The Degree of a Rational Matrix Function

Duffin, R. J.; Hazony, D.

doi:10.1137/0111049

Cited by 19 publications

(12 citation statements)

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“…This method also realises the network with the minimum number of reactive elements, this number being equal to the degree of Z{s) or Y(s). 27 Since the procedure is identical for Z(s) and Y{s) matrixes, only the realisation of Y(s) is considered. From the given Y(s), the corresponding state model in eqn.…”

Section: Jl-«i --• °8 )mentioning

confidence: 99%

Synthesis of LC networks—a state-model approach

Yarlagadda

Tokad

1966

Proc. Inst. Electr. Eng. UK

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SynopsisIn the paper, a general procedure for realising the state model of an LC «-port network is presented. Realisation procedures of w-port LC networks are well established if the specification is given in the complex-frequency domain. The procedure presented here represents an alternative to these well established methods, and is believed to give new insight into the problem of transformerless realisation of LC networks.List of principal symbols G = network graph T = a tree T = complement of a tree T, cotree B,j = submatrixes in the fundamental circuit matrix C h -diagonal matrix-the element values of tree-branch capacitors C c = diagonal matrix-the element values of link capacitors L h = diagonal matrix-the element values of tree-branch inductors L c -diagonal matrix-the element values of link inductors / 0 = current vector of voltage sources I hc = current vector of tree-branch capacitors J cc = current vector of link capacitors I hl = current vector of tree-branch inductors J cl = current vector of link inductors I x -current source vector V o = voltage source vector V hc = voltage vector of tree-branch capacitors V cc = voltage vector of link capacitors V hl = voltage vector of tree-branch inductors V ct = voltage vector of link inductors V x = voltage vector of current sources Y c -admittance matrix of a capacitor network Z L -impedance matrix of an inductor network IntroductionRecently, it has been recognised by many investigators that the topological approach to the synthesis problem might offer new insight. Indeed, the problem of synthetising n-terminal resistive networks characterised by the impedance or admittance matrixes of order « -1 is completely solved. 1 Although extension of resistive network synthesis to some very special class of multiterminal RLC networks is also known, 2 the solution of the general problem appears, at best, to be very difficult.One logical approach to the RLC network synthesis is through state models, since, in general, such a model of the network provides more direct information about the network topology than do the more conventional network matrixes. Recently, some works have been initiated in this direction. 3 " 6 This paper considers only the synthesis of LC networks.The general form of the state model of an RLC network, in symbolic form, iswhere X is called the state vector and consists of some treebranch capacitor voltages and some link inductor currents, and Y is a vector which contains the specified voltages and currents in the source branches; Y contains the complementary variables; i.e. the complementary currents and voltages, respectively, in the source branches, of those Y.The coefficient matrix A is real and square, and is called the operator matrix. 7 All other matrixes B, C, P, Q and R are real and, in general, rectangular.In the synthesis procedure presented here, it is necessary to establish these coefficients as a function of the network structure. The voltage components in the state vector X in eqn. so that T 2 contains T|. Then T 2 is a tree in G and ...

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Section: Jl-«i --• °8 )mentioning

confidence: 99%

Synthesis of LC networks—a state-model approach

Yarlagadda

Tokad

1966

Proc. Inst. Electr. Eng. UK

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“…In a previous paper Duffin and Hazony [5] studied the properties of the degree of a rational matrix function F(s). It was brought out that the degree may be defined in several equivalent ways.…”

Section: Pole Multiplicity Of a Rational Matrixmentioning

confidence: 99%

The Gyration Operator in Network Theory

Duffin¹,

Hazony²,

Morrison³

1965

Self Cite

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“…McMi I Ian [9,70], and has also been discussed by Duff in and Hazony [3] and Kalman [7]. In attempting to understand its role in these applications to control theory we have been led to an approach which enables the main properties of the degree to be developed very directly.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, since detT = defcffdet? by (3), this realisation is irreducible if and only if the realisation (l) is irreducible.…”

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confidence: 99%

Matrices of rational functions

Coppel

1974

Bull. Austral. Math. Soc.

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The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions. . IntroductionIn the application to control problems of spectral factorisation techniques two slightly different approaches have been used. The first, introduced by Popov [73] and further developed by Jakubovic [6], uses a controllability hypothesis to impose three conditions on the state space matrix A . The second, employed by Anderson [7] and Molinari [77], [72]for rather less general problems, uses the controllability hypothesis to impose only two of these conditions on A . The third condition, that A have distinct eigenvalues, is avoided by using properties of the degree of a matrix of rational functions. The concept of degree was introduced byMcMi I Ian [9, 70], and has also been discussed by Duff in and Hazony [3] and Kalman [7]. In attempting to understand its role in these applications to control theory we have been led to an approach which enables the main properties of the degree to be developed very directly. (By means of Theorem 3 below the third condition on A can be avoided also in the general problem of Popov-Jakubovic.) However, the main merit of this approach is that it is no longer necessary to restrict attention to matrices of rational functions. With trivial changes the definition of determinants! denominators and Theorems

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The Degree of a Rational Matrix Function

Cited by 19 publications

References 9 publications

Synthesis of LC networks—a state-model approach

Synthesis of LC networks—a state-model approach

The Gyration Operator in Network Theory

Matrices of rational functions

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