Static and Dynamic Analysis of Imprecisely Defined Systems with Uncertain Parameters

Abstract

Practical problems of various disciplines viz. structural mechanics, robotics, control theory, image processing, stationary analysis of circuits etc. deal with mathematical models often governed by differential equations. Usually, the material properties (or the coefficients of the differential equations) are considered as crisp. But, in actual practice only bounds of values are obtained as a result of errors in measurements, observations and calculations or due to maintenance induced errors etc. These uncertain bounds may be modeled using probabilistic methods, interval computations and fuzzy set theory. In probabilistic approach, the uncertain parameters are considered as random variables whereas in interval and fuzzy computations, the parameters are treated as closed intervals of real line and fuzzy numbers respectively. When only lower and upper bounds of parameters are considered, interval computations may be advantageous as compared to probabilistic approach (which requires sufficient experimental data). Moreover, interval analysis is a tool for studying propagation of fuzzy uncertainties in term of fuzzy intervals using r-cut. As regards, interval computing acts as a robust tool for handling uncertainty propagation in systems having imprecise parameters.
Few authors have developed methods in handling imprecisely defined systems with interval and fuzzy uncertainties. But, existing methods are either problem dependent or result to weak solutions. Also, the existing methods and techniques are sometimes silent about the overestimation problems that occur due to `dependency effect' in interval computations. In this regard, concepts of classical Interval Arithmetic (IA), contractors and parametric approach have been used in this investigation for handling interval uncertainties. In particular, imprecisely defined systems with interval and fuzzy parameters have been modeled here using system of linear equations, eigenvalue problems, constraint satisfaction problems for localization of robots, nonlinear oscillators and system identification problems......