Best possible componentwise parameter inclusions computable from a priori estimates, measurements, and bounds for the measurement errors

摘要

We consider problems of parameter estimation which can be described as follows:The available information about the parameter x̄∈Rn to be estimated isx̄∈f−1(Y)∩X0,where f:D→Rm,D⊂Rn, is a given two times continuously differentiable mapping, Y a given box in Rm, and X0 a given box in D.Then the best possible componentwise inclusion of x̄ is the smallest box X⊂X0 containing f−1(Y)∩X0. Therefore we try to compute boxes X(i),X(o) sufficiently close to X and such thatX(i)⊂X⊂X(o).It will be shown that this can be done under conditions usually fulfilled in practice. Then the problem can be reduced to the solvable problem to compute inner and outer approximations of the interval hull of a given tolerance polyhedron.As a typical practical case, a problem of deriving best inclusions of coordinates of points from a priori estimates, distance measurements, and bounds for the measurement errors is considered.Computed solutions for some illustrating numerical examples are presented.