基于符号计算的系统同时镇定及安全性验证

发布时间：2018-03-27 15:33

本文选题：符号计算　切入点：同时镇定问题　出处：《华东师范大学》2016年博士论文

【摘要】：系统的同时镇定问题和安全性验证是系统科学与控制理论中的基本问题,有着非常重要的理论意义和应用价值。同时镇定问题考虑如何设计单个的控制器,使其可以同时镇定给定的多个对象,即要求系统的闭环传递函数是稳定的。目前学术界仅对线性系统有一些初步的理论结果,特别地,即使对于三个线性对象的同时镇定问题,也存在大量有待探讨的理论问题。而系统的安全性验证问题则是从实际应用出发,探讨如何验证系统是否满足实际需求的某些约束条件性质,避免系统发生某些不期望的演化过程；这些问题往往转化为若干代数不等式的命题是否成立的验证问题,而求解或者证明这些不等式构成的半代数系统,通常依赖于一些特殊的人工技巧结合计算机进行机器自动证明。本文研究了线性系统同时镇定和非线性系统安全性检验的若干相关问题,主要研究内容和工作如下：1.研究了线性系统同时镇定中著名的“比利时巧克力问题”的系统参数的理论上界确定问题。系统地讨论了比利时巧克力问题连续系统情形与对应离散系统情形下参数的关系,这个离散情形下的参数与复分析理论中著名的“Goldberg常数”有着深刻的联系,基于符号计算的方法和椭圆模函数的概念,本文对复平面挖去两点剩下区域的同伦双曲测地线进行了有效的估计,并将该结果应用于比利时巧克力问题的上界估计中,我们得到的参数估计值改进了Hempel和Smith计算的数值结果。2.对于比利时巧克力问题的数值下界,讨论了控制器设计中多项式方程根的分布与系统参数的关系,指出Boston在2012年宣布找到的“巧克力问题的新界限(0.976462)”文中提到的("a new world record")是不可信的,经过验证,他找到的控制器并不能保证系统闭环多项式是稳定的。另外本文作者给出了一种参数扰动和Hurwitz稳定验证结合的方法,对Boston提出的临界多项式进行辅助参数扰动,提出了一种镇定控制设计算法。实验例子表明,在某些低阶情形下我们的算法得到的数值下界显著地改进了对应阶次的最好结果。3.研究了非线性系统的安全性验证问题。基于符号计算工具、结合微分动力系统定性理论、采用构造分段判别函数的思想以及微分方程解的幂级数截断估计技巧,将问题转化为一组代数不等式的验证问题,并以Van der Pol系统为例进行了安全性验证,可按预先设定的精度,精确地捕捉“安全”与“不安全”之间的临界点。可以认为由上述方法所得到的条件是Sharp的。4.考虑了非线性微分方程的安全系统验证的区域变换问题。通过构造从初始区域出发到非安全区域外部的共形映射,将这个映射转换成二维平面上的实变量代换；对已有的非线性系统进行坐标变换,将原来系统的安全性验证问题转化为新坐标下的安全性验证问题。讨论了经过变换后的系统与原系统的保持不变的一些性质。5.利用符号计算的方法,对于著名的Mordell不等式,使用归一化的方法将含有约束条件的问题转化为无约束的不等式验证问题；当n=3时的情况使用柱形代数分解的方法验证不等式是成立的,采用差分代换的方法验证了n=4不等式也是成立的。
[Abstract]:At the same time the system stabilization and safety verification is the basic problem of science and control theory in the system, has a very important theoretical significance and application value. At the same time stabilization problem to consider how to design a single controller, which can make the multiple simultaneous stabilization of a given object, which requires the system closed-loop transfer function is stable at present. The academic circles only for the linear system with some preliminary theoretical results, in particular, even for three linear objects at the same time stabilization problem also exists many problems need to be discussed. And the verification of safety system is based on practical application, to explore how to verify whether the system satisfies some constraint conditions the nature of the actual demand. To avoid the evolution of some unexpected system; these problems are often transformed into several verification problems of algebraic inequality proposition, and solving or That a semi algebraic system composed of these inequalities, usually rely on some special techniques combined with computer automatic artificial proof machine. This paper studies the simultaneous stabilization of linear systems and nonlinear systems of safety tests and some problems, the main research contents and work are as follows: 1. research on simultaneous stabilization of linear systems in the famous "Belgian chocolate" the theoretical upper bound of system parameters to determine the problem. The relationship between parameters of continuous system and the Belgian chocolate problem corresponding to discrete system conditions are discussed systematically, the discrete situation parameters and complex analysis theory in the famous "constant Goldberg" has a profound connection, the concept of symbolic computation method and elliptic modular function based on the complex plane to dig two remaining homotopy geodesic hyperbolic region is estimated, and the node The fruit should be used to estimate the upper bound of the Belgian chocolate problem, we obtain the parameters estimation results of Hempel and Smith improved the numerical calculation of the.2. value for the lower bound of the Belgian chocolate problem, discussed the relationship between the distribution and the system parameters of the root of polynomial equation in controller design, the new line pointed out that chocolate problem Boston announced in 2012 "(0.976462)" mentioned in the text ("a new world record") is not credible, after verification, he found the controller does not guarantee that the closed-loop system is stable. In addition, the polynomial is given a parameter perturbation method and Hurwitz stability verification based on critical polynomial, the auxiliary parameters Boston disturbance, presents a stabilization control algorithm is designed. The experimental examples show that the improved significantly lower bounds of our numerical algorithm in some low order situation The best results of the corresponding order of the.3. on the safety verification problem of nonlinear system. Based on the symbolic computation tool, combined with the qualitative theory of differential dynamical system, using piecewise discriminant function of the thought and the differential equations of power series truncated estimation techniques, the problem was converted into a set of algebraic inequalities of the verification problem, and to Van der Pol system as an example for the safety verification, according to the preset accuracy, accurately capture the critical point between safe "and" unsafe ". That can be obtained by the method is Sharp.4. considered the regional security system transformation problem of nonlinear differential equations. The verification of conformal mapping according to the non security outside the area from the initial region through the structure, the mapping into a real variable substitution in the 2D plane; nonlinear system of coordinate transformation, The safety verification problem of the original system into the safety verification problem is discussed. The new method after keeping some properties of.5. are calculated by using the same symbol system after transformation with the original system, the famous Mordell inequality method, using the normalized containing constraint problem into unconstrained inequality verification problem the method of n=3; when using the cylindrical algebraic decomposition verification inequality is established, verified by n=4 inequality is established for substitution.

【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP13;TP309

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