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  本文選題：符號計算　切入點：同時鎮(zhèn)定問題　出處：《華東師范大學》2016年博士論文

【摘要】：系統(tǒng)的同時鎮(zhèn)定問題和安全性驗證是系統(tǒng)科學與控制理論中的基本問題,有著非常重要的理論意義和應用價值。同時鎮(zhèn)定問題考慮如何設計單個的控制器,使其可以同時鎮(zhèn)定給定的多個對象,即要求系統(tǒng)的閉環(huán)傳遞函數(shù)是穩(wěn)定的。目前學術界僅對線性系統(tǒng)有一些初步的理論結果,特別地,即使對于三個線性對象的同時鎮(zhèn)定問題,也存在大量有待探討的理論問題。而系統(tǒng)的安全性驗證問題則是從實際應用出發(fā),探討如何驗證系統(tǒng)是否滿足實際需求的某些約束條件性質,避免系統(tǒng)發(fā)生某些不期望的演化過程；這些問題往往轉化為若干代數(shù)不等式的命題是否成立的驗證問題,而求解或者證明這些不等式構成的半代數(shù)系統(tǒng),通常依賴于一些特殊的人工技巧結合計算機進行機器自動證明。本文研究了線性系統(tǒng)同時鎮(zhèn)定和非線性系統(tǒng)安全性檢驗的若干相關問題,主要研究內容和工作如下：1.研究了線性系統(tǒng)同時鎮(zhèn)定中著名的“比利時巧克力問題”的系統(tǒng)參數(shù)的理論上界確定問題。系統(tǒng)地討論了比利時巧克力問題連續(xù)系統(tǒng)情形與對應離散系統(tǒng)情形下參數(shù)的關系,這個離散情形下的參數(shù)與復分析理論中著名的“Goldberg常數(shù)”有著深刻的聯(lián)系,基于符號計算的方法和橢圓模函數(shù)的概念,本文對復平面挖去兩點剩下區(qū)域的同倫雙曲測地線進行了有效的估計,并將該結果應用于比利時巧克力問題的上界估計中,我們得到的參數(shù)估計值改進了Hempel和Smith計算的數(shù)值結果。2.對于比利時巧克力問題的數(shù)值下界,討論了控制器設計中多項式方程根的分布與系統(tǒng)參數(shù)的關系,指出Boston在2012年宣布找到的“巧克力問題的新界限(0.976462)”文中提到的("a new world record")是不可信的,經(jīng)過驗證,他找到的控制器并不能保證系統(tǒng)閉環(huán)多項式是穩(wěn)定的。另外本文作者給出了一種參數(shù)擾動和Hurwitz穩(wěn)定驗證結合的方法,對Boston提出的臨界多項式進行輔助參數(shù)擾動,提出了一種鎮(zhèn)定控制設計算法。實驗例子表明,在某些低階情形下我們的算法得到的數(shù)值下界顯著地改進了對應階次的最好結果。3.研究了非線性系統(tǒng)的安全性驗證問題�；诜栍嬎愎ぞ摺⒔Y合微分動力系統(tǒng)定性理論、采用構造分段判別函數(shù)的思想以及微分方程解的冪級數(shù)截斷估計技巧,將問題轉化為一組代數(shù)不等式的驗證問題,并以Van der Pol系統(tǒng)為例進行了安全性驗證,可按預先設定的精度,精確地捕捉“安全”與“不安全”之間的臨界點�？梢哉J為由上述方法所得到的條件是Sharp的。4.考慮了非線性微分方程的安全系統(tǒng)驗證的區(qū)域變換問題。通過構造從初始區(qū)域出發(fā)到非安全區(qū)域外部的共形映射,將這個映射轉換成二維平面上的實變量代換；對已有的非線性系統(tǒng)進行坐標變換,將原來系統(tǒng)的安全性驗證問題轉化為新坐標下的安全性驗證問題。討論了經(jīng)過變換后的系統(tǒng)與原系統(tǒng)的保持不變的一些性質。5.利用符號計算的方法,對于著名的Mordell不等式,使用歸一化的方法將含有約束條件的問題轉化為無約束的不等式驗證問題；當n=3時的情況使用柱形代數(shù)分解的方法驗證不等式是成立的,采用差分代換的方法驗證了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.

【學位授予單位】：華東師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：TP13;TP309

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