Analytical method for finding unknown constant parameters of linear regression inequalities

Abstract

A system of linear regression inequalities with unknown constant parameters, whose number is assumed to be given and finite, is considered. The problem of constructing the domain of the components of the vector of unknown parameters that ensure the validity of the prescribed inequalities is addressed. A method is proposed, based on the procedure of dynamic regressor extension and the selection of active constraints, which reduces the original problem to solving a square system of linear equations. The application of Cramer’s rule and Hadamard’s inequality to the resulting system makes it possible to obtain an analytical upper bound for the components of the vector of unknown parameters. The correctness of the proposed method is illustrated by numerical simulation. Unlike numerical optimization methods, the presented approach does not require iterative computations and provides a rigorous guaranteed bound valid for the entire class of admissible data. A theorem establishing this bound in the general case is formulated and proved. The conclusion discusses the prospects for further development of the proposed approach.