CSTR振荡体系的动力学行为及控制
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摘要
自B-Z反应中的振荡现象被揭示以来,各种化学振荡体系中的非线性现象一直是国内外研究的前沿课题之一。由于化学反应在实际生产中的巨大应用价值,许多学者在该领域做出了大量的工作。近年来,迅速发展的现代非线性理论为各种复杂现象的深入探索提供了有力的工具。本文运用现代非线性分析方法,探讨了CSTR振荡体系的复杂现象及其机理,分析了不同物理参数、初始条件、时滞和耦合等因素对系统的动力学行为的影响,进而揭示了其复杂运动的本质。此外,对CSTR振荡体系的反同步与控制问题也进行了初步探讨。主要内容包括以下几个方面的工作:
首先,给出了三变量CSTR自催化反应系统随着参数的变化经由Hopf分岔走向混沌的道路,分析了时滞反馈对原系统动力学行为的影响。分别探讨了时滞反馈对原系统稳定状态,周期运动和混沌状态的影响。数值仿真结果表明,时滞反馈可以用于控制反应过程,既可使系统从有序走向混沌,也可以从混沌走向有序。
其次,在同时考虑溶液流动造成的时滞与入料溶液流速的扰动时,讨论了两种典型外激励频率下耦合CSTR振荡体系的复杂动力学行为。当外激励频率为原自治系统的固有频率时,反应过程中出现了结构复杂的混沌吸引子和概周期运动,存在混沌由倍周期分岔演变为新的混沌吸引子的过程;当外激励频率与自治系统的固有频率存在量级上差别时,Hopf分岔提前发生,反应过程中存在概周期运动,且存在概周期解的空间轨迹不断扩大而后演化为周期解的过程。数值模拟结果表明,在两种情况下系统的动力学行为差别很大,扰动频率的加入会明显改变系统的动力学演化过程。
再次,设计了一种入料溶液流速随CSTR中溶液浓度的改变而改变的反应系统,并考虑由控制系统造成的时滞,给出了相应的数学模型。通过数值模拟发现,入料溶液的流速及浓度参考值均对系统的动力学行为有很大的影响,系统呈现出与原系统不同的动力学行为,反应过程中出现了概周期运动和倒倍周期分岔等丰富的动力学现象。
最后,讨论了耦合CSTR振荡体系的延迟反同步控制问题。在反应系统中,将反应误差重新定义为实际反应状态与控制目标的差值,并基于Lyapunov稳定性理论和反同步思想,通过构造适当的控制器和控制输入信号可以使整个反应系统中各成分的浓度处于某种状态。该方法对系统无特殊要求,并且无需构造Lvapunov函数。最后用数值模拟验证了方法的有效性。
The nonlinear phenomenon in chemical oscillating systems has been an extensively investigated topic over the past decades since the B-Z oscillating reaction was discovered. Many theoretical and experimental results on chemical reaction have been published due to the great application in practical production. With the development of modern nonlinear theories, various complex phenomena in chemical oscillating systems have been deeply studied. For the CSTR oscillating systems or coupling systems, the complex mechanism is investigated by applying the modern nonlinear theory when parameters and initial conditions changed, and the anti-synchronization control and the delayed effect are also explored. The main contents are as follows:
Firstly, for the autocatalytic chemical reaction of 3-variable CSTR system, the transiton to chaos can occur via a Hopf bifurcation solution when parameter is changed. Delayed effect on the original system's dynamical behaviors is analyzed. The steady state, periodic motion and chaos of the original system are affected by time delay. It is shown that the reaction process can be controlled with time delay. The states of the system can be changed between order and chaos under the influence of time-delayed feedback.
Secondly, complicated dynamical behaviors of the coupled CSTR system have been obtained under two typical external excited frequency when time delay caused by liquor flow and perturbation caused by the flow rate of feeding liquor are taken into consideration. If external excited frequency is the same as the natural frequency of the original autonomous system, nonlinear phenomena such as chaos and quasi-periodic oscillation can be observed. Further more, disturbance may cause another type of chaotic attractor via period-doubling bifurcations. If there exists order gap between the two frequencies, Hopf bifurcation occurs earlier, which leads to quasi-periodic oscillation. The attractors expend in space with the variation of the parameters and finally settle down to periodic movement. Numerical simulation results prove that the dynamic behavior of the system have great differences between two cases. Perturbation frequency will significantly change the evolution forms of dynamical behaviors.
Thirdly, a new reaction system is designed on the assumption that the change of feeding liquor's flow rate comes with the change of concentration. A new model is proposed considering time delay which is caused by the controller. Numerical simulations suggest that the variation of the flow rate as well as the concentrations of the solution may cause great changes of the dynamics. Rich dynamical behaviors are obtained such as quasi-periodic occillations and cascading of period-doubling bifurcations.
Finally, the anti-synchronization problem for the coupled CSTR system is explored when delayed effect is taken into consideration. In autocatalytic chemical reaction system, the reaction error is defined as the difference between actual reaction state and control objective. Based on Lyapunov stability theory and anti-synchronization thought, a general scheme is presented which can be used to control the concentration of the whole reaction system in some special states by selecting the controller and the input signal. The scheme has no special requirements to system, and without structuring Lyapunov functions. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme.
首先,给出了三变量CSTR自催化反应系统随着参数的变化经由Hopf分岔走向混沌的道路,分析了时滞反馈对原系统动力学行为的影响。分别探讨了时滞反馈对原系统稳定状态,周期运动和混沌状态的影响。数值仿真结果表明,时滞反馈可以用于控制反应过程,既可使系统从有序走向混沌,也可以从混沌走向有序。
其次,在同时考虑溶液流动造成的时滞与入料溶液流速的扰动时,讨论了两种典型外激励频率下耦合CSTR振荡体系的复杂动力学行为。当外激励频率为原自治系统的固有频率时,反应过程中出现了结构复杂的混沌吸引子和概周期运动,存在混沌由倍周期分岔演变为新的混沌吸引子的过程;当外激励频率与自治系统的固有频率存在量级上差别时,Hopf分岔提前发生,反应过程中存在概周期运动,且存在概周期解的空间轨迹不断扩大而后演化为周期解的过程。数值模拟结果表明,在两种情况下系统的动力学行为差别很大,扰动频率的加入会明显改变系统的动力学演化过程。
再次,设计了一种入料溶液流速随CSTR中溶液浓度的改变而改变的反应系统,并考虑由控制系统造成的时滞,给出了相应的数学模型。通过数值模拟发现,入料溶液的流速及浓度参考值均对系统的动力学行为有很大的影响,系统呈现出与原系统不同的动力学行为,反应过程中出现了概周期运动和倒倍周期分岔等丰富的动力学现象。
最后,讨论了耦合CSTR振荡体系的延迟反同步控制问题。在反应系统中,将反应误差重新定义为实际反应状态与控制目标的差值,并基于Lyapunov稳定性理论和反同步思想,通过构造适当的控制器和控制输入信号可以使整个反应系统中各成分的浓度处于某种状态。该方法对系统无特殊要求,并且无需构造Lvapunov函数。最后用数值模拟验证了方法的有效性。
The nonlinear phenomenon in chemical oscillating systems has been an extensively investigated topic over the past decades since the B-Z oscillating reaction was discovered. Many theoretical and experimental results on chemical reaction have been published due to the great application in practical production. With the development of modern nonlinear theories, various complex phenomena in chemical oscillating systems have been deeply studied. For the CSTR oscillating systems or coupling systems, the complex mechanism is investigated by applying the modern nonlinear theory when parameters and initial conditions changed, and the anti-synchronization control and the delayed effect are also explored. The main contents are as follows:
Firstly, for the autocatalytic chemical reaction of 3-variable CSTR system, the transiton to chaos can occur via a Hopf bifurcation solution when parameter is changed. Delayed effect on the original system's dynamical behaviors is analyzed. The steady state, periodic motion and chaos of the original system are affected by time delay. It is shown that the reaction process can be controlled with time delay. The states of the system can be changed between order and chaos under the influence of time-delayed feedback.
Secondly, complicated dynamical behaviors of the coupled CSTR system have been obtained under two typical external excited frequency when time delay caused by liquor flow and perturbation caused by the flow rate of feeding liquor are taken into consideration. If external excited frequency is the same as the natural frequency of the original autonomous system, nonlinear phenomena such as chaos and quasi-periodic oscillation can be observed. Further more, disturbance may cause another type of chaotic attractor via period-doubling bifurcations. If there exists order gap between the two frequencies, Hopf bifurcation occurs earlier, which leads to quasi-periodic oscillation. The attractors expend in space with the variation of the parameters and finally settle down to periodic movement. Numerical simulation results prove that the dynamic behavior of the system have great differences between two cases. Perturbation frequency will significantly change the evolution forms of dynamical behaviors.
Thirdly, a new reaction system is designed on the assumption that the change of feeding liquor's flow rate comes with the change of concentration. A new model is proposed considering time delay which is caused by the controller. Numerical simulations suggest that the variation of the flow rate as well as the concentrations of the solution may cause great changes of the dynamics. Rich dynamical behaviors are obtained such as quasi-periodic occillations and cascading of period-doubling bifurcations.
Finally, the anti-synchronization problem for the coupled CSTR system is explored when delayed effect is taken into consideration. In autocatalytic chemical reaction system, the reaction error is defined as the difference between actual reaction state and control objective. Based on Lyapunov stability theory and anti-synchronization thought, a general scheme is presented which can be used to control the concentration of the whole reaction system in some special states by selecting the controller and the input signal. The scheme has no special requirements to system, and without structuring Lyapunov functions. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme.
引文
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