Finite-Horizon near Optimal Adaptive Control of Uncertain Linear Discrete-Time Systems

In this paper, the finite-horizon near optimal adaptive regulation of linear discrete-time systems with unknown system dynamics is presented in a forward-in-time manner by using adaptive dynamic programming and Q-learning. An adaptive estimator (AE) is introduced to relax the requirement of system dynamics, and it is tuned by using Q-learning. The time-varying solution to the Bellman equation in adaptive dynamic programming is handled by utilizing a time-dependent basis function, while the terminal constraint is incorporated as part of the update law of the AE. The Kalman gain is obtained by using the AE parameters, while the control input is calculated by using AE and the system state vector. Next, to relax the need for state availability, an adaptive observer is proposed so that the linear quadratic regulator design uses the reconstructed states and outputs. For the time-invariant linear discrete-time systems, the closed-loop dynamics becomes non-autonomous and involved but verified by using standard Lyapunov and geometric sequence theory. Effectiveness of the proposed approach is verified by using simulation results. The proposed linear quadratic regulator design for the uncertain linear system requires an initial admissible control input and yields a forward-in-time and online solution without needing value and/or policy iterations