Left Matrix Fraction Research Articles

Minimal order realization with a special coordinate for matrix fraction descriptions

This paper presents a generalized and minimal realization algorithm for constructively determining generalized state-space representation from a so-called column (row)- pseudoproper or a column(row)-proper or a nonstrictly-proper right(left) matrix fraction description (MFD). The realized state-space representation form with a special coordinate is proved to be controllable and observable in the sense of Rosenbrock and Cobb; besides, the dimension of the system after realization is equal to the determinantal degree of the given right(left) MFD in the normal sense.

Generalized matrix Euclidean algorithms for solving diophantine equations and associated problems

Based on the matrix fraction descriptions of multi-variable control systems, this paper presents the generalized matrix Euclidean algorithms for solving Diophantine equations and associated problems. A chain rule is developed for finding the greatest common right(left) divisor of two square(non-square) polynomial matrices and the irreducible right(left) matrix fraction description from the reducible or irreducible left(right) matrix fraction description. Also, the development of multi-variable pole-assignment controllers are discussed for engineering applications.

Generalized realizations with a special coordinate for matrix fraction descriptions

A generalized and feasible realization algorithm for constructively determining generalized state-space representation from a so-called column (row)-pseudo-proper or a column (row)-proper right (left) matrix fraction description (MFD) is proposed. The realized state-space representation form with a special coordinate is proved to be controllable and observable in the sense of Rosenbrock and Cobb; moreover, the dimension of the system after realization is equal to the deter-minantal degree of the given right (left) MFD in the normal sense.

Partial-fraction expansion in system analysis

Abstract A procedure is given for finding the partial-fraction expansion of strictly proper rational transfer functions that are commonly found in systems analysis. Eight formulae for the expansions are given to include the cases: (a) scalar transfer functions, (b) matrix transfer functions, (c) left and right matrix fraction descriptions, (d) transfer function matrices, and (e) block expansion of left and right matrix fraction descriptions. These formulae are derived from system theory using the observability canonical form and do not require the use of derivatives, the particular nature of the Jordan form, or interpolating polynomials; rather, it is required that the Vandermonde matrix be formed from the roots, latent roots, eigenvalues or solvents of the denominator. The main features of the procedure include that all the residues are found at the same time by simple matrix manipulation and can be used for the repeated and/or non-repeated case. The structured form of the formulae is very simple and fil...

A new method for block partial fraction expansion of matrix fraction descriptions

The block-observability form is used to develop a method for finding the block partial fraction expansion of an irreducible right or left matrix fraction description with distinct and/or repeated solvents of the denominator matrix polynomials. The new formulas ar very simple and only require that the block Vandermonde matrix be formed from the right or left solvents. All the matrix residues of the expansion are found simultaneously by a simple matrix manipulation and the formulas are suitable for digital computer application.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A multivariable self-tuning controller based on pole-placement design

The design problem of controller for discrete-time linear multi input-output systems is discussed. The basic idea is to use pole-placement approach. A new pole-placement tracking problem formulation and its solution is given. The design of controller requires no a priori information either about stability or minimum phase of controlled plants. The designed controller consists of a feedback part, a feedforward part and a new precompensator and is suitable for self-tuning control of continuous systems with differing and unknown dead-times in each path. The controller can have integral action. The number of outputs can be different from the number of inputs of the controlled system. A reference vector sequence and an effect of the initial state on the system output are given as a left matrix fraction. The self-tuning algorithm is based on identification of explicit plant model. The self-tuning algorithm is illustrated by examples.

Canonical model transformations via long division and continued fraction methods

A long division method based on a matrix euclidean algorithm and a continued fraction method based on a matrix Routh algorithm are developed. These are used for constructing a similarity transformation matrix which would transform an observer (controller) block companion form to a controller (observer) block companion form, and for converting a reducible or an irreducible left (right) matrix fraction description to an irreducible right (left) matrix fraction description. The relationships between the state-space models and the corresponding rational matrices of a class of multivariable systems are also explored. The new procedures enable us to enhance the computational aspects of designing self-tuning controllers for on-line adaptive control of a class of multivariable systems.

State-space self-tuning regulators for general multivariable stochastic systems

The paper presents a state-space approach for the self-tuning control of general linear multivariable discrete-time stochastic systems with the number of inputs (controllability indices) equal to or different from the number of outputs (observability indices). The dynamic system is represented in the state-space innovation form with the Luenberger&apos;s canonical structure. The model parameters, as well as the Kalman gain, are identified via the least-squares ladder algorithm, without utilising the standard stateestimation algorithm. Also, to avoid the direct use of the Luenberger&apos;s canonical transformations, a long division method is introduced for quickly converting a reducible or irreducible left matrix fraction description (LMFD) to an irreducible right matrix fraction description (RMFD) and for constructing the Luenberger&apos;s transformation matrices. In conjunction with the state-space selftuning control, an integral control is used so as to eliminate the steady-state errors and render the closed-loop system less sensitive to modelling errors. The proposed method will enhance the application of the state-space self-tuning concepts to a general class of multivariable stochastic systems.

Generalized Bezoutians and invariant subspaces

Given a controllable and observable linear time-invariant system for which the transfer matrix has a right and left coprime matrix fraction description, the generalized Bezoutian of the matrix polynomials of the matrix fraction description is shown to be related to the right (A,B)-invariant vectors and the left (C,A)-invariant vectors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A simple division method for solving Diophantine equations and associated problems

Based on the transfer function matrices or the matrix fraction descriptions of multivariable systems, a simple division method in conjunction with the method owing to Wolovich and Antsaklis (1984) is presented in this paper for obtaining the observability and controllability indices, the irreducible right (left) matrix fraction descriptions from reducible or irreducible left (right) matrix fraction descriptions, the greatest common divisors of two polynomial matrices and the solution of a Diophantine equation, without utilizing the state-space realization of the matrix Traction description.

Multivariable pole-zero placement self-tuning controller†

Abstract A computationally efficient pole-zero placement self-tuning control algorithm for multi-input-multi-output servo systems is developed. It is effectively an implicit algorithm in the sense that the controller design step is trivial. Moreover, the controller is obtained as a left matrix-fraction and so can be immediately implemented. Although the algorithm is restricted to stable plants, it is applicable to non-minimum-phase plants and can also handle unknown but bounded pure time delays. Results obtained from computer simulations of the algorithm, as well as by trials on an actual stirred-tank process, are presented.

Recursive identification algorithms for right matrix fraction description models

Multivariable identification algorithms are usually designed only for left matrix fraction description (LMFD) models. In this paper we consider recursive identification algorithms for right matrix fraction description (RMFD) models with diagonal denominator matrices. The algorithms are of prediction error (PE) and model reference (MR) type. Results from simulations illustrate the performance of the algorithms.

Canonical forms and strictly positive real functions for discrete multivariable systems

In this paper, we show that, for the discrete multivariable systems, the solution of the Lyapunov equation in the controllable canonical form transforms a companion matrix to another companion matrix. We also show that the first (last) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> columns of the inverse of this solution matrix, where the block size is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times m</tex> , give the left (right) matrix fraction decomposition of a strictly positive real function.

Stability of a class of multivariable control systems described by nested matrix continued-fractions

This paper deals with the stability of a class of multivariable control systems which can be described by a right (left) matrix fraction description and associated right (left) matrix continued-fraction description in the nested Cauer form. Some simple sufficient conditions and some simple necessary conditions are developed for stability determination of the class of multivariable systems via the Lyapunov stability and instability theorems. By simple testing of the (positive or negative) definiteness of a set of matrix quotients obtained from the matrix Sturm algorithms, the stability of the class of multivariable systems can be determined. The matrix fraction descriptions of interest may be unsymmetric and reducible.