# Eigenvalue embedding of undamped piezoelectric structure system with no-spillover

In this paper, the eigenvalue embedding problem of the undamped piezoelectric structure system with no-spillover (EEP-PS) is considered. It aims to update the original system to a new undamped piezoelectric structure system, such that some eigenpairs are replaced by newly given or measured ones, while the remaining eigenpairs are kept unchanged. A set of parametric solutions to the EEP-PS are provided and the performance of the proposed algorithms are illustrated by numerical examples.

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## 1 Introduction

The technology of smart materials and structures especially piezoelectric smart structures has become mature over the last decade. One promising application of piezoelectric structures is the control and suppression of unwanted structural vibrations Sunar-Ps-1999 . Piezoelectric materials such as lead zirconate titanate are extensively used in vibration damping applications AAD-2001 . The three-dimensional piezoelectric constitutive law can be written as MV-book-smart ; IT-book-1990 :

 {{σ}=[c]{ϵ}−[e]T{E},{D}=[e]{ϵ}+[ε]{E}, (1.1)

where mechanical variable and denote the stress and the strain, respectively; electrical variable and denote the electric displacement and the electric field, respectively. Matrices and represent material properties: is the elasticity matrix, is the piezoelectric matrix, is the dielectric matrix. The indirect piezoelectric effect is given by the first equation of (1.1), while the second equation characterizes the direct piezoelectric effect. Variational principles can be used to establish the finite element equations for piezoelectric structures. Modeling of piezoelectric smart structures by the finite element method and its implementation for active vibration control problems has been presented Meng-Ps-2006 ; Naraya-2003 . The global equation of motion governing a undamped structure system with degrees of freedom can be written as Tzou-Ps-1990

 M[¨u¨ϕ]+K[uϕ]=[FuFϕ], (1.2)

where

 M=[Mu000],  K=[KuKuϕKTuϕKϕ]∈Rn×n, (1.3)

with and being symmetric, ; denotes structural displacement, denotes electric potential; and are the structural mass and stiffness matrices, respectively, is the piezoelectric coupling matrix and is the dielectric stiffness matrix; denotes the structural load and denotes the electric load. The modality of undamped piezoelectric structure (1.2

) is characterized by eigenvalues and eigenvectors of its associated generated eigenvalue problem

 P(λ)x:=(λM+K)x=0. (1.4)

We will refer to (1.4) with of the form (1.3) as an undamped piezoelectric structure system.

Eigenvalue embedding, also known as model updating in some literature, has recently been an active topic, for example, Qiang-2017 ; Chu-2007 ; Chu-2008 ; Chu-2009 ; Kang-2018-apm ; Lancaster-2008 ; Friswell-1998 Carvalho-2006 and the references therein. The main purpose of eigenvalue embedding is to update the original system to a new ne, such that some ”troublesome” or ”unwanted” eigenvalues and the corresponding eigenvectors are replaces or some desired eigenvalues and eigenvectors are achieved. Among current developments for model updating, the problem of maintaining the remaining or unknown eigenvalues and eigenvectors unchanged is of practical importance, which is known as no-spillover phenomena in literature, see Carvalho-2006 ; Chu-2007 .

In this paper, we consider the eigenvalue embedding problem of the undamped piezoelectric smart system with no-spillover, which is to update the original system to a new undamped piezoelectric smart system, such that some nonzero eigenvalues are replaced with some desired ones, while the remaining eigenvalues and eigenvectors are kept unchanged. The eigenvectors corresponding to the updated eigenvalues of the new system can be different from those corresponding to the eigenvalues to be updated of the original system. But there may be some restrictions on them. For example, in the model updating problems for the second order system, it has been shown in Chu-2009 ; Chu-2007 that, under some mild conditions, the newly measured eigenvectors must lie in the same subspace as those spanned by eigenvectors corresponding to the eigenvalues to be updated of the original system. Hence, in this paper, we restrict the eigenvalue embedding problem in that the newly given or measured eigenvectors span the same subspaces as the original eigenvectors. The eigenvalue embedding problem for the undamped piezoelectric smart system with no-spillover (EEP-PS) can be stated as follows.

EEP-PS: Given an undamped piezoelectric system with of the form (1.3), some of its eigenpairs with being closed under complex conjugate, and a set of nonzero numbers , closed under complex conjugate, update the original system to a new undamped piezoelectric smart system

 (λ˜M+˜K)x=0

with and of the form (1.3), such that the part of eigenvalues of the original system are replaced by , the eigenvectors corresponding to of the updated system span the same subspaces as , and the remaining eigenvalues and eigenvectors are kept unchanged.

We will provide a set of solutions depending on some parameter matrices to the EEP-PS. Obviously, if , the coefficient matrix in (1.3) is singular, which implies that the matrix polynomial contain both finite and infinity eigenvalues. Thus, the existing results on the Eigenvalue embedding/model updating of quadratic systems cannot be directly applied to the undamped piezoelectric system. In this paper, we update eigenvalues of the finite part.

This paper is organized as follows. In section 2, we give a set of solutions to the EEP-PS depending on several parametric matrices, and propose an algorithm to compute the solution. Numerical examples are given in section 3 to illustrate the performance of the proposed algorithm. Some conclusions are finally given in section 4.

## 2 Eigenvalue embedding with no-spillover

Throughout this paper, we assume that and are nonsingular. Then, it follows from the following Lemma 2.1 that is a regular matrix polynomial, even though the coefficient matrix of (1.4) is singular.

###### Lemma 2.1.

If is nonsingular, then is regular, i.e., is not identically zero for all .

###### Proof.

It follows from being nonsingular that

 det(λM+K)=det(λ[Mu000]+[KuKuϕKTuϕKϕ])=det(λMu+KuKuϕKTuϕKϕ)=det(λMu+Ku−KuϕK−1ϕKTuϕ0KTuϕKϕ)=det(Kϕ)det(λMu+Ku−KuϕK−1ϕKTuϕ).

Since is nonsingular, is not identically zero for all , which implies that is regular. ∎

Clearly, has both finite and infinity eigenvalues, since the lead coefficient matrix is singular. For simplicity, we assume that the finite eigenvalues of are all simple and nonzero. Taking a Jordan pair for every eigenvalue of , we define finite Jordan pair Lancaster-book of as

 XF=[X(λ1),X(λ2),…,X(λq)],  JF=% diag(λ1,λ2,…,λq).

From Lemma 2.1, we known that , which implies that Lancaster-book and the sizes of and are and , respectively.

###### Lemma 2.2.

Lancaster-book Let be a pair of matrices, where is an matrix and is an nonsingular diagonal matrix. Then be a Jordan pair of if and only if and

 M^X^J+K^X=0.

Let

 φ(i)0,…,φ(i)si−1,  i=1,…,r, (2.1)

be a canonical set of Jordan chains of the analytic (at infinity) matrix function corresponding to , if this point is an eigenvalue of . By definition, a canonical set of Jordan chains of an analytic matrix-valued function at infinity is just a canonical set of Jordan chains of the matrix function at zero Lancaster-book . Thus, the Jordan chain (2.1) form a canonical set of Jordan chains of the matrix polynomial corresponding to the eigenvalue zero. Since the matrix in (1.3) is nonsingular, the algebraic multiplicity of zero eigenvalue of is . For simplicity, we assume that the algebraic multiplicity of every Jordan block with zero eigenvalue of is . Then, we shall use the following notation

 X∞=[φ1,…,φnϕ],  J∞=0nϕ×nϕ.

We call an infinite Jordan pair of .

###### Lemma 2.3.

Lancaster-book Let be a pair of matrices, where is an matrix and is an zero matrix. Then be an infinite Jordan pair of if and only if and

 M^X=0.

By Lemma 2.2 and Lemma 2.3, we have

###### Lemma 2.4.

Let be a pair of matrices, where is an matrix and with being a nonsingular diagonal matrix. Then is a Jordan pair of if and only if and

 M^X+K^X[^J−11000]=0.

And in this case, the matrix must be of the following form

 ^X=[^X110^X21^X22], (2.2)

where , and .

###### Proof.

The first part of this Lemma can be directly obtained by Lemma 2.2 and Lemma 2.3. So we just prove the second part, i.e., has the form (2.2). Partition into

 ^X=[^X1  ^X2]=[^X11^X12^X21^X22],

where , . Since is Jordan pair of , we can see from the block structure of that is an infinite Jordan pair of . From Lemma 2.3, it follows that

 M^X2=0,

which implies that since is nonsingular. Therefore, has the form (2.2) and then is nonsingular since . ∎

For simplicity, we assume that are simple and nonzero eigenvalues of the updated system. Without loss of generality, we assume that the eigenpairs are ordered such that for

 λ2j−1=¯¯¯λ2j=αj+iβj,αj∈R,βj>0,x2j−1=¯¯¯x2j=xjR+ixjI,xjR,xjI∈Rn,

and for , . Define

 Λ1=diag([α1β1−β1α1],…,[αsβs−βsαs],λ2s+1,…,λp), (2.3)
 X1=[x1R,x1I,…,xsR,xsI,x2s+1,…,xp], (2.4)

which is referred to as the real representations of Kang-2018-apm ; Kang-amc-2014 ; Kang-2015-COAM . Similarly, assume that the real representation of is

 ~Λ1=diag([~α1~β1−~β1~α1],…,[~α~s~β~s−~β~s~α~s],~λ2~s+1,…,~λp), (2.5)

and those of remaining eigenpairs are and , where

 Λ3=diag([α′1β′1−β′1α′1],…,[α′tβ′t−β′tα′t],λ2t+1,…,λnu−p). (2.6)

We further assume that the sets of eigenvalues in and are disjoint, i.e.,

 λ(Λ1)∩λ(Λ2)=∅, (2.7)

where is the set of all eigenvalues of a matrix.

With notations above, the MUP-PS can be mathematically reformulated as: given and being symmetric, , which satisfy

 [Mu000][X1  X2]+K[X1  X2]⎡⎢ ⎢⎣Λ−11000Λ−130000⎤⎥ ⎥⎦=0, (2.8)

find and being symmetric such that

 [˜Mu000][˜X1  X2]+˜K[˜X1  X2]⎡⎢ ⎢⎣˜Λ−11000Λ−130000⎤⎥ ⎥⎦=0, (2.9)

holds for some , where is nonsingular. Note that and are remaining eigenvalues and eigenvectors of the original undamped piezoelectric system to be kept unchanged after the updating and they are generally unknown in applications. So it is necessary to characterize the solution to the EEP-PS without the information of and .

To give the parametric solutions to the EEP-PS, we will need a spectral decomposition of the undamped piezoelectric smart system of (1.4) which is motivated by Qiang-2017 ; Chu-2009-spectral . We give the sufficient and necessary conditions for the matrices and in (1.3) such that (1.4) is satisfied and express and in terms of eigenvalues and eigenvectors of (1.4). The following theorem then gives the spectral decomposition of the undamped piezoelectric system (1.4), characterizing the structure of the Jordan pair and the relationship between the coefficient matrices , in (1.3) and .

###### Theorem 2.5.

Given nonsingular matrices and . Let and partition as

 X=[XuXϕ],  Xu∈Rnu×n, Xϕ∈Rnϕ×n. (2.10)

Then there exist nonsingular symmetry matrices and such that

 MX+KXJ=0 (2.11)

where , if and only if there exist symmetric matrix and nonsingular symmetric matrix , where is nonsingular, such that

 T−1:=Γ+XTϕΦXϕ (2.12a) is nonsingular and JT1Γ11=Γ11J1, (2.12b) XuTXTϕ=0, (2.12c) XϕTXTϕ=Φ−1. (2.12d)

If such matrices and exist, then the coefficient matrices and can be expressed as

 Mu=(XuTXTu)−1, (2.13a) K=X−T[−Γ11J−1100K′22]X−1, (2.13b) where K′22∈Rnϕ×nϕ is an arbitrary nonsingular symmetric matrix.
###### Proof.

(Necessity) Pre-multiplying on (2.11) we get

 XTMX+XTKXJ=0, (2.14)

Substituting in (1.3) and in (2.10) into (2.14) gives

 XTuMuXu+XTKXJ=0, (2.15)

We define . Let be a nonsingular and symmetric matrix. Since and are nonsingular, it is easy to see that

 Γ+XTϕΦXϕ=[XTu  XTϕ][Mu00Φ][XuXϕ] (2.16)

is also nonsingular. Let . From (2.16) it follows that

 [Mu00Φ]=([XuXϕ]T[XTu  XTϕ])−1=([XuTXTuXuTXTϕXϕTXTuXϕTXTϕ])−1,

which implies that , and , which are exactly the formulas as in (2.12c), (2.12d) and (2.13a), respectively.

Substituting into (2.15) , we have

 Γ+XTKX[J1000]=0. (2.17)

Let

 Γ=[Γ11Γ12ΓT12Γ22],  XTKX=[K11K12KT12K22],

where . Clearly, (2.17) holds if and only if

 Γ12=0,  Γ22=0, (2.18)

and in this case, we have

 K11=−Γ11J−11,  K12=0. (2.19)

Since is symmetric, we must have

 JT1Γ11=Γ11J1. (2.20)

Therefore, we have

 K=X−T[−Γ11J−1100K′22]X−1, (2.21)

where is an arbitrary nonsingular and symmetric matrix. From (2.20), we can see that is nonsingular and .

(Sufficiency) Conversely, if there exist two symmetric matrices and satisfying (2.12), where is nonsingular, then

 XTXT=[XuXϕ]T[XTu  XTϕ]=[XuTXTu00Φ−1], (2.22)

is also nonsingular, which means that is nonsingular and symmetric. Thus is well defined by (2.13a). From (2.22), it follows that

 X−TT−1X−1=[(XuTXTu)−100Φ],

which implies that

 T−1=XTu(XuTXTu)−1Xu+XTϕΦXϕ. (2.23)

By the definitions of and , we can see from (2.23) that . Noting the symmetry of and (2.12b), it is easy to see that defined by (2.13b) is symmetric. By (2.12b) and (2.13b), we have

 XTMX+XTKXJ=XTuMuXu+XTKXJ=Γ+XTKXJ=[Γ11000]+XTKX[J1000]=0,

i.e., since is nonsingular. This completes the proof of the theorem. ∎

Now, with Theorem 2.5, we are ready to characterize the solutions to the EEP-PS. Let in (2.4) and in (2.3) be the eigendata which will be replaced by and in (2.5, and those of remaining eigendata be given by and . From Lemma 2.4 and Theorem 2.5, without loss of generality, we may always assume that and are nonsingular. Partition as

 X1=[X1uX1ϕ] (2.24)

where . Clearly, is a Jordan pair of .The following theorem is our main result which gives the parametric solutions to the EEP-PS.

###### Theorem 2.6.

For any nonsingular matrices and with

 ˜Γ1=diag([a1b1b1−a1],…,[a~sb~sb~s−a~s],c2~s+1,…,cp), (2.25)

the matrices

 ˜Mu=(M−1u−X1uΓ−11XT1u+X1uΘ˜Γ−11ΘTXT1u)−1, (2.26)
 ˜K=(K−1+X1Λ1Γ−11XT1−X1Θ˜Λ1˜Γ−11ΘTXT1)−1, (2.27)

where

 Γ1=XT1uMuX1u, (2.28)

form a solution to the EEP-PS, provided that and in (2.26), (2.27), respectively, are well defined.

###### Proof.

Since and is nonsingular, we can see from Lemma 2.4 that . Then, there exists a nonsingular matrix such that

 X1=Q[X1u0],   QT[Mu000]Q=[Mu000]. (2.29)

Partition as where , and then (2.8) can be equivalently rewritten into

 [Mu000][X1uX2u0X2ϕ]+QTKQ[X1uX2u0X2ϕ]⎡⎢ ⎢⎣Λ−11000Λ−130000⎤⎥ ⎥⎦=0. (2.30)

By (2.30) and Theorem 2.5, we first give the expressions of the coefficient matrices and by the eigenvalue matrix , and the eigenvectors matrix

 [X1uX2u0X2ϕ].

Let , . Similar to the proof of Theorem 2.5, we define

 Γ=XTuMuXu,   T−1=Γ+XTϕΦXϕ,

where is a nonsingular and symmetric matrix. Then

 Γ=[XT1uXT2u]Mu[X1uX2u], (2.31)

and

 T−1=Γ+[0p×p00XT2ϕΦX2ϕ]. (2.32)

Using (2.30), we can see from the proof of Theorem 2.5 that , where , and the matrices satisfy

 [Λ−1100Λ−13]TΓ11=Γ11[Λ−1100Λ−13] (2.33)
 XuTXTϕ=[X1u  X2u]T[0XT2ϕ]=0, (2.34)
 XϕTXTϕ=[0  X2ϕ]T[0XT2ϕ]=Φ−1. (2.35)

From the assumption (2.7) and (2.33), it follows that must be of the block diagonal form , where , and

 Γ1Λ−11=Λ−T1Γ1,  Γ2Λ−13=Λ−T3Γ2.

Substituting into (2.31), it follows from the comparison of two sides of the (2.31) that , i.e., is of form (2.28).

By (2.32), we can see that must be of the block diagonal form , where , and

 T−11=Γ1,  T−12=diag(Γ2,0nϕ×nϕ)+XT2ϕΦX2ϕ.

Hence (2.34) and (2.35) can be equivalently rewritten as

 X2uT2XT2ϕ=0, (2.36)
 X2ϕT2XT2ϕ=Φ−1. (2.37)

and the matrices and of (2.30) can be expressed as

 Mu=(X1uΓ−11XT1u+X2uT2XT2u)−1, (2.38)
 QTKQ=([X1uX2u0X2ϕ][−Λ1Γ−1100ˆK22][XT1u0XT2uX2ϕ])−1=(Q−1[X1X2][−Λ1Γ−1100ˆK22][XT1XT2]Q−T)−1, (2.39)

where , is an arbitrary nonsingular symmetry matrix, which implies that the coefficient matrices of original undamped system can be expressed as

 M=[(X1uΓ−11XT1u+X2uT2XT2u)−1000], (2.40)
 K=([X1X2][−Λ1Γ−1100ˆK22][XT1XT2])−1. (2.41)

Let and is nonsingular. Since is nonsingular and satisfies (2.29), it is easy to verify that (2.9) is equivalent to

 [˜Mu000][Q−1˜X1  Q−1