Convergence of Dynamic Programming on the Semidefinite Cone for Discrete-Time Infinite-Horizon LQR

Abstract

The goal of this article is to investigate new and simple convergence analysis of dynamic programming for the linear–quadratic regulator problem of discrete-time linear time-invariant systems. In particular, bounds on errors are given in terms of both matrix inequalities and matrix norm. Under a mild assumption on the initial parameter, we prove that the Q -value iteration exponentially converges to the optimal solution. Moreover, a global asymptotic convergence is also presented. These results are then extended to the policy iteration. We prove that in contrast to the Q -value iteration, the policy iteration always converges exponentially fast. An example is given to illustrate the results.