Delta算子系统若干控制问题研究
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摘要
Delta算子系统控制是近年来控制理论与应用研究中受到广泛关注的一个问题。采用Delta算子方法离散化连续时间系统,得到的离散时间系统称为Delta算子系统。对于高频采样控制,采用Delta算子离散化方法可以避免传统前向移位算子离散化方法引起的数值不稳定问题。当采样周期趋于零时,Delta算子离散化模型趋近于原来的连续时间系统,这使得连续时间系统和离散时间系统的分析与综合可以统一到Delta算子系统框架中进行研究。
本文对Delta算子系统控制问题进行研究,主要工作如下。
研究Delta算子线性不确定系统的鲁棒协方差控制问题:①讨论基于动态输出反馈的鲁棒协方差控制器设计,给出系统满足协方差指标要求的一个充分条件,在此基础上,采用消元处理方法,推导得到满足性能要求的鲁棒协方差控制器存在条件的矩阵不等式刻画,由此提出相应控制器设计方法;②运用同样处理手段,给出了基于动态输出反馈的具有圆形区域极点约束的鲁棒协方差控制器设计方法。这样的处理手段,克服了控制器求解时须人为选取参数的问题,避免了由人为选择参数可能带来的保守性。借助数学与工程软件MATLAB的LMI工具箱对数值算例进行演算,所得结果验证了设计方法的可行性。
较为系统地研究了Delta算子系统的可靠控制问题。即设计可靠控制器,确保当执行器和(或)传感器发生故障时,控制系统仍具有要求的性能。本文研究中针对的故障模型是更具一般性描述的连续故障模型。运用线性矩阵不等式方法,分别研究了Delta算子线性不确定系统的可靠鲁棒圆形区域极点配置、可靠鲁棒镇定、可靠鲁棒H∞控制等三个问题。针对控制系统含执行器故障、含传感器故障、同时含执行器和传感器故障这三种情形,分别推导得出了相应可靠控制器的存在条件,并由此提出控制器设计方法。一个不同于以往研究的特点是,充分利用了连续故障模型的结构信息,得到了具有更小保守性的控制器存在条件。数值算例验证了这一点。此外,用同样方法可以处理Delta算子标称系统相关可靠控制问题,本文以可靠鲁棒圆形区域极点配置为例,给出了Delta算子标称系统可靠圆形区域极点配置状态反馈控制器的存在条件。
较为系统的研究了Delta算子系统的非脆弱控制问题。在控制器设计时,考虑了控制器含有不确定性的因素。这一部分的研究中,采用的仍是线性矩阵不等式处理方法,在假设控制器含有乘性(加性)增益不确定性的情况下:①研究了Delta算子系统的非脆弱鲁棒镇定问题,提出非脆弱二次稳定性概念,推导了Delta算子不确定系统非脆弱鲁棒二次稳定的充分必要条件,由此提出非脆弱鲁棒稳定控制器设计方法;②通过问题的等价转换,构造新的Delta算子系统,利用①的结论,得到Delta算子不确定系统非脆弱二次D-稳定的充分必要条件。这样的处理方法揭示了Delta算子系统稳定和D-稳定之间的特殊关系,具有一定的通用性;③在根据Lyapunov稳定性理论分析系统性能的基础上,推导得到Delta算子不确定系统非脆弱鲁棒保性能控制器存在的条件,由此提出相应控制器设计方法;④推导给出Delta算子不确定系统具有方差约束的非脆弱鲁棒D-镇定控制器存在条件和设计方法。对于上述四个问题,均通过算例验证了设计方法的可行性和有效性。利用文中的方法可以同样处理Delta算子标称系统相关非脆弱控制问题。
研究了Delta算子线性时不变系统鲁棒极点配置的正规化方法,即在满足系统暂态性能和稳态性能要求的极点可配置的前提下,鉴于多输入情况下实现极点配置的控制器的不唯一性,考虑选择适当控制器,使得闭环系统状态矩阵为正规矩阵,藉此确保闭环控制系统具有更强的鲁棒性。利用正规矩阵性质和矩阵广义逆理论,分别得到期望极点可状态反馈正规配置和可静态输出反馈正规配置的充分必要条件。
Control problems of the delta operator formulated discrete-time systems have recently drawn considerable attention in the field of systems and control. Delta operator based discretization method can avoid ill-conditioned problems caused by the conventional shift operator when fast sampling is used, and the delta operator model converges to corresponding continuous-time model as the sampling interval goes to zero, thus analysis and synthesis for continuous-time and discrete-time systems can be treated in the unified delta operator framework.
This dissertation makes some researchs on the delta operator formulated systems. The main works are stated in brief as follows.
Firstly, robust covariance control problems of the delta operator formulated uncertain linear discrete-time systems are studied. This work includes two aspects:①Design approach of robust covariance controllers based on dynamic output feedback is investigated. A sufficient condition for the delta operator system which satisfying the expected covariance constraints is proposed, and then a sufficient condition for the existence of the robust covariance controllers is derived based on the linear matrix inequality(LMI) approach and , a parametrized characterization of the controllers(if they exist) is given in term of the feasible solution to a certain LMIs.②Similarly, a design method of robust covariance output feedback controllers for the delta operator uncertain system with D-stability constraints is also presented. In both processing, the elimination method is introduced to aviod parameter turning when the controllers are solving, and to reduce the conservativeness. The numerical examples solved by MATLAB shows the usefulness of the controller design method.
Secondly, reliable control problems of the delta operator systems are studied. The aim is to design reliable controller which can meet the desired performance and tolerate actuator and(or) sensor failure. In this section, a more general failures model, so called continuous failure model is adopted for actuator and(or) sensor failure. Based on the LMI approach, reliable robust stabilization, reliable robust D-stabilization and reliable robust H∞control for the delta operator uncertain system are researched respectively. The conditions of the existence of corresponding controllers are deduced, and then the design methods of corresponding controllers are suggested respectively. A distinctive feature of the proposed method is that the structural information of the continuous failure model is fully utilized, which brings less conservativeness in controller design. This is demonstrated by some numerical examples.
Thirdly, non-fragile control problems are studied. The cotrollers to be designed are assumed to have multiplicative(or additive) gain variations. Based on the LMI approach, four problems are discussed.①A criterion of non-fragile quadratic stabilizability is derived, and then the design method of the state feedback cotrollers for non-fragile robust stabilization is presented.②Using the above conclusion, a sufficent and necessary condition of non-fragile quadratic D-stabilizability is obtained via construct an auxiliary delta operator system.③A non-fragile robust guaranteed cost controller design method for uncertain delta operator systems is presented.④An existence condition of the non-fragile robust state feedback control law satisfying disk pole and variance constraints for the delta operator systems is derived, and then the design method of relevant controller is suggested. For all cases, numerical examples are provided to illustrate the corresponding design methods.
Finally, robust pole assignment for the delta operator linear time-invariant system is discussed. A method so call pole normal assignment is evolved. The objective is to find a control law, such that the closed-loop system has desired poles and the closed-loop system matrix is a normal matrix, and then robustness of the control system is enhanced. Using the properties of normal matrix and generalized inverse theory of matrix, necessary and sufficient conditions are given to state feedback pole normal assignment and static output feedback pole normal assignment respectively. When the condition holds, the unified expression of the control laws are showed.
本文对Delta算子系统控制问题进行研究,主要工作如下。
研究Delta算子线性不确定系统的鲁棒协方差控制问题:①讨论基于动态输出反馈的鲁棒协方差控制器设计,给出系统满足协方差指标要求的一个充分条件,在此基础上,采用消元处理方法,推导得到满足性能要求的鲁棒协方差控制器存在条件的矩阵不等式刻画,由此提出相应控制器设计方法;②运用同样处理手段,给出了基于动态输出反馈的具有圆形区域极点约束的鲁棒协方差控制器设计方法。这样的处理手段,克服了控制器求解时须人为选取参数的问题,避免了由人为选择参数可能带来的保守性。借助数学与工程软件MATLAB的LMI工具箱对数值算例进行演算,所得结果验证了设计方法的可行性。
较为系统地研究了Delta算子系统的可靠控制问题。即设计可靠控制器,确保当执行器和(或)传感器发生故障时,控制系统仍具有要求的性能。本文研究中针对的故障模型是更具一般性描述的连续故障模型。运用线性矩阵不等式方法,分别研究了Delta算子线性不确定系统的可靠鲁棒圆形区域极点配置、可靠鲁棒镇定、可靠鲁棒H∞控制等三个问题。针对控制系统含执行器故障、含传感器故障、同时含执行器和传感器故障这三种情形,分别推导得出了相应可靠控制器的存在条件,并由此提出控制器设计方法。一个不同于以往研究的特点是,充分利用了连续故障模型的结构信息,得到了具有更小保守性的控制器存在条件。数值算例验证了这一点。此外,用同样方法可以处理Delta算子标称系统相关可靠控制问题,本文以可靠鲁棒圆形区域极点配置为例,给出了Delta算子标称系统可靠圆形区域极点配置状态反馈控制器的存在条件。
较为系统的研究了Delta算子系统的非脆弱控制问题。在控制器设计时,考虑了控制器含有不确定性的因素。这一部分的研究中,采用的仍是线性矩阵不等式处理方法,在假设控制器含有乘性(加性)增益不确定性的情况下:①研究了Delta算子系统的非脆弱鲁棒镇定问题,提出非脆弱二次稳定性概念,推导了Delta算子不确定系统非脆弱鲁棒二次稳定的充分必要条件,由此提出非脆弱鲁棒稳定控制器设计方法;②通过问题的等价转换,构造新的Delta算子系统,利用①的结论,得到Delta算子不确定系统非脆弱二次D-稳定的充分必要条件。这样的处理方法揭示了Delta算子系统稳定和D-稳定之间的特殊关系,具有一定的通用性;③在根据Lyapunov稳定性理论分析系统性能的基础上,推导得到Delta算子不确定系统非脆弱鲁棒保性能控制器存在的条件,由此提出相应控制器设计方法;④推导给出Delta算子不确定系统具有方差约束的非脆弱鲁棒D-镇定控制器存在条件和设计方法。对于上述四个问题,均通过算例验证了设计方法的可行性和有效性。利用文中的方法可以同样处理Delta算子标称系统相关非脆弱控制问题。
研究了Delta算子线性时不变系统鲁棒极点配置的正规化方法,即在满足系统暂态性能和稳态性能要求的极点可配置的前提下,鉴于多输入情况下实现极点配置的控制器的不唯一性,考虑选择适当控制器,使得闭环系统状态矩阵为正规矩阵,藉此确保闭环控制系统具有更强的鲁棒性。利用正规矩阵性质和矩阵广义逆理论,分别得到期望极点可状态反馈正规配置和可静态输出反馈正规配置的充分必要条件。
Control problems of the delta operator formulated discrete-time systems have recently drawn considerable attention in the field of systems and control. Delta operator based discretization method can avoid ill-conditioned problems caused by the conventional shift operator when fast sampling is used, and the delta operator model converges to corresponding continuous-time model as the sampling interval goes to zero, thus analysis and synthesis for continuous-time and discrete-time systems can be treated in the unified delta operator framework.
This dissertation makes some researchs on the delta operator formulated systems. The main works are stated in brief as follows.
Firstly, robust covariance control problems of the delta operator formulated uncertain linear discrete-time systems are studied. This work includes two aspects:①Design approach of robust covariance controllers based on dynamic output feedback is investigated. A sufficient condition for the delta operator system which satisfying the expected covariance constraints is proposed, and then a sufficient condition for the existence of the robust covariance controllers is derived based on the linear matrix inequality(LMI) approach and , a parametrized characterization of the controllers(if they exist) is given in term of the feasible solution to a certain LMIs.②Similarly, a design method of robust covariance output feedback controllers for the delta operator uncertain system with D-stability constraints is also presented. In both processing, the elimination method is introduced to aviod parameter turning when the controllers are solving, and to reduce the conservativeness. The numerical examples solved by MATLAB shows the usefulness of the controller design method.
Secondly, reliable control problems of the delta operator systems are studied. The aim is to design reliable controller which can meet the desired performance and tolerate actuator and(or) sensor failure. In this section, a more general failures model, so called continuous failure model is adopted for actuator and(or) sensor failure. Based on the LMI approach, reliable robust stabilization, reliable robust D-stabilization and reliable robust H∞control for the delta operator uncertain system are researched respectively. The conditions of the existence of corresponding controllers are deduced, and then the design methods of corresponding controllers are suggested respectively. A distinctive feature of the proposed method is that the structural information of the continuous failure model is fully utilized, which brings less conservativeness in controller design. This is demonstrated by some numerical examples.
Thirdly, non-fragile control problems are studied. The cotrollers to be designed are assumed to have multiplicative(or additive) gain variations. Based on the LMI approach, four problems are discussed.①A criterion of non-fragile quadratic stabilizability is derived, and then the design method of the state feedback cotrollers for non-fragile robust stabilization is presented.②Using the above conclusion, a sufficent and necessary condition of non-fragile quadratic D-stabilizability is obtained via construct an auxiliary delta operator system.③A non-fragile robust guaranteed cost controller design method for uncertain delta operator systems is presented.④An existence condition of the non-fragile robust state feedback control law satisfying disk pole and variance constraints for the delta operator systems is derived, and then the design method of relevant controller is suggested. For all cases, numerical examples are provided to illustrate the corresponding design methods.
Finally, robust pole assignment for the delta operator linear time-invariant system is discussed. A method so call pole normal assignment is evolved. The objective is to find a control law, such that the closed-loop system has desired poles and the closed-loop system matrix is a normal matrix, and then robustness of the control system is enhanced. Using the properties of normal matrix and generalized inverse theory of matrix, necessary and sufficient conditions are given to state feedback pole normal assignment and static output feedback pole normal assignment respectively. When the condition holds, the unified expression of the control laws are showed.
引文
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