奇异摄动系统的稳定反馈控制
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摘要
奇异摄动理论几十年来在数学领域得以迅速发展，同样在控制领域中也取得了突破性进展，并一直伴随着控制理论的发展而不断完善。同时，线性矩阵不等式方法在控制领域中几年来应用越来越广泛，但在奇异摄动系统中的应用还比较少见。本论文主要针对奇异摄动系统，尝试结合线性矩阵不等式方法做了相关的研究。具体工作主要集中在如下几个方面：
(1) 将正常连续系统的二次稳定性概念推广到奇异摄动系统，并证明了奇异摄动系统与其快慢子系统关于二次稳定的等价性。同时，利用矩阵不等式方法，推导了奇异摄动系统二次稳定性的充分条件，并给出了二次可镇定并可解的充分条件，以及二次可镇定的状态反馈控制器的一种迭代求法。另外，给出了离散奇异摄动系统二次稳定和二次可镇定的条件。
(2) 应用线性矩阵不等式方法设计了非标准离散奇异摄动系统的二次型次优调节器。根据一般线性离散系统得到了二次型次优的Riccati不等式条件，通过引入一个特殊分块带有摄动参数的解矩阵形式，进而将条件化为可解的矩阵不等式，从而求取其解，设计次优控制器。与繁琐的最优控制方法进行仿真分析比较，结果表明这种方法的有效性和运算的方便快捷性。
(3) 讨论了带有干扰抑制的奇异摄动系统的二次型次优调节器问题，沿着逆最优调节器问题处理思路，将干扰转化到二次性能指标中，通过适当放大指标，将干扰控制在一定的范围之内。而后将问题转化为一组不含有摄动参数的矩阵不等式问题处理。
(4) 用矩阵不等式方法讨论了连续和离散奇异摄动系统的H_∞控制，通过广义系统途径，结合广义系统的有界实引理，将含有小参数的Riccati不等式等价转化为与小参数无关的不等式问题，分别得到了连续和离散奇异摄动系统H_∞控制的充分条件。同时，讨论了奇异摄动系统的正实性问题，给出了正实反馈控制的充要条件。
(5) 讨论了Delta算子域上的奇异摄动系统的状态反馈。用直接法分别设计了快慢子系统的状态反馈控制，使其达到预期的极点配置结果。所得结论将连续与离
散系统的相关结果统一于Delta算子框架。仿真结果证明了方法的有效性，弥补了
连续系统在高速采样下离散化的不足。
此外，本论文还进一步讨论了离散奇异摄动系统的摄动稳定条件问题，得到了
两种摄动形式稳定的充要条件，并将结果推广到了2一D情形。
关键词:奇异摄动系统，慢变子系统，快变子系统，线性矩阵不等式(LMI)，
次优控制，正实控制，Del七a算子，2一D系统。
Singular perturbation techniques for time scale systems have found wide applications in the area of analysis and synthesis of control processes, which has been studied recently in different set-ups by many researchers. On the other hand, the linear matrix inequality (LMI) approach has been attracting more attention due to its extensive applications in solving various control problems. However, up to the present, the problem of robust control for singularly perturbed systems through LMI approach has not been solved. Considering this, in this dissertation, attention is focused on the development of an LMI approach to solving robust control problems for singularly perturbed systems.The main results obtained in this dissertation are as follows:i) Quadratic stability is extended to singularly perturbed continuous and discrete systems, respectively. It is shown that quadratic stability of a singularly perturbed system is equivalent to that of the slow and fast system. Using the LMI approach, we obtain sufficient conditions of quadratic stability and quadratic stabilizability for singularly perturbed systems. An iterative algorithm is proposed for the design of desired quadratic stabilizing feedback controllers.ii) A sub-optimal regulator is designed for nonstandard discrete-time singularly perturbed systems through the LMI method; it is different from the general fast-slow decomposition method. By introducing a solution matrix with perturbed parameters, sub-optimal controllers are designed. A simulation result is presented to demonstrate the effectiveness of the proposed design method.iii) By a similar method to the inverse optical quadratic control, the problem of sub-optical disturbance attenuation subject to a quadratic performance index is discussed by the LMI approach.iv) The ε-independent H∞ control problem for singularly perturbed continuous and discrete systems is investigated, respectively. Sufficient conditions are obtained and sub-optimal controllers are designed, respectively. By the positive real lemmas for singular systems and regular systems, we present a sufficient and necessary condition for the solvability of positive real control problem for singularly perturbed systems.v) State feedback design for singularly perturbed systems is discussed in the Delta
(d) domain. State feedback controllers of slow and fast singularly perturbed subsystems are designed, respectively, such that the closed-loop system has desired poles. The derived result can be viewed as a unified one for continuous and discrete time systems in the unified Delta framework. An example is presented to demonstrate the validity and effectiveness of our design method.Furthermore, the problem of stability analysis for singularly perturbed discrete systems is considered. A necessary and sufficient condition is obtained for singularly perturbed discrete systems with two kinds perturbations. This is also extended to the case of two-dimensional singularly perturbed Roesser model.
(1) 将正常连续系统的二次稳定性概念推广到奇异摄动系统，并证明了奇异摄动系统与其快慢子系统关于二次稳定的等价性。同时，利用矩阵不等式方法，推导了奇异摄动系统二次稳定性的充分条件，并给出了二次可镇定并可解的充分条件，以及二次可镇定的状态反馈控制器的一种迭代求法。另外，给出了离散奇异摄动系统二次稳定和二次可镇定的条件。
(2) 应用线性矩阵不等式方法设计了非标准离散奇异摄动系统的二次型次优调节器。根据一般线性离散系统得到了二次型次优的Riccati不等式条件，通过引入一个特殊分块带有摄动参数的解矩阵形式，进而将条件化为可解的矩阵不等式，从而求取其解，设计次优控制器。与繁琐的最优控制方法进行仿真分析比较，结果表明这种方法的有效性和运算的方便快捷性。
(3) 讨论了带有干扰抑制的奇异摄动系统的二次型次优调节器问题，沿着逆最优调节器问题处理思路，将干扰转化到二次性能指标中，通过适当放大指标，将干扰控制在一定的范围之内。而后将问题转化为一组不含有摄动参数的矩阵不等式问题处理。
(4) 用矩阵不等式方法讨论了连续和离散奇异摄动系统的H_∞控制，通过广义系统途径，结合广义系统的有界实引理，将含有小参数的Riccati不等式等价转化为与小参数无关的不等式问题，分别得到了连续和离散奇异摄动系统H_∞控制的充分条件。同时，讨论了奇异摄动系统的正实性问题，给出了正实反馈控制的充要条件。
(5) 讨论了Delta算子域上的奇异摄动系统的状态反馈。用直接法分别设计了快慢子系统的状态反馈控制，使其达到预期的极点配置结果。所得结论将连续与离
散系统的相关结果统一于Delta算子框架。仿真结果证明了方法的有效性，弥补了
连续系统在高速采样下离散化的不足。
此外，本论文还进一步讨论了离散奇异摄动系统的摄动稳定条件问题，得到了
两种摄动形式稳定的充要条件，并将结果推广到了2一D情形。
关键词:奇异摄动系统，慢变子系统，快变子系统，线性矩阵不等式(LMI)，
次优控制，正实控制，Del七a算子，2一D系统。
Singular perturbation techniques for time scale systems have found wide applications in the area of analysis and synthesis of control processes, which has been studied recently in different set-ups by many researchers. On the other hand, the linear matrix inequality (LMI) approach has been attracting more attention due to its extensive applications in solving various control problems. However, up to the present, the problem of robust control for singularly perturbed systems through LMI approach has not been solved. Considering this, in this dissertation, attention is focused on the development of an LMI approach to solving robust control problems for singularly perturbed systems.The main results obtained in this dissertation are as follows:i) Quadratic stability is extended to singularly perturbed continuous and discrete systems, respectively. It is shown that quadratic stability of a singularly perturbed system is equivalent to that of the slow and fast system. Using the LMI approach, we obtain sufficient conditions of quadratic stability and quadratic stabilizability for singularly perturbed systems. An iterative algorithm is proposed for the design of desired quadratic stabilizing feedback controllers.ii) A sub-optimal regulator is designed for nonstandard discrete-time singularly perturbed systems through the LMI method; it is different from the general fast-slow decomposition method. By introducing a solution matrix with perturbed parameters, sub-optimal controllers are designed. A simulation result is presented to demonstrate the effectiveness of the proposed design method.iii) By a similar method to the inverse optical quadratic control, the problem of sub-optical disturbance attenuation subject to a quadratic performance index is discussed by the LMI approach.iv) The ε-independent H∞ control problem for singularly perturbed continuous and discrete systems is investigated, respectively. Sufficient conditions are obtained and sub-optimal controllers are designed, respectively. By the positive real lemmas for singular systems and regular systems, we present a sufficient and necessary condition for the solvability of positive real control problem for singularly perturbed systems.v) State feedback design for singularly perturbed systems is discussed in the Delta
(d) domain. State feedback controllers of slow and fast singularly perturbed subsystems are designed, respectively, such that the closed-loop system has desired poles. The derived result can be viewed as a unified one for continuous and discrete time systems in the unified Delta framework. An example is presented to demonstrate the validity and effectiveness of our design method.Furthermore, the problem of stability analysis for singularly perturbed discrete systems is considered. A necessary and sufficient condition is obtained for singularly perturbed discrete systems with two kinds perturbations. This is also extended to the case of two-dimensional singularly perturbed Roesser model.
引文
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