Approximate inversion formula for structural dynamics and control

In a recent paper the authors presented an iterative method and an approximate formula for predicting the response of a system modified by a finite set of rank-one modifications. This method was developed based on the successive application of the Sherman–Morrison matrix inversion formula to calculate the inverse of a matrix changed by a rank-k modification and provides an easier method to calculate the change in the transfer matrix of a dynamic system when it undergoes a number of k simply connected modifications. This paper presents a new and more interesting justification of the method which allows an extension of the approximate formula from a formula for frequency response approximation to a more general approximate matrix inversion formula applicable to particularly shaped matrices generally encountered in structural dynamics and control. This new approach extends the area of the application of the previously presented method. An example of a simple application of this method to the problem of feedback control of structures known only through their estimated receptances is presented.