Minimum Frobenius Norm Research Articles

Some iterative methods related to Frobenius norm minimization

A modification of the work in [1] is established in a way that allows to suppress the assumption of matrix-symmetry. Similar results to the ones obtained in [1], for the spectral radius of the iteration matrix I − M −1 A associated to a splitting of the matrix A, are obtained. Numerical examples show that the computational cost is equivalent. Besides, our method requires to solve only diagonal linear systems.

Fully vectorizable block preconditionings with approximate inverses for non-symmetric systems of equations

We used a conjugate gradient type method, with preconditioning, to solve the sparse linear systems arising from the discretization of PDEs. With such methods, the main obstacles for complete vectorization have been the preconditioning calculation and its application step within the iteration: for the matrices obtained using 5- or 9-point discretization operators, some well known existing preconditionings (like ILU) require a block-recursive procedure which prevents vectorization. Preconditioners based on nested incomplete factorization, which require the calculation of approximate inverses of tridiagonal matrices, allow complete vectorization of the application step. We present a formulation of such a preconditioning, using a Frobenius norm minimization to calculate the inverses, which also allows complete vectorization of the inverses' calculation, thus making the iterative solver completely vectorizable. Numerical experiments show that the method is robust over a range of symmetric and non-symmetric problems, and up to 4 times faster than other existing methods, such as ILU, depending on the computer and compiler being used. We also show the importance of diagonal scaling used in conjunction with other preconditionings and present some theoretical results concerning the approximate inverses of tridiagonal matrices, calculated using the Frobenius norm minimization.

Minimum-gain minimum-time deadbeat controllers

This paper considers the design of minimum-gain minimum-time deadbeat controllers (MGMTDC) for linear discrete-time systems. A new direct method for constructing the MGMTDC of minimum Frobenius norm is developed. Compared with many existing techniques, the proposed method gives analytic expressions for the constructed MGMTDC, requires less computational effort, and does not require that the transition matrix be nonsingular.

Optimal structural design with control gain norm constraint

A structure/cont rol system optimization problem has been formulated with constraints on the closed-loop eigenvalue distribution, structural frequencies, and the minimum Frobenius norm of the required control gains. Suggested is simultaneous optimization where, at each iteration, the control objective function is minimized first with the closed-loop eigenvalue constraints and then structural optimization is performed to satisfy the constraints on the optimal control gain norm and structural frequencies. The feasibility of the approach is demonstrated on a two-bar truss structure. For each locally optimal design, response simulations have been made and control efforts observed. Qualitative aspects of the optimal designs are also included and general conclusions drawn.

On the relationship between controllability indexes, Eigenvector assignment, and deadbeat control

The subspaces, to which closed-loop generalized eigenvectors are restricted, are described in terms of the controllability indexes of the pair ( A, B ) and the polynomials of minimal degree that span ker[ A- \lambda I, B ]. This characterization of the eigenspaces is then used to calculate the deadbeat controller of minimum Frobenius norm.

New extremal characterizations of generalized inverses of linear operators

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A + B (where A + is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX − B. In particular, A + is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A + b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces. In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A + and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A + as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥ A +∥ 2 < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.