区间变时滞广义系统的鲁棒稳定与镇定
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摘要
时滞是客观世界及工程实际中普遍存在的现象,时滞的存在常常导致系统不稳定或性能恶化。因此,对时滞系统的研究具有重要的理论意义与应用价值,近几十年来已引起人们极大的关注。然而,现有大部分文献考虑的时滞变化范围都是从0到某个上界,实际上,区间变时滞常常出现,即时滞在某个下界不限制到零的区间里变化。一个典型的区间变时滞系统的例子就是网络控制系统。由于考虑时滞下界的信息,其复杂性和难度比时滞系统大,近十年来才逐渐发展起来。另一方面,广义系统是一类形式更一般化,并有着广泛应用背景的动力系统。关于区间变时滞广义系统稳定性和镇定问题的研究至今仍少有人涉及,是一个公开但富有挑战性的课题。
本文首先介绍了正常时滞系统、区间时滞系统和广义系统的研究背景及意义,详细分析了时滞相关稳定研究的主要方法的本质特征,揭示了这些方法之间的联系及其局限性。然后从Lyapunov-Krasovskii泛函(LKF)的选取、研究方法以及研究对象三方面,讨论区间变时滞系统的研究现状,并指出了这一领域存在的问题。
研究了不确定区间变时滞广义系统的鲁棒稳定性问题。利用Jensen积分不等式方法,构造一个广义的LKF,引入一些自由权矩阵变量,以线性矩阵不等式(LMI)的形式给出了系统时滞相关的稳定准则。和张先明提出的积分不等式广义形式的方法所得结果做了比较,从数学理论上证明了用这两种方法所得结果是等价的这一结论,但Jensen积分不等式方法无需引入多余自由权矩阵变量,因此运用起来更方便、简单。针对时滞连续不可微(Case1)和连续且可微(Case 2)两种情况,分别获得了保证系统对所有容许的不确定性均正则、无脉冲且鲁棒稳定的时滞相关充分条件,同时建立了这两类条件之间的关系。
在区间变时滞广义系统时滞相关稳定性分析的基础上,设计系统的时滞相关镇定的四种形式的状态反馈控制器,即无记忆状态反馈控制器、有记忆状态反馈控制器以及增益摄动为加性和乘性的弹性(非脆弱)无记忆状态反馈控制器,给出了相应的闭环系统对所有容许的不确定性均为正则、无脉冲且鲁棒稳定的充分条件,采用参数调整方法,把充分条件中的矩阵不等式含有的关于未知变量的非线性项转化为线性项。
针对具有区间变状态时滞和输入时滞不确定广义系统的鲁棒镇定问题进行了研究。在所考虑的系统模型中,除了矩阵E,其余的系数矩阵均含有范数有界不确定性。基于得到的区间变时滞广义系统时滞相关稳定性判据,利用广义二次镇定的思想,在无记忆状态反馈作用下,给出了无记忆鲁棒镇定控制器存在的时滞相关型充分条件。同样利用参数调整方法,把非LMIs形式的充分条件转化为LMI.接下来还考虑了系统仅具有区间变输入时滞的特殊情况。与还原法相比,用此方法设计的控制器结构更简单,更容易实现。
考虑基于T-S模糊模型的区间变时滞模糊广义系统,利用Jensen积分不等式方法,对系统的时滞相关稳定性进行分析,并将其推广到具有时变结构不确定性系统,同时在稳定性分析所得结论的基础上,采用参数调整方法,分别设计无记忆、有记忆和增益摄动为加性和乘性的模糊弹性状态反馈控制器,给出了LMIs形式的不确定模糊闭环广义系统一致正则、一致无脉冲且鲁棒稳定的时滞相关的充分条件。
文中每章后面都有数值实例,以说明所用方法的有效性。
Time delays often occur in many practical and engineering fields and they are often sources of instability and degradation in control systems, so the study of time-delay systems is important both in theory and in practice, and has thus been of great interest to a large number of researchers in the last few decades. However, the range of time-varying delay considered in most exsting literature is from 0 to an upper bound. In practice, a time-varying interval delay is often encounterd, that is, the range of delay varies in an interval for which the lower bound is not restricted to be 0. A typical example of dynamical systems with interval time-varying delay is networked control systems. Since the information of the lower bound of delay must be considered in systems with interval time-varying delay, it is much more complex and difficult to study systems with interval time-varying delays than general time-delay systems and this type of systems has been gradually developed during the last decade. On the other hand, singular system is a kind of dynamic system of more general form and abroad applied background. Up to now, very few papers have investigated the stability and stabilization problem of singular systems with interval time-varying delays, which remains open but challenging.
In this dissertation, the background and significance of normal systems with time delays, systems with interval time-varying delays and singular systems are first introduced. The basic characteristics of main methods used to handle delay-dependent stability problem are analyzed in detail, and then the connection among these methods and the limitations of these methods are revealed. Then, the present study situation for systems with interval time-varying delays is introduced from three respects, i.e. selection of Lyapunov-Krasovskii functional (LKF), research methods and research objects, and some open problems in this field are pointed out as well.
The robust stability problem for uncertain singular systems with interval time-varying delays is investigated. By using Jensen integral inequality method, constructing the general LKF and introducing some free weighting matrices, a delay-dependent stability criterion for the systems is obtained in terms of linear matrix inequality (LMI). The result is compared with the result by using general integral inequality approach proposed by Zhang xianming, and the conclusion is proved in methematics theory that the results in terms of the two methods are equivalent, but redundant free-weighting matrices are not introduced by using Jensen integral inequality method, so it is more convenient and simpler. Two types of time delay are discussed: continuous but not differential (Case 1) and continuous and differential (Case 2); and two classes of delay-dependent stability conditions are respectively given for the systems to be regular, impulse-free and robustly stable for all admissible uncertainties. Moreover, the relationship between these two classes is also expained.
Based on the delay-dependent stability criterion for uncertain singular systems with interval time-varying delays thus derived, four types of state feedback controllers to stabilize the systems are designed, which are memoryless controllers, memorical controllers, resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations. The delay-dependent robust stability conditions are obtained for the corresponding closed-loop systems to be regular, impulse-free and robust stable for all admissible uncertainties. In addition, by adjusting the parameters, the nonlinear terms in the sufficient conditions are transformed into linear ones.
The problem of robust stabilization for uncertain singular systems with interval time-varying state and input delays is considered. All the coefficient matrices except the matrix E include norm-bounded uncertainties. Based on the delay-dependent stability criterion for nominal singular systems with interval time-varying delays obtained, by using the idea of generalized quadratic stabilization, the sufficient conditions for the existence of the robust stabilizing memoryless state feedback controllers are given. In the same way, by adjusting the parameters, the conditions in terms of nonlinear LMIs are converted to ones in terms of LMIs. Following that, the special case in which the systems have only the interval time-varying input delays is considered. Compared with the reduction method, the controllers designed by using this method have simpler structures and can be more easily implemented.
For uncertain fuzzy singular systems with interval time-varying state delays based on T-S fuzzy model, the Jensen integral inequality method is used to study the delay-dependent stability problem, and the results are extended to systems with time-varying structured uncertainties. Then, based on the stability analysis, memoryless controllers, memorical controllers and resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations are designed. The delay-dependent robust stability conditions in terms of LMIs are obtained for the corresponding closed-loop fuzzy systems to be uniformly regular, uniformly impulse-free and robust stable for all admissible uncertainties.
Numerical examples are given to demonstrate the effectiveness of the proposed methods at the end of each chapter in this dissertation.
本文首先介绍了正常时滞系统、区间时滞系统和广义系统的研究背景及意义,详细分析了时滞相关稳定研究的主要方法的本质特征,揭示了这些方法之间的联系及其局限性。然后从Lyapunov-Krasovskii泛函(LKF)的选取、研究方法以及研究对象三方面,讨论区间变时滞系统的研究现状,并指出了这一领域存在的问题。
研究了不确定区间变时滞广义系统的鲁棒稳定性问题。利用Jensen积分不等式方法,构造一个广义的LKF,引入一些自由权矩阵变量,以线性矩阵不等式(LMI)的形式给出了系统时滞相关的稳定准则。和张先明提出的积分不等式广义形式的方法所得结果做了比较,从数学理论上证明了用这两种方法所得结果是等价的这一结论,但Jensen积分不等式方法无需引入多余自由权矩阵变量,因此运用起来更方便、简单。针对时滞连续不可微(Case1)和连续且可微(Case 2)两种情况,分别获得了保证系统对所有容许的不确定性均正则、无脉冲且鲁棒稳定的时滞相关充分条件,同时建立了这两类条件之间的关系。
在区间变时滞广义系统时滞相关稳定性分析的基础上,设计系统的时滞相关镇定的四种形式的状态反馈控制器,即无记忆状态反馈控制器、有记忆状态反馈控制器以及增益摄动为加性和乘性的弹性(非脆弱)无记忆状态反馈控制器,给出了相应的闭环系统对所有容许的不确定性均为正则、无脉冲且鲁棒稳定的充分条件,采用参数调整方法,把充分条件中的矩阵不等式含有的关于未知变量的非线性项转化为线性项。
针对具有区间变状态时滞和输入时滞不确定广义系统的鲁棒镇定问题进行了研究。在所考虑的系统模型中,除了矩阵E,其余的系数矩阵均含有范数有界不确定性。基于得到的区间变时滞广义系统时滞相关稳定性判据,利用广义二次镇定的思想,在无记忆状态反馈作用下,给出了无记忆鲁棒镇定控制器存在的时滞相关型充分条件。同样利用参数调整方法,把非LMIs形式的充分条件转化为LMI.接下来还考虑了系统仅具有区间变输入时滞的特殊情况。与还原法相比,用此方法设计的控制器结构更简单,更容易实现。
考虑基于T-S模糊模型的区间变时滞模糊广义系统,利用Jensen积分不等式方法,对系统的时滞相关稳定性进行分析,并将其推广到具有时变结构不确定性系统,同时在稳定性分析所得结论的基础上,采用参数调整方法,分别设计无记忆、有记忆和增益摄动为加性和乘性的模糊弹性状态反馈控制器,给出了LMIs形式的不确定模糊闭环广义系统一致正则、一致无脉冲且鲁棒稳定的时滞相关的充分条件。
文中每章后面都有数值实例,以说明所用方法的有效性。
Time delays often occur in many practical and engineering fields and they are often sources of instability and degradation in control systems, so the study of time-delay systems is important both in theory and in practice, and has thus been of great interest to a large number of researchers in the last few decades. However, the range of time-varying delay considered in most exsting literature is from 0 to an upper bound. In practice, a time-varying interval delay is often encounterd, that is, the range of delay varies in an interval for which the lower bound is not restricted to be 0. A typical example of dynamical systems with interval time-varying delay is networked control systems. Since the information of the lower bound of delay must be considered in systems with interval time-varying delay, it is much more complex and difficult to study systems with interval time-varying delays than general time-delay systems and this type of systems has been gradually developed during the last decade. On the other hand, singular system is a kind of dynamic system of more general form and abroad applied background. Up to now, very few papers have investigated the stability and stabilization problem of singular systems with interval time-varying delays, which remains open but challenging.
In this dissertation, the background and significance of normal systems with time delays, systems with interval time-varying delays and singular systems are first introduced. The basic characteristics of main methods used to handle delay-dependent stability problem are analyzed in detail, and then the connection among these methods and the limitations of these methods are revealed. Then, the present study situation for systems with interval time-varying delays is introduced from three respects, i.e. selection of Lyapunov-Krasovskii functional (LKF), research methods and research objects, and some open problems in this field are pointed out as well.
The robust stability problem for uncertain singular systems with interval time-varying delays is investigated. By using Jensen integral inequality method, constructing the general LKF and introducing some free weighting matrices, a delay-dependent stability criterion for the systems is obtained in terms of linear matrix inequality (LMI). The result is compared with the result by using general integral inequality approach proposed by Zhang xianming, and the conclusion is proved in methematics theory that the results in terms of the two methods are equivalent, but redundant free-weighting matrices are not introduced by using Jensen integral inequality method, so it is more convenient and simpler. Two types of time delay are discussed: continuous but not differential (Case 1) and continuous and differential (Case 2); and two classes of delay-dependent stability conditions are respectively given for the systems to be regular, impulse-free and robustly stable for all admissible uncertainties. Moreover, the relationship between these two classes is also expained.
Based on the delay-dependent stability criterion for uncertain singular systems with interval time-varying delays thus derived, four types of state feedback controllers to stabilize the systems are designed, which are memoryless controllers, memorical controllers, resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations. The delay-dependent robust stability conditions are obtained for the corresponding closed-loop systems to be regular, impulse-free and robust stable for all admissible uncertainties. In addition, by adjusting the parameters, the nonlinear terms in the sufficient conditions are transformed into linear ones.
The problem of robust stabilization for uncertain singular systems with interval time-varying state and input delays is considered. All the coefficient matrices except the matrix E include norm-bounded uncertainties. Based on the delay-dependent stability criterion for nominal singular systems with interval time-varying delays obtained, by using the idea of generalized quadratic stabilization, the sufficient conditions for the existence of the robust stabilizing memoryless state feedback controllers are given. In the same way, by adjusting the parameters, the conditions in terms of nonlinear LMIs are converted to ones in terms of LMIs. Following that, the special case in which the systems have only the interval time-varying input delays is considered. Compared with the reduction method, the controllers designed by using this method have simpler structures and can be more easily implemented.
For uncertain fuzzy singular systems with interval time-varying state delays based on T-S fuzzy model, the Jensen integral inequality method is used to study the delay-dependent stability problem, and the results are extended to systems with time-varying structured uncertainties. Then, based on the stability analysis, memoryless controllers, memorical controllers and resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations are designed. The delay-dependent robust stability conditions in terms of LMIs are obtained for the corresponding closed-loop fuzzy systems to be uniformly regular, uniformly impulse-free and robust stable for all admissible uncertainties.
Numerical examples are given to demonstrate the effectiveness of the proposed methods at the end of each chapter in this dissertation.
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