Non-Frobenius Spectrum-Transformation Method

Abstract

A method allowing a desirable matrix spectrum to be constructed as an alternative to the method using matrix transformation to the Frobenius form is stated. It can be applied to implement control algorithms for technical systems without executing the variables transformation procedures that are needed for deriving a Frobenius matrix. The method can be used for simulation of systems with different spectrums for choosing an alternative that satisfies to the distinct demands.

A Simple Iterative Method for Exact Solution of Systems of Linear Algebraic Equations

“…An iterative method for exact solution of a SLAE in a finite number of steps has the computational procedure of iterative methods, in the conventional sense, like the Jacobi and Seidel methods, can be called a finite-iterative method. As in control theory, it is based on setting a unit spectrum for the SLAE matrix or a zero spectrum for the matrix of the iteration equation [8,9]. The main difference is possibility to solve not only a homogeneous equation, but also a non-homogeneous one.…”

“…A method to find the exact solution of SLAE in a finite number of iterations is reported for the first time. Some points of the method are stated in [8][9][10], in particular, the iterative equations for a secondorder SLAE represented in analytical form are given in [10]. Here we present a theorem and prove it.…”

A Direct Transformation of a Matrix Spectrum

“…By simulating the system behavior with different spectrums, it is possible to find a suitable alternative, which can be further implemented as a direct digital control algorithm. The paper is an outgrowth of the work [5].…”