非线性系统的分数阶控制理论研究
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摘要
在确定的非线性动力系统中常常被发现存在着混沌这种类似随机的复杂行为。分数阶微积分是研究任意阶微积分的理论,是整数阶微积分向任意阶的推广,分数阶系统与整数阶系统相比更能反应系统呈现的工程物理现象。分数阶系统因其阶次可调,在密码学、保密通信等领域有着更大的应用空间,近年来分数阶系统的混沌同步成为研究热点。
滑模变结构控制通过控制量的切换使系统状态沿着滑模面运动,从而使系统在受到参数摄动和外干扰时具有不变性。正是由于它的鲁棒性好、算法简单、可靠性高,被广泛应用于运动控制中。
本文研究内容如下：
首先研究了具有不匹配参数的分数阶Coullet系统的混沌同步的滑模控制。把分数阶导算子引入滑模变结构控制的指数趋近律中,即分数阶指数趋近律,通过分数阶指数趋近律来设计滑模面和变结构控制,调节相应参数,提高了控制效果,减小了抖动问题。
其次,论文深入研究了一个新的分数阶分段线性系统。根据平衡点个数的不同改变系统本身的参数进行了分类讨论,又基于分段系统初始位置不同而系统表达式不同的特点,继续讨论了不同初值下的系统运动行为,给出了大量的系统相轨迹图,从图中可看出分数阶分段线性系统呈现出了复杂多变的动力学行为,也发现了一些系统的运动规律。
最后研究了分数阶分段线性系统的混沌同步。结合系统本身的特点,设计了两个控制器,一个为分数阶滑模变结构控制,另一个为简单反馈控制。两个控制器最终实现了驱动系统和响应系统的混沌同步,为验证同步效果,给出了混沌状态下的不同参数和不同初值的同步效果图,通过调节分数阶趋近律的阶次α和系数λ,基本消除了由于分段系统切换所造成的毛刺现象,这也验证了该控制方法的灵活性。
The chaos, a complicated motion which is similar to random, has often been found in the determined nonlinear dynamic systems. Fractional calculus is the theory which researches on arbitrary order calculus, and it is the extension of integer order calculus to arbitrary order calculus. Compare with integer-order system, fractional-order system has a better reaction to engineering physics. As the fractional order can be adjusted, the fractional-order chaotic system has a wide application to cryptography, secure communications and other fields. In recent years, the chaotic synchronization in fractional-order nonlinear system has become a researching hotspot.
Sliding mode control can make the system state move on the sliding surface by switching the control variable, so that the system can maintain invariant under parametric perturbation and external disturbances. Sliding mode control is widely used in motion control because of its good robustness, simple algorithm and high reliability.
The research contents in this paper are as follows:
Firstly, the chaotic synchronization in fractional order Coullet system with unmatched parameters using sliding mode control has been studied. The fractional order derivative operator has been introduced into exponential approach law of sliding mode control, which forms the fractional order exponential approach law. Finally, the control effort was increased and the jarring was eliminated with the design of sliding mode surface and the variable structure controller through the fractional order exponential approach law and the adjustment of parameters.
Secondly, a new fractional piecewise linear system has been studied in depth. Classified discussion has been done by changing the system parameters according to the number of equilibrium points, then the system behavior with different initial values is discussed on the basis of characteristics that system expression differs as the different initial position of piecewise systems. Therefore, a large number of figures of phase trajectory have been given. From these figures, the phenomenon that fractional piecewise linear system presents a complex dynamic behavior can be seen, and some motion laws of this system can also be found.
Finally, the chaotic synchronization of fractional-order piecewise linear system has been studied. Combined with the characteristics of the system itself, two controllers have been designed, one is fractional-order sliding mode controller, and the other is a simple feedback controller. Ultimately, two controllers realized chaos synchronization between the drive system and response system. In order to verify the synchronization effect, some chaotic synchronization diagrams have been given with different parameters and different initial values. By adjusting the order a and coefficientλof the fractional-order approach law, the flash burr caused by switching in system has been eliminated largely, which also verifies the flexibility of the control method.
滑模变结构控制通过控制量的切换使系统状态沿着滑模面运动,从而使系统在受到参数摄动和外干扰时具有不变性。正是由于它的鲁棒性好、算法简单、可靠性高,被广泛应用于运动控制中。
本文研究内容如下：
首先研究了具有不匹配参数的分数阶Coullet系统的混沌同步的滑模控制。把分数阶导算子引入滑模变结构控制的指数趋近律中,即分数阶指数趋近律,通过分数阶指数趋近律来设计滑模面和变结构控制,调节相应参数,提高了控制效果,减小了抖动问题。
其次,论文深入研究了一个新的分数阶分段线性系统。根据平衡点个数的不同改变系统本身的参数进行了分类讨论,又基于分段系统初始位置不同而系统表达式不同的特点,继续讨论了不同初值下的系统运动行为,给出了大量的系统相轨迹图,从图中可看出分数阶分段线性系统呈现出了复杂多变的动力学行为,也发现了一些系统的运动规律。
最后研究了分数阶分段线性系统的混沌同步。结合系统本身的特点,设计了两个控制器,一个为分数阶滑模变结构控制,另一个为简单反馈控制。两个控制器最终实现了驱动系统和响应系统的混沌同步,为验证同步效果,给出了混沌状态下的不同参数和不同初值的同步效果图,通过调节分数阶趋近律的阶次α和系数λ,基本消除了由于分段系统切换所造成的毛刺现象,这也验证了该控制方法的灵活性。
The chaos, a complicated motion which is similar to random, has often been found in the determined nonlinear dynamic systems. Fractional calculus is the theory which researches on arbitrary order calculus, and it is the extension of integer order calculus to arbitrary order calculus. Compare with integer-order system, fractional-order system has a better reaction to engineering physics. As the fractional order can be adjusted, the fractional-order chaotic system has a wide application to cryptography, secure communications and other fields. In recent years, the chaotic synchronization in fractional-order nonlinear system has become a researching hotspot.
Sliding mode control can make the system state move on the sliding surface by switching the control variable, so that the system can maintain invariant under parametric perturbation and external disturbances. Sliding mode control is widely used in motion control because of its good robustness, simple algorithm and high reliability.
The research contents in this paper are as follows:
Firstly, the chaotic synchronization in fractional order Coullet system with unmatched parameters using sliding mode control has been studied. The fractional order derivative operator has been introduced into exponential approach law of sliding mode control, which forms the fractional order exponential approach law. Finally, the control effort was increased and the jarring was eliminated with the design of sliding mode surface and the variable structure controller through the fractional order exponential approach law and the adjustment of parameters.
Secondly, a new fractional piecewise linear system has been studied in depth. Classified discussion has been done by changing the system parameters according to the number of equilibrium points, then the system behavior with different initial values is discussed on the basis of characteristics that system expression differs as the different initial position of piecewise systems. Therefore, a large number of figures of phase trajectory have been given. From these figures, the phenomenon that fractional piecewise linear system presents a complex dynamic behavior can be seen, and some motion laws of this system can also be found.
Finally, the chaotic synchronization of fractional-order piecewise linear system has been studied. Combined with the characteristics of the system itself, two controllers have been designed, one is fractional-order sliding mode controller, and the other is a simple feedback controller. Ultimately, two controllers realized chaos synchronization between the drive system and response system. In order to verify the synchronization effect, some chaotic synchronization diagrams have been given with different parameters and different initial values. By adjusting the order a and coefficientλof the fractional-order approach law, the flash burr caused by switching in system has been eliminated largely, which also verifies the flexibility of the control method.
引文
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