不确定时滞系统的鲁棒稳定与镇定研究
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摘要
实际工业过程、生物和社会经济等控制系统中往往会存在时滞现象,而时滞特性常常又会严重影响到控制系统的性能指标甚至使得系统不稳定。所以时滞系统的研究多年来得到了众多学者的广泛关注。另一方面,控制系统的设计所基于的数学模型一般仅仅是被控对象的近似,因为被控对象的复杂性使得我们得到的模型跟实际对象的特性之间存在一定的差距,这种差距可以视为系统模型的不确定性。而且多数情况下我们可以获知此不确定性大小的某种度量,鲁棒控制正是研究在系统模型存在不确定性时如何设计控制器使闭环系统是稳定的,且满足一定的动态性能。
由于时滞依赖的条件比时滞独立具有更小的保守性,本文研究了一些不确定定常和时变时滞系统的时滞依赖鲁棒稳定性分析与控制问题。以线性矩阵不等式为工具,采用奇异系统模型变换的方法或引入自由权值矩阵变量来表示模型变换中的相关量,获得系统时滞依赖的稳定性条件。分别对于不确定网络化控制系统,采用Lyapunov理论及奇异系统模型变换的方法,研究其鲁棒控制问题。对于奇异系统,则引入自由权值矩阵变量研究其鲁棒控制问题。而对于Lurie系统,则主要基于时滞分段的Lyapunov函数方法得到这类系统鲁棒绝对稳定的时滞依赖新判据。
具体研究内容包括如下几个方面:
(1)针对一类具有范数有界不确定参数的线性和非线性网络化控制系统,通过构造适当的Lyapunov泛函和奇异系统模型变换方法导出该系统鲁棒可镇定的充分条件并给出相应的状态反馈控制律设计算法,结论表示为线性矩阵不等式。
(2)针对一类具有范数有界不确定参数的连续线性奇异时滞系统,通过引入自由权值变量来描述快变子系统和慢变子系统之间的代数关系,得到该系统时滞依赖鲁棒稳定和镇定的充分条件,结论表示为线性矩阵不等式。同其他方法相比较,本文给出的方法简洁明了,而且保守性低。
(3)针对一类具有范数有界不确定参数的中立型时滞Lurie直接控制系统,且时滞为时变情形,通过引入自由权值矩阵变量和S过程的方法研究了该类系统的时滞依赖鲁棒绝对稳定性问题,所得结论由线性矩阵不等式描述易于求解。
(4)针对一类具有范数有界不确定参数的中立型时滞Lurie间接控制系统,通过引入Jenson不等式和构造适当的Lyapunov泛函方法得到了系统时滞依赖鲁棒绝对稳定性判据,结论由线性矩阵不等式描述。
(5)基于时滞分段的Lyapunov函数方法得到一般Lurie系统的鲁棒绝对稳定的时滞依赖新判据,结论表示为线性矩阵不等式形式。
论文最后对引入自由权值矩阵变量方法以及时滞分段的Lyapunov函数方法在网络控制系统、奇异系统以及中立型Lurie非线性控制系统的时滞依赖鲁棒稳定和镇定的应用进行了总结,并且对今后的研究作了展望。
Time delays are frequently encountered in various systems, such as industrial engineering systems, biological systems and social economics models. The stability and performance of such systems are often influenced seriously by the time delay. Therefore, the study of time-delay systems has attracted a great deal of attention over the past decades by many researchers. On the other hand, the mathematical model for control system design is only an approximate description to the real plant owing to the complexity of controlled plant. So it is always impossible to obtain accurate mathematics model for the practical controlled plant. There exists some difference between the mathematical model and the real system. And the difference of which some measure is able to get usually can be described as the uncertainties in the model arguments. The objective of robust control theory is to design a control law such that the resultant closed-loop system is stable and satisfies some prescribed performance for all admissible uncertainties.
Since delay-dependent criteria are less conservative than delay-independent ones, the robust stability and stabilization problem for some uncertain systems with time-invariant and time-variant delays is studied in this dissertation. On the basis of linear matrix inequality (LMI) technique, the method of descriptor system model transformation and the free-weighting matrix approach are employed to obtain delay-dependent criteria for the stability of the systems. For uncertain networked control systems, the problem of robust control for uncertain continuous linear time-delay systems based on Lyapunov theory and descriptor system model transformation method is studied. For singular systems, the problems of robust stabilization is studied based on free-weighting matrix approach. And for Lurie systems, by using a new Lyapunov-krasovskii functional which splits the whole delay interval into two subintervals and defines a different energy function on each subinterval, some new delay-dependent criteria are presented to guarantee the robust absolute stability for the systems.
The main contents of this dissertation are outlined as follows,
(1) The problem of robust stabilization for a class of linear and nonlinear networked control systems with norm-bounded uncertain parameters is studied. Based on Lyapunov theory and descriptor system model transformation ap- proach, some sufficient conditions for the stabilization and state feedback control law are given in terms of LMI.
(2) The problem of delay-dependent robust stabilization for norm-bounded uncertain continuous-time linear singular time-delay systems is investigated. By decomposing the nominal system into slow and fast subsystems, a delay-dependent condition is presented for the system to be regular, impulse free and stable, based on which, the robust stabilization problem is studied and the explicit expression for the corresponding state-feedback control law is given in terms of LMI. Compared with other methods, these criteria are easy to compute and less conservative than previous ones.
(3) The problem of absolute stability and robust stability of a class of neutral Lurie direct control systems with mixed time-varying delays is investigated. By introducing the free-weighting matrix approach and s procedure, some delay-dependent criteria for absolute stability of the systems are gained. And the conditions are formulated by LMI which is easy to solve.
(4) The problem of absolute stability and robust stability of a class of neutral Lurie indirect control systems with time delays is investigated. By using Jenson inequality and introducing some free-weighting matrices, some delay-dependent robust absolute stability criteria are presented in terms of strict linear matrix inequalities (LMIs).
(5) The problem of robust absolute stability for a class of general Lurie systems with time delay is addressed. By using a new Lyapunov-krasovskii functional which splits the whole delay interval into two subintervals and defines a different energy function on each subinterval, some new delay-dependent criteria are presented and formulated by LMI.
Finally, how the free-weighting matrix and new Lyapunov-krasovskii functional approach are used to get the delay-dependent robust stability and stabilization problem for networked control systems, singular systems and neutral Lurie control systems with delays is outlined. And the perspective of future studies is refered at the end of the dissertation.
由于时滞依赖的条件比时滞独立具有更小的保守性,本文研究了一些不确定定常和时变时滞系统的时滞依赖鲁棒稳定性分析与控制问题。以线性矩阵不等式为工具,采用奇异系统模型变换的方法或引入自由权值矩阵变量来表示模型变换中的相关量,获得系统时滞依赖的稳定性条件。分别对于不确定网络化控制系统,采用Lyapunov理论及奇异系统模型变换的方法,研究其鲁棒控制问题。对于奇异系统,则引入自由权值矩阵变量研究其鲁棒控制问题。而对于Lurie系统,则主要基于时滞分段的Lyapunov函数方法得到这类系统鲁棒绝对稳定的时滞依赖新判据。
具体研究内容包括如下几个方面:
(1)针对一类具有范数有界不确定参数的线性和非线性网络化控制系统,通过构造适当的Lyapunov泛函和奇异系统模型变换方法导出该系统鲁棒可镇定的充分条件并给出相应的状态反馈控制律设计算法,结论表示为线性矩阵不等式。
(2)针对一类具有范数有界不确定参数的连续线性奇异时滞系统,通过引入自由权值变量来描述快变子系统和慢变子系统之间的代数关系,得到该系统时滞依赖鲁棒稳定和镇定的充分条件,结论表示为线性矩阵不等式。同其他方法相比较,本文给出的方法简洁明了,而且保守性低。
(3)针对一类具有范数有界不确定参数的中立型时滞Lurie直接控制系统,且时滞为时变情形,通过引入自由权值矩阵变量和S过程的方法研究了该类系统的时滞依赖鲁棒绝对稳定性问题,所得结论由线性矩阵不等式描述易于求解。
(4)针对一类具有范数有界不确定参数的中立型时滞Lurie间接控制系统,通过引入Jenson不等式和构造适当的Lyapunov泛函方法得到了系统时滞依赖鲁棒绝对稳定性判据,结论由线性矩阵不等式描述。
(5)基于时滞分段的Lyapunov函数方法得到一般Lurie系统的鲁棒绝对稳定的时滞依赖新判据,结论表示为线性矩阵不等式形式。
论文最后对引入自由权值矩阵变量方法以及时滞分段的Lyapunov函数方法在网络控制系统、奇异系统以及中立型Lurie非线性控制系统的时滞依赖鲁棒稳定和镇定的应用进行了总结,并且对今后的研究作了展望。
Time delays are frequently encountered in various systems, such as industrial engineering systems, biological systems and social economics models. The stability and performance of such systems are often influenced seriously by the time delay. Therefore, the study of time-delay systems has attracted a great deal of attention over the past decades by many researchers. On the other hand, the mathematical model for control system design is only an approximate description to the real plant owing to the complexity of controlled plant. So it is always impossible to obtain accurate mathematics model for the practical controlled plant. There exists some difference between the mathematical model and the real system. And the difference of which some measure is able to get usually can be described as the uncertainties in the model arguments. The objective of robust control theory is to design a control law such that the resultant closed-loop system is stable and satisfies some prescribed performance for all admissible uncertainties.
Since delay-dependent criteria are less conservative than delay-independent ones, the robust stability and stabilization problem for some uncertain systems with time-invariant and time-variant delays is studied in this dissertation. On the basis of linear matrix inequality (LMI) technique, the method of descriptor system model transformation and the free-weighting matrix approach are employed to obtain delay-dependent criteria for the stability of the systems. For uncertain networked control systems, the problem of robust control for uncertain continuous linear time-delay systems based on Lyapunov theory and descriptor system model transformation method is studied. For singular systems, the problems of robust stabilization is studied based on free-weighting matrix approach. And for Lurie systems, by using a new Lyapunov-krasovskii functional which splits the whole delay interval into two subintervals and defines a different energy function on each subinterval, some new delay-dependent criteria are presented to guarantee the robust absolute stability for the systems.
The main contents of this dissertation are outlined as follows,
(1) The problem of robust stabilization for a class of linear and nonlinear networked control systems with norm-bounded uncertain parameters is studied. Based on Lyapunov theory and descriptor system model transformation ap- proach, some sufficient conditions for the stabilization and state feedback control law are given in terms of LMI.
(2) The problem of delay-dependent robust stabilization for norm-bounded uncertain continuous-time linear singular time-delay systems is investigated. By decomposing the nominal system into slow and fast subsystems, a delay-dependent condition is presented for the system to be regular, impulse free and stable, based on which, the robust stabilization problem is studied and the explicit expression for the corresponding state-feedback control law is given in terms of LMI. Compared with other methods, these criteria are easy to compute and less conservative than previous ones.
(3) The problem of absolute stability and robust stability of a class of neutral Lurie direct control systems with mixed time-varying delays is investigated. By introducing the free-weighting matrix approach and s procedure, some delay-dependent criteria for absolute stability of the systems are gained. And the conditions are formulated by LMI which is easy to solve.
(4) The problem of absolute stability and robust stability of a class of neutral Lurie indirect control systems with time delays is investigated. By using Jenson inequality and introducing some free-weighting matrices, some delay-dependent robust absolute stability criteria are presented in terms of strict linear matrix inequalities (LMIs).
(5) The problem of robust absolute stability for a class of general Lurie systems with time delay is addressed. By using a new Lyapunov-krasovskii functional which splits the whole delay interval into two subintervals and defines a different energy function on each subinterval, some new delay-dependent criteria are presented and formulated by LMI.
Finally, how the free-weighting matrix and new Lyapunov-krasovskii functional approach are used to get the delay-dependent robust stability and stabilization problem for networked control systems, singular systems and neutral Lurie control systems with delays is outlined. And the perspective of future studies is refered at the end of the dissertation.
引文
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