【作者】 段志生；

【导师】 黄琳；

【作者基本信息】 北京大学 ， 一般力学与力学基础， 2000， 博士

【摘要】 对于单输入单输出系统,首先用α+βi（α²+β²≤γ²）取代控制器摄动块Δ₂（‖Δ₂‖_∞≤γ）,然后利用有界实引理及传统二次稳定的方法研究了控制器与对象同时以范数有界模式摄动的鲁棒控制问题,给出了线性矩阵不等式设计控制器的方法,这一问题的鲁棒性分析也归结为一个二阶矩阵的无穷范数问题。当控制器与对象都以常数加权加性模式摄动时,本文给出了设计控制器的充分必要条件,一般情况下设计控制器的方法有一定的保守性,但相对来说这一保守性小于处理鲁棒性能问题的传统方法所带来的保守性。此外还研究了一类与控制器和对象同时摄动问题紧密相关的圆盘多项式的稳定性问题。这一方法可以推广到多输入多输出系统,离散系统,时滞系统。 最后,用类似于二次稳定的方法研究了多对象串联反馈系统,说明了结构奇异值综合理论（如D-K搜索法）与指标二次稳定是紧密相关的,这样就在结构奇异值分析、综合与鲁棒稳定、二次稳定之间初步建立起了关系,给传统的鲁棒稳定与二次稳定注入了新的内容。 本文的主要成果在于,基于替换Δ（s）（‖Δ（s）‖_∞≤γ）→α+βi（α²+β²≤γ²）,给出了处理传统基于频域模型描述的鲁棒控制问题的新方法,伴随着这一方法出现了许多新问题、新结果,这也为处理存在多摄动源的鲁棒控制问题提供了新思路。更多还原

【Abstract】 In this dissertation, the robust control problems of linear systems under both controller and plant norm-bounded perturbations are studied. The necessary and sufficient conditions are given for robustness analysis of these problems. And robust stability of a class of disc polynomials is studied in the same framework. In addition, for SISO systems, a new LMI-based controller design method is presented by substituting controller uncertainties Δ（s）（||Δ（s）||∞≤γ） by complex numbers α+βi（α²+β²≤γ²） and using Bounded Real Lemma. Compared with structured singular value synthesis theory, this method is simpler and more direct. Generally this controller design method has some conservatism, but this conservatism is less than the one caused by transforming |||W₁S| + |W₂T|||∞ < 1 into |||W₁S|² + |w₂T|²||∞ <1/2 in the traditional robust performance problems. And in the case of both controller and plant additive perturbations without weighting functions, this method has not any conservatism. This method can be generalized to MIMO systems, discrete systems, time-delay systems.Finally, A class of negative feedback systems under more than two multiplicative perturbations is studied, it shows that there are some natural relations between quadratic stability and stuctured singular value synthesis theory. Some new contents are established in traditional robust stability and quadratic stability.The main contribution in this dissertation is that a new method is presented for traditional robust control problems based on the substitution Δ（s）（||Δ（s）||∞≤γ） —α+βi（α²+β²≤^γ2） . Many new problems arise with this method, and this method gives some new ideas for robust control problems under more than one perturbations.更多还原