Symmetric Tridiagonal Inverse Quadratic Eigenvalue Problems with Partial Eigendata

In this paper we concern the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(¸) := ¸2I + ¸C + K (where I is the identity matrix) possesses the given partial eigendata. We ¯rst provide the su±cient and necessary conditions for the existence of an exact solution to the inverse problem from the self-conjugate set of prescribed four eigenpairs. To ¯nd a physical solution for the inverse problem where the matrices C and K are weakly diagonally dominant and have positive diagonal elements and negative o®-diagonal elements, we consider the inverse problem from the partial measured noisy eigendata. We propose a regularized smoothing Newton method for solving the inverse problem. The global and quadratic convergence of our approach is established under some mild assumptions. Some numerical examples and a practical engineering application in vibrations show the e±ciency of our method

Exact Solution of a Linear-Quadratic Inverse Eigenvalue Problem on a Certain Hamiltonian Symmetric Matrices 1, 2,

This paper investigate the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamiltonian symmetric matrices namely singular symmetric matrices of rank 1 and non-singular symmetric matrices in the neighborhood of the first type of matrices via the Newton’s iterative method

Analysis of structured polynomial eigenvalue problems

This thesis considers Hermitian/symmetric, alternating and palindromic matrix polynomials which all arise frequently in a variety of applications, such as vibration analysis of dynamical systems and optimal control problems. A classification of Hermitian matrix polynomials whose eigenvalues belong to the extended real line, with each eigenvalue being of definite type, is provided first. We call such polynomials quasidefinite. Definite pencils, definitizable pencils, overdamped quadratics, gyroscopically stabilized quadratics, (quasi)hyperbolic and definite matrix polynomials are all quasidefinite. We show, using homogeneous rotations, special Hermitian linearizations and a new characterization of hyperbolic matrix polynomials, that the main common thread between these many subclasses is the distribution of their eigenvalue types. We also identify, amongst all quasihyperbolic matrix polynomials, those that can be diagonalized by a congruence transformation applied to a Hermitian linearization of the matrix polynomial while maintaining the structure of the linearization. Secondly, we generalize the notion of self-adjoint standard triples associated with Hermitian matrix polynomials in Gohberg, Lancaster and Rodman's theory of matrix polynomials to present spectral decompositions of structured matrix polynomials in terms of standard pairs (X,T), which are either real or complex, plus a parameter matrix S that acquires particular properties depending on the structure under investigation. These decompositions are mainly an extension of the Jordan canonical form for a matrix over the real or complex field so we investigate the important special case of structured Jordan triples. Finally, we use the concept of structured Jordan triples to solve a structured inverse polynomial eigenvalue problem. As a consequence, we can enlarge the collection of nonlinear eigenvalue problems [NLEVP, 2010] by generating quadratic and cubic quasidefinite matrix polynomials in different subclasses from some given spectral data by solving an appropriate inverse eigenvalue problem. For the quadratic case, we employ available algorithms to provide tridiagonal definite matrix polynomials.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Constructing the Physical Parameters of a Damped Vibrating System From Eigendata

In this paper we consider the inverse problem for a discrete damped mass-spring system where the mass, damping, and sti®ness matrices are all symmetric tridiagonal. We ¯rst show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector. Then, we study the general under-determined and over-determined problems. In particular, we provide the su±cient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under-determined problem has a physical solution. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we propose the the minimum norm solution over the under-determined noisy data and the least squares solution to the over-determined measured data. We also discuss the physical realizability of the required model by the positivity-constrained regularization method for the ill-posed under-determined problem and the least squares optimization problems with positivity-constraints for the ill-posed over-determined problem. Finally, we give simple numerical examples to illustrate the e®ectiveness of our methods

Closed form solutions to the optimality equation of minimal norm actuation

This research focused on the problem of minimal norm actuation in the context of partial natural frequency or pole assignment applied to undamped vibrating systems by state feedback control. The result of the research was the closed form solutions for the minimal norm control input and gain vectors. These closed form solutions should took open loop eigenpairs and the desired frequencies of the controlled system and outputted the optimal controller parameters. This optimization technique ensures that the system’s dynamics will be effectively controlled while keeping the controller effort minimal. The controller must then be able to shift only the desired the system poles anywhere in the complex s-plane in order to give the system certain desired characteristics with no spillover. The open loop system dynamics were found by applying a discrete model of the studied vibrating system and then finding the eigenvalue problem associated with the second-order open loop system equations. A first order realization was then performed on the system in order to know its response to certain initial conditions. The system’s dynamics where to be modified via closed loop control. Partial natural frequency assignment was chosen as the control technique so that certain system frequencies could be left untouched to ensure that the system will not respond in an unexpected manner. The control was to be optimized by minimizing the norm of the control input and gain vectors. A closed form solution for these vectors was found in so that these vectors could be simply calculated using an algorithm that takes the open loop eigenpairs and the desired eigenvalues of the system and outputs the two vectors. This closed form solution was successful implemented and the controller parameters found were applied to a vibrational system. A simulation for the un-optimized and optimized cases was performed applying both controllers to the same system. The response and controller forces for both cases were plotted in MATLAB and compared. Both systems showed the desired system response meaning that they both had the same effect on the system. Inspecting both controller efforts showed that the optimal control case simulation showed less controller effort than the arbitrary case thus showing successful implementation of minimal norm actuation

The transcendental eigenvalue problem and its application in system identification

An accurate mathematical model is needed to solve direct and inverse problems related to engineering analysis and design. Inverse problems of identifying the physical parameters of a non-uniform continuous system based on the spectral data are still unsolved. Traditional methods, for the system identification purpose, describe the continuous structure by a certain discrete model. In dynamic analysis, finite element or finite difference approximation methods are frequently used and they lead to an algebraic eigenvalue problem. The characteristic equation associated with the algebraic eigenvalue problem is a polynomial. Whereas, the spectral characteristic of a continuous system is represented by certain transcendental function, thus it cannot be approximated by the polynomials efficiently. Hence, finite dimensional discrete models are not capable of identifying the physical parameters accurately regardless of the model order used. In this research, a new low order analytical model is developed, which approximates the dynamic behavior of the continuous system accurately and solves the associated inverse problem. The main idea here is to replace the continuous system with variable physical parameters by another continuous system with piecewise uniform physical properties. Such approximations lead to transcendental eigenvalue problems with transcendental matrix elements. Numerical methods are developed to solve such eigenvalue problems. The spectrum of non-uniform rods and beams are approximated with fair accuracy by solving associated transcendental eigenvalue problems. This mathematical model is extended to reconstruct the physical parameters of the non-uniform rods and beams. There is no unique solution for the inverse problem associated with the continuous system. However, based on several observations a conjecture is established by which the solution, that satisfies the given data by its lowest spectrum, is considered the unique solution. Physical parameters of non-uniform rods and beams were identified using the appropriate spectral data. Modal analysis experiments are conducted to obtain the spectrum of the realistic structure. The parameter estimation technique is validated by using the experimental data of a piecewise beam. Besides the applications in system identification of rods and beams, this mathematical model can be used in other areas of engineering such as vibration control and damage detection

Numerical Solution of Linear and Nonlinear Eigenvalue Problems

Given a real parameter-dependent matrix, we obtain an algorithm for computing the value of the parameter and corresponding eigenvalue for which two eigenvalues of the matrix coalesce to form a 2-dimensional Jordan block. Our algorithms are based on extended versions of the implicit determinant method of Spence and Poulton [55]. We consider when the eigenvalue is both real and complex, which results in solving systems of nonlinear equations by Newton’s or the Gauss-Newton method. Our algorithms rely on good initial guesses, but if these are available, we obtain quadratic convergence. Next, we describe two quadratically convergent algorithms for computing a nearby defective matrix which are cheaper than already known ones. The first approach extends the implicit determinant method in [55] to find parameter values for which a certain Hermitian matrix is singular subject to a constraint. This results in using Newton’s method to solve a real system of three nonlinear equations. The second approach involves simply writing down all the nonlinear equations and solving a real over-determined system using the Gauss-Newton method. We only consider the case where the nearest defective matrix is real. Finally, we consider the computation of an algebraically simple complex eigenpair of a nonsymmetric matrix where the eigenvector is normalised using the natural 2-norm, which produces only a single real normalising equation. We obtain an under-determined system of nonlinear equations which is solved by the Gauss-Newton method. We show how to obtain an equivalent square linear system of equations for the computation of the desired eigenpairs. This square system is exactly what would have been obtained if we had ignored the non uniqueness and nondifferentiability of the normalisation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo