Model-Based and Model-Free Algorithms for Stochastic $H_{\infty}$ Tracking Control

$H_{\infty}$ tracking control involves a complex form of generalized algebraic Riccati equation (GARE), which is difficult to solve directly. In this paper, we derive the $H_{\infty}$ tracking problem for the linear system with $x$-dependent noise and present its formal solution. For solving its GARE, we develop modelbased and model-free algorithms to solve it. The former is designed based on the iterative relation of the GARE. The latter adopts the reinforcement learning framework. Based on the constructed $Q$-function, the $H_{\infty}$ tracking controller can be solved online through the collected system states. The equivalence of the two algorithms is illustrated. Finally, the effectiveness of the two algorithms is verified by simulation experiments.