Optimal trajectory tracking for uncertain linear discrete-time systems using time-varying Q-learning

This article introduces a novel optimal trajectory tracking control scheme designed for uncertain linear discrete-time (DT) systems. In contrast to traditional tracking control methods, our approach removes the requirement for the reference trajectory to align with the generator dynamics of an autonomous dynamical system. Moreover, it does not demand the complete desired trajectory to be known in advance, whether through the generator model or any other means. Instead, our approach can dynamically incorporate segments (finite horizons) of reference trajectories and autonomously learn an optimal control policy to track them in real time. To achieve this, we address the tracking problem by learning a time-varying $Q$-function through state feedback. This $Q$-function is then utilized to calculate the optimal feedback gain and explicitly time-varying feedforward control input, all without the need for prior knowledge of the system dynamics or having the complete reference trajectory in advance. Additionally, we introduce an adaptive observer to extend the applicability of the tracking control scheme to situations where full state measurements are unavailable. We rigorously establish the closed-loop stability of our optimal adaptive control approach, both with and without the adaptive observer, employing Lyapunov theory. Moreover, we characterize the optimality of the controller with respect to the finite horizon length of the known components of the desired trajectory. To further enhance the controller's adaptability and effectiveness in multitask environments, we employ the Efficient Lifelong Learning Algorithm, which leverages a shared knowledge base within the recursive least squares algorithm for multitask $Q$-learning. The efficacy of our approach is substantiated through a comprehensive set of simulation results by using a power system example.