脉冲系统的有限时间稳定性与滤波
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摘要
摘要脉冲系统是将连续的发展过程和状态跳变结合起来的混合动态系统.在物理、药物动力学、生态系统、自动控制等诸多领域内都有广泛的应用.在实际工程中,对于那些工作时间短暂的系统(例如导弹系统、通信网络系统、机器人操控系统等),人们更关心的常常是系统应满足一定的暂态性能要求(例如满足系统轨线对于平衡点的一定偏离范围的要求).由于有限时间稳定性很好的刻画了系统的暂态性能,因而受到了越来越多学者的关注,也取得了丰硕的成果.另一方面,在通信、信号处理和控制领域,如何从被噪声污染的观测信号中过滤噪声,尽可能消除噪声的影响,求未知真实信号或系统状态的估计也是控制界研究的热点之一.由于对有限时间稳定性越来越多的关注以及H∞滤波器的广泛应用,本文研究了几类脉冲系统的有限时间稳定和H∞滤波问题,利用Lyapunov函数和线性矩阵不等式方法给出了系统有限时间稳定和满足H∞性能要求的充分条件.
全文共分五章.
第一章介绍了有关稳定性、有限时间稳定性、H∞滤波以及有限时间滤波的基本知识,这些知识是阅读后续内容所必须的,是概述性的.
第二章首先在介绍了Xu J.等人所研究的离散时间线性脉冲系统的有限时间滤波问题,然后通过一个数值算例和数值模拟来表明其结论是不正确的,最后通过理论分析找出问题所在,并给出了正确的结论.由于结论中的变量是耦合的,因而不能用Matlab直接求解.为了避免上述情况,我们给出了两个相对实用的结果.然后对于连续时间线性脉冲系统的有限时间滤波问题也建立了类似的结果.数值模拟表明,滤波方法是可行的和有效的.
第三章分别给出了离散时间和连续时间广义线性脉冲系统有限时间稳定以及有限时间滤波问题定义,然后利用Lyapunov方法给出了系统有限时间稳定以及满足性能要求的充分条件,并给出了滤波器的设计方法.数值模拟表明了结论的可行性及有效性.
第四章针对线性时变奇异脉冲系统,采用Lyapunov方法研究了它的有限时间稳定和L2增益问题.给出了奇异系统有限时间稳定和具有L2增益的充分条件.与定常的扰动相比,具有时变扰动的系统更能真实反映现实世界,因而针对上述问题研究了具有时变扰动脉冲奇异系统的有限时间稳定性问题.对于不同特点的扰动,分别给出了相应的结论.如果扰动是定常的,利用松弛变量降低了结论的保守性.数值算例表明了结论的可行性和更大的适用范围.
第五章分别研究了离散时间和连续时间分段脉冲仿射系统的有限时间滤波问题,利用Lyapunov函数和线性矩阵不等式方法给出了系统有限时间稳定的几个充分条件；并且给出了有限时间滤波问题可解的充分条件以及滤波器的设计方法.最后通过数值模拟表明了本章方法的可行性.
Abstract Impulsive system is a hybrid dynamic system which combines contin-uous process and jumping in the state. It is widely applied in the areas of physics, pharmacodynamic, ecosystem, automatic control etc. In actual engineering, for those systems whose working time is short (such as missile system, communication network system, robot control system, etc.), people are concerned that systems should satisfy certain transient performance requirements(such as meeting certain deviation range of the system trajectory for balance requirements). Since the finite-time stability(FTS) well depicts the transient performance of the system, it has received more and more attentions from scholars and has achieved fruitful results. On the other hand, in the fields of the communication, signal processing and con-trol, how to filter noises from the signal polluted by noises, eliminate the influence of noises, and strive for the unknown real signal or estimate system states is also one of a hot topics in the control field. Because of the increasing attention to the finite-time stability problem and the wide application of H∞, filter, several types of impulse systems'finite-time stability and H∞filtering problems are studied in this paper, using the Lyapunov function and linear matrix inequality(LMI) methods to establish some sufficient conditions.
The present paper is divided into five chapters.
The first chapter overviews some basic knowledge about stability, finite-time stability, H∞filtering and finite-time filtering, which are preparatory for the latter chapters.
The second chapter firstly introduces the finite-time filtering problem of discrete-time linear impulse system researched by Xu J. et al, then shows that the conclusions are not correct by using a numerical example and simulation, finally, finds out the reasons using theoretical analysis and presents correct conclusions. Since the vari-ables are coupled in this conclusion, it cannot be solved by using Matlab Toolbox directly. In order to avoid the above situation, two relatively practical results are given. Then similar results for the finite-time filtering problem of the continuous-time linear impulse system are established. Numerical simulation shows that the filtering method is feasible and effective.
The definitions for finite-time stability and finite-time filtering problem of the discrete-time and continuous-time singular linear impulse system are given in third chapter, respectively. Sufficient conditions and the design methods of filter, which meet the finite-time stability the H∞performance requirements, are given for the singular impulsive system by using Lyapunov method. Numerical simulation shows the feasibility and effectiveness of the conclusion.
Taking advantage of Lyapunov method, the finite-time stability and L2-gain problems for linear time-varying singular impulse system are investigated in forth chapter. Sufficient conditions, which can ensure that singular system is finite-time stable and can satisfies L2-gain performance, are given with Lyapunov method. Compared with the constant disturbance, systems with time-varying disturbance can reflect the real world, so the finite-time stability problem for the impulse sin-gular systems with time-varying disturbance is studied. Based on the different characteristics of disturbances, corresponding conclusions are given, respectively. If the disturbance is a constant, the conservatism of the conclusion is reduced by using of the slack variable. Numerical examples show that the conclusions is feasible and this method has a larger scope.
The finite-time filtering problem for the discrete-time and continuous-time im-pulses piecewise affine system is studied in fifth chapter, respectively. Some sufficient conditions for the filter error system that is finite-time stable and meets H∞per-formance requirements are given by using of the method of Lyapunov function and linear matrix inequality; And sufficient conditions for the solvability of the finite-time filtering problem and designing methods of the filter are also given. Finally, the numerical simulation shows the feasibility of this method.
全文共分五章.
第一章介绍了有关稳定性、有限时间稳定性、H∞滤波以及有限时间滤波的基本知识,这些知识是阅读后续内容所必须的,是概述性的.
第二章首先在介绍了Xu J.等人所研究的离散时间线性脉冲系统的有限时间滤波问题,然后通过一个数值算例和数值模拟来表明其结论是不正确的,最后通过理论分析找出问题所在,并给出了正确的结论.由于结论中的变量是耦合的,因而不能用Matlab直接求解.为了避免上述情况,我们给出了两个相对实用的结果.然后对于连续时间线性脉冲系统的有限时间滤波问题也建立了类似的结果.数值模拟表明,滤波方法是可行的和有效的.
第三章分别给出了离散时间和连续时间广义线性脉冲系统有限时间稳定以及有限时间滤波问题定义,然后利用Lyapunov方法给出了系统有限时间稳定以及满足性能要求的充分条件,并给出了滤波器的设计方法.数值模拟表明了结论的可行性及有效性.
第四章针对线性时变奇异脉冲系统,采用Lyapunov方法研究了它的有限时间稳定和L2增益问题.给出了奇异系统有限时间稳定和具有L2增益的充分条件.与定常的扰动相比,具有时变扰动的系统更能真实反映现实世界,因而针对上述问题研究了具有时变扰动脉冲奇异系统的有限时间稳定性问题.对于不同特点的扰动,分别给出了相应的结论.如果扰动是定常的,利用松弛变量降低了结论的保守性.数值算例表明了结论的可行性和更大的适用范围.
第五章分别研究了离散时间和连续时间分段脉冲仿射系统的有限时间滤波问题,利用Lyapunov函数和线性矩阵不等式方法给出了系统有限时间稳定的几个充分条件；并且给出了有限时间滤波问题可解的充分条件以及滤波器的设计方法.最后通过数值模拟表明了本章方法的可行性.
Abstract Impulsive system is a hybrid dynamic system which combines contin-uous process and jumping in the state. It is widely applied in the areas of physics, pharmacodynamic, ecosystem, automatic control etc. In actual engineering, for those systems whose working time is short (such as missile system, communication network system, robot control system, etc.), people are concerned that systems should satisfy certain transient performance requirements(such as meeting certain deviation range of the system trajectory for balance requirements). Since the finite-time stability(FTS) well depicts the transient performance of the system, it has received more and more attentions from scholars and has achieved fruitful results. On the other hand, in the fields of the communication, signal processing and con-trol, how to filter noises from the signal polluted by noises, eliminate the influence of noises, and strive for the unknown real signal or estimate system states is also one of a hot topics in the control field. Because of the increasing attention to the finite-time stability problem and the wide application of H∞, filter, several types of impulse systems'finite-time stability and H∞filtering problems are studied in this paper, using the Lyapunov function and linear matrix inequality(LMI) methods to establish some sufficient conditions.
The present paper is divided into five chapters.
The first chapter overviews some basic knowledge about stability, finite-time stability, H∞filtering and finite-time filtering, which are preparatory for the latter chapters.
The second chapter firstly introduces the finite-time filtering problem of discrete-time linear impulse system researched by Xu J. et al, then shows that the conclusions are not correct by using a numerical example and simulation, finally, finds out the reasons using theoretical analysis and presents correct conclusions. Since the vari-ables are coupled in this conclusion, it cannot be solved by using Matlab Toolbox directly. In order to avoid the above situation, two relatively practical results are given. Then similar results for the finite-time filtering problem of the continuous-time linear impulse system are established. Numerical simulation shows that the filtering method is feasible and effective.
The definitions for finite-time stability and finite-time filtering problem of the discrete-time and continuous-time singular linear impulse system are given in third chapter, respectively. Sufficient conditions and the design methods of filter, which meet the finite-time stability the H∞performance requirements, are given for the singular impulsive system by using Lyapunov method. Numerical simulation shows the feasibility and effectiveness of the conclusion.
Taking advantage of Lyapunov method, the finite-time stability and L2-gain problems for linear time-varying singular impulse system are investigated in forth chapter. Sufficient conditions, which can ensure that singular system is finite-time stable and can satisfies L2-gain performance, are given with Lyapunov method. Compared with the constant disturbance, systems with time-varying disturbance can reflect the real world, so the finite-time stability problem for the impulse sin-gular systems with time-varying disturbance is studied. Based on the different characteristics of disturbances, corresponding conclusions are given, respectively. If the disturbance is a constant, the conservatism of the conclusion is reduced by using of the slack variable. Numerical examples show that the conclusions is feasible and this method has a larger scope.
The finite-time filtering problem for the discrete-time and continuous-time im-pulses piecewise affine system is studied in fifth chapter, respectively. Some sufficient conditions for the filter error system that is finite-time stable and meets H∞per-formance requirements are given by using of the method of Lyapunov function and linear matrix inequality; And sufficient conditions for the solvability of the finite-time filtering problem and designing methods of the filter are also given. Finally, the numerical simulation shows the feasibility of this method.
引文
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