Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems

Ming Chen,Gang Li

doi:10.1186/s13662-014-0342-1

Abstract

The aim of this paper is to investigate the infinite horizon linear quadratic (LQ) optimal control for stochastic time-delay difference systems with both state and control dependent noise. To do this, the notion of exact observability of a stochastic time-delay deference system is introduced and its PBH criterion is presented by the spectrum of an operator related with stochastic time-delay deference systems. Under the assumptions of stabilization and exact observability, it is shown that the optimal control law and optimal value exist, and also the properties of the associated general algebraic Ricatti equation (GARE) are discussed.

Highlights

1 Introduction As is well known, the optimal linear quadratic regulation (LQR) problem was initiated by Kalman in [ ], which is one of the most important optimal control problems
By exploiting the dynamic programming approach, the authors presented a solution to the stochastic LQR problem for systems with input delay and stochastic parameter uncertainties in [ ]
We introduce the definition of exact observability of time-delay systems, and under the exact observability, we give an equivalent condition for the stabilizability of stochastic time-delay systems

Summary

Introduction

The optimal linear quadratic regulation (LQR) problem was initiated by Kalman in [ ], which is one of the most important optimal control problems. This paper will discuss the infinite horizon linear quadratic regulation problem for discrete-time stochastic systems with input delay and state delay. In order to guarantee the well posedness of the quadratic performance and the existence of the feedback stabilizing control law, we shall introduce some concepts such as stabilizability and exact observability, as regards which similar definitions have been well defined in [ ] for stochastic Itô systems. It is worth pointing out that, similar to the continuous context [ ], stabilizability and exact observability will play an important role in discussing other problems, such as stochastic time-delay difference H /H∞ control. For stochastic time-delay difference systems, we concentrate our attention upon infinite horizon linear quadratic optimal control. In Section , under assumptions of stabilization and exact observability, we prove that the optimal control law and optimal value exist of stochastic time-delay difference systems.

Now we introduce an operator

Since the matrix

Lyapunov function as a quadratic form

Since fact an

Then the GARE

Then system stable if if the Lyapunov equation