Matrix Inequality Conditions For Stability Research Articles

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 A. 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 ℏ, and then show that (i) all of the I-GPI methods with the same ℏ can be considered equivalent and that (ii) the value function approximated in the policy evaluation step monotonically converges to the exact one as 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→∞. 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→0). From these results, a new classification of the integral reinforcement learning is formed with respect to ℏ. 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.

Switched periodic systems in discrete time: stability and input–output norms

This paper deals with the analysis of stability and the characterisation of input–output norms for discrete-time periodic switched linear systems. Such systems consist of a network of time-periodic linear subsystems sharing the same state vector and an exogenous switching signal that triggers the jumps between the subsystems. The overall system exhibits a complex dynamic behaviour due to the interplay between the time periodicity of the subsystem parameters and the switching signal. Both arbitrary switching signals and signals satisfying a dwell-time constraint are considered. Linear matrix inequality conditions for stability and guaranteed H2 and H∞ performances are provided. The results heavily rely on the merge of the theory of linear periodic systems and recent developments on switched linear time-invariant systems.