复动力系统分形的辨识控制
摘要
现实世界的几何体——分形是非线性科学研究中非常活跃的一个分支,它的研究对象是自然界和非线性系统中出现的不规则和不光滑的几何形体,它的应用几乎涉及自然科学的各个领域甚至于社会科学。
动力系统中的分形集是近些年分形几何中十分活跃的一个研究领域,其中一类分形集产生于复平面上解析映射的迭代。解析映射迭代把复平面划分成两部分,一部分为朱利亚集(Julia集),另一部分为法图集(Fatou集)。人们已经发现分形集中具有重要地位的广义Mandelbrot-Julia (M-J)集深藏着精细而复杂的结构。
在理论研究上,分形集——Julia集及其推广形式的维数、性质及动力学特征成为科学工作者们关注的一个研究方向。另外,牛顿变换的Julia集的理论和Julia集的稳定域也是科学工作者研究的热点问题。然而根据实际情况的需要,人们往往对分形集合非线性吸引域的区域大小以及集合参数有要求,因此对分形集的同步控制和参数辨识的研究是十分必要和有意义的工作。目前,关于分形同步控制的工作已有了一些成果,如非线性耦合方法、梯度控制法,以及最优控制等方法实现复系统Julia集的稳定域的控制和驱动响应系统的同步控制。
但是遗憾的是,这些同步控制方法仅研究最基本的Julia集的同步控制和广义同步控制,并且只有在驱动系统参数已知的情况下才可行。然而,实际情况往往是驱动系统参数不可知,那么这些控制方法就无法解决驱动响应系统的同步控制问题。
本文针对复动力系统广义Julia集、最基本Julia集和三角函数Julia集,以及一类空间Julia集,创新性地解决了复动力系统分形的辨识控制问题。我们提出了新的实现系统同步控制和参数辨识的方法。该方法能够辨识驱动系统的参数,并且成功的解决了在驱动系统参数未知的情况下,无法实现驱动响应系统同步控制的问题。具体内容如下：
1基于非线性反馈控制器方法和差分方程稳定性理论,研究了一类相当广泛的复动力系统广义Julia集和正弦函数Julia集的参数辨识问题。设计了普遍适用的自适应同步控制器,给出了参数自适应律的解析表达式。并在理论上证明了设计的控制器可使得此类复动力系统广义Julia集、正弦函数Julia集达到同步,并且在渐近同步过程中能够辨识Julia集的未知参数。该方法也适用于最基本的Julia集。
2基于变结构控制理论,将滑动变量零渐近的性质应用到复动力系统分形的辨识控制理论中,提出了一种新的实现驱动响应系统同步控制和参数辨识的方法。该方法成功解决了复动力系统广义Julia集和三角函数Julia集在驱动系统参数未知的情况下无法实现同步控制的问题,并且在实现渐近同步的过程中,能够辨识出驱动系统的未知参数。该方法同样适用于最基本的Julia集。
3针对一类空间Julia集提出了一种新的实现驱动响应系统同步控制的方法,并创新性地解决了空间Julia集的参数辨识问题。该方法设计了普遍适用的自适应同步控制器,给出了参数自适应律的解析表达式,解决了在驱动系统参数未知的情况下无法实现同步控制的问题,并且在实现渐近同步的过程中能够辨识出驱动系统的未知参数。该方法是复动力系统广义Julia集辨识控制的自然推广。
这些研究为空间Julia集和复动力系统广义Julia集更好的应用于实际提供了一定的理论基础。
Fractal is the geometric solid in the real world. And the fractal theory is much attention in the nonlinear science. Its study object is the irregular and unsmooth ge-ometric solid in nature or nonlinear systems. Its applications are involved in almost all of the natural science, and even in the social science.
The fractal set of dynamic system is a very active research field in the fractal geometry in the past few years, and one class of them is gotten from the analytic mapping iteration on the complex plane. The complex plane is divided into two parts due to the analytic mapping iteration. One part is called Julia set, and the other part is called Fatou set. Mandelbrot-Julia set which is found to have a fine and complex structure plays an important role in fractal.
In theoretical study, many researchers are interested in the dimensions, prop-erties, and the kinetic characteristics of the Julia sets and their generalized forms. Moreover, the properties of the Newton's transformation Julia sets and stable re-gions of Julia sets are paid much attention. However, the nonlinear attractive domain ranges and parameters of the fractal sets are demanded according to the actual sit-uations. It is very necessary and significant to study the synchronous control and parameter identification of the fractal sets. At present, some significant results of the fractal synchronous control have been reported, such as the nonlinear coupling method, the gradient control, and the optimum control, which realize the control of the stable region and synchronous control for the Julia sets of complex systems.
It is pointed that these methods are only applied to the synchronous control and generalized synchronous control of basic Julia sets. And it is only feasible when the parameters of the drive system are obtained. The parameters are usually not derived actually, so the synchronous control in the drive-response system is not solved by the method above.
In this paper, the identification control on fractal of the complex dynamic sys-tems which include the generalized Julia sets, basic Julia sets, trigonometric function Julia sets and one class of spatial Julia sets, is innovatively achieved. A new method to realize synchronous control and parameter identification of the drive-response system is obtained. And it is applied to synchronous control in the drive-response system when parameters of drive system are unknown, and parameters of the drive system are identified. The main content is following.
1Based on the nonlinear feedback controller method and the stability theory in difference equations, the parameter identification on generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems is studied. The generally applicable adaptive synchronous controller and parameter adaptive ana-lytic expression are designed. It is proved that the controllers make generalized Julia sets and trigonometric function Julia sets achieve the synchronization, and the un-known parameters of the Julia sets can be identified. Particularly, this method is also applied to the basic Julia sets.
2Based on variable structure control theory, the zero asymptotic sliding vari-ables are applied to the fractal identification control for the complex dynamic sys-tems. A new method to realize drive-response system synchronous control and parameter identification is derived. And the problems of synchronous control, for generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems, are solved when the drive system parameters are unknown. The unknown parameters of the drive system can be identified in the asymptotic synchronization process. Moreover, this method is also applied to the basic Julia sets.
3A new method to realize drive-response system synchronous control for the spatial Julia sets is derived. And the parameter identification of the spatial Julia sets is innovatively solved. The widely used adaptive synchronous controller and the analytical expression of the parameter adaptive law are designed by this method. Meanwhile, the unknown parameters of the drive system can be identified in the asymptotic synchronization process. It is successful to realize the synchronization control in the case of the unknown parameters. Particularly, the method is also ap-plied to the basic Julia sets.
These studies are provided certain theoretical bases to the spatial Julia sets and the generalized Julia sets of complex dynamic systems for better applications.
动力系统中的分形集是近些年分形几何中十分活跃的一个研究领域,其中一类分形集产生于复平面上解析映射的迭代。解析映射迭代把复平面划分成两部分,一部分为朱利亚集(Julia集),另一部分为法图集(Fatou集)。人们已经发现分形集中具有重要地位的广义Mandelbrot-Julia (M-J)集深藏着精细而复杂的结构。
在理论研究上,分形集——Julia集及其推广形式的维数、性质及动力学特征成为科学工作者们关注的一个研究方向。另外,牛顿变换的Julia集的理论和Julia集的稳定域也是科学工作者研究的热点问题。然而根据实际情况的需要,人们往往对分形集合非线性吸引域的区域大小以及集合参数有要求,因此对分形集的同步控制和参数辨识的研究是十分必要和有意义的工作。目前,关于分形同步控制的工作已有了一些成果,如非线性耦合方法、梯度控制法,以及最优控制等方法实现复系统Julia集的稳定域的控制和驱动响应系统的同步控制。
但是遗憾的是,这些同步控制方法仅研究最基本的Julia集的同步控制和广义同步控制,并且只有在驱动系统参数已知的情况下才可行。然而,实际情况往往是驱动系统参数不可知,那么这些控制方法就无法解决驱动响应系统的同步控制问题。
本文针对复动力系统广义Julia集、最基本Julia集和三角函数Julia集,以及一类空间Julia集,创新性地解决了复动力系统分形的辨识控制问题。我们提出了新的实现系统同步控制和参数辨识的方法。该方法能够辨识驱动系统的参数,并且成功的解决了在驱动系统参数未知的情况下,无法实现驱动响应系统同步控制的问题。具体内容如下：
1基于非线性反馈控制器方法和差分方程稳定性理论,研究了一类相当广泛的复动力系统广义Julia集和正弦函数Julia集的参数辨识问题。设计了普遍适用的自适应同步控制器,给出了参数自适应律的解析表达式。并在理论上证明了设计的控制器可使得此类复动力系统广义Julia集、正弦函数Julia集达到同步,并且在渐近同步过程中能够辨识Julia集的未知参数。该方法也适用于最基本的Julia集。
2基于变结构控制理论,将滑动变量零渐近的性质应用到复动力系统分形的辨识控制理论中,提出了一种新的实现驱动响应系统同步控制和参数辨识的方法。该方法成功解决了复动力系统广义Julia集和三角函数Julia集在驱动系统参数未知的情况下无法实现同步控制的问题,并且在实现渐近同步的过程中,能够辨识出驱动系统的未知参数。该方法同样适用于最基本的Julia集。
3针对一类空间Julia集提出了一种新的实现驱动响应系统同步控制的方法,并创新性地解决了空间Julia集的参数辨识问题。该方法设计了普遍适用的自适应同步控制器,给出了参数自适应律的解析表达式,解决了在驱动系统参数未知的情况下无法实现同步控制的问题,并且在实现渐近同步的过程中能够辨识出驱动系统的未知参数。该方法是复动力系统广义Julia集辨识控制的自然推广。
这些研究为空间Julia集和复动力系统广义Julia集更好的应用于实际提供了一定的理论基础。
Fractal is the geometric solid in the real world. And the fractal theory is much attention in the nonlinear science. Its study object is the irregular and unsmooth ge-ometric solid in nature or nonlinear systems. Its applications are involved in almost all of the natural science, and even in the social science.
The fractal set of dynamic system is a very active research field in the fractal geometry in the past few years, and one class of them is gotten from the analytic mapping iteration on the complex plane. The complex plane is divided into two parts due to the analytic mapping iteration. One part is called Julia set, and the other part is called Fatou set. Mandelbrot-Julia set which is found to have a fine and complex structure plays an important role in fractal.
In theoretical study, many researchers are interested in the dimensions, prop-erties, and the kinetic characteristics of the Julia sets and their generalized forms. Moreover, the properties of the Newton's transformation Julia sets and stable re-gions of Julia sets are paid much attention. However, the nonlinear attractive domain ranges and parameters of the fractal sets are demanded according to the actual sit-uations. It is very necessary and significant to study the synchronous control and parameter identification of the fractal sets. At present, some significant results of the fractal synchronous control have been reported, such as the nonlinear coupling method, the gradient control, and the optimum control, which realize the control of the stable region and synchronous control for the Julia sets of complex systems.
It is pointed that these methods are only applied to the synchronous control and generalized synchronous control of basic Julia sets. And it is only feasible when the parameters of the drive system are obtained. The parameters are usually not derived actually, so the synchronous control in the drive-response system is not solved by the method above.
In this paper, the identification control on fractal of the complex dynamic sys-tems which include the generalized Julia sets, basic Julia sets, trigonometric function Julia sets and one class of spatial Julia sets, is innovatively achieved. A new method to realize synchronous control and parameter identification of the drive-response system is obtained. And it is applied to synchronous control in the drive-response system when parameters of drive system are unknown, and parameters of the drive system are identified. The main content is following.
1Based on the nonlinear feedback controller method and the stability theory in difference equations, the parameter identification on generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems is studied. The generally applicable adaptive synchronous controller and parameter adaptive ana-lytic expression are designed. It is proved that the controllers make generalized Julia sets and trigonometric function Julia sets achieve the synchronization, and the un-known parameters of the Julia sets can be identified. Particularly, this method is also applied to the basic Julia sets.
2Based on variable structure control theory, the zero asymptotic sliding vari-ables are applied to the fractal identification control for the complex dynamic sys-tems. A new method to realize drive-response system synchronous control and parameter identification is derived. And the problems of synchronous control, for generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems, are solved when the drive system parameters are unknown. The unknown parameters of the drive system can be identified in the asymptotic synchronization process. Moreover, this method is also applied to the basic Julia sets.
3A new method to realize drive-response system synchronous control for the spatial Julia sets is derived. And the parameter identification of the spatial Julia sets is innovatively solved. The widely used adaptive synchronous controller and the analytical expression of the parameter adaptive law are designed by this method. Meanwhile, the unknown parameters of the drive system can be identified in the asymptotic synchronization process. It is successful to realize the synchronization control in the case of the unknown parameters. Particularly, the method is also ap-plied to the basic Julia sets.
These studies are provided certain theoretical bases to the spatial Julia sets and the generalized Julia sets of complex dynamic systems for better applications.
引文
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