分数阶混沌系统的鲁棒同步研究
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摘要
伴随着分数阶理论的发展与混沌系统同步的广泛应用,分数阶混沌系统的同步问题引起了越来越多学者的关注。由于混沌系统对参数扰动与外部干扰特别敏感,同时分数阶系统的稳定性与整数阶系统稳定性又存在很大的区别。所以,分数阶混沌系统的鲁棒同步问题一直是一个棘手的问题。
本文围绕分数阶系统的稳定性与分数阶混沌系统的鲁棒同步方法进行了深入研究。主要内容包括：
1.针对具有外部扰动的不确定分数阶混沌系统,基于终端滑模控制理论设计了一种鲁棒同步控制方案。其原理是基于分数阶Lyapunov稳定性理论对分数阶混沌系统引入了鲁棒滑模控制律,确保了滑模运动在有限时间内发生。对具有外部扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统进行了鲁棒同步仿真,结果验证了方法的有效性。特别指出,该分数阶终端滑模控制器针对一大类具有外部扰动的不确定异结构分数阶混沌系统都是有效的。
2.针对参数不确定与外部扰动整体有界的分数阶混沌系统基于模型逼近方法提出了一种新的鲁棒修正投影方案。通过针对具有扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统的完全同步与修正投影同步两个仿真实例来验证了所设计的基于模型逼近的同步控制器的鲁棒性。从该控制器的设计过程可以得出,该控制方案针对一大类包含外部扰动的不确定异结构分数阶混沌系统均可以实现鲁棒同步。
3.通过分析时变分数阶系统的稳定性,给出了一种微分阶次0<α<1时的时变分数阶系统的稳定性判定定理。基于此定理对分数阶Lu混沌系统的同步进行了仿真研究,仿真结果验证了所提出的时变分数阶系统稳定性定理在实际控制器设计过程中是有效的。
4.首先给出了永磁同步电机(PMSM)混沌系统的分数阶数学模型；然后基于分数阶Lyapunov矩阵微分方程稳定性理论,给出了分数阶Lyapunov鲁棒稳定性定理和推论,理论的提出使时域内分数阶系统的稳定性判定更加便捷。此后,应用此定理和推论分别设计了不同的控制器来实现分数阶PMSM混沌系统的混沌控制与同步,数值仿真曲线表明了控制与同步方案的有效性。
综上,本文针对分数阶系统的稳定性与分数阶混沌系统的鲁棒同步问题的创新性成果主要有以下几点：
1.基于终端滑模控制方法,设计分数阶滑模控制实现了具有扰动的不确定异结构分数阶混沌系统鲁棒同步。
2.针对具有扰动的不确定异结构分数阶混沌系统提出了一种基于模型逼近误差上界的鲁棒同步方法。
3.提出了时变分数阶系统的稳定性定理并将其成功应用到分数阶混沌系统的同步。
4.给出了线性分数阶系统的稳定性定理并将其成功应用于分数阶PMSM混沌系统的同步。
The robust synchronization problem of fractional-order chaotic system, which is pre-sented along with the development of fractional-order theory and extensive use of chaotic synchronization, has attracted more and more attention of scholars. But at the same time, it is also a thorny problem in that chaotic system is particularly sensitive to parameter perturbation and external disturbance, and there is significant difference between the sta-bility of fractional-order chaotic system and integer order chaotic system.
This dissertation focuses on the stability of fractional-order system and the robust synchronization method of fractional-order chaotic systems. Main contributions are given as follows:
1. A robust synchronization control scheme for uncertain fractional-order chaotic sys-tems with external disturbances is designed based on the terminal sliding mode control theory. On the basis of fractional-order Lyapunov stability theory, a robust sliding mode control scheme, which is used to ensure the sliding mode motion occurs in limited time, is introduced to the fractional-order chaotic system. The proposed control scheme is applied to synchronize the fractional order Lorenz chaotic system and fractional-order Chen's cha-otic system with uncertainty and external disturbance parameters, simulation results show the applicability and efficiency of the proposed scheme. It should be pointed out in partic-ular that the introduced fractional-order terminal sliding mode controller is applicable for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
2. Based on model approximation method, a new robust modified projective synchro-nization control scheme is presented for the fractional-order chaotic system which is global bounded for parameter uncertainty and external disturbances. Then the proposed method is used to the complete synchronization and revised projection synchronization of frac-tional-order Lorenz and Chen's chaotic systems with parameter uncertainty and external disturbances, simulation results show the robustness of the designed controller. It can also be concluded from the scheme's design process that the proposed scheme is effective for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
3. By analyzing the stability of a time-varying fractional-order system, a stability the-orem is proposed for a time varying fractional-order system with order0<α<1. Then the presented stability theorem is used in the synchronization of the fractional-order Lii chaotic system, simulation results verify again that it is also effective when used in the process of controller design.
4. First, a fractional-order mathematical model of permanent magnet synchronous motor (PMSM) is given. Then a fractional-order Lyapunov robust stability theorem and deduction, which makes it is more convenient to judge the stability of fractional-order system in time domain, are derived based on the fractional-order Lyapunov matrix differ-ential equation stability theory. Finally, different controllers which are designed respec-tively according to the proposed theorem and deduction are used to achieve the chaotic control and synchronization of the fractional-order PMSM system, numerical simulation curves show the effectiveness of the method.
In conclusion, the main innovative achievements for fractional-order systems stability and fractional-order chaotic system robust synchronization in this paper are as follows:
1. Based on the terminal sliding mode control scheme, fractional-order sliding mode surface and controller are designed to realize the robust synchronization of different un-certain fractional-order chaotic systems under external disturbances.
2. Based on the upper bound of model approximation error method, a robust synchronization scheme is proposed for different uncertain fractional-order chaotic sys-tems under external disturbances.
3. A stability theorem for time-varying fractional-order system is proposed and is successfully applied to the synchronization of fractional-order chaotic systems.
4. A stability theorem for linear fractional-order system is proposed and is successfully applied to the synchronization of fractional-order PMSM chaotic system.
本文围绕分数阶系统的稳定性与分数阶混沌系统的鲁棒同步方法进行了深入研究。主要内容包括：
1.针对具有外部扰动的不确定分数阶混沌系统,基于终端滑模控制理论设计了一种鲁棒同步控制方案。其原理是基于分数阶Lyapunov稳定性理论对分数阶混沌系统引入了鲁棒滑模控制律,确保了滑模运动在有限时间内发生。对具有外部扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统进行了鲁棒同步仿真,结果验证了方法的有效性。特别指出,该分数阶终端滑模控制器针对一大类具有外部扰动的不确定异结构分数阶混沌系统都是有效的。
2.针对参数不确定与外部扰动整体有界的分数阶混沌系统基于模型逼近方法提出了一种新的鲁棒修正投影方案。通过针对具有扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统的完全同步与修正投影同步两个仿真实例来验证了所设计的基于模型逼近的同步控制器的鲁棒性。从该控制器的设计过程可以得出,该控制方案针对一大类包含外部扰动的不确定异结构分数阶混沌系统均可以实现鲁棒同步。
3.通过分析时变分数阶系统的稳定性,给出了一种微分阶次0<α<1时的时变分数阶系统的稳定性判定定理。基于此定理对分数阶Lu混沌系统的同步进行了仿真研究,仿真结果验证了所提出的时变分数阶系统稳定性定理在实际控制器设计过程中是有效的。
4.首先给出了永磁同步电机(PMSM)混沌系统的分数阶数学模型；然后基于分数阶Lyapunov矩阵微分方程稳定性理论,给出了分数阶Lyapunov鲁棒稳定性定理和推论,理论的提出使时域内分数阶系统的稳定性判定更加便捷。此后,应用此定理和推论分别设计了不同的控制器来实现分数阶PMSM混沌系统的混沌控制与同步,数值仿真曲线表明了控制与同步方案的有效性。
综上,本文针对分数阶系统的稳定性与分数阶混沌系统的鲁棒同步问题的创新性成果主要有以下几点：
1.基于终端滑模控制方法,设计分数阶滑模控制实现了具有扰动的不确定异结构分数阶混沌系统鲁棒同步。
2.针对具有扰动的不确定异结构分数阶混沌系统提出了一种基于模型逼近误差上界的鲁棒同步方法。
3.提出了时变分数阶系统的稳定性定理并将其成功应用到分数阶混沌系统的同步。
4.给出了线性分数阶系统的稳定性定理并将其成功应用于分数阶PMSM混沌系统的同步。
The robust synchronization problem of fractional-order chaotic system, which is pre-sented along with the development of fractional-order theory and extensive use of chaotic synchronization, has attracted more and more attention of scholars. But at the same time, it is also a thorny problem in that chaotic system is particularly sensitive to parameter perturbation and external disturbance, and there is significant difference between the sta-bility of fractional-order chaotic system and integer order chaotic system.
This dissertation focuses on the stability of fractional-order system and the robust synchronization method of fractional-order chaotic systems. Main contributions are given as follows:
1. A robust synchronization control scheme for uncertain fractional-order chaotic sys-tems with external disturbances is designed based on the terminal sliding mode control theory. On the basis of fractional-order Lyapunov stability theory, a robust sliding mode control scheme, which is used to ensure the sliding mode motion occurs in limited time, is introduced to the fractional-order chaotic system. The proposed control scheme is applied to synchronize the fractional order Lorenz chaotic system and fractional-order Chen's cha-otic system with uncertainty and external disturbance parameters, simulation results show the applicability and efficiency of the proposed scheme. It should be pointed out in partic-ular that the introduced fractional-order terminal sliding mode controller is applicable for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
2. Based on model approximation method, a new robust modified projective synchro-nization control scheme is presented for the fractional-order chaotic system which is global bounded for parameter uncertainty and external disturbances. Then the proposed method is used to the complete synchronization and revised projection synchronization of frac-tional-order Lorenz and Chen's chaotic systems with parameter uncertainty and external disturbances, simulation results show the robustness of the designed controller. It can also be concluded from the scheme's design process that the proposed scheme is effective for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
3. By analyzing the stability of a time-varying fractional-order system, a stability the-orem is proposed for a time varying fractional-order system with order0<α<1. Then the presented stability theorem is used in the synchronization of the fractional-order Lii chaotic system, simulation results verify again that it is also effective when used in the process of controller design.
4. First, a fractional-order mathematical model of permanent magnet synchronous motor (PMSM) is given. Then a fractional-order Lyapunov robust stability theorem and deduction, which makes it is more convenient to judge the stability of fractional-order system in time domain, are derived based on the fractional-order Lyapunov matrix differ-ential equation stability theory. Finally, different controllers which are designed respec-tively according to the proposed theorem and deduction are used to achieve the chaotic control and synchronization of the fractional-order PMSM system, numerical simulation curves show the effectiveness of the method.
In conclusion, the main innovative achievements for fractional-order systems stability and fractional-order chaotic system robust synchronization in this paper are as follows:
1. Based on the terminal sliding mode control scheme, fractional-order sliding mode surface and controller are designed to realize the robust synchronization of different un-certain fractional-order chaotic systems under external disturbances.
2. Based on the upper bound of model approximation error method, a robust synchronization scheme is proposed for different uncertain fractional-order chaotic sys-tems under external disturbances.
3. A stability theorem for time-varying fractional-order system is proposed and is successfully applied to the synchronization of fractional-order chaotic systems.
4. A stability theorem for linear fractional-order system is proposed and is successfully applied to the synchronization of fractional-order PMSM chaotic system.
引文
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