Optimal Vibration Control of a Class of Nonlinear Stochastic Systems with Markovian Jump

R H Huan,W Q Zhu,D Pu,R C Hu

doi:10.1155/2016/9641075

Abstract

The semi-infinite time optimal control for a class of stochastically excited Markovian jump nonlinear system is investigated. Using stochastic averaging, each form of the system is reduced to a one-dimensional partially averaged Itô equation of total energy. A finite set of coupled dynamical programming equations is then set up based on the stochastic dynamical programming principle and Markovian jump rules, from which the optimal control force is obtained. The stationary response of the optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Itô equation. Two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.

Highlights

Markovian jump system (MJS) is a class of hybrid systems, which has different operation modes or forms governed by a Markov process
A nonlinear optimal control strategy of stochastic excited nonlinear systems has been proposed by Zhu based on the stochastic averaging method [13] and stochastic dynamical programming principle
An infinite time optimal control problem for single degree-of-freedom (SDOF), stochastically excited, nonlinear, Markovian jump system is investigated in this paper

Summary

Introduction

Markovian jump system (MJS) is a class of hybrid systems, which has different operation modes or forms governed by a Markov process. For the problem of optimal control, Krasosvkii and Lidskii studied the LQR control of Markovian jump linear systems firstly [5]. Most of the published results are applicable to linear MJSs. Far less is known about the optimal control of nonlinear MJSs, for stochastically excited nonlinear MJSs. Recently, a nonlinear optimal control strategy of stochastic excited nonlinear systems has been proposed by Zhu based on the stochastic averaging method [13] and stochastic dynamical programming principle. The goal of this paper is to extend the strategy for nonlinear stochastic system to the one for nonlinear stochastic MJSs. An infinite time optimal control problem for single degree-of-freedom (SDOF), stochastically excited, nonlinear, Markovian jump system is investigated in this paper.

Formulation of Control Problem

Partially Averaged Equation

Optimal Control Law

Numerical Example

Optimally controlled

Conclusions