Optimal Tracking Control for Unknown Linear Systems with Finite-Time Parameter Estimation

The optimal control input for linear systems can be solved from algebraic Riccati equation (ARE), from which it remains questionable to get the form of the exact solution. In engineering, the acceptable numerical solutions of ARE can be found by iteration or optimization. Recently, the gradient descent based numerical solutions has been proven effective to approximate the optimal ones. This paper introduces this method to tracking problem for heterogeneous linear systems. Differently, the parameters in the dynamics of the linear systems are all assumed to be unknown, which is intractable since the gradient as well as the allowable initialization needs the prior knowledge of system dynamics. To solve this problem, the method named dynamic regressor extension and mix (DREM) is improved to estimate the parameter matrices in finite time. Besides, a discounted factor is introduced to ensure the existence of optimal solutions for heterogeneous systems. Two simulation experiments are given to illustrate the effectiveness.