【作者】 贾晓影；

【导师】 李春吉；

【作者基本信息】 东北大学 ， 运筹学与控制论， 2009， 硕士

【摘要】 多项式非线性系统在现实世界中广泛存在,如生化领域,化工过程,电子电路等系统,其中许多控制问题均可建模,转化或近似成多项式非线性系统。又因多项式非线性系统在非线性系统家族中具有普遍性,故多项式非线性系统的研究对非线性系统具有重要意义。由此可见,如何分析及综合多项式非线性系统对非线性理论发展以及工程应用是件非常有意义的工作。近年来,通过数值计算的方法对多项式非线性系统进行研究受到相当多的关注,并取得了一系列成果。这些方法在多项式非线性系统稳定性分析上取得了重大进展。本论文研究了一类特殊的时不变多项式非线性系统,对其进行了稳定性分析和控制器综合。论文主要研究内容如下：1.介绍了李亚普诺夫稳定性。在稳定性分析方面,李亚普诺夫稳定性判据是控制理论中的一大基石,它是Lyapunov在19世纪末提出,后经Malkin, Chetaev, Zubov, Krasovskii, Razumikhin等人的发展,成为成熟的体系。2.介绍了Sum-of-Squares (SOS)最优化算法和Canonical Quadratic Distance Problems (CQDP)最优化算法的相关的概念,并且介绍了基于SOS估计吸引域的算法3.1并给出了在算法3.1基础上改进的算法3.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法3.2可以得到比较好的结果。3.介绍了基于Sum-of-Squares(SOS)最优化方法的扰动分析,并且介绍了基于Sum-of-Squares(SOS)的设计状态反馈控制器的算法4.1并且给出了在此算法基础上改进的设计控制器的算法4.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法4.2可以得到比较好的结果。更多还原

【Abstract】 Polynomial nonlinear systems appear widely in the real world. Many control problems in the field of biochemical, chemical process, electric circuits and so on can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Researching polynomial nonlinear systems has significance for the investigation of nonlinear systems, because polynomial nonlinear systems have universality in the nonlinear systems family. Therefore, how to analyze and synthesize polynomial nonlinear systems is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial nonlinear systems in numerical approach. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches.This thesis explores of a class of special time-invariant polynomial nonlinear systems. And it researches their stability analysis and controller synthesis.The main research content are showed as follows:1. The Lyapunov stability is introduced. In the field of stability analysis, the Lyapunov stability criterions are a major cornerstone of the Control Theory. It was put forward at the end of the 19th century by Lyapunov. And then it was developed by Malkin, Chetaev, Zubov, Krasovskii, Razumikhin and so on. After that it became a sophisticated system.2. The related concepts of Sum-of-Squares (SOS) optimization algorithm and canonical quadratic distance problems(CQDP) optimization algorithm are introduced, and algorithm 3.1 bases on SOS to estimates the domain of attraction. And then the improved algorithm 3.2 is given at the basis of algorithm 3.1. Finally, it is illustrated with an example. The example shows that good results can be got by using algorithm 3.2 for some two-dimensional systems.3. The disturbance analysis based on Sum-of-Squares (SOS) optimization algorithm is introduced. And then the state feedback controller in algorithm 4.1 based on Sum-of-Squares (SOS) is introduced. Furthermore, the improved algorithm 4.2 is given at the basis of algorit-hm 4.1 to design the state feedback controller. At last, it is illustrated with an case. The case denominates the better results can be obtained by using algorithm 4.2 for some two-dimensio-nal systems.更多还原