Data-Driven LQR with Uncertainty Refinement and Domain Randomization---Part~II: Fisher-Majorized Certified Policy Iteration

Overview & Synopsis

Building on the uncertainty description developed in Part~I, this paper introduces a Fisher-majorized domain-randomized generalized policy iteration framework for data-driven linear quadratic regulator (LQR) synthesis. At each iteration, partial Bellman evaluation over a fixed batch of uncertainty realizations produces a structured Sylvester policy-improvement step. A residual-weighted Fisher curvature controls the policy-induced variation in the stationary covariance and converts the Sylvester model into a certified upper bound on the cost change, thereby guaranteeing descent for every accepted step. A check-and-correct mechanism then certifies stability over the entire uncertainty support, rather than only over the sampled realizations. Consequently, every iterate is robustly stabilizing and monotonically decreases the fixed-batch objective. The Fisher truncation and stability correction become inactive after finitely many times. The objective gap then contracts linearly to a floor determined by the dispersion of the realization-wise Riccati targets and by measurement noise. In the absence of measurement noise, this floor vanishes when the realization-wise targets coincide.