线性时滞系统有限时间稳定性分析与综合
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摘要
相比于传统的渐近稳定性,有限时间稳定性能够更好地刻画系统在一段特定时间区间内的暂态行为。给定系统初始条件的界,如果系统的状态在一段特定时间区间内始终不超出某个设定区域,则称此系统是有限时间稳定的。因此,对于一些工作时间短暂的系统或必须要求系统状态在给定界限内的实际场合,可以基于有限时间稳定性来进行系统分析和控制。
本文从有限时间稳定相关概念出发,利用类Lyapunov函数和时滞微分不等式等分析方法,讨论线性时滞系统、线性离散时滞奇异系统和线性脉冲时滞切换系统的有限时间稳定分析与综合问题,并在此基础上,将有限时间稳定相关结果应用于线性网络控制系统和线性结构系统的控制器设计中。本文的主要工作包括以下几个方面：
·研究了线性离散变时滞系统的有限时间稳定分析与综合及线性连续常时滞系统的输入—输出有限时间稳定分析与综合问题。基于类Lyapunov函数的方法,分别获得了线性离散变时滞系统有限时间稳定及镇定的充分条件和线性连续常时滞系统输入—输出有限时间稳定及镇定的充分条件。数值仿真说明了所得结果的有效性。
·将有限时间稳定的概念推广到离散奇异系统中,给出了可容许有限时间稳定的定义。针对无时滞和具有时变时滞的不确定线性离散奇异系统,采用类Lyapunov函数方法,分别获得了可容许有限时间稳定的充分条件。在此基础上,分别设计出状态反馈鲁棒控制器以保证两类闭环系统的可容许有限时间稳定性。
·讨论了线性脉冲变时滞切换系统的有限时间稳定分析问题。与常用的类Lyapunov函数分析方法不同,本文基于时滞微分不等式分析方法得出了系统有限时间稳定的充分条件,此条件可表示为一些代数不等式。相比于类Lyapunov函数方法,采用时滞微分不等式方法所得的有限时间稳定条件较容易验证,且不要求每个子系统都是有限时间稳定的。
·将线性时滞系统有限时间稳定相关结果应用于网络控制系统和结构系统的控制器设计中。对于网络控制系统,设计出一种混合控制器,保证闭环系统在正常情况下是渐近稳定的,在异常情况下(即在某些特殊时间区间内出现了较大的网络诱导时延或丢包)是有限时间稳定的。对于带输入时滞的结构系统,综合考虑了渐近稳定性和输入—输出有限时间稳定性来设计控制器从而达到了限制某些变量幅值的目的。针对这两类系统,分别通过仿真例子验证了所设计控制器的有效性。
Finite-time stability can better describe the transient behavior of a system over a cer-tain time interval than the traditional asymptotic stability. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exit a certain domain dur-ing a specified time interval. Therefore, for systems that are known to operate only over a short time interval or whenever, from practical considerations, the system state is required to remain within a prescribed bound, finite-time stability (FTS) can be used.
On the basis of the FTS-related concepts, by employing the Lyapunov-like function method or the delay differential inequality method, FTS analysis and synthesis problems are discussed for linear time-delay systems, linear discrete singular time-delay systems and linear impulsive switched time-delay systems. Then, the obtained FTS-related results are further applied to linear networked control systems and linear structural systems for controller design issues. The main contributions are summarized as follows:
·The problems of FTS analysis and synthesis are investigated for linear discrete sys-tems with time-varying delay. Correspondingly, the problems of input-output finite-time stability (IO-FTS) analysis and synthesis are studied for linear continuous systems with time-invariant delay. Some pertinent conditions are obtained based on the Lyapunov-like function method. Numerical examples are provided to illustrate the validity of the obtained results.
·A new FTS concept, which is defined as admissible finite-time stability (AFTS), is introduced into discrete singular systems. The problems of AFTS analysis and synthesis are addressed for two types of uncertain discrete singular systems (that is, without delay or with time-varying delay), respectively. By using the Lyapunov-like function method, sufficient conditions are proposed for the AFTS of the concerned two uncertain singular systems. Based on the AFTS analysis results, robust state-feedback controllers are designed respectively such that the correspondingly closed-loop systems are admissible finite-time stable for all admissible uncertainties.
·The FTS analysis problem is considered for a class of linear impulsive switched sys-tems with time-varying delay. The DDI method, rather than the commonly used Lyapunov-like function method, is employed to establish a sufficient condition for the system to be finite-time stable. This condition can be expressed in terms of some algebraic inequalities. Compared with the Lyapunov-like function method, the FTS conditions based on the DDI method are easier for checking and do not require FTS of each subsystem.
·The obtained FTS-related results on linear time-delay systems are applied to net-worked control systems (NCSs) and structural systems for controller design issues. For the NCS, a mixed controller design method, which guarantees the asymptotic stability of the closed-loop system in the usual case and the FTS of the closed-loop system in the un-usual case (that is, in some particular time intervals, large network-induced delay or packet dropout occurs), is presented. For the structural system with input delay, the controller is designed by taking account of both asymptotic stability and IO-FTS, which can result in limited amplitudes of some variables. For the above two types of systems, some sim-ulation examples are given respectively to demonstrate the effectiveness of the designed controllers.
本文从有限时间稳定相关概念出发,利用类Lyapunov函数和时滞微分不等式等分析方法,讨论线性时滞系统、线性离散时滞奇异系统和线性脉冲时滞切换系统的有限时间稳定分析与综合问题,并在此基础上,将有限时间稳定相关结果应用于线性网络控制系统和线性结构系统的控制器设计中。本文的主要工作包括以下几个方面：
·研究了线性离散变时滞系统的有限时间稳定分析与综合及线性连续常时滞系统的输入—输出有限时间稳定分析与综合问题。基于类Lyapunov函数的方法,分别获得了线性离散变时滞系统有限时间稳定及镇定的充分条件和线性连续常时滞系统输入—输出有限时间稳定及镇定的充分条件。数值仿真说明了所得结果的有效性。
·将有限时间稳定的概念推广到离散奇异系统中,给出了可容许有限时间稳定的定义。针对无时滞和具有时变时滞的不确定线性离散奇异系统,采用类Lyapunov函数方法,分别获得了可容许有限时间稳定的充分条件。在此基础上,分别设计出状态反馈鲁棒控制器以保证两类闭环系统的可容许有限时间稳定性。
·讨论了线性脉冲变时滞切换系统的有限时间稳定分析问题。与常用的类Lyapunov函数分析方法不同,本文基于时滞微分不等式分析方法得出了系统有限时间稳定的充分条件,此条件可表示为一些代数不等式。相比于类Lyapunov函数方法,采用时滞微分不等式方法所得的有限时间稳定条件较容易验证,且不要求每个子系统都是有限时间稳定的。
·将线性时滞系统有限时间稳定相关结果应用于网络控制系统和结构系统的控制器设计中。对于网络控制系统,设计出一种混合控制器,保证闭环系统在正常情况下是渐近稳定的,在异常情况下(即在某些特殊时间区间内出现了较大的网络诱导时延或丢包)是有限时间稳定的。对于带输入时滞的结构系统,综合考虑了渐近稳定性和输入—输出有限时间稳定性来设计控制器从而达到了限制某些变量幅值的目的。针对这两类系统,分别通过仿真例子验证了所设计控制器的有效性。
Finite-time stability can better describe the transient behavior of a system over a cer-tain time interval than the traditional asymptotic stability. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exit a certain domain dur-ing a specified time interval. Therefore, for systems that are known to operate only over a short time interval or whenever, from practical considerations, the system state is required to remain within a prescribed bound, finite-time stability (FTS) can be used.
On the basis of the FTS-related concepts, by employing the Lyapunov-like function method or the delay differential inequality method, FTS analysis and synthesis problems are discussed for linear time-delay systems, linear discrete singular time-delay systems and linear impulsive switched time-delay systems. Then, the obtained FTS-related results are further applied to linear networked control systems and linear structural systems for controller design issues. The main contributions are summarized as follows:
·The problems of FTS analysis and synthesis are investigated for linear discrete sys-tems with time-varying delay. Correspondingly, the problems of input-output finite-time stability (IO-FTS) analysis and synthesis are studied for linear continuous systems with time-invariant delay. Some pertinent conditions are obtained based on the Lyapunov-like function method. Numerical examples are provided to illustrate the validity of the obtained results.
·A new FTS concept, which is defined as admissible finite-time stability (AFTS), is introduced into discrete singular systems. The problems of AFTS analysis and synthesis are addressed for two types of uncertain discrete singular systems (that is, without delay or with time-varying delay), respectively. By using the Lyapunov-like function method, sufficient conditions are proposed for the AFTS of the concerned two uncertain singular systems. Based on the AFTS analysis results, robust state-feedback controllers are designed respectively such that the correspondingly closed-loop systems are admissible finite-time stable for all admissible uncertainties.
·The FTS analysis problem is considered for a class of linear impulsive switched sys-tems with time-varying delay. The DDI method, rather than the commonly used Lyapunov-like function method, is employed to establish a sufficient condition for the system to be finite-time stable. This condition can be expressed in terms of some algebraic inequalities. Compared with the Lyapunov-like function method, the FTS conditions based on the DDI method are easier for checking and do not require FTS of each subsystem.
·The obtained FTS-related results on linear time-delay systems are applied to net-worked control systems (NCSs) and structural systems for controller design issues. For the NCS, a mixed controller design method, which guarantees the asymptotic stability of the closed-loop system in the usual case and the FTS of the closed-loop system in the un-usual case (that is, in some particular time intervals, large network-induced delay or packet dropout occurs), is presented. For the structural system with input delay, the controller is designed by taking account of both asymptotic stability and IO-FTS, which can result in limited amplitudes of some variables. For the above two types of systems, some sim-ulation examples are given respectively to demonstrate the effectiveness of the designed controllers.
引文
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