A model updating method for undamped structural systems

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given a full column rank matrix X ∈ R n × p , a diagonal matrix Λ ∈ R p × p and matrices K 0 ∈ R r × r , M 0 ∈ R r × r , find n × n matrices K , M such that ∥ KX - MX Λ ∥ = min , s.t. K ( [ 1 , r ] ) = K 0 , M ( [ 1 , r ] ) = M 0 , where K ( [ 1 , r ] ) and M ( [ 1 , r ] ) are, respectively, the r × r leading principal submatrices of K and M . We then consider a best approximation problem: Given n × n matrices K a , M a with K a ( [ 1 , r ] ) = K 0 , M a ( [ 1 , r ] ) = M 0 , find ( K ^ , M ^ ) ∈ S E such that ∥ K a - K ^ ∥ 2 + ∥ M a - M ^ ∥ 2 = inf ( K , M ) ∈ S E ( ∥ K a - K ∥ 2 + ∥ M a - M ∥ 2 ) , where S E is the solution set of LSP. We show that the best approximation solution ( K ^ , M ^ ) is unique and derive an explicit formula for it.