Data-based L 2 gain optimal control for discrete-time system with unknown dynamics

This paper considers the L 2 gain optimal problem for a class of discrete-time linear time-invariant with state-disturbance feedback controller and unknown system dynamic. Firstly, for a given stabilizing control policy, we establish the relation between the L 2 gain and a sequence of lower triangle Toeplitz matrices. Meanwhile, we show that the upper bound of optimal L 2 gain is proportional to the linear correlation degree between the input and disturbance matrices. Secondly, to overcome the obstacle arising from the unknown system dynamics, a data-based reinforcement learning scheme is developed for the optimal control policy by using linear matrix inequality technique and Q-learning with policy iteration. Under certain conditions, we prove that either the reinforcement learning process ends in a finite number of iterations, or the L 2 gain sequence is strictly monotonically convergent along the iteration axis provided that the disturbance data set can fully activate the closed-loop system. Finally, simulations are given to illustrate the effectiveness of our findings.(c) 2023 The Franklin Institute. Published by Elsevier Inc. All rights reserved.