Data-Driven LQR with Finite-Time Experiments via Extremum-Seeking Policy Iteration

In this paper, we address Linear Quadratic Regulator (LQR) problems through a novel iterative algorithm named EXtremum-seeking Policy iteration LQR (EXP-LQR). The peculiarity of EXP-LQR is that it only needs access to a truncated approximation of the infinite-horizon cost associated to a given policy. Hence, EXP-LQR does not need the direct knowledge of neither the system and cost matrices. In particular, at each iteration, EXP-LQR refines the maintained policy using a truncated LQR cost retrieved by performing finite-time virtual or real experiments in which a perturbed version of the current policy is employed. Such a perturbation is done according to an extremum-seeking mechanism and makes the overall algorithm a time-varying nonlinear system. By using a Lyapunov-based approach exploiting averaging theory, we show that EXP-LQR exponentially converges to an arbitrarily small neighborhood of the optimal gain matrix. We corroborate the theoretical results with numerical simulations involving the control of an induction motor.