Biswa Nath Datta

Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

Abstract The eigenvectors of a symmetric matrix can be chosen to form an orthogonal 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 orthogonal 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 orthogonal. One of these is used to derive an explicit solution of the 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.

Numerically robust pole assignment for second-order systems

We propose two new methods for solution of the eigenvalue assignment problem associated with the second-order control system \global\hsize=30pc Specifically, the methods construct feedback matrices F 1 and F 2 such that the closed-loop quadratic pencil has a desired set of eigenvalues and the associated eigenvectors are well conditioned. Method 1 is a modification of the singular value decomposition-based method proposed by Juang and Maghami which is a second-order adaptation of the well-known robust eigenvalue assignment method by Kautsky et al. for first-order systems. Method 2 is an extension of the recent non-modal approach of Datta and Rincon for feedback stabilization of second-order systems. Robustness to numerical round-off errors is achieved by minimizing the condition numbers of the eigenvectors of the closed-loop second-order pencil. Control robustness to large plant uncertainty will not be explicitly considered in this paper. Numerical results for both the two methods are favourable. A compara...

Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment

Abstract We propose an Arnoldi-based numerical method for solving a Sylvester-type equation arising in the construction of the Luenberger observer. Given an N × N matrix A and an N × m matrix G , the method simultaneously constructs an m × m Hessenberg matrix H with a preassigned spectrum and an N × m orthonormal matrix X such that AX − XH = G . We consider the case when A is large and sparse, so that the standard techniques such as the well-known Hessenberg-Schur method for solving a Sylvester equation cannot be easily applied. As a byproduct, we propose an algorithm for the partial pole-assignment problem for large matrices.

An algorithm to assign eigenvalues in a Hessenberg matrix: Single input case

We propose a conceptually simple algorithm to assign eigenvalues in a Hessenberg matrix. The method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer. Numerical experiments compare favorably to those of other methods recently proposed for the problem.

Feedback stabilization of a second-order system: a nonmodal approach

A novel approach for feedback stabilization of a second-order model is proposed. Specifically, two nonmodal algorithms are described, and it is shown mathematically that under some mild assumptions on the damping matrix, the feedback matrix obtained by each algorithm indeed stabilizes a closed-loop system. The first algorithm requires the solution of a symmetric positive definite linear system and the inversion of a small matrix. A remarkable feature of this algorithm is that it does not require knowledge of the stiffness and damping for implementation, which makes it very feasible for practical application. The second algorithm requires the solution of a small linear least-squares problem and an estimate of the stability index, which can be obtained just by computing the extremal eigenvalues of the data matrices, which are symmetric for almost all practical applications. These minimal computational requirements make the proposed algorithms suitable for practical implementations, even for large and sparse systems, using the state-of-the-art techniques of matrix computations. This is in sharp contrast with the traditional modal approach ordinarily used by practicing engineers, which requires the solution of a quadratic eigenvalue problem or, equivalently, the solution of a 2n x 2n generalized eigenvalue problem, and therefore may not be practical for very large and sparse problems. Besides the proposed algorithms, the paper also contains some mathematical results on the bounds of the eigenvalues of a second-order pencil, which may be useful for investigating stability of second-order systems, and are of independent interest.

Stability and inertia

Abstract The purpose of this paper is to present a brief overview of matrix stability and inertia theory. A few applications of inertia and stability theorems, and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented. Inter-relationships between different theorems are explicitly stated, whenever appropriate. The paper concludes with some problems for future research in this area.

Spillover Phenomenon in Quadratic Model Updating

Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils, whereas spillover for damped quadratic pencils is generally unpreventable.

Systems and control in the twenty-first century

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Krylov subspace methods for large-scale matrix problems in control

This paper presents a brief but state-of-the-art survey of some of existing Krylov subspace methods for large-scale matrix problems in control. Based on the discussions and observations, some research problems are suggested.

A Parallel Algorithm for the Sylvester Observer Equation

We present a new algorithm for solving the Sylvester observer equation arising in the context of the Luenberger observer. The algorithm embodies two main computational phases: the solution of several independent equation systems and a series of matrix--matrix multiplications. The algorithm is, thus, well suited for parallel and high-performance computing. By reducing the coefficient matrix