突变控制机制及其应用研究
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摘要
突变理论的创立是非线性科学的重要成就之一，随着航空航海工程、电力大系统及复杂生产过程等被控对象的日益复杂化，加之人们对控制系统的性能指标的要求越来越高，复杂系统的控制问题，特别是带有突变现象的复杂系统的控制，使传统的控制理论面临着严峻的挑战，所以迫切需要为解决非线性突变系统的控制问题寻找新的方法。
为适应这种实际需要，突变控制理论应运而生，对突变控制理论的探索研究处于起步阶段，本文从研究系统的突变现象入手，把突变理论引入到系统的控制和稳定性分析中，研究突变控制机制及其应用。
突变理论与控制技术结合主要包含两方面内容：一是在保持系统原来平衡点不变的条件下，为系统建立有益的突变，称为突变控制；二是消除或抑制系统中的有害突变，称为反突变控制。提出了状态反馈突变控制法，对于连续系统和离散系统分别进行了推导，并给出了具体的仿真实例。提出了基于冲失滤波器辅助反馈控制的突变控制法，它不但能够在系统模型不确定的条件下，保持原系统的平衡点不随控制的施加而改变，而且能简化控制器的设计，更大限度地扩展系统控制参数的操作范围。
针对施加控制器能增加被控系统的维数，提出了基于非线性多项式函数的状态反馈突变控制法。这种方法能够保持原系统的平衡点不变，方程的线性部分能够消除已有突变或者使之延迟，以改变系统的线性稳定性；方程的非线性部分能够改变突变点的稳定性，而且不增加系统的维数。
通过对潜艇垂直面和水平面非线性运动模型进行分析，导出垂直面运动的突变控制模型，并进行了突变稳定性分析；在水平面运动的非线性分析中，通过中心流形定理简化方程，得出临界点处极限环的稳定性与微差占的符号有关的结论，这样在无法或很难得到李亚普诺夫函数的情况下，可以通过这种方法判别临界点的稳定性。
最后，提出并研究了突变预测控制模型，主要有：SCPC(synergeticscatastrophe prediction control)模型，该模型能够反映系统在失稳过程中各个因素的协同作用及突变现象，预测系统的失稳时间；对于强混沌系统，建立了胞映射—突变预测模型，该模型充分考虑了系统的非线性动力学本质及多变
哈尔滨工程大学博士学位论文
量的整体综合作用，利用胞映射理论巧妙地消除了观测初值的随机性和系统
本身的内在随机性对预测结果的影响，因此能够更客观地描述了系统的动力
学机制;建立了统计尖点突变预测模型，并进行了参数估计、可接受性假设
检验及预测等统计学分析应用的研究，为统计尖点突变预测模型在预测控制
中的应用奠定了基础。
关键词:突变控制;冲失滤波器;非线性多项式函数;统计尖点突变模型
The found of Catastrophe theory is one of important achievements in nonlinear science. With the complication of controlled object on aeronautic and marine engineering, heavy electric power system and complex productive progress, together with the desire of performance index on controlled system becomes higher and higher. The controlling problem of complex system, especially the complex system with catastrophic characteristic, is an austere challenge the traditional control theory has to be faced with. So that new control methods are urgently required for the control problem of nonlinear catastrophe system.To adapt this actual demand, Catastrophe control theory emerges as the times require. The research of catastrophe control theory is just in its first stages, this dissertation starts with the investigation of catastrophe phenomena, catastrophe theory is introduced to control and stability analysis of system, to research catastrophe control mechanism and it's application.There are two aspects in the combination of catastrophe theory and control technology:one is to create new beneficial catastrophe while preserve all initial equilibria, named catastrophe control; another is to eliminate or restrain harmful catastrophe, called averse-catastrophe. State feedback catastrophe control method is introduced, deducing study is given both in continuous and discrete system, and simulation results confirm these methods. Washout filter based on feedback catastrophe control method is put forward, it is not only preserve the original equilibrium even under model uncertainty, but also can simplify the controller, to expand the range of control parameter.For controller always adds the dimension of controlled system, nonlinear polynomial function feedback catastrophe control method is addressed. This method can also preserve the equilibrium. The linear part can eliminate or delay the exist catastrophe to change the linear stability, while the nonlinear part can change the stability of catastrophic point, while this method can't increase system dimension.
By the analysis of dive and horizontal plane nonlinear model, the catastrophe model on de dive plane is deduced, and catastrophe stability is analysized. In the horizontal plane, simplified equation is gained by center manifold theory, then draws the conclusion that the stability of limit circle arises from critical point is relate to small difference of move point and critical point. Thus,when Lypunov function cannot or is difficult to gain, one can distinguish the stability of critical point through this method.Finally, catastrophe prediction model is put forward and studied. The main catastrophe prediction model in this dissertation including: SCPC (synergetics-catastrophe prediction control) model, it can reflect the cooperative effect of all factors and catastrophe phenomenon, then predictive the time of losing stability. For strong chaos system, cell-to-cell mapping catastrophe predictive model is built, it considers the integrate effect of the nature of nonlinear dynamic and multi-variable, through the cell-to-cell map theory skillfully eliminate the effect of random initial value and inherence random of system, so it can describe the mechanism of system dynamic. Statistical cusp catastrophe predictive model is established, and the statistical analysis is applied, including parameter estimation and acceptability hypothesis test and prediction applied research, it establishs the foundation for the statistical cusp catastrophe model used in predictive control.
为适应这种实际需要，突变控制理论应运而生，对突变控制理论的探索研究处于起步阶段，本文从研究系统的突变现象入手，把突变理论引入到系统的控制和稳定性分析中，研究突变控制机制及其应用。
突变理论与控制技术结合主要包含两方面内容：一是在保持系统原来平衡点不变的条件下，为系统建立有益的突变，称为突变控制；二是消除或抑制系统中的有害突变，称为反突变控制。提出了状态反馈突变控制法，对于连续系统和离散系统分别进行了推导，并给出了具体的仿真实例。提出了基于冲失滤波器辅助反馈控制的突变控制法，它不但能够在系统模型不确定的条件下，保持原系统的平衡点不随控制的施加而改变，而且能简化控制器的设计，更大限度地扩展系统控制参数的操作范围。
针对施加控制器能增加被控系统的维数，提出了基于非线性多项式函数的状态反馈突变控制法。这种方法能够保持原系统的平衡点不变，方程的线性部分能够消除已有突变或者使之延迟，以改变系统的线性稳定性；方程的非线性部分能够改变突变点的稳定性，而且不增加系统的维数。
通过对潜艇垂直面和水平面非线性运动模型进行分析，导出垂直面运动的突变控制模型，并进行了突变稳定性分析；在水平面运动的非线性分析中，通过中心流形定理简化方程，得出临界点处极限环的稳定性与微差占的符号有关的结论，这样在无法或很难得到李亚普诺夫函数的情况下，可以通过这种方法判别临界点的稳定性。
最后，提出并研究了突变预测控制模型，主要有：SCPC(synergeticscatastrophe prediction control)模型，该模型能够反映系统在失稳过程中各个因素的协同作用及突变现象，预测系统的失稳时间；对于强混沌系统，建立了胞映射—突变预测模型，该模型充分考虑了系统的非线性动力学本质及多变
哈尔滨工程大学博士学位论文
量的整体综合作用，利用胞映射理论巧妙地消除了观测初值的随机性和系统
本身的内在随机性对预测结果的影响，因此能够更客观地描述了系统的动力
学机制;建立了统计尖点突变预测模型，并进行了参数估计、可接受性假设
检验及预测等统计学分析应用的研究，为统计尖点突变预测模型在预测控制
中的应用奠定了基础。
关键词:突变控制;冲失滤波器;非线性多项式函数;统计尖点突变模型
The found of Catastrophe theory is one of important achievements in nonlinear science. With the complication of controlled object on aeronautic and marine engineering, heavy electric power system and complex productive progress, together with the desire of performance index on controlled system becomes higher and higher. The controlling problem of complex system, especially the complex system with catastrophic characteristic, is an austere challenge the traditional control theory has to be faced with. So that new control methods are urgently required for the control problem of nonlinear catastrophe system.To adapt this actual demand, Catastrophe control theory emerges as the times require. The research of catastrophe control theory is just in its first stages, this dissertation starts with the investigation of catastrophe phenomena, catastrophe theory is introduced to control and stability analysis of system, to research catastrophe control mechanism and it's application.There are two aspects in the combination of catastrophe theory and control technology:one is to create new beneficial catastrophe while preserve all initial equilibria, named catastrophe control; another is to eliminate or restrain harmful catastrophe, called averse-catastrophe. State feedback catastrophe control method is introduced, deducing study is given both in continuous and discrete system, and simulation results confirm these methods. Washout filter based on feedback catastrophe control method is put forward, it is not only preserve the original equilibrium even under model uncertainty, but also can simplify the controller, to expand the range of control parameter.For controller always adds the dimension of controlled system, nonlinear polynomial function feedback catastrophe control method is addressed. This method can also preserve the equilibrium. The linear part can eliminate or delay the exist catastrophe to change the linear stability, while the nonlinear part can change the stability of catastrophic point, while this method can't increase system dimension.
By the analysis of dive and horizontal plane nonlinear model, the catastrophe model on de dive plane is deduced, and catastrophe stability is analysized. In the horizontal plane, simplified equation is gained by center manifold theory, then draws the conclusion that the stability of limit circle arises from critical point is relate to small difference of move point and critical point. Thus,when Lypunov function cannot or is difficult to gain, one can distinguish the stability of critical point through this method.Finally, catastrophe prediction model is put forward and studied. The main catastrophe prediction model in this dissertation including: SCPC (synergetics-catastrophe prediction control) model, it can reflect the cooperative effect of all factors and catastrophe phenomenon, then predictive the time of losing stability. For strong chaos system, cell-to-cell mapping catastrophe predictive model is built, it considers the integrate effect of the nature of nonlinear dynamic and multi-variable, through the cell-to-cell map theory skillfully eliminate the effect of random initial value and inherence random of system, so it can describe the mechanism of system dynamic. Statistical cusp catastrophe predictive model is established, and the statistical analysis is applied, including parameter estimation and acceptability hypothesis test and prediction applied research, it establishs the foundation for the statistical cusp catastrophe model used in predictive control.
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