Tracking Problem for Itô Stochastic System with Input Delay

In this article, linear quadratic tracking (LQT) problem for Itô stochastic system with time delay in control input is studied. The difference between LQT and linear quadratic regulation (LQR) is that the state and control input are desired to track a reference trajectory. It is noteworthy that the separation principle is not valid for Itô stochastic system. In this paper, we design an optimal controller based on the maximum principle, in which the optimal controller minimizes the linear quadratic performance index of state and control input tracking errors. Based on the coupled Riccati-like equation, the explicit expression of the optimal controller is given. The key technology of this paper is to solve forward and backward stochastic differential equations (FBSDEs). Finally, a numerical example is provided to validate the results.

1 Introduction

Linear quadratic (LQ) optimal control problem is one of the most mature parts of optimal control theory. It is widely used in engineering fields because of its simple control law, easy realization and reasonable control energy. Up to now, LQ optimal control theory has made rapid development. They often design controllers to ensure that system state and control input are as close to zero as possible, while minimizing quadratic performance indicator of state and control input. However, in many practical problems such as robot trajectory planning, missile guidance [1], economic stabilization [2] and so on, they hope that the state of system and control input can track a desired reference trajectory which is usually not equal to zero. Thus, it can be seen that linear quadratic tracking (LQT) is an important branch of optimal control theory and application. To meet this requirement, we study the LQT problem and take the square integral of state error and control input error as the performance index. The tracking problem has been investigated in the past few decades; see e.g. [3–6] and the references therein.