非线性动力学分析与控制的若干理论问题及应用研究
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摘要
世界的本质是非线性的,以前我们研究的线性理论只是对客观的非线性世界的一种近似。因此,研究非线性动力学中的混沌运动和分岔现象是非常有趣的事情,它们帮助我们更深刻的认识自然界的非线性本质。在工程中的水电站系统,是典型的耦合了水利系统、机械系统和电气系统的复杂非线性系统。生产实践中,也发现国内外大中型水电站中水力发电机组均存在一定的稳定性问题。其表现出水轮机调节系统的控制十分困难,与之矛盾的是：水电站机组装机容量、其所占电力系统容量的比率越来越大和自动化水平要求的不断提高。所以,本文从非线性动力学理论入手,研究新的物理系统、物理现象、物理规律和物理概念,并尝试将其应用到水电站系统稳定性分析与控制。
论文的主要内容和结论如下：
(1)提出了一个新三维的混沌系统,若对这个三维混沌系统中的一个线性项进行绝对值运算,即可得到该系统的羽翼倍增的新混沌系统。它让我们认识到次要部分的微小改变的确也能重大改变事物的整体特征,因为,我们知道“非线性是产生混沌的必要条件”,换句话说,非线性项是非线性动力学系统中的主要部分。同时,还对这个新系统进行了耗散性分析、平衡点特征分析等动力学分析。最后,给出了实现该系统的电路图,电路实验观测结果和数值模拟结果彼此一致且互相印证。
(2)通过提出一个新的分段线性函数,实现了能够产生二到n个涡卷的新混沌系统,并且给出了其电路设计方案。电路实验和数值模拟结果相互统一。最重要的是,在进行非线性动力学的过程中发现：系统的相轨迹图、庞加莱图和分岔图是统一在一起的。具体的说就是：①庞加莱图与相轨迹图在随着涡卷数量的变化时,是彼此一致的。更确切的为,在庞加莱图上的分形的圆圈数量是随着涡卷数量的增多而逐渐增多的。因此,我们可以说,庞加莱图和相轨迹图两者在这个方面上是一致的。②不同涡卷的分岔图上的密集点数目随着涡卷数量的增大而递增。换句话说,他们彼此是一致的。这就意味着,我们可以通过相轨迹图中吸引子的数目来判定分岔图中密集点的数目,而判定分岔图的正确与否,反之亦然。最后,我们可以说,相轨迹图、分岔图和庞加莱映射图,这些非线性动力学分析工具是统一的一个整体,彼此能互相校核。当然,各自都有自己的独特一面,能反映非线性动力学系统的一个侧面。
(3)提出了一个新的四维的分数阶混沌系统,给出了分数阶混沌系统的数值模拟的算法迭代过程,根据分数阶稳定性定理,分析了分数阶动力学系统的平衡点稳定性。给出了分数阶0.8,0.9阶的单元电路的形式,和完整的新分数阶混沌系统的电路图。提出了一个新的三维分数阶混沌系统,通过对其第二个方程的一个线性项进行绝对值运算,可以得到该分数阶系统的羽翼倍增系统。并设计了其实现电路。
(4)在混沌控制方面,首先研究了在考虑有界噪声时的滑模变结构控制方法对混沌系统的控制与实践,探讨了用饱和函数解决滑模变结构控制的抖振问题,深入研究了不同趋近率对被控系统过渡过程的影响；其次研究了一类三维分数阶混沌系统的控制问题,提出了一类三维分数阶混沌系统的一般形式,并通过三个典型例子证明所提方法的有效性；最后,提出了用只含一个控制项的控制器控制一类四维超混沌系统,该控制方法对非线性输入和随机噪声都具有很好的鲁棒性。
(5)在混沌同步方面,首先基于LMI (Linear Matrix Inequality)变换采用模糊控制方法实现了分数阶与整数阶混沌系统的同步,以此来建立两者之间的桥梁,还同时实现了混沌系统的反同步,指出同步与反同步是统一在一起的,可以用同一方法实现,最后实现了不同维混沌系统之间的同步,提出控制是特殊的同步,同步是更广义的控制。最后,基于模糊滑模变结构控制实现一类n维混沌系统的同步,如果将每一维方程看作一个节点,该方法可以应用于复杂网络同步。
(6)在同步电路方面,首先基于错位同步理论实现了整数阶混沌系统的同步电路设计。其次,基于非线性反馈方法实现了分数阶混沌系统的同步。最后,提出了一个全新的最小单元响应电路,该电路能够同步任一对称或不对称的整数阶或分数阶混沌电路,给出两个典型例子来证明该电路的正确性和可行性。这也是第一次在工程上实现分数阶与整数阶混沌系统的同步,此实验的成功为两者建立了完美的桥梁。在此工作之后,基于混沌电路的保密通信也将面临新的挑战。
(7)在简单式调压井模型基础上,考虑弹性水击效应、发电机转矩角的影响等,建立了单机单管时的水轮机调节系统非线性数学模型。在此基础上,利用非线性理论,分析了系统的非线性动力学特征；建立了带缓冲罆的水轮机调节系统的非线性数学模型,在考虑电网频率波动对系统的非线性动力学行为的影响；在考虑调压设备及尾水管道的基础上建立了水轮机调节系统弹性水击非线性数学模型。并在此模型基于上,运用非线性动力学理论对某一具体水电站进行了稳定性分析和具体运动特性研究：最后,提出了模糊滑模变结构控制,并将其应用到水轮机调节系统中,消除系统的不稳定状态。
As we all know, the world is nonlinear in essence. Actually, what we have got based on linear theory is approximate average value of the world. Therefore, it is very interesting to study chaos and bifurcation of nonlinear systems, for that we can understand the nature of the things more deeply. As for Hydropower system, it is a complex nonlinear system including hydro system, mechanical system and electrical system. In production practice, stability is a universal problem for hydropower system in all over the world. Also, it is hard to control. Correspondingly, the capacity of the hydropower is becoming larger and larger, and the requirement of the system is increasing higher and higher. Therefore, we study new physical system, physical phenomenon, new law and physical concepts by using nonlinear dynamics theory, and try to apply it to the stability analysis of the hydropower system.
The contents and conclusions of this paper include:
(1) A new three dimentional chaotic system was presented here. Then, we can get a new double-wing chaotic system by the absolute operation of a linear term. It shows that a slight change on the secondary part can change the mainly characters of the whole system, for that we all know that nonlinear is a requirement to the occurance of chaos, in other words, nonlinear term is the main part of the system. Finally, the circuit diagram was also derived. Its experimental results are agreement with the numerical simulation.
(2) A new multi-scroll chaotic system was presented by using a new piecewise function. Moreover, the circuit diagram was also designed, the experimental results in which are agreement with numerical simulation. We can get conclusion that the phase trajectories, Poincare maps and bifurcation diagram are all uniform and can describe different aspects of the dynamical system.①the numbers of circles in fractal structure in Poincare maps is gradually increasing with the increase of the scroll number. Therefore, the phase trajectories and Poincare maps are agreement with each other. In other words, they are consistent in essence.②The number of dense points is consistent with the number of the scrolls. Obviously, the number of dense points region is increasing with the number of the scrolls. In other words, they are equal to each other. In the future, we can judge the number of the attractors by confirming the number of the dense points region in bifurcation diagram, while the attractors are more intuitive from the phase trajectories maps.At last, we cam get that the phase trajectories, Poincare maps and bifurcation diagram are all uniform and can describe different aspects of the dynamical system.
(3) A new four-dimentional fractional-order chaotic system and a new double-wing fractional-order chaotic system were derived including circuit diagram.
(4) As for chaos control, we first study the sliding mode control with bounded noise, and discuss the no-chattering sliding mode control by using saturation funciton. Furthermore, we also study the effection of different reaching rate. Second, we also study the control of a class of chaotic systems by using three typical examples. Finally, we implemented a class of four dimentional hyperchaotic systems with only one controller term.
(5) As for chaos synchronization, I try to bridge the fractional order chaotic system and integer order chaotic system by reaching the synchronization of fractional-order chaotic system and integer order chaotic sytem based on LMI. We also point that synchronization and anti-synchronization are unified with each other. They could be realized by the same method. Furthermore, we get a conclusion that chaos control is a special case of chaos synchronization, and chaos synchronization is a generalized concept of chaos control. Finally, a class of n-dimentional chaotic systems was realized via fuzzy sliding mode control, which is also suitable for complex networks.
(6) As for circuit design, a circuit diagram of chaos synchronization via mismatch synchronization. Also, a circuit diagram of fractional order chaotic synchronization via nonlinear feedback control. Finally, a unit circuit of response system was presented, which is suitable for any fractional-order or integer order chaotic system, which was proved by two typical examples. It is the first time to reach synchronization of fractional-order and integer order chaotic systems. After this work, the secure communication will meet new change based on chaotic circuit.
(7) A new dynamical model for the hydro-turbine governing system with elastic water hammer-impact and the second order model of generator was established, which is based on the model of simple surge shaft. By virtue of bifurcation diagram, Poincare maps, the power spectrum diagram, the time domain diagram, the orbits diagram and the spectrum diagram, the nonlinear dynamical responses of generator's relative deviation of rotational seed and the water pressure at the entrance of surge shaft were analyzed. Second, a new nonlinear mathematical model of a hydro-turbine governing system with a surge tank was presented. Finally, we introduce a new model of hydro-turbine system with the effect of surge tank based on state-space to study the dynamical behaviors of a hydro-turbine system.
论文的主要内容和结论如下：
(1)提出了一个新三维的混沌系统,若对这个三维混沌系统中的一个线性项进行绝对值运算,即可得到该系统的羽翼倍增的新混沌系统。它让我们认识到次要部分的微小改变的确也能重大改变事物的整体特征,因为,我们知道“非线性是产生混沌的必要条件”,换句话说,非线性项是非线性动力学系统中的主要部分。同时,还对这个新系统进行了耗散性分析、平衡点特征分析等动力学分析。最后,给出了实现该系统的电路图,电路实验观测结果和数值模拟结果彼此一致且互相印证。
(2)通过提出一个新的分段线性函数,实现了能够产生二到n个涡卷的新混沌系统,并且给出了其电路设计方案。电路实验和数值模拟结果相互统一。最重要的是,在进行非线性动力学的过程中发现：系统的相轨迹图、庞加莱图和分岔图是统一在一起的。具体的说就是：①庞加莱图与相轨迹图在随着涡卷数量的变化时,是彼此一致的。更确切的为,在庞加莱图上的分形的圆圈数量是随着涡卷数量的增多而逐渐增多的。因此,我们可以说,庞加莱图和相轨迹图两者在这个方面上是一致的。②不同涡卷的分岔图上的密集点数目随着涡卷数量的增大而递增。换句话说,他们彼此是一致的。这就意味着,我们可以通过相轨迹图中吸引子的数目来判定分岔图中密集点的数目,而判定分岔图的正确与否,反之亦然。最后,我们可以说,相轨迹图、分岔图和庞加莱映射图,这些非线性动力学分析工具是统一的一个整体,彼此能互相校核。当然,各自都有自己的独特一面,能反映非线性动力学系统的一个侧面。
(3)提出了一个新的四维的分数阶混沌系统,给出了分数阶混沌系统的数值模拟的算法迭代过程,根据分数阶稳定性定理,分析了分数阶动力学系统的平衡点稳定性。给出了分数阶0.8,0.9阶的单元电路的形式,和完整的新分数阶混沌系统的电路图。提出了一个新的三维分数阶混沌系统,通过对其第二个方程的一个线性项进行绝对值运算,可以得到该分数阶系统的羽翼倍增系统。并设计了其实现电路。
(4)在混沌控制方面,首先研究了在考虑有界噪声时的滑模变结构控制方法对混沌系统的控制与实践,探讨了用饱和函数解决滑模变结构控制的抖振问题,深入研究了不同趋近率对被控系统过渡过程的影响；其次研究了一类三维分数阶混沌系统的控制问题,提出了一类三维分数阶混沌系统的一般形式,并通过三个典型例子证明所提方法的有效性；最后,提出了用只含一个控制项的控制器控制一类四维超混沌系统,该控制方法对非线性输入和随机噪声都具有很好的鲁棒性。
(5)在混沌同步方面,首先基于LMI (Linear Matrix Inequality)变换采用模糊控制方法实现了分数阶与整数阶混沌系统的同步,以此来建立两者之间的桥梁,还同时实现了混沌系统的反同步,指出同步与反同步是统一在一起的,可以用同一方法实现,最后实现了不同维混沌系统之间的同步,提出控制是特殊的同步,同步是更广义的控制。最后,基于模糊滑模变结构控制实现一类n维混沌系统的同步,如果将每一维方程看作一个节点,该方法可以应用于复杂网络同步。
(6)在同步电路方面,首先基于错位同步理论实现了整数阶混沌系统的同步电路设计。其次,基于非线性反馈方法实现了分数阶混沌系统的同步。最后,提出了一个全新的最小单元响应电路,该电路能够同步任一对称或不对称的整数阶或分数阶混沌电路,给出两个典型例子来证明该电路的正确性和可行性。这也是第一次在工程上实现分数阶与整数阶混沌系统的同步,此实验的成功为两者建立了完美的桥梁。在此工作之后,基于混沌电路的保密通信也将面临新的挑战。
(7)在简单式调压井模型基础上,考虑弹性水击效应、发电机转矩角的影响等,建立了单机单管时的水轮机调节系统非线性数学模型。在此基础上,利用非线性理论,分析了系统的非线性动力学特征；建立了带缓冲罆的水轮机调节系统的非线性数学模型,在考虑电网频率波动对系统的非线性动力学行为的影响；在考虑调压设备及尾水管道的基础上建立了水轮机调节系统弹性水击非线性数学模型。并在此模型基于上,运用非线性动力学理论对某一具体水电站进行了稳定性分析和具体运动特性研究：最后,提出了模糊滑模变结构控制,并将其应用到水轮机调节系统中,消除系统的不稳定状态。
As we all know, the world is nonlinear in essence. Actually, what we have got based on linear theory is approximate average value of the world. Therefore, it is very interesting to study chaos and bifurcation of nonlinear systems, for that we can understand the nature of the things more deeply. As for Hydropower system, it is a complex nonlinear system including hydro system, mechanical system and electrical system. In production practice, stability is a universal problem for hydropower system in all over the world. Also, it is hard to control. Correspondingly, the capacity of the hydropower is becoming larger and larger, and the requirement of the system is increasing higher and higher. Therefore, we study new physical system, physical phenomenon, new law and physical concepts by using nonlinear dynamics theory, and try to apply it to the stability analysis of the hydropower system.
The contents and conclusions of this paper include:
(1) A new three dimentional chaotic system was presented here. Then, we can get a new double-wing chaotic system by the absolute operation of a linear term. It shows that a slight change on the secondary part can change the mainly characters of the whole system, for that we all know that nonlinear is a requirement to the occurance of chaos, in other words, nonlinear term is the main part of the system. Finally, the circuit diagram was also derived. Its experimental results are agreement with the numerical simulation.
(2) A new multi-scroll chaotic system was presented by using a new piecewise function. Moreover, the circuit diagram was also designed, the experimental results in which are agreement with numerical simulation. We can get conclusion that the phase trajectories, Poincare maps and bifurcation diagram are all uniform and can describe different aspects of the dynamical system.①the numbers of circles in fractal structure in Poincare maps is gradually increasing with the increase of the scroll number. Therefore, the phase trajectories and Poincare maps are agreement with each other. In other words, they are consistent in essence.②The number of dense points is consistent with the number of the scrolls. Obviously, the number of dense points region is increasing with the number of the scrolls. In other words, they are equal to each other. In the future, we can judge the number of the attractors by confirming the number of the dense points region in bifurcation diagram, while the attractors are more intuitive from the phase trajectories maps.At last, we cam get that the phase trajectories, Poincare maps and bifurcation diagram are all uniform and can describe different aspects of the dynamical system.
(3) A new four-dimentional fractional-order chaotic system and a new double-wing fractional-order chaotic system were derived including circuit diagram.
(4) As for chaos control, we first study the sliding mode control with bounded noise, and discuss the no-chattering sliding mode control by using saturation funciton. Furthermore, we also study the effection of different reaching rate. Second, we also study the control of a class of chaotic systems by using three typical examples. Finally, we implemented a class of four dimentional hyperchaotic systems with only one controller term.
(5) As for chaos synchronization, I try to bridge the fractional order chaotic system and integer order chaotic system by reaching the synchronization of fractional-order chaotic system and integer order chaotic sytem based on LMI. We also point that synchronization and anti-synchronization are unified with each other. They could be realized by the same method. Furthermore, we get a conclusion that chaos control is a special case of chaos synchronization, and chaos synchronization is a generalized concept of chaos control. Finally, a class of n-dimentional chaotic systems was realized via fuzzy sliding mode control, which is also suitable for complex networks.
(6) As for circuit design, a circuit diagram of chaos synchronization via mismatch synchronization. Also, a circuit diagram of fractional order chaotic synchronization via nonlinear feedback control. Finally, a unit circuit of response system was presented, which is suitable for any fractional-order or integer order chaotic system, which was proved by two typical examples. It is the first time to reach synchronization of fractional-order and integer order chaotic systems. After this work, the secure communication will meet new change based on chaotic circuit.
(7) A new dynamical model for the hydro-turbine governing system with elastic water hammer-impact and the second order model of generator was established, which is based on the model of simple surge shaft. By virtue of bifurcation diagram, Poincare maps, the power spectrum diagram, the time domain diagram, the orbits diagram and the spectrum diagram, the nonlinear dynamical responses of generator's relative deviation of rotational seed and the water pressure at the entrance of surge shaft were analyzed. Second, a new nonlinear mathematical model of a hydro-turbine governing system with a surge tank was presented. Finally, we introduce a new model of hydro-turbine system with the effect of surge tank based on state-space to study the dynamical behaviors of a hydro-turbine system.
引文
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