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This paper presents a model-free reinforcement learning (RL) algorithm to solve the risk-averse optimal control (RAOC) problem for discrete-time nonlinear systems. While successful RL algorithms have been presented to learn optimal control solutions under epistemic uncertainties (i.e., lack of knowledge of system dynamics), they do so by optimizing the expected utility of outcomes, which ignores the variance of cost under aleatory uncertainties (i.e., randomness). Performance-critical systems, however, must not only optimize the expected performance, but also reduce its variance to avoid performance fluctuation during RL's course of operation. To solve the RAOC problem, this paper presents the following three variants of RL algorithms and analyze their advantages and preferences for different situations/systems: 1) a one-shot static convex program -based RL, 2) an iterative value iteration (VI) algorithm that solves a linear programming (LP) optimization at each iteration, and 3) an iterative policy iteration (PI) algorithm that solves a convex optimization at each iteration and guarantees the stability of the consecutive control policies. Convergence of the exact optimization problems, which are infinite-dimensional in all three cases, to the optimal risk-averse value function is shown. To turn these optimization problems into standard optimization problems with finite decision variables and constraints, function approximation for value estimations as well as constraint sampling are leveraged. Data-driven implementations of these algorithms are provided based on Q-function which enables learning the optimal value without any knowledge of the system dynamics. The performance of the approximated solutions is also verified through a weighted sup-norm bound and the Lyapunov bound. A simulation example is provided to verify the effectiveness of the presented approach. |
