Differential equation models for damped vibrating systems are associated with quadratic matrix eigenvalue problems. The matrices in these systems are typically real and symmetric. The design and stabilisation of systems modelled by these equations require the determination of solutions to the inverse problem which are themselves real, symmetric and possibly with extra structure. In this paper we present a new method for pole assignment to a quadratic pencil by using affine sums. The method extends the work of H. Dai and P. Lancaster in which a similar problem for the generalized inverse eigenvalue problem is solved.
Co-authors: Y.M. Ram
year = {2003},
volume = {},
pages = {},
note = {To Appear CTAC2003 Proceedings}
Affine inverse eigenvalue problems are usually solved using iterations where the object is to diminish the difference between a set of prescribed eigenvalues and those calculated during iteration. Such an approach requires a scheme for pairing the eigenvalues consistently throughout the iterative process. There appears to be no obvious criterion for such pairing for problems with complex eigenvalues. Consequently the methods previously proposed in the literature are restricted to symmetric eigenvalue problems with real eigenvalues. Real eigenvalues can be paired using their natural increasing order. This paper presents a new Newton's iteration based method where the subject of iteration is the affine coefficients set. With the new method proposed the non-symmetric inverse eigenvalue problem, with inherent complex eigenvalues can be solved, as well as problems associated with symmetric pencils of high order. An immediate application presented in the paper deals with the reconstruction and passive control of damped vibratory systems.
Co-authors: Y.M. Ram
journal = {Inverse Problems},
year ={2002},
volume = {18},
number = {2},
pages = {455-466}
Co-authors: G.H. Golub and Y.M. Ram
Suppose the spectrum of a symmetric definite linear pencil is known. This paper addresses the question of what can be said about the spectrum when scalar multiples of a rank-one update are added to each matrix in the pencil. The secular equation for this problem is derived, and from it, a certain separation property is found which gives insight into the connection between the eigenvalues before and after modification. In the context of structural dynamics the result characterises the behaviour of a finite dimensional vibrating system undergoing mass and stiffness modifcations. The result also leads to applications such as a divide and conquer algorithm for the eigenvalues of the modified system (so called matrix tearing) and spectral shifting. An illustrative example is also given.
journal = {Computers and Maths with Appl.},
year ={2003},
volume = {46},
pages = {1413-1426}
The problem of reassigning some poles of a vibratory system, while keeping the other poles unchanged, is considered. The problem may be solved uniquely by single-input state feedback control. A family of solutions to the partial pole assignment problem may be obtained by applying multi-input control forces. An algorithm for determining a multi-input control which is small in some sense is presented. The non-iterative algorithm proposed, defines a closed form solution to the partial pole assignment problem in its natural second order form, and no first order realisation is used. The reduction in the control effort achieved by the proposed method is demonstrated by numerical examples.
journal = {Journal of Sound Vibration},
volume = {230},
year = {2000},
pages = {309-321}
Co-authors: G.M.L. Gladwell, G.H. Golub and Y.M. Ram
year = {1999},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {20},
number = {3},
pages = {563-574}
Co-authors: B.N. Datta and Y.M. Ram and D.R. Sarkissian
It is shown in this paper that, by the appropriate choice of gain and input influence matrices, certain eigenpairs of a vibrating system may be assigned while the other eigenpairs remain unchanged. The system under considertion is modelled by a set of second order differential equations and the assignment is carried by multi-input state feedback control. The solution may be of particular interest in the stabilization and control of flexible structures using smart materials, where only a small part of the eigenstructure is to be reassigned and the rest is required to remain unchanged. The method presented is illustrated with a numerical example.
journal = {Journal of Sound and Vibration},
year = {2000},
volume = {230},
pages = {101-110}
Co-authors: B.N. Datta and Y.M. Ram
The eigenvectors of a symmetric matrix can be chosen to form a biorthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric definite linear pencil can be chosen to be biorthogonal with respect to the pair.This paper presents the three sets of matrix weights, with respect to which the eigenvectors of the symmetric definite quadratic pencil are biorthogonal. One of these relations is used to derive an explicit solution of the {\em partial pole assignment problem} by state feedback control for a control system modeled by a system of second order differential equations. The solution may be of particular interest in the stabilization and control of flexible, large, space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged.
journal = {Linear Algebra and Appl},
year = {1997},
volume = {257},
pages = {29-48}
Co-author: Y.M. Ram
The discretization of the differential equation governing the axial vibration of a rod with varying cross-section leads to a specially structured matrix pencil. This paper deals with the reconstruction of this pencil from its spectrum. An iterative algorithm for this problem and an analytic characterization of complementary solutions are given. The method is demonstrated on some examples.
year = {July 1998},
volume = {14},
number = {7},
publisher = {John Wiley & Sons Ltd},
pages = {597-608},
note = {ISSN: 1069-8299},
journal = {Communications in Numerical Methods in Engineering}
Co-authors: B.N. Datta and Y.M. Ram
In this paper, we present a new algorithm for the multi-input pole assignment problem of a control system modeled by a system of second-order differential equations. Specifically, given the damped symmetric definite second-order control system Mv''+Cv'+Kv=0 and a set of 2p numbers, closed under complex conjugation, the algorithm finds matrices F and G such that the spectrum of the closed-loop pencil P(t)=t^2M+t(C-BF')+(K-BG') contains the given set and the complementary part of the spectrum has non-positive real part; that is, no spill-over occurs.
The algorithm does not require explicit knowledge of the eigenvalues and eigenvectors of the associated open-loop quadratic pencil. This is in sharp contrast with the traditional approach. It is composed of numerically effective tools of matrix computations such as the Cholesky-factorization, singular value decomposition, and solutions of linear systems.
booktitle = {Proceedings of the 35th IEEE Conference on Decision and Control},
year = {1996},
volume = {2},
pages = {2025-2029},
Co-author: Y.M. Ram
A method is presented which constructs an $n$ by $n$ tridiagonal, symmetric, quadratic pencil which has its $2n$ eigenvalues and the $2n-2$ of its $n-1$ dimensional leading principal subpencil prescribed. It is shown that if the given eigenvalues are distinct, there are at most $2^n(2n-3)!/(n-2)!$ different solutions. In the degenerate case, where some of the given eigenvalues are common, there are an infinite number of solutions. Apart from finding the roots of certain polynomials, the problem is solved in a finite number of steps. Where the problem has only a finite number of solutions, they can all be found in a systematic manner. The method is demonstrated with a simple example and its use is illustrated with a practical engineering application in vibrations.
journal = {SAIM J. Appl. Math.},
volume = {56},
number = {1},
year = {1996},
pages = {232-244}
Co-author: Y.M. Ram
It is shown that the simple dynamic absorber is a special case of a more general phenomenon surrounding the behaviour of multi degree of freedom vibratory systems. In particular, the motion of a prescribed mass in a multiply connected mass spring system may be absorbed by attaching an appropriately chosen system to it. Using this result, the frequency response function of a simply connected system is completely characterized. The results in this paper can be used in engineering vibrations design and may have application in system identification and control.
journal = {Journal of Sound and Vibration},
year = {1996},
volume = {195},
number = {4},
pages = {607-615}
Co-author: Y.M. Ram
Consider the axially vibrating rod with constant material properties but with varying cross sectional area. There exist two rods, with generally different cross sections, which have the same spectrum. The connection between the eigenfunctions of one rod and the other is given. It is shown that this duality carries across from the continuous to the analogous discrete model.
Similar results are established for a certain model of the vibrating beam.
journal = {Journal of Sound and Vibration},
year = {1995},
volume = {184},
pages = {759-766}