Stable Measurement of Temperature Errors During Testing of Quenchants by Methods of the Theory of Ill-Posed Problems

In this work a new approach is proposed for timely determination of work-piece temperature errors at various stages of quenching. The essence of the approach is the special type of construction of a diagnostic matrix C=A^-1B, where A matrix elements are computation-identification parameters, but B matrix elements are technologically measured parameters of the manufactured work-piece at various stages, and also parameters of technical means, involved into the quenching process. It has to be noticed that the above mentioned computation-identification matrix is very sparse, that is, the computation-identification matrix A is an ill-posed matrix, and moreover, the determinant of this matrix can be equal to zero. Therefore, the inverse matrix A^-1 does not exist in the classic sense, and the problem of constructing the matrix C becomes ill-posed. So there appears a problem of finding its stable inverse matrix (not finding the pseudoinverse). This problem becomes even more acute and sophisticated, when elements of the computation-identification matrix are characterized by the errors, where maximal ranges of their change are usually known a priori.

In this work a new regularization algorithm is offered for determination of matrix C for stable finding of computation-identification parameters of work-pieces at various stages of quenching process. The proposed approach is based on the Tikhonov algorithm and method of regularization. However, the proposed approach has a principal difference from the classic Tikhonov approach when choosing a regularization parameter in the Tikhonov functional.

Present work was executed within the framework of the European Social Fund Project No. 1DP/1.1.1.2.0/09/APIA/VIAA/142.