Theory And Application Analysis Of Interval And Affine Method For Unascertained Parameters

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.