【作者】 谢永强；

【导师】 陈建军；

【作者基本信息】 西安电子科技大学 ， 机械制造及其自动化， 2014， 博士

【摘要】 不确定性问题，特别是不确定参数问题，在理论研究和工程应用中广泛存在。本论文以区间参数问题为研究对象，以含区间参数的函数界限计算的研究为基础，以区间和仿射算法为研究工具，探索性地研究了复杂函数的全局优化问题、结构参数和外部激励为区间变量时系统的静力响应问题、复杂非凸不连续可行域系统的可靠度计算问题以及基于仿射逆方法的结构分析问题。主要内容如下：1．含区间参数的函数界限计算将函数中的不确定性参数用区间数表示，利用区间算法对函数界限进行分析。讨论了区间细分方法和不同型式的区间方法以提高计算精度，为含区间参数函数界限的计算提供了多种选择。并将区间方法应用于结构的非概率可靠度分析及静力分析中。2．复杂函数的全局优化问题研究分析了确定性和随机性优化、局部和全局优化之间的区别与联系，针对传统区间算法求解全局优化问题、耗时长、空间复杂度较高及收敛速度较慢的缺点，将仿射算法及局部优化算法引入了全局优化问题，给出了一种全局优化求解的仿射算法。由局部优化算法和各求解区间上待优化函数的仿射运算得到全局最优解的一个上界，再依据对各区间仿射运算的下界与全局最优解上界的比较来确定相应区间的去留，通过对不含全局最优解的子区间的删除来确定出最优解所在的子区间，并最终找到全局最优解。数值试验表明，该算法相对于传统的区间优化算法有较高的收敛速度，且占用了较少的系统资源。3．含区间参数的系统响应界限分析针对仿射运算时新符号噪声的引入必然造成误差放大的不足，在函数上下界计算中引入了矩阵形式的上下界的仿射计算公式，提出了一种计算上下界的改进仿射算法。该算法在仿射变量进行乘法运算时不会引入新的噪声，相对与传统的仿射算法能得到更紧凑的界限；并通过实例计算演示了该公式的计算过程及计算方法的有效性。将有界不确定性变量的仿射型及改进的仿射运算引入不确定系统响应上下界的计算。仿真结果表明，相对于区间算法及传统的仿射算法，该算法得到解的界限更为紧凑。4．基于域分析的控制系统稳定域求解针对不确定系统的区间表示不能描述变量间的相关性、相应的区间算法容易导致误差爆炸的问题，提出了不确定系统的仿射表示法及系统稳定性的仿射不等式判断方法。首先将系统中的不确定信息用仿射参数来表示，得到不确定控制系统传递函数的仿射形式，然后通过求解含仿射参数的不等式组求得了满足系统的稳定性条件时各噪声允许的范围。由于考虑了变量间的相关性，相对于区间算法，所提出的方法可以在更大的不确定范围内判断出系统的稳定性。5．复杂非凸不连续可行域系统的可靠度计算提出了一种复杂函数或电路系统的可行域计算的仿射区间方法，利用仿射区间方法对设计域内函数的函数界限进行分析，利用分支定界法将该区域分类为：可行、不可行域及不确定域；再将不确定区域进行细分，并对每个细分后的子区域进行进一步的函数界限分析及分类，直至子区域半径达到设计要求，然后进行可行域统计计算，将每个可行域的面积进行求和可得到函数的总可行域。该方法可以对非凸函数甚至可行域不连续函数的可行域进行估计。通过算例演示了该方法的计算过程并验证了该方法的有效性。6．基于区间逆阵求解及仿射逆阵求解的结构分析对于一些复杂问题，更常见的是多个变量组合起来，形成向量或者矩阵出现在方程中，当这些区间/仿射变量本身在某个区间内变动时，这些区间/仿射变量可以组合成区间/仿射向量或区间/仿射矩阵。以区间/仿射矩阵及其逆阵的解法为工具，对不确定工程结构的动力与静力分析进行研究。介绍了区间/仿射向量、区间/仿射矩阵及相关的一些概念，重点讨论了区间矩阵与仿射矩阵的逆阵求解方法，以实例演示了文中的方法并对其有效性进行了验证。更多还原

【Abstract】 Uncertain problems, especially for parameter uncertain problems,is Widely exist inthe actual scientific research and engineering research.In this thesis,the parameteruncertain problems are taken as research objects,the function with uncertain parameterlimits problem are taken as research base,the interval and affine arithmetic are tacken asresearch tools. The exploratory research including:the global optimization problems;static and dynamic response problems of systems with uncertain parameters; reliabilitycauculation of non-convex and not simply connected fuction;structural analasis problemsbased on the solution of uncertain interval or affine matrix inverse problems.The main research works can be described as follows:1. The bounds analysis of fuction with uncertain parameterBy representing the uncertain parameters as interval numbers, the uncertaintymodels are obtained.The interval arithmetic and affine arithmetic are adopted to get thebounds of uncertainty models.In order to get more higher precision, subdivisions andrefinements method and several form of interval extensions are applied. An exampleswere provided to illustrate the validity and feasibility of the present method.Anextended beam and a three-truss example are provided to illustrate the validity andfeasibility of the presented procedures.2. The study on the global optimization problemsThe affine algorithm for global optimization was proposed by introducing theaffine algorithm and local optimization into the problem of global optimization, whichaim at solving disadvantage of the time consuming, higher space co mplexity and slowconvergence speed for the traditional interval algorithm in the solution of globaloptimization problem. The upper bound of global optimal solution was obtained bylocal optimization algorithm and affine arithmetic for objective function in eachsubinterval. And then, whether corresponding interval was discarded or not depends oncomparing the lower bound of affine arithmetic for objective function in eachsubinterval with the upper bound of global optimization solution. The subinterval whichcontains the optimal value was obtained by deleting the subinterval which did notcontain the optimal value. Lastly, the global optimal solution was found. NumericalSimulation results show that the proposed algorithm has higher convergence speedcompare with traditional interval optimization algorithm. At the same time, it alsooccupies less system resource.3. Uncertain system response bounds analysis with interval arithmetic andaffinearithmetic The introduction of new noise symbols causes error amplification in affinearithmetic inevitably. To avoid this disadvantage, this paper presents a modified affinearithmetic in matrix form for bounds computation of functions. The modified affinearithmetic does not introduce new noises during multiplication operation of affinevariables, and it can obtain compacter bounds compared to conventional affinearithmetic. The formulas computing processes and the validity of proposed method aredemonstrated by an example. The affine form of bounded uncertain variables andmodified affine arithmetic are brought in to calculate response bounds of uncertainsystem. The simulations show that, the proposed approach can obtain closer responsebounds than interval arithmetic and conventional affine arithmetic.4. Stability region analysis using interval and affine methodAs the interval form of uncertainty system can not express the pertinence ofuncertain variable, and interval algorithm may cause error explosion. The affine form ofuncertainty system and the affine inequality judge method for uncertainty system arepresented, at first, the certain parameter in certain system was substituted by affineparameter, we get the affine form transmit function of uncertain system, secondly, bysolving Matrix inequalities,we can get tolerable noise range with which the systemstability condition hold. An example shows that as take the pertinence of uncertainvariable into consideration, this method can judge the stability in a larger region, showsthe validity and the advantage of this method.5. Reliability cauculation of non-convex and not simply connected fuctionIn this chapter an affine-interval arithmetic-based method for the feasible regionevaluation of function or electronic circuits is presented. This method use affine-intervalarithmetic analyze the bounds of the function, and use branch and bound methoddivided these intervals into three kinds: accept regions, refuse regions and those ofuncertain regions; and the next, all the uncertain regions are re-divided and the boundscalculation and classification performed again until the subintervals small enough. Thestatistics on each of accept regions are performed next to get the sum of the acceptregions. The proposed technique guarantees an efficient, reliable and accurateevaluation of the yield, even for non-convex and not simply connected feasible region.The example presented shows the features of the approach.6. The static analysis of interval structures based on interval matrix inversion andaffine matrix inversionIt is commonly used that many variables combined together, forms array or matixin engineering problems.The interval or affine array and matix was formed when one or more variable change in a certain range. The static analysis of interval structures basedon interval matrix inversion and affine matrix inversion.The basic conception wasdiscussed and the work focus on how to get interval matrix inversion and affine matrixinversion in a efficient method. The example presented shows the features and validityof this method.更多还原