Markov跳跃系统的鲁棒控制
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摘要
Markov跳跃系统是一类同时包含相互作用的离散事件和连续变量的特殊混杂系统,它的提出具有很强的工程背景。随着科学技术的飞速发展,在实际系统中,如制造系统、生化系统、电力系统甚至经济系统等,常常会因内部部件故障、维修,或者受到环境扰动等因素影响而使系统结构发生变化,因而可以抽象成Markov跳跃系统模型。正是由于跳跃系统所具有的特殊混合信息结构,使得传统的单由时间驱动的动态系统的控制理论或者由单一针对离散事件动态系统的控制理论与控制方法已经难以适应Markov跳跃系统的多模态逻辑切换结构。因此研究跳跃系统的鲁棒控制具有十分重要的理论意义和实用价值。
本文基于Lyapunov稳定性理论,利用线性矩阵不等式(LMI)方法,结合弱无穷小算子,Dynkin公式,Grownwall-Bellman引理,Shur补引理、投影引理等数学工具,针对Markov跳跃系统的鲁棒控制问题做了深入的研究,建立了有效的系统性能准则,并基于该准则得到了一系列有意义的结果。文章由浅入深,从线性不确定Markov跳跃系统入手,逐渐深入解决更为普遍的模态依赖时滞Markov跳跃系统的控制、滤波器设计及模型降阶问题。本论文的工作包括以下几个方面:
第一,针对线性不确定Markov系统,研究了随机闭环系统在无干扰输入和不确定满足可容许条件下的随机鲁棒镇定问题,进而通过引入非逃逸集和可达集的概念,建立了系统的L_1性准则,研究了系统的混合L_1/H_∞控制问题。首先基于Lyapunov稳定性理论,得到了系统随机可镇定的条件,然后通过构建新的Lyapunov-Krasovskii泛函,得到了系统的L_1性准则。并在该性能准则基础上,利用LMI技术,得到了保证闭环系统随机稳定且具有L_1/H_∞性能的状态反馈控制器的设计方法。
第二,针对模态依赖的变时滞Markov跳跃系统,研究了系统的L_1控制及滤波器设计问题。采用以状态非逃逸集逼近可达集获得诱导L∞范数的方法,首次推导出模态依赖时滞跳跃系统依赖于模态的鲁棒L_1性能准则,并基于该性能准则,利用LMI技术得到依赖于模态的鲁棒L_1控制器存在的充分条件。进而研究了系统的滤波问题,给出了使系统随机稳定且具有L_1性能约束的滤波器存在的充分条件。
第三,将Markov跳跃参数引入到2D时滞系统中,针对一类基于Rosseor模型的离散时滞2D跳跃系统,分别对系统的鲁棒镇定,鲁棒H_∞控制,H_∞模型降阶问题进行了研究。首先基于Lyapunov函数研究使系统鲁棒可镇定的条件,进而给出使系统均方稳定且具有一定的干扰抑制水平的H_∞控制器存在的充分条件,并将控制器的设计问题转化为受LMI约束的凸优化问题。在以上基础上,进一步研究了系统的H_∞模型降阶问题。推导误差系统均方稳定且满足H_∞性能的充分条件,基于此条件采用投影定理给出了降阶系统的求解条件。因所给充分条件为非严格的线性矩阵不等式,根据锥补线性化(CCL)方法将矩阵不等式可解性问题转化为受LMI约束的凸优化问题,并给出了降阶模型参数化的迭代算法。
第四,将含有Markov跳跃参数的不确定时滞系统的鲁棒控制问题推广到含有Markov跳跃参数的不确定时滞中立系统中,研究了一类含有离散时滞和分布时滞的不确定中立型跳跃系统的H_2/H_∞控制问题,目的是设计一个状态反馈控制器使系统在满足H_∞指标的前提下,H_2性能最小。采用LMI推导出H_2/H_∞控制器存在的充分条件,并将控制器的设计问题转化为具有LMI约束的凸优化问题。
本文对主要设计方案都进行了数值仿真,仿真结果表明了所提出方案的有效性。
最后,对所做的工作进行了总结,并给出了Markov跳跃系统的鲁棒控制理的进一步的研究方向。
Markov jump systems as a special class of hybrid systems with interacting discrete event systems and continuous time dynamic systems are introduced for the strong engineering background. As the development of science and technology, it is quite common for practical systems, such as manufacturing systems, biology and chemistry systems, electric power systems as well as economic systems, having variable structures due to interconnection failures and repair, or environment disturbances, etc. Markovian jump models are often introduced to describe these characters. Due to the complicated structure,the research of jump systems with multi-model switching logic structure cannot be treated as simple combination of traditional control theories for the continuous systems or discrete events systems,respectively. Thus,the study on robust control for Markov jump systems has important theoretic significance and practical value.
This dissertation concerns with the problem of robust control for Markov jump systems, constructs some effective performance criteria and obtains a serial of significant results according to Lyapunov stability theory, Linear Matrix inequality(LMI), weak infinitesimal operator, Dynkin formula, Grownwall-Bellman lemma, Shur complement lemma, and projection lemma. This dissertation goes from the easy to the difficult and complicated, first treats with uncertain Markovian jump systems, then a more general and complex case, the problems of robust control, filter design and model reduction related to the Markov jump systems with mode dependent time-delays are intensive researched. The main research work of this dissertation is as follows:
a) Under the condition of without disturbance input, for all admissible uncertainties, the stochastic robust stabilization problem for a kind of linear uncertain Markov jump systems is studied, the L_1 performance criterion is established by the introduction of inescapable set and reachable set, and then research the mixed L_1/H_∞control problem. Based on Lyapunov stability theory, the stochastic robust stabilization condition is first given, then the L_1 performance criterion is obtained by choosing appropriate stochastic Lyapunov-Krasovskii function. Upon the proposed performance criterion, a controller design method that guarantees the closed-loop system is stochastically stable and has L_1/H_∞performance constraint is given in terms of a group of LMIs.
b) Consider a kind of Markov jump systems with mode-dependent time-varying delays, investigate the L_1 control and filtering problems. Based on the method of approximating the state reachable set with inescapable set to obtain the induced L∞norm, mode-dependent robust L_1 performance criterion is first proposed for mode-dependent time-delay jumping systems. Upon the proposed performance criterion, a sufficient condition for the existence of mode-dependent robust L_1 controller is given in terms of LMIs technology. Furthermore, the problem of robust L_1 filtering is investigated, a sufficient condition for the existence of robust L_1 filter that guarantees the filtering error system is stochastically stable and has L_1 performance constraint is given.
c) The markov jump parameters are intyoduced into 2D sysyems. The problems of robust stabilization, robust H_∞control and H_∞model reduction are studied for a class of discrete time 2D jump systems with state delays in Roesser model respectively. At The robust stabilization conditions based on Lyapunov function is proposed firstly, then a sufficient condition for the existence of H_∞controller that guarantees the close-loop system is stochastically stable and certain disturbance inhibition level is obtained, and the controller design problem is cast into a convex optimization problem subjected to LMIs. On the basis of the above, this paper further investigates the H_∞model reduction problem of 2D delayed jump systems. A delay-independent condition of stochastically stability with H_∞performance constraint for 2D delayed jump systems, and then the projection approach is exploited to solve the reduced-order systems. Since the obtained conditions are not expressed in LMI form, the cone complementary linearization method is applied to cast them into convex optimization problems subject to LMI constraints. An explicit parameterization iterative algorithm of the desired reduced-order models is also presented.
d) Expand the robust control problem for uncertain time-delay systems with Markov jump parameters to uncertain time-delay neutral systems with Markov jump parameters. The problem of robust H_2/H_∞control is investigated for a class of uncertain neutral jumping systems with discrete and distributed delays. It is desired to design a linear state feedback controller such that the H_2 performance measure is minimized while guaranteeing a prescribed H_∞norm bound on the controlled system. A sufficient condition for the existence of robust H_2/H_∞control law is given in terms of a group of LMIs. Furthermore, the controller design problem is converted into a convex optimization problem subjected to LMI constraints.
Besides,simulations are made for main design schemes,and simulation results show the effectiveness of the proposed approaches.
Finally,a brief review of this dissertation is given,then some future research areas and open problems in theory and practice are highlighted.
本文基于Lyapunov稳定性理论,利用线性矩阵不等式(LMI)方法,结合弱无穷小算子,Dynkin公式,Grownwall-Bellman引理,Shur补引理、投影引理等数学工具,针对Markov跳跃系统的鲁棒控制问题做了深入的研究,建立了有效的系统性能准则,并基于该准则得到了一系列有意义的结果。文章由浅入深,从线性不确定Markov跳跃系统入手,逐渐深入解决更为普遍的模态依赖时滞Markov跳跃系统的控制、滤波器设计及模型降阶问题。本论文的工作包括以下几个方面:
第一,针对线性不确定Markov系统,研究了随机闭环系统在无干扰输入和不确定满足可容许条件下的随机鲁棒镇定问题,进而通过引入非逃逸集和可达集的概念,建立了系统的L_1性准则,研究了系统的混合L_1/H_∞控制问题。首先基于Lyapunov稳定性理论,得到了系统随机可镇定的条件,然后通过构建新的Lyapunov-Krasovskii泛函,得到了系统的L_1性准则。并在该性能准则基础上,利用LMI技术,得到了保证闭环系统随机稳定且具有L_1/H_∞性能的状态反馈控制器的设计方法。
第二,针对模态依赖的变时滞Markov跳跃系统,研究了系统的L_1控制及滤波器设计问题。采用以状态非逃逸集逼近可达集获得诱导L∞范数的方法,首次推导出模态依赖时滞跳跃系统依赖于模态的鲁棒L_1性能准则,并基于该性能准则,利用LMI技术得到依赖于模态的鲁棒L_1控制器存在的充分条件。进而研究了系统的滤波问题,给出了使系统随机稳定且具有L_1性能约束的滤波器存在的充分条件。
第三,将Markov跳跃参数引入到2D时滞系统中,针对一类基于Rosseor模型的离散时滞2D跳跃系统,分别对系统的鲁棒镇定,鲁棒H_∞控制,H_∞模型降阶问题进行了研究。首先基于Lyapunov函数研究使系统鲁棒可镇定的条件,进而给出使系统均方稳定且具有一定的干扰抑制水平的H_∞控制器存在的充分条件,并将控制器的设计问题转化为受LMI约束的凸优化问题。在以上基础上,进一步研究了系统的H_∞模型降阶问题。推导误差系统均方稳定且满足H_∞性能的充分条件,基于此条件采用投影定理给出了降阶系统的求解条件。因所给充分条件为非严格的线性矩阵不等式,根据锥补线性化(CCL)方法将矩阵不等式可解性问题转化为受LMI约束的凸优化问题,并给出了降阶模型参数化的迭代算法。
第四,将含有Markov跳跃参数的不确定时滞系统的鲁棒控制问题推广到含有Markov跳跃参数的不确定时滞中立系统中,研究了一类含有离散时滞和分布时滞的不确定中立型跳跃系统的H_2/H_∞控制问题,目的是设计一个状态反馈控制器使系统在满足H_∞指标的前提下,H_2性能最小。采用LMI推导出H_2/H_∞控制器存在的充分条件,并将控制器的设计问题转化为具有LMI约束的凸优化问题。
本文对主要设计方案都进行了数值仿真,仿真结果表明了所提出方案的有效性。
最后,对所做的工作进行了总结,并给出了Markov跳跃系统的鲁棒控制理的进一步的研究方向。
Markov jump systems as a special class of hybrid systems with interacting discrete event systems and continuous time dynamic systems are introduced for the strong engineering background. As the development of science and technology, it is quite common for practical systems, such as manufacturing systems, biology and chemistry systems, electric power systems as well as economic systems, having variable structures due to interconnection failures and repair, or environment disturbances, etc. Markovian jump models are often introduced to describe these characters. Due to the complicated structure,the research of jump systems with multi-model switching logic structure cannot be treated as simple combination of traditional control theories for the continuous systems or discrete events systems,respectively. Thus,the study on robust control for Markov jump systems has important theoretic significance and practical value.
This dissertation concerns with the problem of robust control for Markov jump systems, constructs some effective performance criteria and obtains a serial of significant results according to Lyapunov stability theory, Linear Matrix inequality(LMI), weak infinitesimal operator, Dynkin formula, Grownwall-Bellman lemma, Shur complement lemma, and projection lemma. This dissertation goes from the easy to the difficult and complicated, first treats with uncertain Markovian jump systems, then a more general and complex case, the problems of robust control, filter design and model reduction related to the Markov jump systems with mode dependent time-delays are intensive researched. The main research work of this dissertation is as follows:
a) Under the condition of without disturbance input, for all admissible uncertainties, the stochastic robust stabilization problem for a kind of linear uncertain Markov jump systems is studied, the L_1 performance criterion is established by the introduction of inescapable set and reachable set, and then research the mixed L_1/H_∞control problem. Based on Lyapunov stability theory, the stochastic robust stabilization condition is first given, then the L_1 performance criterion is obtained by choosing appropriate stochastic Lyapunov-Krasovskii function. Upon the proposed performance criterion, a controller design method that guarantees the closed-loop system is stochastically stable and has L_1/H_∞performance constraint is given in terms of a group of LMIs.
b) Consider a kind of Markov jump systems with mode-dependent time-varying delays, investigate the L_1 control and filtering problems. Based on the method of approximating the state reachable set with inescapable set to obtain the induced L∞norm, mode-dependent robust L_1 performance criterion is first proposed for mode-dependent time-delay jumping systems. Upon the proposed performance criterion, a sufficient condition for the existence of mode-dependent robust L_1 controller is given in terms of LMIs technology. Furthermore, the problem of robust L_1 filtering is investigated, a sufficient condition for the existence of robust L_1 filter that guarantees the filtering error system is stochastically stable and has L_1 performance constraint is given.
c) The markov jump parameters are intyoduced into 2D sysyems. The problems of robust stabilization, robust H_∞control and H_∞model reduction are studied for a class of discrete time 2D jump systems with state delays in Roesser model respectively. At The robust stabilization conditions based on Lyapunov function is proposed firstly, then a sufficient condition for the existence of H_∞controller that guarantees the close-loop system is stochastically stable and certain disturbance inhibition level is obtained, and the controller design problem is cast into a convex optimization problem subjected to LMIs. On the basis of the above, this paper further investigates the H_∞model reduction problem of 2D delayed jump systems. A delay-independent condition of stochastically stability with H_∞performance constraint for 2D delayed jump systems, and then the projection approach is exploited to solve the reduced-order systems. Since the obtained conditions are not expressed in LMI form, the cone complementary linearization method is applied to cast them into convex optimization problems subject to LMI constraints. An explicit parameterization iterative algorithm of the desired reduced-order models is also presented.
d) Expand the robust control problem for uncertain time-delay systems with Markov jump parameters to uncertain time-delay neutral systems with Markov jump parameters. The problem of robust H_2/H_∞control is investigated for a class of uncertain neutral jumping systems with discrete and distributed delays. It is desired to design a linear state feedback controller such that the H_2 performance measure is minimized while guaranteeing a prescribed H_∞norm bound on the controlled system. A sufficient condition for the existence of robust H_2/H_∞control law is given in terms of a group of LMIs. Furthermore, the controller design problem is converted into a convex optimization problem subjected to LMI constraints.
Besides,simulations are made for main design schemes,and simulation results show the effectiveness of the proposed approaches.
Finally,a brief review of this dissertation is given,then some future research areas and open problems in theory and practice are highlighted.
引文
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