A Variational Inequality-Based Control Strategy for Autonomous Systems Under External Disturbances

Abstract

Engineering systems that autonomously operate in uncertain environment have to be able to meet control constraints, perform prescribed trajectory-tracking task and at the same time be robust to external disturbances. Conventional feedback control systems typically optimize tracking performance of nominal system and provide only limited robustness guarantees if the external disturbances are adversarial and belong to bounded sets of possible disturbances. This paper proposes minimax control methodology for autonomous engineering systems described by dynamical system equations subjected to external disturbances. The proposed methodology consists of the minimax control problem formulation expressed as a constrained variational inequality. Specifically, represent controller and disturbance as minimizing and maximizing decision-makers, respectively, and transform robust control problem into the saddle point problem. A projected variational inequality operator that satisfies control constraints imposed on admissible controls is developed. Existence and uniqueness of the solution are established using strong monotonicity property of the operator and boundedness of the tracking error under persistent disturbances is proven using Lyapunov function. The projected minimax algorithm for online implementation is developed. The approach is illustrated using the trajectory-tracking problem of autonomous mobile robot that is subjected to an external disturbance.