An iterative updating method for damped gyroscopic systems

The problem of updating damped gyroscopic systems using measured modal data can be mathematically formulated as following two problems. Problem I: Given Ma ∈ Rn×n,Λ = diag{λ1, ?, λp} ∈ Cp×p,X = [x1, ?, xp] ∈ Cn×p, where p 2j = -λ2j-1∈ C, x2j = -x2j-1 ∈ Cn for j = 1, ?, l, and λk ∈ R, xk ∈ Rn for k = 2l+1, ?, p, find real-valued symmetric matrices D,K and a real-valued skew-symmetric matrix G (that is, GT = -G) such that MaXΛ2 +(D+G)XΛ+KX = 0. Problem II: Given real-valued symmetric matrices Da,Ka ∈ Rn×n and a real-valued skew-symmetric matrix Ga, find (Dˆ, Ĝ, Kˆ) ∈ SE such that ||Dˆ -Da||2+||Ĝ-Ga||2+|| Kˆ -Ka||2 = min(D,G,K)∈SE(||D- Da||2 + ||G - Ga||2 + ||K - Ka||2), where SE is the solution set of Problem I and || · || is the Frobenius norm. This paper presents an iterative algorithm to solve Problem I and Problem II. By using the proposed iterative method, a solution of Problem I can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum Frobenius norm solution of Problem I can be obtained by choosing a special kind of initial matrices. Moreover, the optimal approximation solution (Dˆ, Ĝ, Kˆ) of Problem II can be obtained by finding the minimum Frobenius norm solution of a changed Problem I. A numerical example shows that the introduced iterative algorithm is quite efficient.