Realization Of Matrix Research Articles

Optimization realization of a distance matrix †

In this paper we present an algorithm for optimum realization of n × n distance matrix by a weighted digraph. The method has two major steps. In the first step the index pairs of the nonzero finite elements of the given matrix are partitioned such that the elements corresponding to each class have the same value. The second step involves a systematic graph construction process that uses the information from the first step and an algorithm for computing a distance matrix for a digraph. We have proved the correctness of our algorithm and shown that the optimum realization is unique. An example is presented for illustrating the algorithm.

Comments on ‘Efficient algorithm for irreducible realization of a rational matrix ’

Comments on ‘Efficient algorithm for irreducible realization of a rational matrix ’

On the number of conductances required for realizing Y and K matrices

AbstractUpper bounds are established on the number of conductances required for realizing a real symmetric matrix Y as the short‐circuit conductance matrix of a resistive n‐port network containing no negative conductances, and for the realization of a real matrix K as the potential factor matrix of a similar network without negative conductances. These results are the consequence of the properties of the modified cut‐set matrix of an n‐port and a theorem in the theory of linear programming.

Minimal realization of a transfer function matrix in canonical form

An algorithm is presented to obtain a minimal-order realization of a linear time invariant multivariable system from a given rational transfer function matrix. The procedure yields the dynamical equations directly in a canonical form, and it is computationally very efficient.

(n+2)-node resistive n-port realizability of Y-matrices

Sufficient conditions for the realization of a real symmetric matrix as the short-circuit conductance matrix of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 2)</tex> -node resistive n-port network are established. The properties of networks of departure and padding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port networks are used to arrive at these results. It is shown that any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n x n)</tex> real symmetric matrix realizable by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port network can also be realized by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 2)</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port network containing at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(n + 1)/2</tex> conductances if all the zero conductances of the corresponding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> - node realization <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , form part of a fundamental cut-set in N with respect to the port tree of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> .

Minimal realizations of transfer-function matrices by means of matrix continued fraction

Abstract The matrix continued fraction technique is utilized to obtain the minimal realization of a transfer-function matrix with various inputs and outputs. If the ratio of the rank and the order of a square transfer-function matrix is an integer and there are no numerically ill-conditional elements in the matrix, then the matrix Routh algorithm is applied for the minimal realization. A method is also presented to deal with ill-conditional cases.

Minimal realization of a transfer function matrix in canonical forms

An algorithm is presented for obtaining a minimal-order realization of a linear time-invariant multivariable system from a given rational transfer function matrix. The procedure yields the dynamical equations directly in a canonical form, while still requiring less amount of computation than the existing methods.

Efficient algorithm for irreducible realization of a rational matrix

Abstract A computationally efficient algorithm is presented for irreducible realization of a given rational transfer function matrix. It is shown that either the Hankel matrix for the system or a reduced form of it can be systematically transformed to the Hermite normal form using outer products. The dynamical equations then follow immediately. This algorithm requires less computation than the existing algorithms.

Part II Minimal realization

A complotely new algorithm for calculating a state space minimal realization of a rational transfer function matrix is given. Tho algorithm consists of four major steps : calculation of the least common denominators of the columns of the transfer function matrix ; calculation of the greatest common right divisor of two polynomial matrices ; factoring the divisor from the two matrices ; and expansion of the resulting relatively right prime matrices into constant matrices. The transfer function matrix need not be proper and the result of the calculations leaves the polynomial part of the transfer function matrix unaltered, thus allowing a useful check on the accuracy of the computations to be incorporated. The results of several examples show that this new algorithm is superior in terms of computing time to other methods and is very accurate.

Irreducible canonical form realization of a rational matrix†

An irreducible realization procedure of a rational matrix is proposed. The procedure not only yields a canonical form dynamical equation but also requires less amount of computation than the existing ones.

Minimal realization of a symmetric transfer-function matrix using moments

It is shown that the method of [1] when applied to the Hankel matrix constructed from moments, in place of Markov parameters, for the case of symmetric transfer-function matrix also results in reciprocal realizations.

Matrices, polynomials, and linear time-variant systems

Some recent developments in the applications of matrices to problems arising in linear systems theory are described. It is shown how companion form matrices can be used to provide a unified framework for dealing with the qualitative analysis of polynomials, including such problems as determination of greatest common divisors. Relationships to classical theorems involving bigradients and to controllability are discussed. When applied to the determinantal stability criteria of Hurwitz and others, the companion matrix approach results in minors of half the original orders. The problem of minimal realization of a transfer function matrix is dealt with in terms of polynomial matrices using methods due to Rosenbrock, and links with the results on scalar polynomials are demonstrated. Some applications of Lyapunov theory to systems in state-space form are briefly reviewed.

The distance matrix of a graph and its tree realization

The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights. Conditions under which such matrices have a tree, hypertree or directed tree realization are given, uniqueness of these realizations is discussed and algorithms for their construction are indicated.

Synthesis of RC multiport impedance functions

This correspondence presents a method of realizing a set of open-circuit RC impedance functions z_{11}, z_{12}, ,, , z_{22} , z_{23} , ,, , z_{rr} ,,. A simple and systematic technique of determining the parameter matrices is developed by using linear transformations. A pertinent network is obtained through the realization of the conductance parameter matrix by a single-element-type network for the tree configuration, and the capacitance matrix is obtained by connecting unit capacitors to the appropriate nodes.

Realization of modified cut-set matrix and applications

The concepts of network of departure and padding n-port network are defined and certain important properties of these networks are stated. A necessary and sufficient criterion for the realization of the modified cut-set matrix of a resistive n-port network having a prescribed configuration is given. A new approach to the synthesis of short-circuit conductance matrix of a resistive n-port network with more than (n + 1) nodes is outlined. Necessary and sufficient conditions for the realization of the potential factor matrix of an (n + 2) node resistive n-port network are also obtained. These conditions lead to a simple t&t for the realizability of the modified cut-set matrix of an (n + 2) node n-port network. Examples are worked out to illustrate the application of these results.

Comments on "Irreducible Jordan form realization of a rational matrix"

Comments on "Irreducible Jordan form realization of a rational matrix"

Irreducible Jordan form realization of a rational matrix

This paper presents a new method for realizing a rational transfer function matrix into an irreducible Jordan canonical form state equation. The method consists of two steps, first, to form a controllable state equation and then secondly, to obtain a controllable as well as observable realization by nonsingular transformations. If every denominator of the rational matrix is given in the factored form, then the proposed method can be carried out quite easily.

The Theory of Generalized Interdigital Networks

A theory for two-port generalized interdigital networks is presented. Initially, after the definition of an arbitrary generalized interdigital network, properties of admittance scaling and transformations of general interdigital lines are formulated. The necessary and sufficient conditions are then presented for the individual realization of any admittance parameter of such a network. The results obtained in this section influence the choice of a compact set of sufficiency conditions for the realization of any arbitrary twoport admittance matrix by this class of networks. These sufficiency conditions are also shown to imply simple sufficiency conditions on the insertion-loss function of a resistively terminated, two-port generalized interdigital network.

Synthesis of a Class of n-Port Networks

The properties of a class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> -node networks, called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -networks, are discussed. The characteristic of a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -network is that when any one of its ports is connected to a voltage source keeping all the other ports short circuited, then all the short-circuited ports are at the same potential. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> -node network with a pair of equal conductances joining any two ports, as obtained by the presently known procedure for the realization of a dominant conductance matrix, is shown to be a special structure belonging to this general class. It is shown that the realization of a real dominant matrix as the short-circuit conductance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port network can be convenientlycarried out using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -networks. Further, the "modified cut-set matrix" of a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -network is of a special form, independent of edge conductances. This property can be made use of in generating a range of equivalent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port networks for a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> . Examples illustrating the realization procedures are included.

Realization of Resistive N-Port Networks without Tree Transformations

ABSTRACTThe class of short circuit conductance matrices realizable as n-port resistive networks is extended in this paper using generalized tree terminal graphs. In Part I, the realization is done in terms of PL tree terminal graphs. Necessary and sufficient conditions are given so that the tree terminal graph corresponding to a given short circuit conductance matrix is a PL tree. These conditions are based entirely on the sign pattern of the entries of the short circuit conductance matrix. Also the necessary and sufficient conditions for the realizability are stated.In Part II, the general question of realizability, determination of an appropriate tree terminal graph and the complete realization of a short circuit conductance matrix are discussed. No tree transformations are used in the realization procedure.