A new approach for the partial eigenvalue and eigenstructure assignment of undamped vibrating systems is developed. This approach deals with the constant output feedback control with the collocated actuator and sensor configuration, and the output matrix is also considered as a design parameter. It only needs those few eigenpairs to be assigned as well as mass and stiffness matrices of the open-loop vibration system and is easy to implement. In addition, this approach preserves symmetry of the systems. Numerical example demonstrates the effectiveness and accuracy of the proposed approach.

1. Introduction

Eigenstructure assignment techniques based on active feedback control have attracted much attention and have been widely used for vibration suppression during the past three decades. The extensive research results can be found in review articles [1, 2] and references therein. As the dynamics of a structural system is naturally described by a set of second-order differential equations and control theory and estimation techniques are established for first-order realization of the systems, the majority of previous researches are made via transferring second-order equations to first-order configuration (see, e.g., [3]). In the past ten years, in order not to increase the dimension of the equations and to preserve the symmetry and sparsity of the structural matrices, many authors proposed their works based directly on the second-order equation models [4–12]. On the other hand, it is needed to change only few undesirable eigenvalues and/or corresponding eigenvectors which are purposefully assigned to desired values, and it is desirable to keep all other eigenpairs unchanged. So some methods have been proposed to implement the partial eigenvalue or eigenstructure assignment [13–18]. For these methods, the process of applying a control, whether state feedback or output feedback, usually produces a closed-loop matrix which is no longer symmetric. However, it is sometimes necessary for the closed-loop system to satisfy the reciprocity principle of the structure. For systems with this requirement one restriction is that the available controls may need to be symmetric, as indicated in [19].

In this respect, Elhay [19] used the symmetric rank-one matrix modification and derived an explicit solution for a symmetry preserving partial eigenvalue assignment method for the generalised eigenvalue problem. But the method is difficult to implement in feedback control. Ram [20] solved the eigenvalue assignment problem for the vibrating rod. More recently, Liu and Li [21] suggested a method for the symmetry preserving partial eigenvalue assignment of undamped structural systems. They adapted the method proposed in [22] to the requirement of the symmetry preserving. Their results involve complex mathematical calculation. In this paper we propose a new approach for the symmetry preserving eigenstructure assignment of undamped vibrating systems, which is also applicable to the partial eigenvalue assignment. This approach is concerned with the constant output feedback control with the collocated actuator and sensor configuration and uses a partial eigenstructure modification formulation for the incremental mass and stiffness matrices to be satisfied. This formulation was recently obtained by Zhang et al. [18]. Our method proposed in this paper is easy to implement and the calculation procedure is relatively simple and clear.

The rest of the paper is organized as follows. Section 2 presents some preliminaries and the problem description. Section 3 gives our approach. Section 4 provides a numerical example to show the accuracy and effectiveness of the proposed approach. Finally, conclusions are drawn in Section 5.

2. Preliminaries and Problem Statement2.1. Preliminaries

Consider an n-degree-of-freedom undamped vibration system that is modelled by the following set of second-order ordinary differential equations:(1)M0q¨t+K0qt=ft,where qt∈Rn is displacement vector,ft∈Rn is the vector of external forces, and M0,K0∈Rn×n are constant mass and stiffness matrices, respectively. In general, M0 is symmetric and positive definite, and K0 is symmetric and positive semidefinite; that is, M0=M0T>0, K0=K0T≥0.

It is well known that if qt=xejωt is a fundamental solution of (1), then the natural frequency ω and the mode shape vector x must satisfy the following generalised eigenvalue equation:(2)K0-λiM0xi=0,i=1,2,…,n,where λi=ωi2 is the square of the ith natural frequency ωi, called the ith eigenvalue, and xi is the corresponding ith eigenvector. Equation (2) can be written in a compact representation as follows:(3)K0X=M0XΛ,where Λ=diagλ1,λ2,…,λn and X=x1,x2,…,xn make up the complete eigenstructure of system (1) and X satisfies the mass-normalised condition XTM0X=In.

Suppose that the system described by (1) is modified by the incremental matrices ΔM∈Rn×n and ΔK∈Rn×n. Then the motion of the modified system is governed by(4)M0+ΔMq¨t+K0+ΔKqt=ftand it satisfies the following eigenmatrix equation:(5)K0+ΔKY=M0+ΔMYΣ,where Σ=diagμ1,μ2,…,μn and Y=y1,y2,…,yn are the complete eigenstructure of modified system (4).

In [18] a necessary and sufficient condition was proposed for the incremental mass and stiffness matrices that modify some eigenvalues or eigenpairs while keeping other eigenpairs unchanged, which is shown in the following:(6)ΔKM0-1-X1X1T-ΔMM0-1K0M0-1-X1Λ1X1T=0,where Λ1 and X1 are submatrices of Λ and X and are composed of eigenvalues and eigenvectors to be modified in system (1), respectively. It implies that if ΔM and ΔK satisfy (6), the following equation then holds:(7)K0+ΔKX2=M0+ΔMX2Λ2,where Λ2 and X2 are submatrices of Λ and X and are composed of unchanged eigenvalues and eigenvectors of system (1). Here Λ=diagΛ1,Λ2 and X=X1,X2. Equation (7) means that Λ2 and X2 are also eigenpairs of the modified system. When only the stiffness modification ΔK is concerned, (6) reduces to(8)ΔKM0-1-X1X1T=0which is crucial to address the partial eigenvalue or eigenstructure assignment problem by static output feedback control in this paper.

2.2. Problem Statement

When considering the feedback control, we set ft=-Bu(t) in (1), where B∈Rn×m is full rank constant control matrix and u(t)∈Rm×1 the control vector. If the real constant static output feedback(9)ut=Gvtis applied to system (1), where G∈Rm×r is an output feedback gain matrix to be determined and vt=Cqt∈Rr is the output or measurement vector, where C∈Rr×n is a real constant output matrix, then the closed-loop system becomes(10)M0q¨t+K0+BGCqt=0which is supposed to have desired eigenvalues or eigenpairs.

In this paper, we consider B and C as design variables as well. Moreover, a special situation, B=CT, is concerned. Namely, it means the use of collocated actuator and sensor pairs, and the number of output measurements r is equal to the number of inputs m. The problem of partial eigenvalue or eigenstructure assignment is then formulated as follows: assuming that system (1) with ft=-Bu(t) and v(t)=BTqt is controllable and observable, given matrices M0 and K0, the subset λii=1p of the open-loop eigenvalues λii=1n, and the corresponding eigenvector set xii=1pp<n, and given a set μii=1p and the suitable vector set yii=1p, find the control matrix B and the feedback gain matrix G such that the closed-loop system (11)M0q¨t+K0+BGBTqt=0has eigenvalues μii=1p and λii=p+1n (i.e., the partial eigenvalue assignment) or has eigenvalues μii=1p, λii=p+1n and the corresponding eigenvectors yii=1p, xii=p+1n (i.e., the partial eigenstructure assignment). Additionally, the coefficient matrices of the closed-loop system are of the symmetry preserving; that is, K0+BGBT is a symmetric matrix. Note that λii=p+1n and xii=p+1n denote unassigned and unchanged eigenvalues and eigenvectors of open-loop system (1).

For convenience, the following partitions and notation are used:

Λ1=diagλ1,λ2,…,λp, whose diagonal elements are the open-loop eigenvalues to be altered.

X1=x1,x2,…,xp, whose columns are the open-loop eigenvectors to be altered.

Λ2=diagλp+1,λp+2,…,λn, whose diagonal elements are the open-loop eigenvalues that did not take part in the assignment.

X2=xp+1,xp+2,…,xn, whose columns are the open-loop eigenvectors that did not take part in the assignment.

Σ1=diagμ1,μ2,…,μp, whose diagonal elements are the assigned closed-loop eigenvalues, corresponding to Λ1.

Y1=y1,y2,…,yp, whose columns are the assigned closed-loop eigenvectors, corresponding to X1.

3. Problem Solution

Let ΔK=BGBT; (8) can be rewritten as(12a)BGBTM0-1-X1X1T=0.Because B is assumed to be of full column rank, (12a) implies that(12b)GBTM0-1-X1X1T=0.Transposing both sides of (12a), it has(12c)M0-1-X1X1TBGT=0.It is apparent that if B satisfies the following matrix equation, then (12c) (i.e., (12a)) holds:(13)M0-1-X1X1TB=0which means that B belongs to the right null space of the matrix M0-1-X1X1T(denoted by NM0-1-X1X1T). Importantly, (13) can be used to determine B; that is, B can be chosen to be composed of the basis vectors of NM0-1-X1X1T. As rankM0-1-X1X1T=n-p, then rankB=p. It implies that the column order of B is constrained to be m≤p under the assumption of B being full column rank. It should be noted that the proposed approach to the determination of B from basis vectors of NM0-1-X1X1T is also applicable to the situation of B not to be full column rank and m>p.

Now the partial eigenvalue or eigenstructure assignment problem involves solving two matrix equations as follows:(14)M0-1-X1X1TB=0,K0+BGBTY1=M0Y1Σ1.To obtain G, we now turn to discuss the second matrix equation of (14). After rearranging it, we have(15)BGBTY1=M0Y1Σ1-K0Y1.Equation (15) is of the form AZE=F, where A and E are given matrices of appropriate dimensions and the matrix Z needs to be determined. The necessary and sufficient condition for the existence of solutions on this type of matrix equation is AA+FE+E=F [23], where the superscript + denotes the Moore-Penrose inverse of a matrix. Supposing here that (15) has solutions, in what follows, we present a symmetric solution for G. Premultiplying (15) by Y1T, (15) can be rewritten as(16)Y1TBGBTY1=Y1TM0Y1Σ1-Y1TK0Y1.However, the symmetry of the solution G from (16) is not guaranteed yet. We will now show the symmetry of G by imposing the mass normalisation condition; that is, Y1 will be such that(17)Y1TM0Y1=D,where D is a diagonal matrix. First note that Σ1 is a diagonal matrix. Then, using (17) and noting that Y1TK0Y1 is a symmetric matrix, we see that the right-hand side of (16) is symmetric. Thus, the left-hand side matrix Y1TBGBTY1 of (16) is also symmetric, implying that G is symmetric.

The minimal norm solution of (16) is given as follows [23]:(18)G=BTY1+TY1TM0Y1Σ1-Y1TK0Y1BTY1+,where BTY1+ is the Moore-Penrose inverse of the matrix BTY1 which is a p×p matrix with a lower order.

As is known, not any given vectors Y1 can be assigned to closed-loop system (11). In order that the second matrix equation of (14) (or (15)) holds for a symmetric matrix G and the partial eigenstructure assignment involved in this paper can be achieved, an additional requirement for Y1 must be satisfied. Because Y1TM0X2=0 based on the orthogonality properties of the generalised eigenvalue problem for the real symmetric matrices, Y1 must be of the form Y1=X1L for some p×p nonsingular matrix L. Furthermore, L can be written as L=VT, where V is nonsingular diagonal matrix and T is orthogonal [24]. It is worthwhile noting that this condition of Y1 is seemingly rather strict; but, in practice, this condition is almost always satisfied [25].

Based on the discussion above, we now state the proposed approach of the partial eigenvalue or eigenstructure assignment in this paper as follows.

Step 1.

Compute the orthonormal basis vectors of NM0-1-X1X1T; use the obtained basis vectors to form the control matrix B.

Step 2.

Check the given eigenvector matrix Y1 to be sure that (17) holds; otherwise, update the matrix Y1 by computing the singular value decomposition of Y1TM0Y1=PSQT, and Y1 should be substituted by Y1PT. When only the partial eigenvalue assignment is required, it is appropriate to set Y1 to be X1.

Step 3.

Check the compatibility condition of (15). If it is satisfied, Step 4 gives an exact solution of the feedback gain matrix G; otherwise, it gives just a least-squares solution.

Step 4.

Compute the feedback gain matrix G from (18).

4. A Numerical Example

In this section, a numerical example in [21] is used to illustrate our approach. For an undamped vibrating system, it has the stiffness and mass matrices as follows:(19)K0=1-10000-12-10000-12-10000-12-10000-12-10000-11,M0=1316000016231600001623160000162316000016231600001623.Its eigenvalues are {0.0, 0.3564, 1.5403, 3.8816, 7.6127, 11.3551}. Two eigenvalues 0.3564,1.5403 among them are to be reassigned to 0.75,1.85, which is also the assignment in [21]. We have Λ1=diag0.3564,1.5403 and Σ1=diag0.75,1.85. The corresponding eigenvectors of Λ1 are(20)X1=-0.62930.6821-0.52350.2641-0.2415-0.47760.1217-0.63400.4439-0.01330.61690.6236.

Case 1.

Only the partial eigenvalue assignment is required. Here m=r=p=2. Let Y1=X1; using our approach, we obtain the control matrix(21)B=-1.42640.6197-1.71670.46420.5051-0.90611.6380-1.17791.0000001.0000.Now we can inspect the controllability and observability conditions with respect to B and Λ1 and present the following results:(22)rankB,K0-λiM0=6,rank=BTK0-λiM0=6,i=2,3.Furthermore, it is believed that (15) has solutions after we check its solvability condition. Thus we have the output feedback gain matrix and the symmetric matrix BGBT as follows:(23)G=0.06920.07870.07870.1457,BGBT=0.05750.0754-0.0053-0.0558-0.0499-0.02200.07540.10980.0196-0.0552-0.0822-0.0675-0.00530.01960.06520.0491-0.0364-0.0922-0.0558-0.05520.04910.08400.0206-0.0427-0.0499-0.0822-0.03640.02060.06920.0787-0.0220-0.0675-0.0922-0.04270.07870.1457.The closed-loop eigenvalues are {0.0000, 0.7500, 1.8500, 3.8816, 7.6127, 11.3551}, and M0X2Λ2-K0+BGBTX2F=4.9665e-015. It can be seen that our approach can accurately solve the partial eigenvalue assignment problem. Interestingly, the symmetric matrix BGBT obtained here is the same as that in [21].

Case 2.

The partial eigenstructure assignment: Λ1, X1, and Σ1 are the same as the above; let the assigned closed-loop eigenvectors Y1 be(24)Y1=-0.04320.6133-0.30530.3582-0.6734-0.1351-0.4376-0.37010.4461-0.19821.19020.0463which satisfies mass normalisation condition (17). Thus we obtain the following:(25)G=0.15620.08450.0845-0.0425,BGBT=0.15210.22440.0470-0.1061-0.1704-0.14690.22440.31640.0337-0.1808-0.2289-0.16480.04700.0337-0.0724-0.09180.00230.0812-0.1061-0.1808-0.09180.03400.15630.1884-0.1704-0.22890.00230.15630.15620.0845-0.1469-0.16480.08120.18840.0845-0.0425and B is the same as that in Case 1. Σ1 and Y1 are accurately assigned in the closed-loop system (for the sake of saving space, they are not listed here). Additionally, we can compute M0Y1Σ1-K0+BGBTY1F=1.2071e-015 and M0X2Λ2-K0+BGBTX2F=5.1580e-015. This also means that our approach can accurately solve the partial eigenstructure assignment problem, which is a distinctive advantage with respect to the approach proposed in [21].

5. Conclusions

Based on a partial eigenstructure modification formulation for the incremental mass and stiffness matrices to be satisfied, an approach is successfully developed to assign the partial eigenvalue and eigenstructure of undamped vibrating systems. It can preserve symmetry of the closed-loop system’s matrices and its calculation steps are simple and clear.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

SobelK. M.ShapiroE. Y.AndryA. N.Eigenstructure assignmentWhiteB. A.Eigenstructure assignment: a surveyRastgaarM.AhmadianM.SouthwardS.A review on eigenstructure assignment methods and orthogonal eigenstructure control of structural vibrationsJuangJ.-N.MaghamiP. G.Robust eigensystem assignment for state estimators using second-order modelsSchulzM. J.InmanD. J.Eigenstructure assignment and controller optimization for mechanical systemsChuE. K.DattaB. N.Numerically robust pole assignment for second order systemsChanH. C.LamJ.HoD. W. C.Robust eigenvalue assignment in second-order systems: a gradient flow approachNicholsN. K.KautskyJ.Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular caseChuE. K.Pole assignment for second-order systemsDuanG. R.Parametric eigenstructure assignment in second-order descriptor linear systemsOuyangH.Pole assignment of friction-induced vibration for stabilisation through state-feedback controlAbdelazizT. H. S.Eigenstructure assignment for second-order systems using velocity-plus-acceleration feedbackDattaB. N.ElhayS.RamY. M.SarkissianD. R.Partial eigenstructure assignment for the quadratic pencilRamY. M.ElhayS.Pole assignment in vibratory systems by multi-input controlZhangJ. F.Partial pole assignment for general vibration systems by multi-input state feedbackQianJ.XuS. F.Robust partial eigenvalue assignment problem for the second-order systemRamadanM. A.El-SayedE. A.Partial eigenvalue assignment problem of high order control systems using orthogonality relationsZhangJ. F.OuyangH.YangJ.Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedbackElhayS.Symmetry preserving partial pole assignment for the standard and the generalized eigenvalue problemsRamY. M.Pole assignment for the vibrating rodLiuH.LiC.Symmetry preserving partial eigenvalue assignment for undamped structural systemsLinW.-W.WangJ.-N.Partial pole assignment for the quadratic pencil by output feedback control with feedback designsBen-IsraelA.GrevilleT. N.ChuM. T.DattaB. N.LinW.-W.XuS. F.Spillover phenomenon in quadratic model updatingCarvalhoJ.DattaB. N.GuptaA.LagadapatiM.A direct method for model updating with incomplete measured data and without spurious modes