Solution of the linear quadratic regulator problem of black box linear systems using reinforcement learning

In this paper, a Q-learning algorithm is proposed to solve the linear quadratic regulator problem of black box linear systems. The algorithm only has access to input and output measurements. A Luenberger observer parametrization is constructed using the control input and a new output obtained from a factorization of the utility function. An integral reinforcement learning approach is used to develop the Q-learning approximator structure. A gradient descent update rule is used to estimate on-line the parameters of the Q-function. Stability and convergence of the Q-learning algorithm under the Luenberger observer parametrization is assessed using Lyapunov stability theory. Simulation studies are carried out to verify the proposed approach.