东北大学 信息科学与工程学院,辽宁 沈阳 110819
收稿日期: 2015-06-12
基金项目: 国家自然科学基金重点资助项目(61573088,61433004).
作者简介: 齐文海(1986-),男,山东泰安人,东北大学博士研究生。
摘要: 研究了带有执行器饱和与转移概率部分已知的随机Markov跳变系统的非脆弱有限时间镇定问题.转移概率部分已知包含转移概率完全已知和转移概率完全未知两类特殊的情况.首先基于参数依赖型Lyapunov函数和自由权矩阵方法,对随机Markov饱和跳变系统的镇定进行了研究,提出了有限时间稳定的充分条件.然后利用线性矩阵不等式的方法实现了非脆弱有限时间状态反馈控制器与吸引域最大估计值的求解.最后通过四模态随机Markov跳变系统的数值例子验证了结论的有效性
关键词:执行器饱和Markov跳变系统转移概率部分已知有限时间镇定非脆弱状态反馈控制器
Non-fragile Finite-Time Stabilization for Stochastic Markov Jump Systems with Actuator Saturation
QI Wen-hai, LI Yue-xiang, CUI Xiu-li
School of Information Science & Engineering,Northeastern University,Shenyang 110819,China
Corresponding author: QI Wen-hai,E-mail: qiwhtanedu@163.com
Abstract: This paper dealed with the problem of finite-time stabilization for a class of stochastic Markov jump systems with both actuator saturation and partly known transition probabilities. Partly known transition probabilities covered completely known transition probabilities and completely unknown transition probabilities as two special cases. Firstly,by use of parameter-dependent Lyapunov function and free-connection weighting matrices,the problem of finite-time stabilization of stochastic Markov jump systems with actuator saturation was analyzed and sufficient conditions for finite-time stability were proposed. The procedure of solution for the non-fragile finite-time state feedback controller and the maximum domain of attraction were built in the form of linear matrix inequalities(LMIs). Finally,a numerical example about four modes with Markov jump parameters was given to show the validity of the results.
Key Words: actuator saturationMarkov jump systemspartly known transition probabilitiesfinite-time stabilizationnon-fragile state feedback controller
Markov跳变系统是一类包含多个模态的重要随机混杂系统,是Krasovskii和Lidskiid于1961年建立的[1].在过去的几十年里,由于能更好描述许多实际系统的特性,例如故障诊断系统[2]、制造系统[3]等,Markov跳变系统引起了很大关注.关于这类问题的研究大都基于转移概率完全已知的情况.然而,考虑实际过程中的复杂因素,转移概率只有部分得到.所以,转移概率部分已知的Markov跳变系统的研究得到了越来越多的关注,包括稳定、镇定[4]、时滞[5]等.
另一方面,出于执行器幅值的限制或者安全因素的原因,执行器饱和的存在严重影响系统性能甚至导致系统不稳定.所以,对执行器饱和的研究有很大的实际意义.近年来,越来越多的学者也研究这类问题,如在稳定性分析与镇定[6]、容错控制[7]、时滞[8]、Markov跳变[9]等方面.同时,随机系统的研究也已成为热点,例如网络控制[10]、扰动[11]和多时滞[12].然而,考虑带有执行器饱和的Markov跳变系统的相关文献却很少.
现有的文献大都建立在Lyapunov稳定基础上,意味着系统在无限时间范围内是稳定的.相对于Lyapunov稳定性,有限时间稳定性意味着状态在固定时间区域内不会超过一定的界限[13 - 15].值得关注的是有限时间稳定并不意味着Lyapunov稳定.如果系统响应的瞬态超过预定的界限,Lyapunov稳定并不包含有限时间稳定.
研究一类转移概率部分已知的随机Markov饱和跳变系统的非脆弱有限时间镇定问题,通过参数依赖型Lyapunov函数和自由权矩阵方法,设计了非脆弱有限时间状态反馈控制器并获得了吸引域的最大估计值.
1 问题描述及相关引理考虑如下带有执行器饱和的随机Markov跳变系统:
(1) |
本文将设计非脆弱状态反馈控制器:
(2) |
文中考虑转移概率部分已知,意味着在矩阵Π={πij}中只有一部分元素能够得到.对于i∈S,集合Si=Ski∪Suki,其中
定义1[15] 对于给定的时间常数T,系统(1)(u(t)=0)是关于(c1,c2,T,R i)有限时间稳定的,如果下列条件成立:x0TRix0≤c1?xT(t)Rix(t)<c2,?t∈[0,T],其中0<c1<c2,Ri>0.
对于任意矩阵Pi>0,定义椭圆
考虑控制器(2),得到如下闭环系统:
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
证明 选择Lyapunov函数为
(9) |
(10) |
对于?j∈Suki,如果i∈Ski,条件(4)~(5)和πij≥0(?i,j∈S,i≠j)得到 Π ij<0 .另一方面,对于?j∈Suki,如果i∈Suki,由条件(4)~(6)和
其次,对ΓV(x(t))<αV(x(t)),t∈[0,T]两边积分得
(11) |
(12) |
(13) |
(14) |
(15) |
证毕.
2.2 有限时间状态反馈控制器设计和吸引域估计下面设计控制器并得到系统均方意义下吸引域估计值.假设如下最优化问题:
(16) |
令β=a-2,Xi=Pi-1,Yi=KiXi,Di=FiXi,Vi=XiQiXi,注意问题(16)中的①等价于a2(x0g)TPix0g≤1(g=1,2,…,ρ).应用Schur补引理,可得到
(17) |
(18) |
对式(4)两边同时乘以对角矩阵diag{Xi,I,I},可以得到
(19) |
情况1 当i∈Ski,应用Schur补引理,不等式(19)中相应参数为
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
3 数值例子四模态随机Markov跳变系统参数如下:
图 1(Fig. 1)
图 1 跳变模态Fig.1 Jumping mode |
图 2(Fig. 2)
图 2 状态轨迹Fig.2 State trajectories |
图 3(Fig. 3)
图 3 xT(t)Rix(t)的轨迹Fig.3 Evolution of xT(t)Rix(t) |
4 结 语本文针对执行器饱和Markov跳变系统的非脆弱有限时间镇定,设计了非脆弱有限时间状态反馈控制器.通过线性矩阵不等式的方式,实现了控制器和吸引域最大化的求解.
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