Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of kaucher arithmetic

摘要

In this paper, theidentification problem, thetolerance problem, and thecontrol problemare treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution setΣ∃∃(A, b)={x∈ ℝn (∃A∈ A)(Ax∈ b)}, of thetolerable solution setΣ∀∃(A, b)={x∈ ℝn (∀A∈ A)(Ax∈ b)}, and of thecontrollable solution setΣ∃∀(A, b)={x∈ ℝn (∀b∈ b)(Ax∈b)} respectively. Analgebraic approachto their solution is developed in which the initial problem is replaced by that of finding analgebraic solutionof some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its c