Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control
The object of the paper is the demonstration that the basic problem of interval computations is feasible for fractionally linear functions. In detail, let a fractionally linear function \(f\) be given by the tuple \((n;a_0,a_1,\dots,a_n;\) \(b_0,b_1,\dots,b_n)\) with real numbers \(a_i\), \(b_i\) \((i=0,\dots,n)\), and thus let its value for \(x=(x_1,\dots,x_n)\) be defined by \(y=f(x)=(a_0+a_1x_1+\cdots+a_nx_n)/(b_0+b_1x_1+\cdots+b_nx_n)\). A basic interval computation problem for \(f\) is denoted by the tuple \((f,X_1,\dots,X_n)\), where \(X_i=[x^-_i,x^+_i]\) are intervals, and this problem is non-degenerate if \(0\not\in b_0+b_1\cdot X_1+\cdots+b_n\cdot X_n\), where \(b_i\cdot X_i=[\min(b_ix^-_i,b_ix_i^+),\max(b_ix^-_i,b_ix_i^+)]\). Furthermore, if an algorithm computes the range \(Y=f(X_1,\dots,X_n)\) it is said to compute the optimal enclosure for the problem \((f,X_1,\dots,X_n)\). The main result of the present paper is the proof of the existence of an algorithm that computes the optimal enclosure for an arbitrary non-degenerate fractionally linear problem in quadratic time. Besides, in great detail, applications are discussed for problems of intelligent control. The bibliography contains 35 entries.
scientific article; zbMATH DE number 915298
scientific article