On integral generalized policy iteration for continuous-time linear quadratic regulations

This paper mathematically analyzes the integral generalized policy iteration (I-GPI) algorithms applied to a class of continuous-time linear quadratic regulation (LQR) problems with the unknown system matrix AA. GPI is the general idea of interacting policy evaluation and policy improvement steps of policy iteration (PI), for computing the optimal policy. We first introduce the update horizon ℏℏ, and then show that (i) all of the I-GPI methods with the same ℏℏ can be considered equivalent and that (ii) the value function approximated in the policy evaluation step monotonically converges to the exact one as ℏ→∞ℏ→∞. This reveals the relation between the computational complexity and the update (or time) horizon of I-GPI as well as between I-PI and I-GPI in the limit ℏ→∞ℏ→∞. We also provide and discuss two modes of convergence of I-GPI; I-GPI behaves like PI in one mode, and in the other mode, it performs like value iteration for discrete-time LQR and infinitesimal GPI (ℏ→0ℏ→0). From these results, a new classification of the integral reinforcement learning is formed with respect to ℏℏ. Two matrix inequality conditions for stability, the region of local monotone convergence, and data-driven (adaptive) implementation methods are also provided with detailed discussion. Numerical simulations are carried out for verification and further investigations.