摘要
数字控制系统是一个十分重要而且活跃的研究领域，其无限精度设计理论已经比较成熟并得到广泛的应用。从理论上讲，一个理想系统可以有许多种完全等价的实现方案，然而，在用物理元器件、模块或子系统实现系统时，由于存在着有限字长效应，实际系统中的性能往往要偏离理想性能。系统的性能与系统的实现结构有着密切的关系。在数字控制器的设计与实现过程中，必须考虑到有限字长对系统性能的影响。
有限字长控制器研究的一个方向是有限字长控制器的实现问题，即对于已经设计好的控制器，得到能最小化有限字长影响的控制器实现；另一个方向是直接设计一个有限字长控制器，使闭环系统能满足一定的性能指标。
本论文首先对数字系统，特别是数字控制器设计与实现中的有限字长问题和非脆弱控制器设计的背景、历史发展情况等作了简单介绍，然后通过分析数字控制器中不同来源的有限字长误差，定义不同的性能指标，较系统地分析并讨论不同的有限字长数字控制器设计和实现问题。主要的工作概括如下：
针对数字控制器参数的有限字长误差，分别定义并分析z算子实现和δ算子实现的FWL系统中闭环传递函数相对于数字控制器实现参数的灵敏度，通过灵敏度最小化来设计控制器最优实现；
针对控制器参数的有限字长误差，分别定义并分析z算子实现和δ算子实现的FWL系统中闭环极点位置相对于数字控制器实现参数的灵敏度，通过灵敏度最小化来设计控制器最优实现；
针对控制器内部信号的有限字长误差，基于随机控制理论，分别定义并分析z算子实现和δ算子实现的FWL系统中的闭环舍入噪声增益，通过最小化舍入噪声增益指标来设计控制器最优实现；
针对控制器内部信号的有限字长误差，分别基于l_1范数和H_∞范数定义并分析z算子实现的FWL系统中的闭环舍入噪声增益，通过最小化舍入噪声增益指标来设计控制器最优实现；并将H_∞舍入噪声增益问题推演转化为一个LMI问
摘要
题，从而利用MATLAB可很方便地进行求解；
针对控制器内部信号的有限字长误差，采用直接方法来设计有限字长控制
器，使有限字长协方差上界控制问题和有限字长见控制问题转化为统一的矩阵
不等式求解问题，求得的有限字长控制器能使闭环系统满足给定的性能指标。
最后，在总结全文的基础上，提出了若干有待进一步深入研究的问题。
Digital control system has been considered as one of the most important and active fields in engineering for many years, and the infinite precision design theory and application have been well studied for a long time. An ideal system can be implemented in different but equivlent realizations theoretically, however, due to the finite word length (FWL) effects resulting from actual devices, modules or subsystems, the system performance will always deviate from the ideal performance in practice. The system performance is closely related with the implementation of the system. Thus, the FWL effects need to be considered when a digital controller is designed and implememted.
One approach of FWL controller study is to obtain the optimal realizations of a given FWL controller to minimize the FWL effects. An alternative approach is to design the FWL controller directly according to some given performance index.
The backgrounds and a historical view of FWL problems and non-fragile controller design are presented at the beginning of the dissertation. In the body of the dissertation, different performance indexes are defined and analyzed based on the analysis of different types of FWL errors in digital controller, and different FWL digital controller design and realization problems are considered and solved. The major contributions of this dissertation are as follows:
Considering the FWL errors of controller coefficients, define and analyze the closed-loop transfer function sensitivity measure with respect to the controller coefficients, obtain the optimal controller realizations via minimizing the transfer function sensitivity measure. Both z -operator and δ -operator parameterized systems are considered.
Considering the FWL errors of controller coefficients, define and analyze the closed-loop pole place sensitivity measure with respect to the controller coefficients, obtain the optimal controller realizations via minimizing the pole sensitivity measure. Both z -operator and δ -operator parameterized systems are considered.
Considering the FWL errors of internal signals in digital controller, based on stochastic control theory, define and analyze the closed-loop roundoff noise gain in both z -operator and 5 -operator parameterized FWL systems, and obtian the optimal controller realizations via minimizing the roundoff noise gain.
Considering the FWL errors of internal signals in digital controller, define and analyze the roundoff noise gain based on l1 -norm and H∞ -norm respectively,
and obtain the optimal controller realization via minimize the newly defined roundoff noise gains. The H∞ roundoff noise gain minimizing problem is
reduced to an linear matrix inequalities problem which can be solved using MATLAB conveniently.
Considering the FWL errors of internal signals in digtial controllers, direct approach is adopted to design FWL controller. A FWL covariant upper bound control problem and a FWL H∞ control problem are reduced to a unified matrix inequalities problem.
Finally, more problems to be studied and investigated are pointed out on the basis of summarization of this dissertation.
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