A structure-preserving iteration method of model updating based on matrix approximation theory

A structure-preserving iteration method of model updating based on matrix approximation theory^*

Dongxiu Xie

School of Science, Beijing Information Science and Technology University, Beijing, 100192, China E-mail: [e-mail]

ABSTRACT

Some theories and a method are discussed on updating a generalized centrosymmetric model. It gives a generalized centrosymmetric modified solution with partial prescribed least square spectra constraints. The emphasis is given on exploiting structure-preserving algorithm based on matrix approximation theory. A perturbation theory for the modified solution is established. The convergence of an iterative solution is investigated. Illustrative examples are provided.

Mathematical subject classification: 15A29, 15A90, 41A29, 65F15, 65L70.

Key words: structure-preserving algorithm, generalized centrosymmetric matrix, model updating, perturbation analysis.

1 Introduction

Structure design with prescribed natural frequencies and main vibration modes is an important research topic in structural engineering [12]. Structure finite element model updating technology is the most common method in structure design. For example, an undamped free vibration model [14] is described by

M + Kx = 0,

where K is a stiffness matrix and M is a mass matrix. The corresponding eigenvalue problem is

(K - λM)φ = 0,

where λ is an eigenvalue and φ is a vibration mode. If M = I, it is a standard eigenvalue problem. In practice, a portion of eigenvalues and vibration modes can be identified and these data are credible, whereas the stiffness or mass matrix is always unknown and usually estimated by finite element method. Finite element model is not very close to the real structure because of some simplified hypothesis and treatment of boundary conditions. There often exists a discrepancy between the eigenvalues of the approximate model and the identified one. To modify the approximate model to minimize the difference becomes a must.

A correction of structural stiffness or mass matrix using vibration tests was solved by nonlinear optimization techniques [2, 3]. But these methods do not guarantee the existence and the uniqueness of the solution and the solution is not doomed to be the best one. To this end, we present a method to correct an approximate model based on structured inverse eigenproblem with two constraints-the spectral constraint referring to the prescribed spectral data, and the structural constraint referring to the desirable structure. They can be formulated as to find A such that AX = XΛ, where A is some desirable structure matrix, X and Λ are given identified modes and eigenvalue matrices, and to find the best approximate matrix  to minimize the Frobenius norm of C - A for any estimate matrix C. But the determinations of eigenvalues and modes from vibration test data involve numerous sources of discrepancies or errors. Thus we consider its least-square problem.

Here desirable structure of matrices is generalized centrosymmetric. For the convenience of description, we translate the model updating problem into the following problems.

Problem I. Given a real n × m matrix X = (x₁, x₂,..., x_m) and a diagonal matrix Λ = diag(λ₁, λ₂,..., λ_m), find all generalized centrosymmetric matrices A such that

Problem II. Given a real n × n matrix C, find  ∈ such that

where is the solution set of Problem I and ||·|| is the Frobenius norm. X, Λ, C are consistent with above description.

Problem I is a structural inverse eigenproblem and Problem II is the best approximate problem with assigned least square spectra constraints. They arise in many areas of important applications [7, 8, 10-12]. Indeed, partial inverse eigenpair problems are used to modify some models [5, 13, 14]. Depending on the applications, inverse eigenproblems may be described in several different forms. Therefore inverse problems are different for different classes of matrices. Problem I and II were studied for some classes of structured matrices. We refer the reader to [1, 16, 19-22] and references therein. For example, Zhou et al. [22] and Zhang et al. [21] considered the problems for the case of centrosymmetric matrices and Hermitian-generalized Hamiltonian matrices, respectively. They established existence theorems for the solutions and derived expressions of the general solutions. Abdalla et al. [1] and Moreno et al. [16] discussed them in the case of the symmetric positive definite eigenproblem and quadratic inverse eigenvalue problem using some projection method, respectively. In this paper we investigate them for the set of all real generalized centrosymmetric matrices defined by the following definition. The solution to corresponding Problem II is the first modified solution. Generally a structure matrix is sparse. The sparse structure of the first modified solution may be destroyed. In this paper we will present a structure-preserving iteration algorithm and analysis a perturbation of the modified solution. We not only give an expression of the solutions but also provide a structure-preserving iterative algorithm of finite element model updating, based on the theory of inverse eigenpair problem.We also study a perturbation of the modified solution, which was not done in [20-22]. Next we introduce the definition of generalized centrosymmetric matrices.

Definition 1. Assume that E, F are real k × k matrices, u and v are k-dimensional real vector, P is some orthogonal k × k matrix and α is a real number. If

then A_2k and A_2k+1 are called generalized centrosymmetric matrices.

The centrosymmetric matrices have wide applications in many fields (see [4, 9-11]). If P = (e_k, e_k-1, ..., e₁) and e_i is the ith column of identity matrix I_k, the inverse eigenpairs problem of centrosymmetric matrices becomes a special case of this paper.

In this paper, we denote by

^n×m the set of all real n × m matrices. The set of all n × n orthogonal matrices is represented by ^n×n. A⁺ stands for the Moore-Penrose generalized inverse and P_A = AA⁺ is the orthogonal projection onto (A), the range of A. We define inner product (A, B) = trace(B^T A) in ^n×m. Thus ^n×m is a Hilbert space and the induced norm is the Frobenius norm ||A|| = ()^1/2. For perturbation analysis, we also use the 2-norm ||A||₂ for matrix A ∈^n×n.

This paper is organized as follows. In Section 2, we first give the expressions of the solutions to Problem I and II. And then we provide a structure-preserving iteration algorithm of model updating problem. In section 3, we study a perturbation bound of the modified solution and analyze the convergence of iteration solutions. Finally, some conclusions are presented in Section 4.

2 The basic theory and a numerical method

2.1 Expressions of the solutions to Problem I and II

Many stiffness or mass matrices from vibration model are not only structured but also large scale. We can reduce their scale to half because generalized centrosymmetric matrices are similar to a block diagonal matrices. Firstly, we consider some properties of generalized centrosymmetric matrices.

Throughout this paper, P is the same as in Definition 1. Let

When n = 2k, we take

when n = 2k + 1, we take

Then D_2k and D_2k+1 are orthogonal and A_n in Definition 1 has the following formula

where G₁∈ ^k×k, G₂∈ ^(n-k)×(n-k), D is the same as (6) or (7) when n = 2k or n = 2k + 1.

Next our goal is to give an expression of the set . First, we introduce the following lemma.

Lemma 1 ([15]). Suppose X, B ∈ ^n×m. Then a matrix A to satisfy ||AX - B|| = min is

Theorem 1.Given X ∈ ^n×m, Λ = diag(λ₁, λ₂,..., λ_m), assume D^T X = , where X₁∈ ^k×m, X₂∈ ^(n-k)×m, k = . Then there is a generalized centrosymmetric matrix A such that ||AX - XΛ|| = min and

and D is the same as (6) or (7).

Proof. By (8),

where G₁∈ ^k×k and G₂∈ ^(n-k)×(n-k). Because

minimization of ||AX - XΛ|| is equivalent to

From Lemma 1 we know that (13) and (14) are solvable and the solutions are

Substituting (15) and (16) into (11) we have (10).

Next we discuss Problem II.

Theorem 2. Given C ∈ ^n×n, the notation of X, Λ and the conditions are the same as those in Theorem 1. Then there is a unique  ∈ to Problem II and

where if n = 2k, D is the same as (6) and

if n = 2k + 1, D is the same as (7) and

Proof. It is easy to verify that is a closed convex set. Therefore there exists a unique solution to Problem II [6, p. 22]. According to (10) any A ∈ can be represented as

where

Let

Since

then ||A - C|| = min, where A is taken over all n × n generalized centrosymmetric matrices, is equivalent to

and

Equations (23) and (24) are equivalent to

Suppose a singular value decomposition of X₁ is

where U = (U₁, U₂) ∈ ^k×k, U₁∈ ^k×r1, r₁ = rank(X₁), V = (V₁, V₂) ∈ ^k×k, V₁∈ ^k×r1. Then it follows from orthogonal invariance of the Frobenius norm that

Therefore, (25) holds if and only if

Z₁ = ₁ = C₁₁ is the solution of (26). Substituting all such Z₁ into (20) the solution of (23) is ₁ = X₁Λ + ₁(I_k - X₁). Similarly, the solution of (24) is ₂ = X₂ Λ + ₂(I_n-k - X₂), where ₂ = C₂₂. By the definition of D_2n or D_2n+1 we have, for n = 2k

for n = 2k + 1

Thus the unique generalized centrosymmetric matrix solution of Problem II is (17).

2.2 A structure-preserving iteration algorithm and numerical examples

The structure constraint is usually imposed due to the realizability of the underlying physical system. In Theorem 2  is the modified solution. Though  satisfies the spectra constraints and is the best approximation of C, it does not preserve desirable structure such as banded, sparse etc. Next we will modify  such that the corrected model preserves sparse structure. If  = (â_ij) in (17) is not sparse, we modify Â. Let à = (ã_ij) where

à is a projection of  in some sense [1, 16]. But à is not the solution of Problem II. We modify it again by Theorem 2. To get a better numerical solution of Problem II we propose a structure-preserving iteration algorithm of model updating as follows.

Algorithm 1:

1) Input , C ∈ ^n×n, X ∈ ^n×m and Λ = diag(λ₁, λ₂,..., λ_m) ∈ ^n×n. Take k = .

2) If C and the exact A have the same zero elements structure we take an initial matrix Ã₀ = C. Otherwise, get Ã₀ from C according to (29).

3) Compute X₁ and X₂ by D^T X = .

4) Compute and by SVD.

5) C is replaced by Ã₀. Compute 1 and 2 by (18) and (19) respectively.

6) Compute Â₁ by (17).

7) Get the projection Ã₁ of Â₁.

8) If ||Ã₁ - Ã₀||/||C|| < , goto 9); otherwise, Ã₀ is replaced by Ã₁, go to 5).

9) Output Ã₁.

10) Stop.

We will prove that the matrix sequence {Ã_k} generated by the algorithm converge to the exact A in the next section. We first investigate its numerical results.

Guided by the algorithm many numerical experiments were carried out, and all of them were performed using Matlab 7.1. Next we report two numerical examples to illustrate our theory.

Example 1. For simulation in a vibrating system with 8 degrees of freedom, we assume an exact stiffness matrix to be

It is a tridiagonal generalized centrosymmetric matrix. In Definition 1

The min and max eigenvalue is, respectively, λ_min = -797.075893, λ_max = 2797.123691, and associated modes matrix is

To illustrate our theory we choose the identified min and max eigenvalue, associated modes matrix to be in accordance with their exact values respectively.

Step 1. Input = 10^-7, X, Λ = diag(-797.075893,2797.123691) and an initial estimate stiffness matrix C is

Step 2.₀ = C. Its min and max eigenvalue is respectively

There are big errors between the initial eigenvalues and the exact ones. Therefore modifying C is necessary.

Step 3.

Step 4.

Step 5.

Step 6.

Step 7.₁ is closer to K than C (||₁ - K|| = 30.7597, ||C - K|| = 75.1665). Though ₁ satisfies the spectra constraints and is the best approximation of C, it is not the structured matrix just as K and C. We get1 from ₁ according to (29). Thus

Step 8. ||Ã₁ - Ã₀||/||C|| = 0.01430621168271 > . ₀ is replaced by ₁. We repeat above steps. We can find desirable _m that is close to K after finite iterations. Here after the 47-th modified solution is

It is nearer to K than ₁ (||₄₇ - K|| = 16.1148). After 47 iterations the modified model ₄₇ and its min and max eigenvalues are close to their exact values. The min and max eigenvalues of updated model are in the following table.

where iter is iteration number.

Next we see an example in the case of large scale matrices.

Example 2. Denote by magic(k) the Magic matrix of order k. For example,

Assume B = magic(k) with k = 400, n = 800. The elements of matrix E is defined by

In Definition 1, P = (e_k , e_k-1, ..., -e₁) and e_i is the ith column of identity matrix I_k. If

then A is a triangular generalized centrosymmetric matrix. The condition number of A is cond(A) = 3928.726516440267. Assume λ_i, x_i are eigenpairs of A. The eigenvalues of the minimum and maximum modulus are λ_min = 0.124120226980 and λ_max = 318.959325051064 , respectively. To illustrate our theory we choose the elements of Λ and the columns of X in Problem I and II to be a part of eigenvalues and associated eigenvectors of A. Let the elements of matrix R be

An initial estimate matrix C = A + R is a triangular matrix. Its eigenvaluesof the minimal and the maximum modulus are respectively

There are big errors between the initial eigenvalues λ'_max, λ'_min and the exacteigenvalues λ_max, λ_min. Therefore modifying C is necessary. Suppose X₁ consists of the eigenvectors associated with λ_max and λ_min;

The diagonal elements of Λ₁, Λ₂, Λ₃ and Λ ₄ contain λ_min and λ_max. By the algorithm we obtain Ã_iter. It approaches the exact A. It is triangular generalized centrosymmetric in structure and satisfies the spectra constraints. The updated eigenvalues of the minimum and maximum modulus are provided by the following table.

The eigenvalues approach their exact values and their relative errors decrease as the size of X increases. In addition, because the pseudo inverse is computed by stable singular value decomposition and from (51) in next section it is easy to see that our algorithm is stable. Numerical examples show that the method is reliable and effective.

3 A perturbation and convergence

In this section, we will study a perturbation of the modified solution  in Theorem 2 and the convergence of the iteration method in Section 2.

Theorem 3. Let , , ₁, ₂, and be perturbed counterparts of X, Λ, X₁, X₂ and C in Theorem 2, respectively. And  and à are the solutions of corresponding Problem II. Then

where

Proof. Because

and

where for n = 2k

and

for n = 2k + 1

and

we have

It follows that

By [18],

By [17],

Substituting (40) and (41) into (39) we obtain (30)-(32).

Even though ||

₁ - X₁||₂ and ||₂ - X₂||₂ are small, || - ||₂ and || - ||₂ may be very large. It is because ||||₂ and ||||₂ may infinitely increase as ||₁ - X₁||₂ and ||₂ - X₂||₂ approach zero respectively. Therefore we provide the following conditions that and continuously change.

Theorem 4. The notations are the same as those in Theorem 3 and assume

Then

where

and

Proof. If the conditions (42) are satisfied, we have [17, 18]

and

where

Substituting (47)-(49) into (39) we obtain (44)-(45).

In Theorem 3 if = X, = Λ, then

where α < 1.

In fact,

And

₁ = C₁₁, ₁ = ₁₁, ₂ = C₂₂, ₂ = ₂₂, where

Let

Then α < 1 for X ≠ 0. Thus

Theorem 5. The matrices sequences {Ã_m} and {Â_m} generated by Algorithm 1 is convergent to its exact matrix A.

Proof. C and are replaced by Ã₀ and Ã₁ in (50) respectively. It follows from (50) that

By (29) and Algorithm 1, the nonzero elements of Ã₂ - Ã₁ are the same as those of Â₂ - Â₁ and the number of nonzero elements of Ã₂ - Ã₁ is fewer than that ones of Â₂ - Â₁. It means that

If C is taken as the exact A then  = A. If C and are placed by A and Ã_m-1 in (50) respectively. We have

Analogously, the nonzero elements of Ã_m - A are the same as those of Â_m - A and the number of nonzero elements of Ã_m - A is fewer than that ones of Â_m - A. Thus

Thus both {Ã_m} and {Â_m} converge to A because of α < 1.

4 Conclusion

In this paper, we have investigated some theories and a numerical method on modifying a generalized centrosymmetric model. These include the structure-preserving algorithm for solving the model updating problem based on matrix approximation with least squares spectra constrains and perturbation analysis of the modified solution. We can draw the following items.

1. Perturbation theory of the modified solution is given.

2. The algorithm is suitable for both sparse and dense matrix C. In particular, if all elements of C are not zero, the iteration number is "iter=1".

3. Convergence speed depends on α. But α is determined by (51). If the ranks of X₁ and X₂ are nearer to k and (n - k) respectively, generally X is a matrix of full column rank and α is close to 0. The iteration number and relative errors of the modified solution decrease as the size of X increases. Thus the modified solutions sequence more quickly approaches the true model if more eigenvalues and modes are provided.

4. In addition, because the pseudo inverse is computed by stable singular value decomposition the algorithm is stable.

5. In this paper we not only give theory but also provide a structure-preserving iterative algorithm on updating a generalized centrosymmetric model, based on the theory of inverse eigenpair problem. The conclusions are correct and the method is very reliable and effective.

Acknowledgement. The author is very grateful to the anonymous referees for their helpful and valuable comments.

Received: 04/VIII/09.

Accepted: 04/II/10.

#CAM-121/09. thanks

• [1] M. Abdalla, K. Grigoririadis and C. Zimmerman, Enhanced structure damage detection using alternating projection methods AIAA Journal, 36 (1998), 1305-1311.
• [2] M. Baruch, Optimization procedure to correct stiffness and flexibility matrices using vibration tests AIAA J., 16(11) (1978), 1208-1210.
• [3] A. Berman, Mass matrix correction using an imcomplete set of measured modes AIAA J., 17 (1979), 1147-1148.
• [4] A. Bunse-Gerstner, Eigenvectors of certain matrices Linear Algebra Appl., 7 (1973), 151-162.
• [5] X. Chen, The Theoretical Method and its Application of Designing of Structured Dynamics in Machine The Mechanical Industry Press(Chinese), Beijing, China (1997).
• [6] E.W. Cheney, Introduction to Approximation Theory New York: McGraw-Hill (1966).
• [7] M.T. Chu and G.H. Golub, Structured inverse eigenvalue problems Acta Numerica, 11 (2002), 1-71.
• [8] M.T. Chu and G.H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms and Applications Oxford University Press (2005).
• [9] L. Datta and S. Morgera, On the reducibility of centrosymmetric matrices-applications in engineering problems Circuits Syst. Signal Process, 3(1) (1989), 71-96.
• [10] F. Diele, T. Laudadio and N. Mastronardi, On some inverse eigenvalue problems with Toeplitz-related structure SIAM J. Matrix Anal. Appl., 26(1) (2004), 285-294.
• [11] F. Diele and I. Sgura, Centrosymmetric isospectral flows and some inverse eigenvalue problems Linear Algebra Appl., 366 (2003), 199-214.
• [12] K.T. Joseph, Inverse eigenvalue problem in structured design AIAA J., 30 (1992), 2890-2896.
• [13] S. Li, T.H. Feng and S.Q. Fan, A method for correction of finite element model by using test data Chinese Journal of Applied Mechanics (in Chinese), 12 (1995), 52-56.
• [14] B.Y. Liao, X.M. Zhou and Z.H. Yi, Modern Machine Dynamics and its Application in Engineering Mechanical Industry Press, Beijing (in Chinese) (2004).
• [15] S.K. Mitra, A pair of simultaneous linear matrix equations A₁X B₁ = C₁, A₂X B₂ = C₂ and a matrix programming problem Linear Algebra and Appl., 131 (1990), 107-123.
• [16] J. Moreno and B. Datta and M. Raydan, A symmetry preserving alternating projection method for matrix model updating Mechanical Systems and Signal Processing, 23 (2009), 1784-1791.
• [17] J.G. Sun, The stability of orthogonal projections J. Graduate School (in Chinese), 1(2) (1984), 123-133.
• [18] P.-Å. Wedin, Perturbation theory for pseudoinverses BIT, 13 (1973), 217-232.
• [19] D.X. Xie, N.J. Huang and Q. Zhang, An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-Hamilton matrices Numer. Algor., 46 (2007), 23-34.
• [20] Y.X. Yuan and H. Dai, Inverse problems for generalized centrosymmetric matrices on the linear manifold (in Chinese) Math. Numer. Sin., 27 (2005), 383-394.
• [21] Z.Z. Zhang, X.Y. Hu and L. Zhang, The solvability conditions for the inverse eigenvalue problem of Hermitian-generalized Hamiltonian matrices Inverse Probl., 18 (2002), 1369-1376.
• [22] F.Z. Zhou, X.Y. Hu and L. Zhang, The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices Linear Algebra Appl., 364 (2003), 147-160.

*

Supported by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR20100611).

Publication Dates

• Publication in this collection
  22 July 2010
• Date of issue
  June 2010

History

• Received
  04 Aug 2009
• Accepted
  04 Feb 2010