Elsevier

Journal of Sound and Vibration

Volume 332, Issue 4, 18 February 2013, Pages 780-794
Journal of Sound and Vibration

Minimum norm partial quadratic eigenvalue assignment with time delay in vibrating structures using the receptance and the system matrices

https://doi.org/10.1016/j.jsv.2012.10.015Get rights and content

Abstract

The partial quadratic eigenvalue assignment problem (PQEAP) is to compute a pair of feedback matrices such that a small number of unwanted eigenvalues in a structure are reassigned to suitable locations while keeping the remaining large number of eigenvalues and the associated eigenvectors unchanged. The problem arises in active vibration control of structures. For real-life applications, it is not enough just to compute the feedback matrices. They should be computed in such a way that both closed-loop eigenvalue sensitivity and feedback norms are as small as possible. Also, for practical effectiveness, the time-delay between the measurement of the state and implementation of the feedback controller should be considered while solving the PQEAP. These problems are usually solved using only system matrices and do not necessarily take advantage of the receptances which are available by measurements.

In this paper, we propose hybrid methods, combining the system matrices and measured receptances, for solutions of the multi-input PQEAP and the minimum-norm PQEAP, both for systems with and without time-delay. These hybrid methods are more efficient than the standard methods which only use the system matrices and not the receptances. These hybrid methods also offer several other computational advantages over the standard methods. Our results generalize the recent work by Ram et al. [Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra and its Applications 434 (2011) 1689–1696]. The results of numerical experiments demonstrate the effectiveness of the proposed methods.

Introduction

Vibrating structures, such as bridges, highways, buildings, automobiles, airplanes, etc., are usually modeled by a system of second-order differential equations of the form:Mx¨(t)+Dx˙(t)+Kx(t)=f(t),where the matrices M,D, and K are, respectively, known as the mass, damping and stiffness matrices. They are very often structured with special properties. They are symmetric and furthermore, M is usually positive definite and diagonal or tridiagonal and K is positive semi-definite and tridiagonal.

The dynamics of the system (1) are governed by the natural frequencies and mode shapes. The natural frequencies are related to the eigenvalues and mode shapes are the eigenvectors of the associated quadratic matrix pencil: P(λ)=λ2M+λD+K.If each of the matrices M,D, and K is of order n, then P(λ) has 2n finite eigenvalues and 2n associated eigenvectors under the assumption that M is nonsingular [1], [2].

One of the fundamental problems in vibration is to control the undesired vibrations, such as those caused by resonances, when vibrating structures are acted upon by some external forces, such as the wind, an earthquake, or human weight.

Resonance is caused when some natural frequencies become close or equal to the external frequencies. Therefore, mathematically, the vibration control problem is to reassign those few unwanted resonant eigenvalues to suitably chosen locations, selected by the engineers, while keeping the large number of remaining eigenvalues and their corresponding eigenvectors unchanged. The latter is known as the no spill-over phenomenon in vibration engineering.

In mathematics and control literature, the above problem is known as the partial quadratic eigenvalue assignment problem (PQEAP). A fundamental computational challenge is to solve the PQEAP using only a small number of eigenvalues and eigenvectors of the pencil P(λ), which are available by computation with the state-of-the-art computational techniques, such as the Jacobi–Davidson method [3] or by measurements from a vibration laboratory using limited hardware facilities.

The PQEAP as stated above is basic. For practical effectiveness, the problem must be solved by addressing several important practical issues. These include:

  • Robustness and minimum-norm feedback: Since the eigenvalues of a matrix may be very sensitive even to small perturbations, the feedback matrices must be computed in such a way that the closed-loop eigenvalues remain as insensitive as possible to small perturbations of the data. Also, for applications, the feedback design should be such that the norms of the feedback matrices are as small as possible. These considerations lead to robust and minimum-norm quadratic partial eigenvalue assignment problems. Solutions of robust and minimum-norm problems give rise to nonlinear optimization problems. There still do not exist viable methods for numerically solving nonlinear optimization problems. Even for local minimization, the computational challenge is to be able to compute the gradient expressions using only a few available eigenvalues and eigenvectors.

  • Time-delay in the system: Time-delay is an inevitable practical phenomenon. There always exists a time-delay in the application of the required control force to the structure.

    Design of feedback controllers is a much more difficult and challenging problem for a time-delay system; because it involves only 2n parameters whereas the closed-loop system in this case has an infinite number of eigenvalues. Fortunately, however, it has been shown earlier (e.g., [4]) that p<2n eigenvalues can be reassigned in the time-delay case.

  • Use of receptances: The receptance matrix corresponding to system (1) is defined by H(s)=(s2M+sD+K)1.The entries of this matrix are available by measurements. It is, therefore, highly desirable that these measurements are used as much as possible, to ease the burden of computations of the feedback matrices.

    Some remarkable progress has been made on the solution of the PQEAP, that has addressed some of the above challenges. The PQEAP was first solved by Datta et al. [5] in the single-input case and later their method was generalized to the multi-input case by Datta and Sarkissian [6] and by Ram and Elhay [7] and Sarkissian [8].

    Robust and minimum-norm problems have been considered by Bai et al. [9], Brahma and Datta [10], Chan et al. [11], Chu and Datta [12], Lam and Tam [13], [14], Lam and Yan [15], and Qian and Xu [16], Datta [1] and the references therein, etc. Meanwhile, Mottershead et al. [17] and Ram and Mottershead [18] studied several aspects of active vibration control using receptance measurements. Recently, Ram et al. [19] proposed a hybrid method, combining receptances and system matrices, to solve the single-input quadratic eigenvalue assignment and extended their method to the time-delay case. An important observation made in the paper is that the quadratic partial eigenvalue assignment problem in the time-delay case cannot be solved by using receptance alone—a hybrid approach is needed.

In this paper, we

  • First, generalize the single-input hybrid method of Ram, Mottershead, and Tehrani [19] to the solution of the multi-input PQEAP.

  • Then, propose a new optimization-based hybrid method for computing minimum feedback norms of the multi-input PQEAP, for both with and without time-delay. The proposed hybrid method offers several computational advantages over the standard methods (without the use of the receptances) that were proposed earlier for the PQEAP [9], [10], [20], [5], [12], [21].

    • First, the need to solve the Sylvester-matrix equation in computation of the feedback matrices is eliminated.

    • Second, the eigenvectors of the closed-loop pencil corresponding to the eigenvalues that are to be reassigned are not needed in this hybrid method. They are readily available from the entries of the receptance matrix (see Eq. (15)).

  • More importantly, the new hybrid method does not involve computation of the parametric matrix. The proper choice of this parametric matrix for the methods in [9], [10], [20] is crucial—it needs to be chosen in such a way that the solution of the associated Sylvester equation becomes nonsingular [see Eq. (5)]. At this time, there is no systematic way to choose this matrix, except by trial-and-error processes (see Remark 3.1 in Section 3).

Results of numerical experiments show that in all cases, the hybrid method was quite effective: (i) the eigenvalues are reassigned quite accurately, (ii) no spill-over is nicely maintained, and (iii) feedback norms are considerably smaller with the hybrid methods than those obtained without the use of receptances.

Section snippets

Problem statements

Suppose a control force of the form f(t)=Bu(t), where B is the control matrix of order n×m (mn), is applied to the structure model by (1). Choosingu(t)=FTx˙(t)+GTx(t),where F and G are two n×m feedback matrices, we have the closed-loop control system:Mx¨(t)+(DBFT)x˙(t)+(KBGT)x(t)=0.The dynamics of this closed-loop system are now governed by the eigenvalues and eigenvectors of the closed-loop quadratic pencil Pc(λ)=λ2M+λ(DBFT)+(KBGT).

Let {λ1,,λp;λp+1,,λ2n} be the spectrum of P(λ) with

Solution of the PQEAP without the use of receptances

In what follows, we assume that M, D and K are real symmetric with M positive definite. Let · and ·2 denote the Euclidean vector norm and the matrix 2-norm, respectively. Denote by A(:,k) and A(k,:) the kth column and the kth row of a matrix A, respectively. In is the identity matrix of order n. Suppose that (i) {μ1,,μp}{λ1,,λ2n}= and {λ1,,λp}{λp+1,,λ2n}=, (ii) the control matrix B has full column rank, and (iii) (P(λ),B) is partially controllable with respect to the eigenvalues λ1,

Partial quadratic eigenvalue assignment using the partial measured receptance and the system matrices

In this section, we propose a hybrid method for solving the PQEAP that make use of both receptance measurements and the system matrices M,D,K. For any sC, the receptance matrix H(s) to the open-loop pencil P(λ) is defined by H(s)=(s2M+sD+K)1,which can be measured without any explicit knowledge of the matrices M,D,K [22]. Let Hc(s) denote the receptance matrix corresponding to the closed-loop pencil Pc(λ), i.e., Hc(s)=(s2M+s(DBFT)+(KBGT))1sC.By the Sherman–Morrison–Woodbury formula [23],

A hybrid method for partial quadratic eigenvalue assignment with time delay

In practice, there exists time delay between the measurement of the state feedback and the implementation of feedback controller. We, therefore, would like to consider the following feedback control system with time delay τ: Mx¨(t)+Dx˙(t)+Kx(t)=Bu(tτ),where τ is the input time delay and u(t) is a state feedback controller defined by (3). The associated closed-loop delayed pencil is given by P˜c(λ)λ2M+λ(DeλτBFT)+(KeλτBGT).The time-delay PQEAP is to find two feedback matrices F and G such

Conclusion

Active control by state feedback gives rise to partial quadratic eigenvalue assignment which concerns reassigning a few unwanted eigenvalues while keeping the remaining large number of them and the corresponding eigenvectors unchanged. For robust active control, feedback must be computed so that both feedback norms and the closed-loop eigenvalue sensitivity are minimized. We have proposed new hybrid algorithms for the partial quadratic eigenvalue assignment problem and minimization of feedback

Acknowledgments

We would like to thank the editor and the anonymous referees for their valuable comments which have considerably improved this paper.

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    1

    The research of this author was partially supported by the National Natural Science Foundation of China Grant 11271308, the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (No. 2010J06002), and NCET.

    2

    The research of this author was supported by NSF Grant #DMS-0505784.

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