Q -Learning Approach to Finite-Horizon H ∞ Tracking With Partial Observation

This article investigates the finite-horizon $H_{\infty } $ tracking control problem for discrete-time (DT) linear systems with partial observation and unknown dynamics from a game-theoretic perspective. Unlike existing reinforcement learning (RL) approaches that primarily address infinite-horizon, time-invariant systems with full state information, our setting requires solving time-varying Riccati equations and developing model-free methods that rely solely on input-output data. To tackle these challenges, we reconstruct the system state from historical input-output trajectories, driving to a data-driven system representation, and we define an input-output-based time-varying $Q$ -function. We then propose two minimax $Q$ -learning algorithms that do not require an initially admissible policy and avoid the use of a discount factor, thereby removing a long-standing obstacle to stability guarantees. Moreover, the framework readily extends to both infinite-horizon and time-varying systems without structural modifications. Convergence is proved theoretically, and the effectiveness of the algorithms is validated through simulations.