复动力系统的混沌控制与同步及其在通信中的应用
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摘要
目前,复动力系统己广泛地应用到通信、金融、生物等多个学科中.复混沌系统就是一个典型的复动力系统.自从1982年复Lorenz方程被提出以来,复混沌系统在物理学的许多领域发挥了重要作用,尤其是通信系统.复变量增加了所传输信息的内容并提高了安全性.复混沌系统的控制和同步成为混沌保密通信的热点问题.因此,本文对复混沌系统的控制和同步及其在安全通信中的应用进行了一系列基础研究,其主要工作和创新点如下：
1.实混沌系统的控制与同步
对于连续系统,提出了累积误差控制器.该控制器采用了累积误差的线性函数,在实际工程上可由RC电路实现.对于离散系统,提出了误差反馈矩阵控制.它类似于线性系统的状态反馈,但是该矩阵是时变的,包含状态变量的耦合信息,根据梯度下降法在线更新,并趋于一个恒定矩阵.结合变量间的耦合关系,本文还提出了部分矩阵控制和全矩阵控制.部分矩阵控制仅采用对部分变量的控制就实现全部变量达到期望值,易于工程实现.但是,上述控制器只能实现不动点控制和自同步.因此,结合速度梯度法和非线性函数控制,本文实现了带有未知参数的连续混沌系统的跟踪任意有界参考信号和参数辨识.
2.复混沌系统的复比例因子投影同步与参数辨识
提出了复比例因子投影同步(CMPS).完全同步(CS),反同步(AS),投影同步(PS)和修正投影同步(MPS)都是CMPS的特殊情况,因此它包含了以前的工作并进行推广.考虑到复混沌系统的未知参数和有界干扰的各种可能情形,本文设计了带有收敛因子和动态控制力量CMPS控制器；同时基于可持续激励和线性独立给出了参数收敛到真值的充分条件和必要条件,给出了改变比例因子的方法以辨识出参数真值.另外,CMPS在实混沌系统和复混沌系统之间建立同步联接.因此,本文首次讨论了实混沌系统和复混沌系统的复比例因子投影同步,基于速度梯度法设计了包含伪梯度条件的同步控制器.
3.复混沌系统的复函数同步及其通信方案
提出复函数投影同步(CFPS). CS, AS, PS, MPS,全状态组合投影同步(FSHPS),函数投影同步(FPS),修正函数投影同步(MFPS),广义函数投影同步(GFPS)和CMPS都是CFPS的特殊情形.CFPS几乎没有被研究过,而且将包括大部分存在的同步情形.针对带有未知参数的复混沌系统,设计了CFPS控制器,并针对耦合复混沌系统,设计了基于CFPS的通信方案.该通信方案本质上是混沌掩盖,但是所传输的信号是信息信号(作为复比例函数)与混沌信号积的导数.因为复比例函数比实函数更加不可预测和复杂,则入侵者从传输信号中提取信息的可能性更小.
4.复混沌系统的差函数投影同步及其通信方案
从两复函数相减的角度提出了差函数投影同步(DFPS),自同步和相位同步均是其特殊情形.设计了DFPS控制器,并将其应用到耦合复混沌系统中,提出了基于DFPS的通信方案.该通信方案本质上仍是混沌掩盖,但是所传输的信号是信息信号和混沌信号的和的导数.它避免了CFPS中由于驱动系统状态(作为除数)接近零时产生的算法误差.
5.时滞复混沌系统的自时滞同步及其通信方案
提出了复混沌系统的自时滞同步(STDS),它是自同步的扩展,进一步拓宽了同步问题的视野,并避免了因时滞而产生的各种问题.时滞复混沌系统能产生高度随机性和不可预测性的时间序列,应用在混沌保密通信中能够提高保密性能.因此,本文研究了时滞复Lorenz系统的混沌特性,并设计控制器实现了时滞复Lorenz系统的自时滞同步.针对耦合时滞复混沌系统,给出了基于STDS的通信方案.该通信方案考虑了信息传输中产生的时滞,更接近实际情况,时滞复混沌系统也进一步增强了保密效果.
综上所述,本文围绕复动力系统的混沌控制与同步及其在通信中的应用展开了研究,提出了复比例因子投影同步,复函数投影同步,差函数投影同步及自时滞同步等概念,并将复变量引入到混沌通信中,促进了复动力系统的发展,为进一步加强通信安全提供了理论依据.
At present, the complex dynamic system is widely used in communications, finance, biology and many other subjects. Complex chaotic system is a typical com-plex dynamic system. Since complex Lorenz equation has been proposed in1982, complex chaotic systems have played an important role in many fields of physics, es-pecially in communication. The complex variables increase the content of transmit-ted information and improve the safety. The control and synchronization of complex chaotic systems is becoming one of hot spots of chaos-communication. Therefore, this thesis focuses on the control and synchronization of complex chaotic system and its application in secure communication. The main work and innovation points are as follows,
1. The control and synchronization of real chaotic system
For continuous system, we propose cumulative error controller. The controller is the linear function of cumulative error in essence, which can be realized by RC circuit in practical engineering. For discrete system, we put forward error feedback dynamic matrix. It is similar to the state feedback matrix for linear systems; but the former is time-varying and converges to a constant matrix. The controller includes coupling information of state variables, and updates online according to the gradi-ent descent method. Using the coupling relationship between variables, we present whole matrix control and partial matrix control. For partial matrix control, all vari-ables can reach the expected value only by the control of some variables, which is easy to implement in practice. However, the above controllers only can realize the stabilization of the fixed point and self-synchronization. Therefore, we combine with the speed gradient method and the nonlinear function control, and realize the smooth tracking of arbitrary bounded reference signal and parameter identification for continuous chaotic systems with unknown parameters.
2. The modified projective synchronization with complex scaling factor and parameter identification
We propose modified projective synchronization with complex scaling factors (CMPS). Since complete synchronization (CS), anti-synchronization (AS), projec-tive synchronization (PS) and modified projective synchronization (MPS) are special cases of CMPS, CMPS contains existing works and extend previous works. Consid-ering all possible cases of unknown parameters and bounded disturbance, we design CMPS controller which contains convergence factor and dynamical control strength, and then we draw on the sufficient condition and necessary condition that the un-known parameters converge to their true values based on the persistency of excitation and linear independence. We also discuss the method to change the scaling factors to identify the true values of the parameters, which is important in the practical control system. In addition, CMPS establishes a link between real chaos and complex chaos. Therefore, based on speed-gradient method, we first design adaptive CMPS schemes including pseudogradient condition.
3. Complex function projective synchronization of complex chaotic system and its application in communication
We present complex function projective synchronization (CFPS). CS, AS, PS, MPS, full states hybrid projective synchronization (FSHPS), function projective syn-chronization (FPS), modified function projective synchronization (MFPS), general-ized function projective synchronization (GFPS) and CMPS are the special cases of CFPS. Therefore, CFPS, which has rarely been explored, will contain most existing works and extend previous works. For complex system with unknown parameters, we report CFPS schemes. Aiming at coupled complex chaotic systems, we design a novel communication scheme based on CFPS. The essence of this scheme is chaotic masking, but the transmitted signal is the derivative of the product of the information signal (used as complex scaling function) and chaotic signal. As the complex scal-ing functions are arbitrary and more unpredictable than real scaling functions, the possibility that an interceptor extracts the information from the transmitted signal is greatly reduced.
4. Difference function projective synchronization of complex chaotic system and its application in communication
We put forward difference function projection synchronization (DFPS) from the view of the difference of two functions. CS and phase synchronization are its special cases. We design DFPS controller, and apply it to the coupling complex chaotic system, and design the communication scheme based on DFPS. This com-munication scheme is still chaos masking in essence, but the transmitted signal is the derivative of the sum of information signal and chaotic signal. It avoids the error which is produced when the state of drive system (as the divisor) is close to zero in the communication scheme based on CFPS.
5. Self-time-delay synchronization of complex chaotic system arid its applica-tion in communication
We put forward self-time-delay synchronization (STDS) of complex chaotic systems. STDS is an extension of self-synchronization, which further broadens the horizons of synchronization issues, and avoids several problems caused by time-delay. Time-delay complex chaotic systems can generate time series with high ran-domness and unpredictability which can improve the security in chaos-communication. Therefore, we study the characteristics of time-delay complex Lorenz system, and design a feedback controller to realize STDS. For coupling time-delay complex chaotic systems, we discuss the communication scheme based on STDS. This com-munication scheme refers to the time-delay produced in the transmission process, which is more close to the real situation. The adoption of time-delay complex chaotic systems further enhances the effect of secrecy.
To sum up, we study chaos control and synchronization of complex dynamical system and its applications in communication, and present the definitions such as modified projective synchronization with complex scaling factors, complex function projective synchronization, difference function projective synchronization and self-time-delay synchronization. We introduce the complex variables into the chaos- communication, which promote the development of the complex dynamic system, and provide the theoretical basis for strengthening security of communications.
1.实混沌系统的控制与同步
对于连续系统,提出了累积误差控制器.该控制器采用了累积误差的线性函数,在实际工程上可由RC电路实现.对于离散系统,提出了误差反馈矩阵控制.它类似于线性系统的状态反馈,但是该矩阵是时变的,包含状态变量的耦合信息,根据梯度下降法在线更新,并趋于一个恒定矩阵.结合变量间的耦合关系,本文还提出了部分矩阵控制和全矩阵控制.部分矩阵控制仅采用对部分变量的控制就实现全部变量达到期望值,易于工程实现.但是,上述控制器只能实现不动点控制和自同步.因此,结合速度梯度法和非线性函数控制,本文实现了带有未知参数的连续混沌系统的跟踪任意有界参考信号和参数辨识.
2.复混沌系统的复比例因子投影同步与参数辨识
提出了复比例因子投影同步(CMPS).完全同步(CS),反同步(AS),投影同步(PS)和修正投影同步(MPS)都是CMPS的特殊情况,因此它包含了以前的工作并进行推广.考虑到复混沌系统的未知参数和有界干扰的各种可能情形,本文设计了带有收敛因子和动态控制力量CMPS控制器；同时基于可持续激励和线性独立给出了参数收敛到真值的充分条件和必要条件,给出了改变比例因子的方法以辨识出参数真值.另外,CMPS在实混沌系统和复混沌系统之间建立同步联接.因此,本文首次讨论了实混沌系统和复混沌系统的复比例因子投影同步,基于速度梯度法设计了包含伪梯度条件的同步控制器.
3.复混沌系统的复函数同步及其通信方案
提出复函数投影同步(CFPS). CS, AS, PS, MPS,全状态组合投影同步(FSHPS),函数投影同步(FPS),修正函数投影同步(MFPS),广义函数投影同步(GFPS)和CMPS都是CFPS的特殊情形.CFPS几乎没有被研究过,而且将包括大部分存在的同步情形.针对带有未知参数的复混沌系统,设计了CFPS控制器,并针对耦合复混沌系统,设计了基于CFPS的通信方案.该通信方案本质上是混沌掩盖,但是所传输的信号是信息信号(作为复比例函数)与混沌信号积的导数.因为复比例函数比实函数更加不可预测和复杂,则入侵者从传输信号中提取信息的可能性更小.
4.复混沌系统的差函数投影同步及其通信方案
从两复函数相减的角度提出了差函数投影同步(DFPS),自同步和相位同步均是其特殊情形.设计了DFPS控制器,并将其应用到耦合复混沌系统中,提出了基于DFPS的通信方案.该通信方案本质上仍是混沌掩盖,但是所传输的信号是信息信号和混沌信号的和的导数.它避免了CFPS中由于驱动系统状态(作为除数)接近零时产生的算法误差.
5.时滞复混沌系统的自时滞同步及其通信方案
提出了复混沌系统的自时滞同步(STDS),它是自同步的扩展,进一步拓宽了同步问题的视野,并避免了因时滞而产生的各种问题.时滞复混沌系统能产生高度随机性和不可预测性的时间序列,应用在混沌保密通信中能够提高保密性能.因此,本文研究了时滞复Lorenz系统的混沌特性,并设计控制器实现了时滞复Lorenz系统的自时滞同步.针对耦合时滞复混沌系统,给出了基于STDS的通信方案.该通信方案考虑了信息传输中产生的时滞,更接近实际情况,时滞复混沌系统也进一步增强了保密效果.
综上所述,本文围绕复动力系统的混沌控制与同步及其在通信中的应用展开了研究,提出了复比例因子投影同步,复函数投影同步,差函数投影同步及自时滞同步等概念,并将复变量引入到混沌通信中,促进了复动力系统的发展,为进一步加强通信安全提供了理论依据.
At present, the complex dynamic system is widely used in communications, finance, biology and many other subjects. Complex chaotic system is a typical com-plex dynamic system. Since complex Lorenz equation has been proposed in1982, complex chaotic systems have played an important role in many fields of physics, es-pecially in communication. The complex variables increase the content of transmit-ted information and improve the safety. The control and synchronization of complex chaotic systems is becoming one of hot spots of chaos-communication. Therefore, this thesis focuses on the control and synchronization of complex chaotic system and its application in secure communication. The main work and innovation points are as follows,
1. The control and synchronization of real chaotic system
For continuous system, we propose cumulative error controller. The controller is the linear function of cumulative error in essence, which can be realized by RC circuit in practical engineering. For discrete system, we put forward error feedback dynamic matrix. It is similar to the state feedback matrix for linear systems; but the former is time-varying and converges to a constant matrix. The controller includes coupling information of state variables, and updates online according to the gradi-ent descent method. Using the coupling relationship between variables, we present whole matrix control and partial matrix control. For partial matrix control, all vari-ables can reach the expected value only by the control of some variables, which is easy to implement in practice. However, the above controllers only can realize the stabilization of the fixed point and self-synchronization. Therefore, we combine with the speed gradient method and the nonlinear function control, and realize the smooth tracking of arbitrary bounded reference signal and parameter identification for continuous chaotic systems with unknown parameters.
2. The modified projective synchronization with complex scaling factor and parameter identification
We propose modified projective synchronization with complex scaling factors (CMPS). Since complete synchronization (CS), anti-synchronization (AS), projec-tive synchronization (PS) and modified projective synchronization (MPS) are special cases of CMPS, CMPS contains existing works and extend previous works. Consid-ering all possible cases of unknown parameters and bounded disturbance, we design CMPS controller which contains convergence factor and dynamical control strength, and then we draw on the sufficient condition and necessary condition that the un-known parameters converge to their true values based on the persistency of excitation and linear independence. We also discuss the method to change the scaling factors to identify the true values of the parameters, which is important in the practical control system. In addition, CMPS establishes a link between real chaos and complex chaos. Therefore, based on speed-gradient method, we first design adaptive CMPS schemes including pseudogradient condition.
3. Complex function projective synchronization of complex chaotic system and its application in communication
We present complex function projective synchronization (CFPS). CS, AS, PS, MPS, full states hybrid projective synchronization (FSHPS), function projective syn-chronization (FPS), modified function projective synchronization (MFPS), general-ized function projective synchronization (GFPS) and CMPS are the special cases of CFPS. Therefore, CFPS, which has rarely been explored, will contain most existing works and extend previous works. For complex system with unknown parameters, we report CFPS schemes. Aiming at coupled complex chaotic systems, we design a novel communication scheme based on CFPS. The essence of this scheme is chaotic masking, but the transmitted signal is the derivative of the product of the information signal (used as complex scaling function) and chaotic signal. As the complex scal-ing functions are arbitrary and more unpredictable than real scaling functions, the possibility that an interceptor extracts the information from the transmitted signal is greatly reduced.
4. Difference function projective synchronization of complex chaotic system and its application in communication
We put forward difference function projection synchronization (DFPS) from the view of the difference of two functions. CS and phase synchronization are its special cases. We design DFPS controller, and apply it to the coupling complex chaotic system, and design the communication scheme based on DFPS. This com-munication scheme is still chaos masking in essence, but the transmitted signal is the derivative of the sum of information signal and chaotic signal. It avoids the error which is produced when the state of drive system (as the divisor) is close to zero in the communication scheme based on CFPS.
5. Self-time-delay synchronization of complex chaotic system arid its applica-tion in communication
We put forward self-time-delay synchronization (STDS) of complex chaotic systems. STDS is an extension of self-synchronization, which further broadens the horizons of synchronization issues, and avoids several problems caused by time-delay. Time-delay complex chaotic systems can generate time series with high ran-domness and unpredictability which can improve the security in chaos-communication. Therefore, we study the characteristics of time-delay complex Lorenz system, and design a feedback controller to realize STDS. For coupling time-delay complex chaotic systems, we discuss the communication scheme based on STDS. This com-munication scheme refers to the time-delay produced in the transmission process, which is more close to the real situation. The adoption of time-delay complex chaotic systems further enhances the effect of secrecy.
To sum up, we study chaos control and synchronization of complex dynamical system and its applications in communication, and present the definitions such as modified projective synchronization with complex scaling factors, complex function projective synchronization, difference function projective synchronization and self-time-delay synchronization. We introduce the complex variables into the chaos- communication, which promote the development of the complex dynamic system, and provide the theoretical basis for strengthening security of communications.
引文
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