线性离散周期系统的极点配置和观测器设计
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摘要
作为连接线性时不变系统和时变系统的桥梁,线性周期时变系统是一类非常重要的系统。一方面,现实世界中的许多问题可以归结为周期动态模型;另一方面,在一些系统的控制过程中,定常反馈控制律失效时,周期反馈控制律往往能够胜任。由于电子计算机技术的迅猛发展,许多连续系统都要进行离散处理,因此,线性离散周期系统的研究就尤为重要。目前,在线性离散周期系统领域,已经取得了相当数量的研究成果,但仍有许多问题有待于进一步探讨。
本文立足于线性离散周期系统中最基本的问题,采用参数化设计方法,研究了线性离散周期系统的周期极点配置问题和周期观测器设计问题,主要内容包括以下几个方面:
第一,研究了线性离散周期系统的周期状态反馈极点配置问题。通过对闭环系统的单值性矩阵进行的一系列处理,使得周期状态反馈增益包含在一类推广的Sylvester矩阵方程的解中,利用这类方程的解,给出了周期状态反馈增益的参数化递推求解算法。利用存在于这些参数化周期状态反馈增益中的充分的自由度,考虑了鲁棒极点配置问题。提出了一个反映周期系统极点对于参数扰动的灵敏度指标,将鲁棒极点配置问题转化为了一个静态优化问题,最终给出了鲁棒极点配置问题的设计算法。
第二,研究了线性离散周期系统的周期输出反馈极点配置问题。在周期系统完全能达和完全能观测的条件下,给出了周期输出反馈极点配置问题的参数化解。并进一步考虑了周期输出反馈鲁棒极点配置问题。分别给出了周期输出反馈参数化极点配置算法和周期输出反馈鲁棒极点配置算法。虽然在线性时不变系统中,不能用定常输出反馈进行任意极点配置,数值结果表明周期输出反馈不仅能够胜任,而且还能提供相当多的自由度。
第三,研究了利用周期动态反馈来对线性离散周期系统进行极点配置的问题。和线性时不变系统相似,事实表明,周期动态补偿器的设计问题可以转化为一个增广周期系统的周期输出反馈控制器设计问题。采用周期输出反馈控制器设计的参数化方法,给出了参数化动态补偿器的设计方案。对于存在多个扰动变量的情形,提出了闭环周期系统极点对于扰动的灵敏度指标,并设计了鲁棒周期动态补偿器。
第四,研究了周期观测器设计问题。将前向和逆向离散周期Sylvester矩阵方程的求解问题转化为一般的Sylvester矩阵方程的求解问题。给出了这两类方程的显式、完全、参数化解。前向离散周期Sylvester矩阵方程可用于极点配置,和现有的方法相比,能够将实现极点的任意配置,且对系统参数没有任何要求。考虑了周期Luenberger函数观测器设计问题,提出了观测器成立的充分必要条件。利用逆向离散周期Sylvester矩阵方程的解,设计出了参数化的周期Luenberger函数观测器。进一步研究了基于观测器的控制系统设计问题,指出在了线性离散周期系统中也存在分离原理,给出了基于全维状态观测器的状态反馈控制器设计算法。
第五,将周期鲁棒极点配置方法应用到卫星姿态控制中去。根据磁力矩器控制卫星姿态时在滚动和偏航通道产生的周期动态特性,对这两个通道设计了周期状态反馈控制器,进一步构造了鲁棒性能的目标函数,并对其进行优化,求得了鲁棒周期状态反馈控制器。采用零阶保持,对滚动和偏航姿态进行了仿真。结果表明,利用周期控制律,能够很好地对卫星的滚动和偏航姿态进行镇定。
As a bridge between the linear time invariant systems and time varying systems,linear periodic time varying systems are a class of systems of great importance. On theone hand, many problems in real world can be reduced into periodic models; on the otherhand, in the control process of some systems, when the constant feedback control lawsdo not work, the periodic ones do perfectly. Due to the rapid development of computertechnology, many continuous systems need to be discretized. Thus, the research on lineardiscrete-time periodic systems is especially important. Up to now, there are already agreat deal of results in the field of discrete-time periodic systems, but there are still manyproblems to be discussed.
Aiming at the basic problem in linear discrete periodic systems, employing paramet-ric design approaches, this thesis investigates pole assignment via periodic control lawsand design of periodic observers. The main contributions given by this thesis are list asfollows.
1. Pole assignment for linear discrete periodic systems via periodic state feedback isdiscussed. The monodromy matrix of the closed-loop system is represented in a specialform. By combining this special form with the recent result on solutions to a class ofgeneralized Sylvester matrix equations, a complete parametric approach for pole assign-ment via periodic state feedback is proposed. The free parameters existing in the solutionsto pole assignment are further used to achieve robustness performance. The robust poleassignment problem is converted into a static optimization problem. The algorithm forrobust pole assignment is proposed.
2. Pole assignment for linear discrete periodic systems via periodic output feedbackis investigated. Under the condition that the periodic system is completely reachable andcompletely observable, parametric solutions to the pole assignment problem for lineardiscrete periodic systems via periodic output feedback are presented. Further, robust poleassignment via periodic output feedback are solved. Both parametric pole assignmentalgorithm and robust pole assignment algorithm are obtained. It is shown that using peri-odic output feedback control laws can not only realize arbitrary pole assignment but alsoprovide many degree of freedom.
3. Pole assignment for linear discrete periodic systems via periodic dynamical feed-back is investigated. Similar to time invariant systems, the design of a periodic dynamicalcompensator can be reduced into the design of a periodic output feedback controller fora augmented periodic system. By employing the parametric approach of periodic outputfeedback controllers, a parametric design method for periodic dynamical compensators isproposed. For the system with several pertubation variables, a robustness index is pro-posed and robust dynamical compensator design problem is solved.
4. Periodic observers design problem for linear discrete periodic systems is re-searched. Both solving forward-time discrete periodic Sylvester matrix equations andsolving reverse-time discrete-time discrete periodic Sylvester matrix equations are re-duced into solving a class of generalized Sylvester matrix equations. The explicit, com-plete and parametric solutions for the two class of matrix equations are presented. Basedon the solutions to reverse-time periodic Sylvester matrix equation, problem for peri-odic Luenberger function observer design is solved. Further, it is shown that there existsseparation principle of observer-based controllers design in this type of systems. An algo-righm for design a periodic state feedback controller based on a full order state observeris provided.
5. Stabilization of satellite attitude by magnetorquer is considered. Under the ac-tion of geomagnetic field, the roll and yaw dynamics result in a linear periodic model.By setting an objective function of robust performance and optimization, a set of robustperiodic state feedback controllers is obtained. A numerical simulation through the zero-order-hold is conducted. The result shows that the periodic controllers are very effectivein regulating roll angle and yaw angle to zero.
本文立足于线性离散周期系统中最基本的问题,采用参数化设计方法,研究了线性离散周期系统的周期极点配置问题和周期观测器设计问题,主要内容包括以下几个方面:
第一,研究了线性离散周期系统的周期状态反馈极点配置问题。通过对闭环系统的单值性矩阵进行的一系列处理,使得周期状态反馈增益包含在一类推广的Sylvester矩阵方程的解中,利用这类方程的解,给出了周期状态反馈增益的参数化递推求解算法。利用存在于这些参数化周期状态反馈增益中的充分的自由度,考虑了鲁棒极点配置问题。提出了一个反映周期系统极点对于参数扰动的灵敏度指标,将鲁棒极点配置问题转化为了一个静态优化问题,最终给出了鲁棒极点配置问题的设计算法。
第二,研究了线性离散周期系统的周期输出反馈极点配置问题。在周期系统完全能达和完全能观测的条件下,给出了周期输出反馈极点配置问题的参数化解。并进一步考虑了周期输出反馈鲁棒极点配置问题。分别给出了周期输出反馈参数化极点配置算法和周期输出反馈鲁棒极点配置算法。虽然在线性时不变系统中,不能用定常输出反馈进行任意极点配置,数值结果表明周期输出反馈不仅能够胜任,而且还能提供相当多的自由度。
第三,研究了利用周期动态反馈来对线性离散周期系统进行极点配置的问题。和线性时不变系统相似,事实表明,周期动态补偿器的设计问题可以转化为一个增广周期系统的周期输出反馈控制器设计问题。采用周期输出反馈控制器设计的参数化方法,给出了参数化动态补偿器的设计方案。对于存在多个扰动变量的情形,提出了闭环周期系统极点对于扰动的灵敏度指标,并设计了鲁棒周期动态补偿器。
第四,研究了周期观测器设计问题。将前向和逆向离散周期Sylvester矩阵方程的求解问题转化为一般的Sylvester矩阵方程的求解问题。给出了这两类方程的显式、完全、参数化解。前向离散周期Sylvester矩阵方程可用于极点配置,和现有的方法相比,能够将实现极点的任意配置,且对系统参数没有任何要求。考虑了周期Luenberger函数观测器设计问题,提出了观测器成立的充分必要条件。利用逆向离散周期Sylvester矩阵方程的解,设计出了参数化的周期Luenberger函数观测器。进一步研究了基于观测器的控制系统设计问题,指出在了线性离散周期系统中也存在分离原理,给出了基于全维状态观测器的状态反馈控制器设计算法。
第五,将周期鲁棒极点配置方法应用到卫星姿态控制中去。根据磁力矩器控制卫星姿态时在滚动和偏航通道产生的周期动态特性,对这两个通道设计了周期状态反馈控制器,进一步构造了鲁棒性能的目标函数,并对其进行优化,求得了鲁棒周期状态反馈控制器。采用零阶保持,对滚动和偏航姿态进行了仿真。结果表明,利用周期控制律,能够很好地对卫星的滚动和偏航姿态进行镇定。
As a bridge between the linear time invariant systems and time varying systems,linear periodic time varying systems are a class of systems of great importance. On theone hand, many problems in real world can be reduced into periodic models; on the otherhand, in the control process of some systems, when the constant feedback control lawsdo not work, the periodic ones do perfectly. Due to the rapid development of computertechnology, many continuous systems need to be discretized. Thus, the research on lineardiscrete-time periodic systems is especially important. Up to now, there are already agreat deal of results in the field of discrete-time periodic systems, but there are still manyproblems to be discussed.
Aiming at the basic problem in linear discrete periodic systems, employing paramet-ric design approaches, this thesis investigates pole assignment via periodic control lawsand design of periodic observers. The main contributions given by this thesis are list asfollows.
1. Pole assignment for linear discrete periodic systems via periodic state feedback isdiscussed. The monodromy matrix of the closed-loop system is represented in a specialform. By combining this special form with the recent result on solutions to a class ofgeneralized Sylvester matrix equations, a complete parametric approach for pole assign-ment via periodic state feedback is proposed. The free parameters existing in the solutionsto pole assignment are further used to achieve robustness performance. The robust poleassignment problem is converted into a static optimization problem. The algorithm forrobust pole assignment is proposed.
2. Pole assignment for linear discrete periodic systems via periodic output feedbackis investigated. Under the condition that the periodic system is completely reachable andcompletely observable, parametric solutions to the pole assignment problem for lineardiscrete periodic systems via periodic output feedback are presented. Further, robust poleassignment via periodic output feedback are solved. Both parametric pole assignmentalgorithm and robust pole assignment algorithm are obtained. It is shown that using peri-odic output feedback control laws can not only realize arbitrary pole assignment but alsoprovide many degree of freedom.
3. Pole assignment for linear discrete periodic systems via periodic dynamical feed-back is investigated. Similar to time invariant systems, the design of a periodic dynamicalcompensator can be reduced into the design of a periodic output feedback controller fora augmented periodic system. By employing the parametric approach of periodic outputfeedback controllers, a parametric design method for periodic dynamical compensators isproposed. For the system with several pertubation variables, a robustness index is pro-posed and robust dynamical compensator design problem is solved.
4. Periodic observers design problem for linear discrete periodic systems is re-searched. Both solving forward-time discrete periodic Sylvester matrix equations andsolving reverse-time discrete-time discrete periodic Sylvester matrix equations are re-duced into solving a class of generalized Sylvester matrix equations. The explicit, com-plete and parametric solutions for the two class of matrix equations are presented. Basedon the solutions to reverse-time periodic Sylvester matrix equation, problem for peri-odic Luenberger function observer design is solved. Further, it is shown that there existsseparation principle of observer-based controllers design in this type of systems. An algo-righm for design a periodic state feedback controller based on a full order state observeris provided.
5. Stabilization of satellite attitude by magnetorquer is considered. Under the ac-tion of geomagnetic field, the roll and yaw dynamics result in a linear periodic model.By setting an objective function of robust performance and optimization, a set of robustperiodic state feedback controllers is obtained. A numerical simulation through the zero-order-hold is conducted. The result shows that the periodic controllers are very effectivein regulating roll angle and yaw angle to zero.
引文
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84 G. R. Duan. Solutions of the Equation ???? ???? ???? and Their Application toEigenstructure Assignment in Linear Systems[J]. IEEE Transactions on AutomaticControl, 1993, 38(2):276–280.
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101 F. Nollet, T. Floquet, W. Perruquetti. Observer-based Second Order Sliding ModeControl Laws for Stepper Motors[J]. Control Engineering Practice, 2008, 16:429–443.
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87 D. Aeyels, J. L. Willems. Pole Assignment for Linear Time-invariant Systems byPeriodic Memeoryless Output Feedback[J]. Automatica, 1992, 28(6):1159–1168.
88 G. R. Duan. Robust Eigenstructure Assignment via Dynamical Compensators[J].Automatica, 1993, 29(2):469–474.
89 Z. Z. Han. Eigenstructure Assignment Using Dynamical Compensator[J]. Interna-tional Journal of Control, 1989, 49:233–245.
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96 G. R. Duan, J. H. Li, L. S. Zhou. Design of Robust Luenberger Observers[J]. ActaAutomatica Sinica, 1991, 18(6):742–747.
97 G. R. Duan. Design of the Luenberger Function Observers with Disturbance De-coupling[J]. Acta Automatica Sinica, 1992, 20(5):548–552.
98 R. J. Patton, J. Chen. Observer-based Fault Detection and Isolation: Robustnessand Applications[J]. Control Engineering Practice, 1997, 5(5):671–682.
99 G. R. Duan, R. J. Patton. Robust Fault Detection Using Luenberger-type UnknownInput Observers—a Parametric Approach[J]. International Journal of Systems Sci-ence, 2001, 32(4):533–540.
100 C. H. Lien. An Efficient Method to Design Robust Observer-based Controlof Uncertain Linear Systems[J]. Applied Mathematics and Computation, 2004,158(1):29–44.
101 F. Nollet, T. Floquet, W. Perruquetti. Observer-based Second Order Sliding ModeControl Laws for Stepper Motors[J]. Control Engineering Practice, 2008, 16:429–443.
102 S. Ibrir, S. Diopt. Novel Lmi Conditions for Observer-based Stabilization of Lips-chitzian Nonlinear Systems and Uncertain Linear Systems in Discrete-time[J]. Ap-plied Mathematics and Computation, 2008, 206:579–588.
103屠善澄.卫星姿态动力学与控制[M].宇航出版社, 1999.
104王庆观,刘敏,刘藻珍.环境力矩在现代小卫星姿态控制中的应用综述[J].战术导弹控制技术,2005,51(4):8–10.
105 M. E. Pittelkau. Frequency Weighted Lqg Control of Spacecraft Atti-tude[C]//Proceeding of the 1st IEEE Conference on Control Applications. NewYork, 1992:336–341.
106刘良栋.航天器的姿态动力学与控制[M].航空工业出版社, 1992.
107 J. Bezos. The Titlesec and Titletoc Packages[M]. 2nd ed., Cityname: University ofSomeName, 2002:10–20.
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