不确定混沌系统控制与同步及应用研究
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摘要
本文研究不确定混沌系统的混沌控制与同步问题,并且探讨了混沌同步在保密通讯中的应用。混沌是一种特殊的非线性现象,由于混沌系统对初始条件极其敏感,因此一度被认为是不可控和不可预测的,过去人们对混沌系统的恐惧远多于兴趣。然而越来越多的的研究表明:混沌系统不仅可以是长期可控和短期可预测的,甚至是可以利用的。混沌系统具有许多特殊的性质,例如吸引子镶嵌有无数个不稳定周期轨道、对初始条件的敏感性、状态的有界性和运动的遍历性等。正是由于这些特性,对混沌系统的研究不仅要考虑系统的运动,而且要结合参数在其所属空间变化一并考虑问题。控制和同步是混沌系统研究中的两个主要方面。混沌系统的控制包括两种思路:一种是控制,目的是稳定吸引子内部的不稳定周期轨道或者是消除系统的混沌行为;另外一种是反控制,目的是加强已有的混沌行为或者使不是混沌的系统产生混沌。混沌系统的同步就是使两个或多个混沌系统按照某种相同的方式演化。
近年来,已有许多的研究者们对混沌系统进行了深入的探讨。物理学、社会学、生物学、化学、信息学等不同领域的专家学者们,基于不同的视野与角度对混沌系统进行了一系列的研究,取得了一定的成果。受到近年来控制领域中新建立的一系列理论与方法的启发,本文将非线性系统设计中的一些成果运用于混沌系统的研究中,对混沌控制与同步提出了一些新的方法和结果。在实际设计中,系统模型必然会带有某种不确定性,这种不确定性可能来自系统参数的蜕变,或者建模误差,因而需要研究不确定混沌系统。上世纪80年代,Artstein和Sontag在研究非线性系统时分别提出了控制Lyapunov函数(CLF)的的概念,使得Lyapunov方法从一种分析工具成为设计工具。但是到目前为止,作者还没有看到将控制Lyapunov函数(CLF)引入到混沌系统的研究文献。本文将CLF引入到混沌控制与同步研究中,并结合有限时间稳定性理论,首次提出了有限时间控制Lyapunov函数(f-CLF)概念,设计出相应的反馈控制实现了有限时间混沌控制与同步。
本文主要的创新点概括如下:
1)本文从实际应用出发,着眼于更简单的控制系统设计方法。分别应用级联系统稳定性理论和输入状态稳定(ISS)理论,对混沌系统的控制与同步,设计了简单的线性控制器。与现有的混沌系统控制与同步方案相比,本文给出的线性控制器设计是先进的,是最简单的设计之一。而且,本文作者率先将CLF引入进混沌系统设计,对统一混沌模型建立了简单有效的CLF。
2)本文从响应快速性要求出发,考虑了有限时间的混沌控制与同步问题。大多数考虑的控制系统设计都是渐近稳定的,就是要经过无穷时间,系统的行为才达到控制目标。本文考虑有限时间达到目标,分别用终端滑模控制与CLF实现了有限时间的镇定与同步。使得控制系统设计理论更有应用效性。
3)本文总结出一种逐步设计技术,这种设计方法适用于一类半解耦型系统。逐步地,将系统状态驱动到零,对于已经设计的状态,在下一步设计中视作零处理。这种设计方法极大地简化了一类非线性系统的设计问题,有着广泛的应用前景。
全文内容可分为两部分,第一部分是不确定混沌系统的混沌控制与同步问题,它包含了第二、三、四、五章;第六章是本文的第二部分,研究了基于奇异模型的混沌同步与保密通讯问题。现将各章的具体内容和研究结果概述如下:
第一章是绪论,介绍混沌定义、混沌特征、CLF的定义、滑模控制、有限时间稳定等概念。综述了混沌系统的研究状况,尤其是各种控制方法在混沌研究中的应用。
第二章对一类不确定混沌系统提出了基于滑模控制进行混沌抑制。通过求解线性矩阵不等式来确定切换面中的参数及控制增益,对Chua电路和Lorenz混沌系统分别给出了具体的反馈控制设计。本章同时提出终端滑模方法来研究混沌控制问题,提出了新的非奇异终端滑模设计方案,有效地解决了反馈控制的奇异问题,实现了有限时间混沌抑制。
第三章基于控制Lyapunov函数(CLF)研究了不确定混沌系统的同步问题。将CLF方法引入到混沌系统研究中来,解决了不确定统一混沌系统的CLF的构造问题,设计了基于CLF的反馈控制实现渐近混沌同步。然后,结合有限时间稳定性理论,提出了有限时间控制Lyapunov函数(f-CLF),并基于f-CLF设计了具有强鲁棒性和快速性的反馈控制来实现有限时间混沌同步。
第四章结合级联系统稳定性理论及输入状态稳定(ISS)理论,对混沌系统提出了一种反馈半解耦模型,进而设计出简单易实现的线性反馈控制来实现混沌控制与同步。
第五章以前几章内容为基础,把所得结论进一步拓展到半解耦系统中,提出了一种新的stepping design的设计方法,此种设计方法简单可行,可以有效解决一类非线性系统的控制器设计问题。
第六章考虑混沌系统的应用:保密通讯问题。将待传输的保密信号视作一个新的系统状态,可将混沌系统转换为一个描述系统模型,利用描述系统的方法,设计出了描述观测器来实现混沌同步与保密通讯,并且观测器中的增益矩阵的选取可以通过求解线性矩阵不等式来获得。
This paper studies the problem of control and synchronization for uncertain chaoticsystems. As a result, this paper also considers the application of chaos synchronization insecure communication. Chaos is a special phenomenon of the nonlinear system. Beingsensitive to initial conditions, for a long time chaos is considered uncontrollable and un-predictable. So people treated the chaos with fear rather than interests. Since the end ofthe last century, more and more research results have shown that chaos is not only control-lable but also useful in many fields. The chaotic systems have many special characters.For example, chaotic attractors contain infinite unstable periodic trajectories. chaotic sys-tem is sensitive to initial conditions and its states are bounded. Just because of thesespecial characters, it is necessary to study chaotic systems in state space when research-ing on the chaotic systems. Chaos control and synchronization are two main aspects.There are two control problems for the chaotic systems. One is chaos control. Its aim isto stabilize the unstable periodic trajectory in the chaotic attractor or suppress the chaoticbehavior. The other is anti-control. Its aim is to strength the existing chaotic behavior ordrive the system which is not chaotic to be chaotic. Chaos synchronization is to drive theslave systems to follow up a chaotic system.
In recent years, many researchers studied the chaotic systems. Specialists in physics,society, biologic, chemistry, information science and other many areas have done manyworks in chaotic systems based on the aspects of the investigation in their own fields.And they have got lots interesting results. Inspired by the theory and techniques that de-veloped recently from the control science, this paper studies the chaotic systems usingnonlinear design methods. It is true that a practical system always contains uncertainty.The uncertainty may come from the parameter varying, or from the modeling error. Soit is necessary to research the design of uncertain chaotic systems. CLF is an effectivedesign method for nonlinear systems. It’s definition was proposed by Arstein and Sontagindependently when they researched the control of nonlinear systems in the 1980s. ThenLyapunov function is turned into a design tool from an analysis tool. Till now, few re-searcher has applied the CLF in the design of chaotic system to the best of the author’s known. This paper introduces the CLF into the design of chaos control and synchroniza-tion. Integrating the finite-time stability theory, we firstly defined the idea of finite-timeControl Lyapunov Function(f-CLF). And we also designed the feedback control to realizefinite-time chaos control and synchronization by using the f-CLF.
The main contributions of this thesis are as follows:
1) Starting form the practical application aspect, we concern on simpler designmethod for control system. Based on cascade system stability theory and Input-to-State-Stability(ISS) theory, we propose a simple linear feedback for chaos control and synchro-nization. Compared with the existing chaos control and synchronization results, the linearcontroller given in this dissertation is advanced and it is one of the most simplest design.Furthermore, we take the lead in applying the CLF in chaotic system design. And weconstruct a simple and effective CLF for the unified chaotic system.
2) To improve the transient response performancet, we consider finite time chaoscontrol and synchronization. Most control system design is to realize asymptotical sta-bility of the close-loop systems. This is to say that the system reach the control target ininfinite time. This dissertation studies realize such goals in finite time. Based on terminalsliding mode and CLF respectively, finite time chaos control and synchronization is real-ized. This makes the design of the control system more effective.
3) This thesis proposes a step design technique. This design method suits to a classof semi-decouple system. Step by step, the system states can be driven to zero. For thestates that have been designed, they can be treated as zero in the next design. This designmethod greatly simplify the design of a class of nonlinear system. It has a wide applica-tion prospect.
This dissertation consists of two parts. The first part is the design of chaos controland synchronization that contains Chapters Two, Three, Four and Five. Chapter Six isthe second part. It is about the secure communication based on descriptor model. Theconcrete contents and results of the thesis are as follows sincerely.
Chapter One is a survey. It introduces the definition of chaos, chaotic characters, thedefinition of CLF, sliding mode control, finite-time stability and other items. It sums upthe research status of the chaotic systems. Different control methods that are introducedin chaotic system are also introduced especially.
Based on the sliding mode control, Chapter Two presents chaos control for a class of uncertain chaotic systems. Both the parameters in the switching surface and the controllergains can be obtained through the solvability of the linear matrix inequality. Feedbackcontrol are designed for the Chua’s circuit and Lorenz chaotic system respectively. At thesame time, this chapter proposes terminal sliding mode control to realize chaos control.A new nonsingular terminal sliding mode is presented to tackle the singularity problemof the controller. The finite-time chaos control is attained through such a design.
Chapter Three studies chaos synchronization for uncertain chaotic systems based onthe Control Lyapunov Functions(CLFs). We have completed the task of construction ofthe CLF for the uncertainty unified chaotic system. Controllers are designed to realizeasymptotic synchronization. Then, by combining the finite-time stability theory with theCLF, we presente finite-time Control Lyapunov Function(f-CLF). Based on the f-CLF,robust and rapid controllers are proposed to realize finite-time chaos synchronization.
By virtue of the stable theory of the cascade system and input-to-state stability(ISS)theory, we can get a semi-decouple form for the chaotic systems by feedback. Ultimately,we can design linear controller which is very simple to realize chaos control and synchro-nization.
Based on the preceding results, Chapter Five extends the anterior results to the semi-decouple systems and puts forward a new stepping design method. This design methodis easy to be implemented and can tackle the design problem for a class of nonlinear sys-tems.
Chapter Six considers secure communication which is an application of chaotic sys-tem. By making use of the descriptor system theory, we can design descriptor observerto realize chaos synchronization and secure communication. In the mean time, the gainmatrices in the observer can be chosen by solving of a linear matrix inequality. Takingthe transmitting signal as a new system state, we transform the chaotic system into adescriptor system.
近年来,已有许多的研究者们对混沌系统进行了深入的探讨。物理学、社会学、生物学、化学、信息学等不同领域的专家学者们,基于不同的视野与角度对混沌系统进行了一系列的研究,取得了一定的成果。受到近年来控制领域中新建立的一系列理论与方法的启发,本文将非线性系统设计中的一些成果运用于混沌系统的研究中,对混沌控制与同步提出了一些新的方法和结果。在实际设计中,系统模型必然会带有某种不确定性,这种不确定性可能来自系统参数的蜕变,或者建模误差,因而需要研究不确定混沌系统。上世纪80年代,Artstein和Sontag在研究非线性系统时分别提出了控制Lyapunov函数(CLF)的的概念,使得Lyapunov方法从一种分析工具成为设计工具。但是到目前为止,作者还没有看到将控制Lyapunov函数(CLF)引入到混沌系统的研究文献。本文将CLF引入到混沌控制与同步研究中,并结合有限时间稳定性理论,首次提出了有限时间控制Lyapunov函数(f-CLF)概念,设计出相应的反馈控制实现了有限时间混沌控制与同步。
本文主要的创新点概括如下:
1)本文从实际应用出发,着眼于更简单的控制系统设计方法。分别应用级联系统稳定性理论和输入状态稳定(ISS)理论,对混沌系统的控制与同步,设计了简单的线性控制器。与现有的混沌系统控制与同步方案相比,本文给出的线性控制器设计是先进的,是最简单的设计之一。而且,本文作者率先将CLF引入进混沌系统设计,对统一混沌模型建立了简单有效的CLF。
2)本文从响应快速性要求出发,考虑了有限时间的混沌控制与同步问题。大多数考虑的控制系统设计都是渐近稳定的,就是要经过无穷时间,系统的行为才达到控制目标。本文考虑有限时间达到目标,分别用终端滑模控制与CLF实现了有限时间的镇定与同步。使得控制系统设计理论更有应用效性。
3)本文总结出一种逐步设计技术,这种设计方法适用于一类半解耦型系统。逐步地,将系统状态驱动到零,对于已经设计的状态,在下一步设计中视作零处理。这种设计方法极大地简化了一类非线性系统的设计问题,有着广泛的应用前景。
全文内容可分为两部分,第一部分是不确定混沌系统的混沌控制与同步问题,它包含了第二、三、四、五章;第六章是本文的第二部分,研究了基于奇异模型的混沌同步与保密通讯问题。现将各章的具体内容和研究结果概述如下:
第一章是绪论,介绍混沌定义、混沌特征、CLF的定义、滑模控制、有限时间稳定等概念。综述了混沌系统的研究状况,尤其是各种控制方法在混沌研究中的应用。
第二章对一类不确定混沌系统提出了基于滑模控制进行混沌抑制。通过求解线性矩阵不等式来确定切换面中的参数及控制增益,对Chua电路和Lorenz混沌系统分别给出了具体的反馈控制设计。本章同时提出终端滑模方法来研究混沌控制问题,提出了新的非奇异终端滑模设计方案,有效地解决了反馈控制的奇异问题,实现了有限时间混沌抑制。
第三章基于控制Lyapunov函数(CLF)研究了不确定混沌系统的同步问题。将CLF方法引入到混沌系统研究中来,解决了不确定统一混沌系统的CLF的构造问题,设计了基于CLF的反馈控制实现渐近混沌同步。然后,结合有限时间稳定性理论,提出了有限时间控制Lyapunov函数(f-CLF),并基于f-CLF设计了具有强鲁棒性和快速性的反馈控制来实现有限时间混沌同步。
第四章结合级联系统稳定性理论及输入状态稳定(ISS)理论,对混沌系统提出了一种反馈半解耦模型,进而设计出简单易实现的线性反馈控制来实现混沌控制与同步。
第五章以前几章内容为基础,把所得结论进一步拓展到半解耦系统中,提出了一种新的stepping design的设计方法,此种设计方法简单可行,可以有效解决一类非线性系统的控制器设计问题。
第六章考虑混沌系统的应用:保密通讯问题。将待传输的保密信号视作一个新的系统状态,可将混沌系统转换为一个描述系统模型,利用描述系统的方法,设计出了描述观测器来实现混沌同步与保密通讯,并且观测器中的增益矩阵的选取可以通过求解线性矩阵不等式来获得。
This paper studies the problem of control and synchronization for uncertain chaoticsystems. As a result, this paper also considers the application of chaos synchronization insecure communication. Chaos is a special phenomenon of the nonlinear system. Beingsensitive to initial conditions, for a long time chaos is considered uncontrollable and un-predictable. So people treated the chaos with fear rather than interests. Since the end ofthe last century, more and more research results have shown that chaos is not only control-lable but also useful in many fields. The chaotic systems have many special characters.For example, chaotic attractors contain infinite unstable periodic trajectories. chaotic sys-tem is sensitive to initial conditions and its states are bounded. Just because of thesespecial characters, it is necessary to study chaotic systems in state space when research-ing on the chaotic systems. Chaos control and synchronization are two main aspects.There are two control problems for the chaotic systems. One is chaos control. Its aim isto stabilize the unstable periodic trajectory in the chaotic attractor or suppress the chaoticbehavior. The other is anti-control. Its aim is to strength the existing chaotic behavior ordrive the system which is not chaotic to be chaotic. Chaos synchronization is to drive theslave systems to follow up a chaotic system.
In recent years, many researchers studied the chaotic systems. Specialists in physics,society, biologic, chemistry, information science and other many areas have done manyworks in chaotic systems based on the aspects of the investigation in their own fields.And they have got lots interesting results. Inspired by the theory and techniques that de-veloped recently from the control science, this paper studies the chaotic systems usingnonlinear design methods. It is true that a practical system always contains uncertainty.The uncertainty may come from the parameter varying, or from the modeling error. Soit is necessary to research the design of uncertain chaotic systems. CLF is an effectivedesign method for nonlinear systems. It’s definition was proposed by Arstein and Sontagindependently when they researched the control of nonlinear systems in the 1980s. ThenLyapunov function is turned into a design tool from an analysis tool. Till now, few re-searcher has applied the CLF in the design of chaotic system to the best of the author’s known. This paper introduces the CLF into the design of chaos control and synchroniza-tion. Integrating the finite-time stability theory, we firstly defined the idea of finite-timeControl Lyapunov Function(f-CLF). And we also designed the feedback control to realizefinite-time chaos control and synchronization by using the f-CLF.
The main contributions of this thesis are as follows:
1) Starting form the practical application aspect, we concern on simpler designmethod for control system. Based on cascade system stability theory and Input-to-State-Stability(ISS) theory, we propose a simple linear feedback for chaos control and synchro-nization. Compared with the existing chaos control and synchronization results, the linearcontroller given in this dissertation is advanced and it is one of the most simplest design.Furthermore, we take the lead in applying the CLF in chaotic system design. And weconstruct a simple and effective CLF for the unified chaotic system.
2) To improve the transient response performancet, we consider finite time chaoscontrol and synchronization. Most control system design is to realize asymptotical sta-bility of the close-loop systems. This is to say that the system reach the control target ininfinite time. This dissertation studies realize such goals in finite time. Based on terminalsliding mode and CLF respectively, finite time chaos control and synchronization is real-ized. This makes the design of the control system more effective.
3) This thesis proposes a step design technique. This design method suits to a classof semi-decouple system. Step by step, the system states can be driven to zero. For thestates that have been designed, they can be treated as zero in the next design. This designmethod greatly simplify the design of a class of nonlinear system. It has a wide applica-tion prospect.
This dissertation consists of two parts. The first part is the design of chaos controland synchronization that contains Chapters Two, Three, Four and Five. Chapter Six isthe second part. It is about the secure communication based on descriptor model. Theconcrete contents and results of the thesis are as follows sincerely.
Chapter One is a survey. It introduces the definition of chaos, chaotic characters, thedefinition of CLF, sliding mode control, finite-time stability and other items. It sums upthe research status of the chaotic systems. Different control methods that are introducedin chaotic system are also introduced especially.
Based on the sliding mode control, Chapter Two presents chaos control for a class of uncertain chaotic systems. Both the parameters in the switching surface and the controllergains can be obtained through the solvability of the linear matrix inequality. Feedbackcontrol are designed for the Chua’s circuit and Lorenz chaotic system respectively. At thesame time, this chapter proposes terminal sliding mode control to realize chaos control.A new nonsingular terminal sliding mode is presented to tackle the singularity problemof the controller. The finite-time chaos control is attained through such a design.
Chapter Three studies chaos synchronization for uncertain chaotic systems based onthe Control Lyapunov Functions(CLFs). We have completed the task of construction ofthe CLF for the uncertainty unified chaotic system. Controllers are designed to realizeasymptotic synchronization. Then, by combining the finite-time stability theory with theCLF, we presente finite-time Control Lyapunov Function(f-CLF). Based on the f-CLF,robust and rapid controllers are proposed to realize finite-time chaos synchronization.
By virtue of the stable theory of the cascade system and input-to-state stability(ISS)theory, we can get a semi-decouple form for the chaotic systems by feedback. Ultimately,we can design linear controller which is very simple to realize chaos control and synchro-nization.
Based on the preceding results, Chapter Five extends the anterior results to the semi-decouple systems and puts forward a new stepping design method. This design methodis easy to be implemented and can tackle the design problem for a class of nonlinear sys-tems.
Chapter Six considers secure communication which is an application of chaotic sys-tem. By making use of the descriptor system theory, we can design descriptor observerto realize chaos synchronization and secure communication. In the mean time, the gainmatrices in the observer can be chosen by solving of a linear matrix inequality. Takingthe transmitting signal as a new system state, we transform the chaotic system into adescriptor system.
引文
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