path realizations by Nature acting on the original problem.

The distinguishing feature of the risk-averse control paradigm is that the performance index is multiobjective in nature, being composed of both risk-neutral integrals and risk-sensitive costs associated with the ubiquitous linear-quadratic-Gaussian (LQG) and rather recent risk-sensitive control problems. Another issue that makes this class of control particularly interesting is the fact that Nature has the ability to exercise all the higher-order characteristics of the uncertain chi-squared performance measure. The efficient controller, having access to Nature’s apriori statistical knowledge and employing dynamic output feedback, seeks to minimize the performance uncertainty that Nature can do over the set of mixed random realizations.

Furthermore, the results herein potentially generalize the existing results for the single-objective H ₂, H _∞, and risk-sensitive control problems to a substantially larger class of systems, wherein Nature selects mixed sample-path realizations that need not be Gaussian. That is, the entire probability density function of Nature’s choices is not necessarily known except for its first two statistics. Finally, the numerical simulations for a two-time-scale longitudinal dynamics of the F-8 jet aircraft demonstrate that the proposed control paradigm has competitive performance in the closed-loop system responses and offers multiple levels of robustness for the system performance.