Interval Methods for Circuit Analysis

Abstract

Interval Methods for Circuit Analysis Downloaded from www.worldscientific.com by 34.217.195.47 on 05/10/18. For personal use only. This page is intentionally left blankInterval Methods for Circuit Analysis Downloaded from www.worldscientific.com by 34.217.195.47 on 05/10/18. For personal use only. PREFACEProgress in the development of a theoretical subject is, to a great extent, predetermined by the adequacy, variety and power of the mathematical models used.Various mathematical tools have been utilised in circuit theory. They differ from one another in their complexity, generality and area of application. Thus, the topological properties of both linear and nonlinear electric circuits are best described by graph theory. Steady-state analysis of large linear circuits is carried out in the most effective way by using sparse matrix techniques. Qualitative properties of nonlinear circuits -existence, uniqueness, boundedness of steady-states and other related topics -seem to be most adequately analyzed in terms of the qualitative theory of nonlinear differential equations.The overwhelming majority of the mathematical models now in use in circuit theory is based on the concept of real number (complex numbers can always, at least conceptually, be viewed as pairs of real numbers in a corresponding two-dimensional space). This approach is quite natural and satisfactory if the initial data about the electric circuit studied (parameters of passive elements, energy sources etc.) can be assumed to be known exactly. Since this is rarely the case, such an idealization is acceptable only when each item of the input data can be represented within resonable accuracy as a real number. As each real number can be geometrically viewed as a point on the real line, all the data relevant to the problem at hand can be visualized as a point in a space of appropriate dimensionality. Therefore, for brevity of expression, a mathematical model based on such an approach will be called a "point" model. Although intrinsically inaccurate the point model is practically the best model for tackling problems in which the input data can reasonably be assumed to be known exactly.However, the input data are, theoretically always and practically most often, known only with some degree of uncertainty. In circuit theory the basic approach to handling such problems is to appeal to a probabilistic description of the problem and to apply a certain statistical method for its solution. This approach is associated with the necessity to introduce experimentally some distribution law describing the probability with which the point representing the input data appears in each element of a corresponding parameter space. Another possibility is to resort to some results from fuzzy sets theory. Once again, some statistical information is needed to describe the "fuzziness" of the sets involved.An alternative approach to treating electric circuits with inaccurate data is to apply the notions and methods of the so-called interval analysis.Interval analysis is a new and int...

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