奇异系统分析与综合方法及其在结构系统控制中的应用
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摘要
奇异系统是一类比正则系统具有更广泛适用度的系统描述形式,可以方便地用于描述实际中的许多系统,如电网系统、奇异摄动系统、Leontief模型、化工过程、核能源反应系统等等。由于奇异系统除了包含微分方程(连续系统)或差分方程(离散系统)描述的慢变子系统,还包含代数方程描述的快变子系统,因此,对奇异系统的稳定性分析与镇定控制器设计比正常系统复杂得多。奇异系统分析不仅要考虑系统的稳定性,还要考虑系统的正则性与无脉冲性(连续系统)或因果性(离散系统)。此外,对于实际存在的大量物理系统,由于信息传递及数据处理等原因,时滞常常是客观存在的。在实际系统的分析与综合中,如果对时滞考虑不恰当,常常会造成系统的性能变差,甚至不稳定。对于奇异系统来说,时滞甚至可能破坏系统的正则性、因果性或无脉冲性,因此研究时滞奇异系统的分析理论与分析方法,特别是时滞相关的理论与方法具有非常重要的现实意义。
针对奇异系统中的时滞与参数不确定性问题,本文通过选择合适的Lyapunov函数及不等式放大技术,研究了常时滞及变时滞奇异系统的时滞相关稳定与控制器存在的充分条件。并将奇异系统分析方法推广至线性结构系统的稳定性分析中,讨论了结构系统的被动控制与主动控制方法。
论文的主要工作包括以下几个方面：
(1)针对定时滞奇异系统,通过选用合适的Lyapunov函数,采用积分不等式及自由权矩阵等方法,研究了连续及离散奇异系统的时滞相关可容许、鲁棒可容许、系统可镇定及鲁棒可镇定充分条件。主要特点是提出了部分非正(PNP)李亚普诺夫函数概念,并将其应用于定时滞奇异系统的分析与综合,得到了保守性有所降低的结果。
(2)针对变时滞奇异系统,本文通过系统模型变换得到了模型变换后的系统描述形式,并结合时滞分割技术分别讨论了连续时间变时滞奇异系统及离散时间变时滞马尔科夫随机跳变奇异系统的时滞相关可容许、鲁棒可容许、系统可镇定及鲁棒可镇定充分条件。并通过实例仿真验证了相关理论的有效性。
(3)将奇异系统的描述形式推广到结构系统的描述中,当系统参数出现不确定性时,采用奇异形式描述的结构系统模型避免了不确定性参数的耦合,降低了参数摄动对系统模型造成的保守性。在结构系统的被动控制中,通过引入遗传算法寻优,获得了不同目标函数下全局最优TMD参数设计方法。在结构系统主动控制中,提出了液压伺服驱动下结构系统的参数依赖控制器设计方法,并通过实例仿真,说明了其有效性。
Singular system, which is more suitable to describe a lot of practical systems than the regular system, has been widely used to describe many practical systems, such as, power grid system, singularly perturbed systems, leontief economic model, chemical process, nuclear reactor system, and so on. Singular system contains not only the slow subsystem which is described by differential equations (for continuous singular system) or difference equations (for discrete singular system), but also the fast subsystem which is described by algebraic equations. Thus, the stability analysis and stabilizing controller design for singular system are much more complicated than those for regular system. The analysis of singular system requires the consideration of not only stability, but also regularity and the absence of impulses (for continuous singular system) or causality (for discrete singular system) simultaneously. In addition, due to the information transmission and data processing, etc., time delay often exists in the physical systems. During the analysis and synthesis of the actual systems, inappropriate handling of time delay often causes performance deterioration, or even leads to instability. For singular system, time-delay may even damage the regularity, causality or the absence of impulses of the system. Therefore, how to obtain the stability analysis and controller synthesis results (especially delay-dependent results) for the time-delay singular system is of very important practical significance.
Aiming at the problems about time delay and uncertainty in singular system, the delay-dependent sufficient conditions for the stability and stabilization of constant and time-varying delay singular systems are developed in this paper. Furthermore, the obtained results for singular systems are extended to the stability analysis for structural systems, and the passive control and active control methods for structure system are discussed in this paper.
The work of this paper mainly contains three parts:
(1) By appropriate choice of the Lyapunov function, the delay-dependent sufficient conditions for admissibility, robust admissibility, stabilizability, and robust stabilizability of continuous and discrete singular systems are studied based on integral inequality and free weight matrix methods. The main contribution of this section is the introduction of a new Lyapunov function, namely, partial non-positive (PNP) Lyapunov function, by which the less conservative results about system analysis and synthesis are obtained.
(2) Aiming at the time-varying delay singular system, the delay-dependent sufficient conditions for admissibility, robust admissibility, stabilizability, and robust stabilizability of continuous singular system and discrete lime-varying delay Markovian jump singular system are studied by using model transformation and delay-partitioning technique. Numerical examples are provided to demonstrate the effectiveness of the proposed methods.
(3) The description form of singular system is extended to describe structural system. When the parameter-uncertainties appear in the structural systems, singular description form can avoid the coupling of the uncertain parameters and reduce the conservatism of the system model. In the passive control of structural system, the global optimal TMD parameters design method is obtained by introducing the optimization of genetic algorithm. In the active control of structural system, the parameter-dependent controller design method is presented for structural system driven by electro-hydraulic servo system, and its effectiveness is illustrated by numerical simulation.
针对奇异系统中的时滞与参数不确定性问题,本文通过选择合适的Lyapunov函数及不等式放大技术,研究了常时滞及变时滞奇异系统的时滞相关稳定与控制器存在的充分条件。并将奇异系统分析方法推广至线性结构系统的稳定性分析中,讨论了结构系统的被动控制与主动控制方法。
论文的主要工作包括以下几个方面：
(1)针对定时滞奇异系统,通过选用合适的Lyapunov函数,采用积分不等式及自由权矩阵等方法,研究了连续及离散奇异系统的时滞相关可容许、鲁棒可容许、系统可镇定及鲁棒可镇定充分条件。主要特点是提出了部分非正(PNP)李亚普诺夫函数概念,并将其应用于定时滞奇异系统的分析与综合,得到了保守性有所降低的结果。
(2)针对变时滞奇异系统,本文通过系统模型变换得到了模型变换后的系统描述形式,并结合时滞分割技术分别讨论了连续时间变时滞奇异系统及离散时间变时滞马尔科夫随机跳变奇异系统的时滞相关可容许、鲁棒可容许、系统可镇定及鲁棒可镇定充分条件。并通过实例仿真验证了相关理论的有效性。
(3)将奇异系统的描述形式推广到结构系统的描述中,当系统参数出现不确定性时,采用奇异形式描述的结构系统模型避免了不确定性参数的耦合,降低了参数摄动对系统模型造成的保守性。在结构系统的被动控制中,通过引入遗传算法寻优,获得了不同目标函数下全局最优TMD参数设计方法。在结构系统主动控制中,提出了液压伺服驱动下结构系统的参数依赖控制器设计方法,并通过实例仿真,说明了其有效性。
Singular system, which is more suitable to describe a lot of practical systems than the regular system, has been widely used to describe many practical systems, such as, power grid system, singularly perturbed systems, leontief economic model, chemical process, nuclear reactor system, and so on. Singular system contains not only the slow subsystem which is described by differential equations (for continuous singular system) or difference equations (for discrete singular system), but also the fast subsystem which is described by algebraic equations. Thus, the stability analysis and stabilizing controller design for singular system are much more complicated than those for regular system. The analysis of singular system requires the consideration of not only stability, but also regularity and the absence of impulses (for continuous singular system) or causality (for discrete singular system) simultaneously. In addition, due to the information transmission and data processing, etc., time delay often exists in the physical systems. During the analysis and synthesis of the actual systems, inappropriate handling of time delay often causes performance deterioration, or even leads to instability. For singular system, time-delay may even damage the regularity, causality or the absence of impulses of the system. Therefore, how to obtain the stability analysis and controller synthesis results (especially delay-dependent results) for the time-delay singular system is of very important practical significance.
Aiming at the problems about time delay and uncertainty in singular system, the delay-dependent sufficient conditions for the stability and stabilization of constant and time-varying delay singular systems are developed in this paper. Furthermore, the obtained results for singular systems are extended to the stability analysis for structural systems, and the passive control and active control methods for structure system are discussed in this paper.
The work of this paper mainly contains three parts:
(1) By appropriate choice of the Lyapunov function, the delay-dependent sufficient conditions for admissibility, robust admissibility, stabilizability, and robust stabilizability of continuous and discrete singular systems are studied based on integral inequality and free weight matrix methods. The main contribution of this section is the introduction of a new Lyapunov function, namely, partial non-positive (PNP) Lyapunov function, by which the less conservative results about system analysis and synthesis are obtained.
(2) Aiming at the time-varying delay singular system, the delay-dependent sufficient conditions for admissibility, robust admissibility, stabilizability, and robust stabilizability of continuous singular system and discrete lime-varying delay Markovian jump singular system are studied by using model transformation and delay-partitioning technique. Numerical examples are provided to demonstrate the effectiveness of the proposed methods.
(3) The description form of singular system is extended to describe structural system. When the parameter-uncertainties appear in the structural systems, singular description form can avoid the coupling of the uncertain parameters and reduce the conservatism of the system model. In the passive control of structural system, the global optimal TMD parameters design method is obtained by introducing the optimization of genetic algorithm. In the active control of structural system, the parameter-dependent controller design method is presented for structural system driven by electro-hydraulic servo system, and its effectiveness is illustrated by numerical simulation.
引文
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