Matrix Fraction Description Research Articles

Coprime Matrix Fraction Description via Sylvester's Resultant Matrix

Two algorithms are presented in this paper in order to compute the coprirne matrix fraction description of a given proper rational transfer function matrix. Generalised Sylvester resultant matrix is formed and coprirne matrix fractions are determined employing Gaussian elimination method. With the help of one of these algorithms both the right and the left coprirne matrix fraction descriptions as well as the greatest common divisor can simultaneously be determined.

Model reduction of nonsquare linear MIMO systems using multipoint matrix continued-fraction expansions

This paper deals with the multipoint Cauer matrix continued-fraction expansion (MCFE) for model reduction of linear multi-input multi-output (MIMO) systems with various numbers of inputs and outputs. A salient feature of the proposed MCFE approach to model reduction of MIMO systems with square transfer matrices is its equivalence to the matrix Padé approximation approach. The Cauer second form of the ordinary MCFE for a square transfer function matrix is generalized in this paper to a multipoint and nonsquare-matrix version. An interesting connection of the multipoint Cauer MCFE method to the multipoint matrix Padé approximation method is established. Also, algorithms for obtaining the reduced-degree matrix-fraction descriptions and reduced-dimensional state-space models from a transfer function matrix via the multipoint Cauer MCFE algorithm are presented. Practical advantages of using the multipoint Cauer MCFE are discussed and a numerical example is provided to illustrate the algorithms.

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.

Characterizations of Transmission Zeros and Fixed Modes Via Generalized Cramer's Rule

This paper presents results on characterization of transmission zeros and fixed modes of linear multivariable systems in matrix fraction descriptions. The derivation of these results are based on a newly obtained formula, which is a natural generalization of well-known Cramer's rule.

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.

Implementation of column-pseudoproper right matrix fraction descriptions with cascaded or parallel active RC networks

An effective and constructive approach for implementing the so-called column-pseudoproper right matrix fraction descriptions defined by Tan and Vandewalle with cascaded or parallel active RC networks is presented. The resulting structure contains m sets of decoupled RC-voltage amplifier networks in a cascaded form or in a parallel form, for structuring accessible intermediate states. A state-feedback control law is applied around the auxiliary network for the realization of the desired overall voltage matrix fraction descriptions. The proposed cascaded and parallel RC networks have shown good sensitivity and stability properties.

Matrix-fraction description from frequency samples

It is shown how the controllability indices in a minimal state-space realization of a real rational transfer matrix may be calculated from evaluations of this transfer matrix at a sufficient number of discrete points in the frequency domain. Subsequently a procedure for obtaining coprime matrix-fraction descriptions of the transfer matrix is given. An example is used to illustrate the results.

System zeros determination from an unreduced matrix fraction description

A square-subsystem concept is used to determine the system zeros of an unreduced matrix fraction description of a linear dynamic system. The concept reveals more clearly the parallelism of SISO and MIMO zeros. >

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.

2-D System Structure and Applications

The structure of 2-D systems is investigated by using the polynomial models of a matrix fraction description and the more general model of a Rosenbrock system matrix.These results are then applied to the case of the unit memory process to characterise the decoupling structure of the system

A new approach for calculating doubly coprime matrix fraction descriptions

Using the concepts of state feedback in generalized state-space, explicit formulas for calculating doubly coprime matrix fraction descriptions (MFDs) of the transfer matrix of a regular state-space system and the polynomial generalized Bezout identity elements corresponding to those coprime MFDs are presented. They can be easily calculated with the help of existing computational algorithms and software packages.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coprime matrix fraction description via orthogonal structure theorem

An algorithm to compute the coprime matrix fraction description from a given transfer function matrix W(s) is presented. The algorithm is based on Householder transformations to compute the minimal realization of W(s) and on the orthogonal version of the structure theorem of W.A. Wolovich (1974) and of Wolovich and P.L. Falb (1969) to determine the coprime form. The proposed method is numerically less expensive than a similar algorithm given by R.V. Patel (1981) which uses singular value decomposition. >

The observability gramian of the observer form and the Schur-Cohn matrix

The observer form is a standard minimal realization of the transfer-function matrix G based on an irreducible left matrix-fraction description G = U -1V . It is shown that (in the case of stable G ) there is a simple closed-form expression for the observability gramian of this realization in term of the polynomial matrix U and its ‘reflection’: in fact, gramian is the inverse of the schur-cohn of U .

Design of observer-based compensators in the frequency domain: the discrete-time case

For nth order systems with m outputs, the design of compensators using state observers of order n − x with 0 ≤ x ≥ m is presented in the z-domain. The design is carried out in equivalence with the usual time-domain approach and the relations between the time- and the frequency-domain results are presented. The state feedback and the observer are characterized by polynomial matrices in the frequency domain and the corresponding compensator matrix fraction descriptions are obtained from the solution of a linear diophantine equation. In view of the optimal state-estimation problem, the use of an updated state estimate is of interest. It is shown how this problem can also be solved in the frequency domain.

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...

Determination of doubly coprime matrix fraction discriptions and corresponding Bezout identity solutions for a regular state-space system

A new approach to calculating doubly coprime matrix fraction descriptions and corresponding Bezout identity solutions for a regular state-space system is presented. The proposed method needs only four constant matrices which can be selected at random. The connections between state-space and matrix fraction descriptions are established in this paper.

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>

The use of the homogeneous form in the study of singular systems

The objective of this paper is to present an algebraic theory for analyzing the infinite structures of singular systems. The key concept used in the development is the so-called homogeneous form for polynomial and rational matrices, which can be seen as a generalization of the well-known notion of homogeneous form for matrix pencils. We show how familiar concepts such as matrix fraction description, infinite poles and zeros, irreducibility, and cancellation at infinity appear naturally in this framework. The characterizations for some of these concepts are derived in terms of homogeneous forms. Considering singular systems to be alternative representations for (possibly nonproper) rational matrices, we are able to relate the fundamental concepts of controllability and reconstructibility for singular systems to the concept of complete coprimeness for rational matrices, and to relate the concept of minimal order of singular systems to the concept of the McMillan degree of rational matrices. As an important result, the equivalence relations among the concepts of joint controllability and reconstructibility, minimal order, and complete irreducibility are established. Moreover, we show that for any rational matrix factorized in matrix fraction description, the determinantal degree of the denominator matrix is the McMillan degree of the rational matrix if and only if the matrix fraction description is completely irreducible.

Optimal solution for discrete systems via spectral factorization

Abstract Based on the matrix fraction description (MFD) consideration and polynomial matrix algebra, interesting results for the multivariable discrete linear quadratic regulator and optimal estimator are derived. The solutions are obtained by the spectral factorization of polynomial matrices in concise forms. The difficulties of solving the discrete algebraic Riccati equation can thus be avoided. In the derivation, some connections between the state space approach and the frequency domain approach are further investigated.

On the Model Order Reduction of Linear Multivariable Systems Using a Block-Schwarz Realization

A block-Schwarz state-space realization is utilized to derive a model order reduction method for linear multivariable systems. It is shown that the proposed method may be regarded as a frequency-domain continued-fraction technique based upon a block "alpha-beta" expansion of matrix fraction descriptions. A new algorithm is presented in order to obtain the block-Schwarz realization from matrix fraction descriptions. Once the block-Schwarz matrix satisfies certain stability conditions then stable reduced models are provided. The proposed method constitutes a block version of the popular Routh approximation.