线性Markov跳变随机系统的Pareto最优控制  

Pareto optimal control of linear Markov jump stochastic systems

作　　者：王乐[1] 崔凯 蒋秀珊 赵东亚[1] 张维海 WANG Le;CUI Kai;JIANG Xiu-shan;ZHAO Dong-ya;ZHANG Wei-hai(College of New Energy,China University of Petroleum(East China),Qingdao Shandong 266580,China;College of Electrical Engineering and Automation,Shandong University of Science and Technology,Qingdao Shandong 266590,China)

机构地区：[1]中国石油大学(华东)新能源学院,山东青岛266580 [2]山东科技大学电气与自动化工程学院,山东青岛266590

出　　处：《控制理论与应用》2025年第1期59-66,共8页Control Theory & Applications

基　　金：国家自然科学基金项目(62103442,12326343,62373229);山东省自然科学基金项目(ZR2021QF080);中央高校基本科研业务费专项资金项目(23CX06024A);山东省高校优秀青年创新团队项目(2023KJ061)资助.

摘　　要：目前针对多个主体、多个目标的带有Markov跳变的线性随机系统的控制问题的研究较少且较为浅显.本文研究了具有乘性噪声的连续时间线性Markov跳变随机系统的Pareto最优控制问题.假设多个主体、多个性能指标由状态和控制变量中的二次部分和线性部分的线性组合而形成,证明了Pareto最优与加权和优化之间的关系,从而将多目标优化问题转化为特殊的单目标加权和最优控制问题.基于Pareto博弈理论与广义Itô公式,系统的Pareto有效策略可以通过一组耦合的广义Riccati微分方程与一组耦合的线性微分方程求解,并且可以得到每一个控制器的Pareto解.最后,本文通过数值仿真验证理论结果的有效性.At present,there is relatively little research on the control problem of linear stochastic systems with Markov jumps for multiple agents and objectives.This paper investigates the Pareto optimal control problem for continuous time linear Markov jump stochastic systems with multiplicative noise.Assuming that multiple entities and performance indicators are formed by a linear combination of the quadratic and linear parts of state and control variables,the relationship between Pareto optimality and weighted sum optimization is proved,thereby transforming multi-objective optimization into a special single objective weighted sum optimal control problem.Based on the Paret ogame theory and the generalized Itô formula,the Pareto effective strategy of the system can be obtained by a set of coupled generalized Riccati differential equations and a set of coupled linear differential equations,and under this strategy,the Pareto solution of each controller can be obtained.Finally,the validity of the theoretical results is verified by a numerical simulation.

关 键 词：MARKOV跳变 随机系统 Pareto控制 最优控制系统 

分 类 号：O232[理学—运筹学与控制论] O211.62[理学—数学]