interconnection of -lossless behaviours

Systems & Control Letters 59 (2010) 323–332

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Systems & Control Letters

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Interconnection of J-lossless behavioursShodhan Rao ∗Control Engineering group, Faculty of Electrical Engineering, University of Twente, EL/CE (Room H.O. 8164), P.O. Box 217, 7500 AE Enschede, The Netherlands

a r t i c l e i n f o

Article history:Received 11 August 2009Received in revised form5 January 2010Accepted 16 March 2010Available online 17 April 2010

Keywords:Behavioural systems theoryQuadratic differential formsLossless electrical networksLossless positive real transfer functionsLinear lossless systemsOscillatory behaviours

a b s t r a c t

In this paper, motivated by the phenomenon of the interconnection of lossless electrical networks, aclass of behaviours known as J-lossless behaviours is introduced, where J is a symmetric two-variablepolynomial matrix. It is shown that for certain values of J , interconnection of J-lossless behaviours leadsto an oscillatory behaviour. Physically this translates to the fact that interconnection of two multi-portlossless electrical networks results in an autonomous lossless electrical network. Finally, the problem ofdecomposition of an oscillatory behaviour with a given characteristic polynomial as an interconnectionof two single-input–single-output behaviours, such that one has a lossless positive real transfer functionand the other has a lossless negative real transfer function is also considered. This problem can be viewedas an inverse problem to the one of interconnection of J-lossless behaviours.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Consider the interconnection of two lossless one-port electricalnetworks as depicted in Fig. 1. It can be inferred that this intercon-nection will result in an autonomous lossless electrical network asit will not have any dissipative component. In this paper, this resultis proved mathematically. The main result of this paper is a gen-eralization of this result for the case of interconnection of multi-port lossless electrical networks which is proved in the frameworkof behavioural systems theory. It is assumed that the reader is fa-miliar with the behavioural framework and with the calculus ofquadratic differential forms, and the interested readers are referredto respectively [1] and [2] for a thorough exposition of the conceptsand mathematical techniques.In [3], losslessness has been studied in the context of nonlinear

network theory. Here, a state representation of a multi-portelectrical network is defined as lossless if the energy required totravel between any two points in the state space is independent ofthe path taken. In [3], it has been also shown that under certainconditions, the interconnection of multi-port networks withlossless state representations has a lossless state representation. Incontrast, the technique used in this paper to prove the oscillatorynature of the interconnection of lossless networks is independentof the representation of its behaviour.In this paper, for a given nonzero finite-dimensional symmetric

two-variable polynomial square matrix J , a class of behaviours

∗ Tel.: +31 534892778; fax: +31 534892223.E-mail addresses: [email protected], [email protected].

0167-6911/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2010.03.007

known as J-lossless behaviours is defined. The main result of thispaper is Theorem 16 where it is proved that for certain J ’s thatare associated with lossless electrical networks, interconnectionof two J-lossless behaviours results in an oscillatory behaviourwhich is a behaviour whose trajectories are linear combinationsof vector sinusoidal functions. Physical examples of oscillatorysystems are mechanical systems consisting of frictionless springsand masses having as external variables the displacements orthe velocities of the masses from the equilibrium positions; andelectrical systems consisting of the interconnection of inductorsand capacitors, having as external variables the voltages across thecapacitors or the currents in the inductance components.In this paper, it is shown that the main result (Theorem 16)

regarding interconnection of J-lossless behaviours is relatedto a well-known result in state-space theory known as theKalman–Yakubovich–Popov (KYP) Lemma, which is also called thepositive real lemma. This lemma holds for systems with transferfunction matrices that are positive real (See Appendix for adefinition). In [4], it has been proved that G is a hybrid transferfunctionmatrix (seeAppendixD) of amulti-port electrical networkconsisting of a finite number of resistors, capacitors, inductors,transformers and gyrators if and only if it is positive real. This resultwas earlier proved for the case of one-port electrical networks byBrune [5]. ThusKYP lemmaholds formulti-port electrical networksconsisting of a finite number of resistors, capacitors, inductors,transformers and gyrators. In this paper, it is shown that theexternal behaviours of certain systems with lossless positive realtransfer functions (see Appendix for a definition) are J-lossless fora certain J . Thus the physical interpretation of the main result ofthis paper is that the interconnection of a certain type of losslesselectrical networks leads to oscillatory systems.

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324 S. Rao / Systems & Control Letters 59 (2010) 323–332

Fig. 1. Interconnection of one-port electrical networks.

The structure of this paper is as follows. In Section 2, some basicconcepts from behavioural systems theory and some propertiesof quadratic differential forms and properties of autonomous,oscillatory and lossless systems are discussed. In Section 3,a special class of behaviours known as J-lossless behavioursis introduced followed by a discussion of properties of theirimage representation and another property of such behaviours.Section 4 presents results pertaining to interconnection ofJ-lossless behaviours and those pertaining to decomposition ofan oscillatory behaviour with a given characteristic polynomial asan interconnection of two SISO behaviours, such that one has alossless positive real transfer function and the other has a losslessnegative real transfer function. Section 5 presents conclusionsbased on the results in Sections 3 and 4.The notation used in this paper is standard: the space of

n-dimensional real, respectively complex vectors is denoted byRn,respectively Cn, the space of m × n real matrices by Rm×n, and thespace of m× m symmetric real matrices, by Rm×m

s . Whenever one ofthe two dimensions is not specified, a bullet • is used; so that forexample,R•×n denotes the set of real matrices with n columns andan unspecified number of rows. In order to enhance readability,when dealing with a vector space R• whose elements are denotedwith w (or x), the notation Rw (or Rx) (note the typewriter fonttype!) is used and when dealing with a vector space R• whoseelements are denoted with `, the notation Rl is used; similarconsiderations hold for matrices representing linear operators onsuch spaces. The ring of polynomials with real coefficients inthe indeterminate ξ is denoted by R[ξ ]; the set of two-variablepolynomials with real coefficients in the indeterminates ζ and η isdenoted byR[ζ , η]. The space of allw×l polynomialmatrices in theindeterminate ξ is denoted byRw×l

[ξ ], and that consisting of allw×l polynomial matrices in the indeterminates ζ and η by Rw×l

[ζ , η].The space of real rational functions in the indeterminate ξ isdenoted by R(ξ) (note the difference in bracket type as comparedto the ring of polynomials) and the space of allmatrices of size w×l,whose entries are real rational functions of the indeterminate ξ aredenoted by Rw×l(ξ). The set of infinitely differentiable functionsfrom R to Rw is denoted by C∞(R,Rw). R+ denotes the set ofpositive real numbers. N denotes the set of positive integers. Ilstands for identity matrix of size l. 0w×l denotes a matrix of sizew × l consisting of zeroes and 0n denotes a vector of dimensionn consisting of zeroes. col(L1, L2) denotes the matrix obtained bystacking the matrix L1 over L2, which has the same number ofcolumns as L1 and row(R1, R2) denotes the matrix obtained bystacking the matrix R2 to the right of R1, which has the samenumber of rows as R2. diag(a1, . . . , an) denotes a block diagonalmatrixwith entries a1, . . . , an along the diagonal in the given orderif a1, . . . , an are real squarematrices. The class of linear differentialbehaviours with infinitely differentiable manifest variable w isdenoted by Lw. det(A) denotes the determinant of a square matrixA. deg(r) denotes the degree of a polynomial r . dim(V ) denotesthe dimension of a vector V . Im(M) denotes the image of a linearmapM .

2. Preliminaries

In this section, the basic definitions and concepts of [1,2,6–8]that are necessary to understand the results given in this paper areillustrated.

2.1. Behaviours

A behaviour B is a subspace of C∞(R,Rw) consisting ofall solutions w of a given system of linear constant-coefficientdifferential equations of the form

R0w + R1dwdt+ · · · + RL

dLwdtL= 0 (1)

where Ri ∈ Rg×w for i = 0, 1, . . . , L. Define the polynomial matrixR ∈ Rg×w

[ξ ] as

R(ξ) := R0 + R1ξ + · · · + RLξ L.

Using the above equation, Eq. (1) can also be written as

R(ddt

)w = 0. (2)

The behaviourB can be defined as

B :=

{w ∈ C∞(R,Rw) | R

(ddt

)w = 0

}.

Thus, considering R( ddt ) as an operator from C∞(R,Rw) toC∞(R,Rg), B = ker

(R( ddt )

). Linearity of the differential operator

R( ddt ) results in linearity of the behaviour B. B is shift-invariantas the coefficients of the polynomial matrix R are constant. Thesystem of linear constant coefficient differential equation (2) iscalled a kernel representation of the behaviourB. It is calledminimalif every other kernel representation ofBhas at leastg rows. The setof behaviours with infinitely often differentiable manifest variablew is denoted by Lw (the superscript w in Lw refers to the dimensionofw ∈ B).When modelling a system, two types of variables namely

manifest variables (denoted by w) and latent variables (denotedby `) are encountered. Manifest variables are the variables whoseevolution with time is of interest, while latent variables are theother variables that come up during the process of modelling. Ifthe system under consideration is a linear differential system, thenthe trajectories belonging to the system can be described by a setof linear constant coefficient ordinary differential equations

R(ddt

)w = M

(ddt

)` (3)

whereR ∈ Rg×w[ξ ] andM ∈ Rg×l

[ξ ]. The above equationdescribesthe full behaviour

Bf := {(w, `) ∈ C∞(R,Rw+l) | (3) holds}

and the projection ofBf on thew variable, i.e.

B := {w | ∃` such that (3) holds}

is called themanifest behaviour associated with (3). If R(ξ) = Iw inEq. (3),

w = M(ddt

)`. (4)

The above equation is called an image representation of B. Itcan be showed that an image representation exists for B iff B iscontrollable in the behavioural sense (see chapter 5 of [1]). Theimage representation (4) is called observable if ` is observable fromw, i.e if [w = M( ddt )` = 0] H⇒ [` = 0] (see chapter 5 of [1] fora description of observability). Using Theorem 5.3.3, p. 174 of [1],it can be proved that this is the case if and only if the matrixM(λ)has full column rank for all λ ∈ C. It can also be proved that ifB iscontrollable, then there exists an observable image representationofB.

S. Rao / Systems & Control Letters 59 (2010) 323–332 325

In this paper the concepts of state and of state representationare also used (see [6] for a thorough discussion). A latent variable` is a state variable for B if B admits a representation (3) of firstorder in ` and zeroth order in w : E d`dt + F` + Gw = 0. Such arepresentation is called a state-space representation ofB. The statevariable ` of B is often denoted by x. By combining the notionof state with that of inputs and outputs one can arrive at theinput/state/output (i/s/o) representation

dxdt= Ax+ Bu

y = Cx+ Du (5)w = col(y, u).

The state representation (5) is said to beminimal if any other staterepresentation of B with state variable x1 is such that dim(x1) ≥dim(x). If the state representation (5) is minimal, then it can beshown that x is observable from w, i.e [w = 0] H⇒ [x = 0]. Insuch a case, there exists X ∈ Rx×w

[ξ ], with x = dim(x), such thatx = X( ddt )w.

2.2. Quadratic differential forms

Consider the set of bilinear functionals acting on an infinitelydifferentiable trajectoryw of the form

QΦ(w) =N∑

h,k=0

(dhwdth

)>Φh,k

(dkwdtk

)(6)

where Φh,k are w × w-dimensional real matrices, and N isa nonnegative integer. Such a functional is called a quadraticdifferential form (QDF). With the QDF given by Eq. (6), is associateda two-variable polynomial matrixΦ(ζ , η), which is given by

Φ(ζ , η) =

N∑h,k=0

Φh,kζhηk.

A QDF QΦ is called symmetric ifΦ(ζ , η) = Φ(η, ζ )>. The notion ofthe derivative of a QDF is defined below.

Definition 1. A QDF QΨ is called the derivative of a QDF QΦ withΦ ∈ Rw×w

s [ζ , η], denoted by QΨ =ddtQΦ if

ddtQΦ(w) = QΨ (w) for

allw ∈ C∞(R,Rw).

In terms of the two-variable polynomial matrices associated withQΨ and QΦ , the relationship in Definition 1 can be expressedas follows: QΨ is the derivative of QΦ if and only if for thecorresponding two-variable polynomial matrices, there holds (ζ +η)Φ(ζ , η) = Ψ (ζ , η) (see [2], p. 1710).Defined below are the notions of nonnegativity and positivity

of QDFs.

Definition 2. Let Φ ∈ Rw×ws [ζ , η]. QΦ is said to be nonnegative,

denoted by QΦ ≥ 0 if QΦ(w) ≥ 0 for all w ∈ C∞(R,Rw); andpositive, denoted by QΦ > 0, if QΦ ≥ 0, and [QΦ(w) = 0] H⇒[w = 0].

The notion of a QDF being nonnegative or positive along aparticular behaviour is defined below.

Definition 3. Let Φ ∈ Rw×ws [ζ , η]. QΦ is said to be nonnegative

along B, denoted by QΦB

≥ 0 if QΦ(w) ≥ 0 for all w ∈ B, and

positive along B, denoted by QΦB> 0, if QΦ

B

≥ 0 and [QΦ(w) = 0]H⇒ [w = 0].

Given below are algebraic conditions on the two-variablepolynomial matrix corresponding to a QDF under which it isnonnegative and positive along a given behaviour.

Proposition 4. Let Φ ∈ Rw×ws [ζ , η] and let B = ker

(R( ddt )

). Then

1. QΦB

≥ 0 iff there exists F ∈ R•×w[ζ , η] and D ∈ R•×w[ξ ], such that

Φ(ζ , η) = D(ζ )>D(η)+ F(η, ζ )>R(η)+ R(ζ )>F(ζ , η).

2. QΦB> 0 iff QΦ

B

≥ 0 and col(D(λ), R(λ)

)has full column rank for all

λ ∈ C.

Proof. See Proposition 3.5 of [2]. �

2.3. Autonomous, oscillatory and lossless systems

An autonomous system is a system with no free variables.For such a system, the future of every trajectory is completelydetermined by its past. A formal definition for an autonomousbehaviour is given below.

Definition 5. A behaviour B is called autonomous if for allw1, w2 ∈ B,

[w1(t) = w2(t) ∀t ≤ 0] H⇒ [w1 = w2].

The following proposition relates the property of autonomy ofa multi-variable behaviour to algebraic properties of a matrix Rinducing a kernel representation of the behaviour.

Proposition 6. Let B = ker(R( ddt )

), with R ∈ Rg×w

[ξ ], be a kernelrepresentation of B ∈ Lw. ThenB is autonomous iff R has full columnrank.

The invariant polynomials of a polynomial matrix P ∈ Rw×w[ξ ]

are the diagonal elements of the Smith form (see Section 6.3-3, [9]for a definition) of P . Let B = ker

(R( ddt )

)be a minimal kernel

representation of an autonomous behaviourB. Then the invariantpolynomials of R are also called the invariant polynomials ofB.An oscillatory behaviour is defined below.

Definition 7. A behaviourB defines an oscillatory system if everysolutionw : R→ Rw ofB is bounded on R.

From the definition, it follows that an oscillatory system isnecessarily autonomous: if there were any input variables in w,then those components of w could be chosen to be unbounded.It was proved in Proposition 2 of [7] that any behaviour B isoscillatory if and only if every nonzero invariant polynomial ofB has distinct and purely imaginary roots. In the following, apolynomial matrix will be called oscillatory if all its invariantpolynomials have distinct and purely imaginary roots.The notion of a conserved quantitywas first defined in [7], and it

is used for defining lossless systems. This definition is given below.

Definition 8. Let B be an autonomous behaviour. A QDF QΦ is aconserved quantity forB if

ddtQΦ(w) = 0 ∀w ∈ B.

Thus, conserved quantity is a QDF, whose derivative is zero alongthe trajectories of a given behaviour. The notion of an autonomouslossless system as in [8], is defined below.

Definition 9. An autonomous behaviourB ∈ Lw is lossless if thereexists a conserved quantityQE associatedwithB, such thatQE > 0.Such a QE is called an energy function for the system.


Themain result of [8]which is used in this paper, is given below.

Theorem 10. An autonomous behaviour B ∈ Lw is lossless if andonly if it is oscillatory.

Proof. See proof of Theorem 3, p. 1529, [8]. �

Given below is a modified version of Theorem 10 which is usedto prove the main result of this paper.

Theorem 11. A behaviour B is oscillatory iff there exists a QDF QE ,

such that QEB> 0 and ddtQΦ(w) = 0∀w ∈ B.

Proof. (Only if ): If B is oscillatory, then from Theorem 10, itfollows that there exists an energy function QE which is conserved

along B, and QE > 0. This implies that QEB> 0 and d

dtQΦ(w) =0∀w ∈ B. Hence the proof.

(If ): Now assume that there exists a QDF QE , such that QEB> 0

and ddtQΦ(w) = 0∀w ∈ B. It is first proved thatB is autonomous.LetB = ker

(R( ddt )

). Let R = U∆V be a Smith form decomposition

of R. Define R1 := ∆V . Since U is unimodular, B = ker(R1( ddt )

)is another kernel representation ofB. Since QE

B> 0, from Proposi-

tion 4, it follows that there exist F ∈ R•×w[ζ , η] and D ∈ R•×w[ξ ],such that

E(ζ , η) = D(ζ )>D(η)+ F(η, ζ )>R1(η)+ R1(ζ )>F(ζ , η)

and col(D(λ), R1(λ)) has full column rank for all λ ∈ C. De-fine V1 := V−1 and D1 := DV1. Then it is easy to see thatcol(D1(λ),∆(λ)) has full column rank for all λ ∈ C. Now assumeby contradiction that B is not autonomous. Then from Proposi-tion 6, it follows that R1 does not have full column rank. It followsthat∆ has its last column full of zeroes. Let Dw ∈ R•[ξ ] denote thelast column of D1. Since col(D1(λ),∆(λ)) has full column rank forall λ ∈ C, it follows that Dw(λ) 6= 0 for all λ ∈ C. Define w1 :=col(0w−1, eλ1t) where λ1 ∈ R is nonzero and w2 := V1( ddt )w1. Ob-serve thatw2 ∈ B, and hence

QE(w2) =(D(ddt

)w2

)> (D(ddt

)w2

)=

(D1

(ddt

)w1

)> (D1

(ddt

)w1

)=

(Dw

(ddt

)eλ1t

)> (Dw

(ddt

)eλ1t

)= Dw(λ1)>Dw(λ1)e2λ1t .

Differentiating the above equation with respect to time, we get

ddtQE(w2) = 2λ1Dw(λ1)>Dw(λ1)e2λ1t .

Since w2 ∈ B, ddtQE(w2) = 0. This implies that Dw(λ1)>Dw(λ1) =

0,which in turn implies thatDw(λ1) = 0 asDw(λ1)>Dw(λ1) is a sumof squares of real numbers. This is a contradiction. This proves thatB is autonomous.Next in order to prove thatB is oscillatory, the case ofB ∈ L1

is first dealt with. Consider the (Only If ) part of the proof of Theo-rem 1, p. 1524, [8]. Here it has been proved that if an autonomousbehaviourB ∈ L1 is not oscillatory then there does not exist a con-

served quantity QE , such that QEB> 0. This proves the theorem for

any autonomousB ∈ L1. The theorem can now be proved for thecase of a multi-variable autonomous behaviour B along the samelines as the (Only If ) part of the proof of Theorem3, p. 1529, [8]. �

3. J-lossless behaviours

In this section, the notion of a J-lossless behaviour is introducedand its properties are discussed.

Definition 12 (J-Lossless Behaviour). Let J ∈ Rw×ws [ζ , η] be such

that J 6= 0. A behaviourB ∈ Lw is said to be J-lossless if there exists

a QDF QEB> 0 with E ∈ Rw×w

s [ζ , η], such that for every trajectoryw ∈ B, QJ(w) = d

dtQE(w).

Remark 13. The concept of J-losslessness is related to the conceptof half-line nonnegativity of QDFs described in pp. 1725–1726of [2]. In [2], the concept of average nonnegativity of a QDF isalso discussed. Here, it is proved that a QDF QΦ being averagenonnegative is equivalent with the existence of another QDF QΨcalled as a storage function, which obeys

ddtQΨ ≤ QΦ

and QΦ being half-line nonnegative is equivalent with theexistence of a storage function QΨ , which obeys

ddtQΨ ≤ QΦ

and QΨ ≥ 0. In Remark 5.9, p. 1723 of [2], a behaviourB is said tobe lossless or conservative with respect to a supply rate QΦ if thereexists a storage functionQΨ , such that ddtQΨ (w) = QΦ(w)∀w ∈ B.Assume that a given behaviour B is conservative as per this defi-nition, and controllable. This implies thatB has an observable im-age representation given by B = Im

(M( ddt )

). Define Ψ ′(ζ , η) :=

M(ζ )>Ψ (ζ , η)M(η) and Φ ′(ζ , η) = M(ζ )>Φ(ζ , η)M(η). Then itfollows thatddtQΨ ′ = QΦ′ .

This implies that QΦ′ is average nonnegative. Extending the defi-nition of conservative behaviours so as to include the concept ofhalf-line nonnegativity, one can say that a behaviour B is half-lineconservativewith respect to a supply rate QΦ if there exists a storage

function QΨB

≥ 0, such that ddtQΨ (w) = QΦ(w)∀w ∈ B. Observethat ifB is half-line conservative and controllable with an observ-able image representation given byB = Im

(M( ddt )

), thenQΦ′ with

Φ ′(ζ , η) := M(ζ )>Φ(ζ , η)M(η), is half-line nonnegative. Accord-

ing to Definition 12, B is Φ-lossless if there exists a QDF QΨB> 0,

such that ddtQΨ (w) = QΦ(w)∀w ∈ B. Observe that definition of aJ-lossless behaviour is related to the definition of a half-line con-servative behaviour with respect to QJ , but not exactly the same.

The next Lemma gives algebraic conditions on an observableimage representation of a controllable behaviour B for it to beJ-lossless.

Lemma 14. Consider a controllable behaviour B ∈ Lw for whichan observable image representation is B = Im

(M( ddt )

)with M ∈

Rw×l[ξ ]. Let J ∈ Rw×w

s [ζ , η] be such that J 6= 0.B is J-lossless if andonly if the following hold:1. M(−ξ)>JM(ξ) = 0.2. Φ(ζ , η) := M(ζ )>JM(η)

ζ+ηis such that QΦ > 0.

Proof. (If): Assume that M(−ξ)>JM(ξ) = 0. Consider a trajec-tory w = M( ddt )`. By assumption, ` is observable from w. Con-sequently, there exists F ∈ Rl×w[ξ ], such that ` = F( ddt )w. Nowdefine E(ζ , η) := F(ζ )>Φ(ζ , η)F(η). Then

QE(w) = QΦ

(F(ddt

)w

)= QΦ(`).


It can be easily verified that ddtQE(w) = QJ(w). Now assume thatQΦ > 0. Then it is easy to see that QE(w) > 0 for any nonzerotrajectory w ∈ B, and QE(w) = 0 implies that w = 0. HenceB isJ-lossless if QΦ > 0.(Only If): Assume that B is J-lossless. Consider a trajectory

w ∈ B given by w = M( ddt )`. Since B is J-lossless, there exists

E ∈ Rw×ws [ζ , η], such that QE

B> 0 and d

dtQE(w) = QJ(w). DefineJ ′(ζ , η) := M(ζ )>JM(η) and Φ(ζ , η) := M(ζ )>E(ζ , η)M(η) andobserve that

QJ ′(`) =ddtQΦ(`). (7)

We have from Eq. (7),

M(ζ )>JM(η) = (ζ + η)Φ(ζ , η).

From the above equation, it follows that M(−ξ)>JM(ξ) = 0 andΦ(ζ , η) =

M(ζ )>JM(η)ζ+η

. We have QE(w) = QΦ(`) for all (w, `), such

that w = M( ddt )`. Since QEB> 0, it follows that QΦ > 0. Hence the

claim. �

Given below is another property of J-lossless behaviours whichwill be useful in proving some of the results of this paper.

Lemma 15. Consider a J-lossless behaviour B ∈ Lw. Let A ∈ Rw×w

be an invertible matrix. Define B := A−1B, and Φ(ζ , η) := A>J(ζ , η)A. Then B isΦ-lossless.

Proof. Consider a trajectory w ∈ B. Define w := A−1w, and notethat w ∈ B. SinceB is J-lossless, there exists E ∈ Rw×w

s [ζ , η], such

that QEB> 0 and d

dtQE(w) = QJ(w). Now observe that

QJ(w) = QJ(Aw) = QΦ(w).

Define E(ζ , η) := A>E(ζ , η)A and observe that QE(w) = QE(Aw)= QE(w). This implies that

ddtQE(w) = QΦ(w). (8)

If w 6= 0, then QE(w) > 0. This implies that QE(w) > 0 if w 6= 0.Since [QE(w) = 0] H⇒ [w = 0] and A is invertible, it follows that[QE(w) = 0] H⇒ [w = 0]. Hence fromDefinition 2, it follows that

QEB> 0. (9)

It now follows from Eqs. (8) and (9) that B isΦ-lossless. �

4. Interconnection of J-lossless behaviours

4.1. Main result

The main result of this paper is given below.

Theorem 16. Define

J :=[0l×l IlIl 0l×l

](10)

Σ :=

[−Il 0l×l0l×l Il

]. (11)

Consider two J-lossless behavioursB1,B2 ∈ L2l. DefineB′2 := ΣB2.Then the behaviour B = B1 ∩B′2 is oscillatory.

Proof. Consider a trajectory w = col(w1, w2) ∈ B, where w1,w2 ∈ C∞(R,Rl). It is easy to see that w ∈ B1 and Σw ∈ B2,or col(−w1, w2) ∈ B2. Since B1 is J-lossless, there exists E1 ∈

R2l×2ls [ζ , η], such that QE1B1> 0, and

QJ(w) = w>1 w2 + w>

2 w1 =ddtQE1(w). (12)

SinceB2 is also J-lossless, there exists E2 ∈ R2l×2ls [ζ , η], such that

QE2B2> 0, and

QJ(Σw) = −w>1 w2 − w>

2 w1 =ddtQE2(Σw) =

ddtQE′2(w) (13)

where E ′2(ζ , η) := Σ>E2(ζ , η)Σ . Define E(ζ , η) := E1(ζ , η) +E ′2(ζ , η). Adding Eqs. (12) and (13), we get

ddtQE(w) = 0. (14)

Note that Σ> = Σ−1 = Σ . Let B2 = ker(R2( ddt )

)denote a

minimal kernel representation forB2. SinceQE2B2> 0, fromProposi-

tion 4, it follows that there exist F ∈ R•×2l[ζ , η] andD2 ∈ R•×2l[ξ ],such that

E2(ζ , η) = D2(ζ )>D2(η)+ F(η, ζ )>R2(η)+ R2(ζ )>F(ζ , η)

and col(D2(λ), R2(λ)) has full column rank for all λ ∈ C. It followsthat

E ′2(ζ , η) =(D2(ζ )Σ

)>(D2(η)Σ

)+(F(η, ζ )Σ

)>(R2(η)Σ

)+(R2(ζ )Σ

)>(F(ζ , η)Σ

).

Observe that B′2 = ker(R2( ddt )Σ

)is a minimal kernel represen-

tation for B′2. Since Σ is nonsingular, col(D2(λ)Σ, R2(λ)Σ) hasfull column rank for all λ ∈ C. From Proposition 4, it follows that

QE′2B′2> 0. Now observe that

QE(w) = QE1(w)+ QE′2(w) (15)

for any trajectoryw ∈ B. Observe that

[QE1B1> 0] H⇒ [QE1

B> 0] and [QE′2

B′2> 0] H⇒ [QE′2

B> 0].

This implies that for any nonzero trajectoryw ∈ B, the right handside of Eq. (15) is positive. If QE(w) = 0, then QE1(w) = QE′2(w) =

0, which imply that w = 0. Hence QE(w)B> 0. Since Eq. (14) holds

for every trajectory w ∈ B, from Theorem 11, it follows that B isoscillatory. �

Remark 17. With reference to Theorem 16, let

B1 =

{col(I1, V1) ∈ C∞(R,R2l) | D1

(ddt

)I1 = N1

(ddt

)V1

}B2 =

{col(I2, V2) ∈ C∞(R,R2l) | D2

(ddt

)I2 = N2

(ddt

)V2

}beminimal kernel representations ofB1 andB2 whereD1,N1,D2,N2 ∈ R•×l. Define

R :=[D1 −N1D2 N2

].

Then a kernel representation forB is

B =

{w ∈ C∞(R,R2l) | R

(ddt

)w = 0

}.


Given below is a corollary of Theorem 16. It will be shownlater that this result can be used to prove that interconnection oftwo lossless networks leads to an oscillatory systemwhenever thenetworks are described by scattering representations.

Corollary 18. Let J and Σ be defined by Eqs. (10) and (11) respec-tively. Consider two Σ-lossless behaviours B1,B2 ∈ L2l. Define B′2:= JB2. Then the behaviour B = B1 ∩B′2 is oscillatory.

Proof. Define Bi := ΞBi for i = 1, 2, where

Ξ :=1√2

[−Il IlIl Il

].

Observe that Ξ> = Ξ−1 = Ξ , and ΞΣΞ = J . From Lemma 15,it follows that B1 and B2 are J-lossless. Define B := ΞB, B′2 :=ΞB′2. It is easy to see that B = B1 ∩ B′2. Now observe that

B′2 = ΞB′2 = Ξ JB2 = Ξ JΞB2 = ΣB2.

It now follows from Theorem 16 that B is oscillatory. This impliesthatB is also oscillatory. �

4.2. Relation with Kalman–Yakubovich–Popov lemma

In this section, it is shown that there is a relationship betweenthe lossless version of Kalman–Yakubovich–Popov (KYP) lemmaand Theorem 16. Throughout this section J and Σ are given byEqs. (10) and (11) respectively. In the following, using KYP lemma,it is shown that certain systems with lossless positive real transferfunctions (see Appendix for a definition) are J-lossless. The losslessversion of KYP lemma (see [4], pp. 221–222) is given below:

Lemma 19. Consider a multi-variable, linear, time-invariant systemdescribed by the following state-space representation:

dx(t)dt= Ax(t)+ Bu(t) (16)

y(t) = Cx(t)+ Du(t)

where x(t) ∈ Rx, u(t), y(t) ∈ Rl with x ≥ l. Suppose that thestate-space representation (16) is minimal. Then the transfer functionmatrix G(s) = C(sIx − A)−1B + D corresponding to the system islossless positive real if and only if there exists P ∈ Rx×x, such thatP = P> > 0 and

A>P + PA = 0B>P = CD+ D> = 0.

The following lemmagives a condition underwhich a behaviourwith a lossless positive real transfer function is J-lossless.

Lemma 20. Consider a behaviour B ∈ L2l with an output–inputpartition col(y, u)

(y, u ∈ C∞(R,Rl)

)and a minimal state-space

representation

dxdt= Ax+ Bu (17)

y = Cx+ Du.

Assume that B has full column rank and the transfer function matrixof B given by G(ξ) = C(ξ Ix − A)−1B + D is lossless positive real.ThenB is J-lossless.

Proof. Since G is lossless positive real, from KYP lemma, it followsthat D+D> = 0 and there exists P ∈ Rx×x, such that P = P> > 0,A>P + PA = 0 and B>P = C . Define S = x>Px. Then it can beverified thatdSdt= u>y+ y>u.

Define w := col(y, u). Since the state-space representation (17) isminimal, the system is state-observable, or that [w = 0] H⇒ [x =0]. This implies that there exists a matrix X ∈ Rx×l

[ξ ], such thatx = X( ddt )w. Define E(ζ , η) := X(ζ )

>PX(η) and observe that

S = x>Px =(X(ddt

)w)>PX(ddt

)w = QE(w).

Observe that ddtQE(w) = QJ(w). Since B has full column rank, itfollows that there exists B1 ∈ Rl×x, such that B1B = Il. Since P ispositive definite, it follows that [S = x>Px = 0] H⇒ [x = 0] H⇒[u = 0, y = 0] H⇒ [w = 0]. From Definition 3, it follows that

QEB> 0. From Definition 12, it now follows thatB is J-lossless. �

In [4,10], it has been proved that G is a hybrid transfer func-tion matrix (see Appendix D) of a multi-port linear electrical net-work consisting of a finite number of ideal capacitors, inductors,transformers and gyrators if and only if it is lossless positivereal. Now consider such a multi-port electrical network for whichV1 = col(V11, V12, . . . , V1l) denotes the vector of port voltagesand I1 = col(I11, I12, . . . , I1l) denotes the vector of respectiveport-currents. Consider partitions of V1 and I1 given by V1 =col(V11, V12) and I1 = col(I11, I12) such that dim(V11) ∈ {0, 1,. . . , l}, dim(I11) = dim(V11) and the transfer function from u1 :=col(V11, I12) to y1 := col(I11, V12) exists and is a hybrid transferfunction for the network. Let G1 ∈ Rl×l(ξ) denote this transferfunctionmatrix. Then G1 is lossless positive real. LetB1 denote thespace of all admissible trajectories col(y1, u1). From KYP lemma, aminimal state-space representation for the network given by

dx1dt= A1x1 + B1u1 (18)

y1 = C1x1 + D1u1

is such that there exists P1 ∈ Rx1×x1 with P1 = P>1 > 0 and

ddt(x>1 P1x1) = u

>

1 y1 + y>

1 u1 = V>

11I11 + I>11V11 + V>

12I12 + I>12V12

= V>1 I1 + I>1 V1.

Define l1 := dim(V11). Define

P1 :=

Il1 0l1×(l−l1) 0l1×l1 0l1×(l−l1)0(l−l1)×l1 0(l−l1)×(l−l1) 0(l−l1)×l1 Il−l10l1×l1 0l1×(l−l1) Il1 0l1×(l−l1)0(l−l1)×l1 Il−l1 0(l−l1)×l1 0(l−l1)×(l−l1)

.DefineH1 :=

[Il 0l

]P1and K1 :=

[0l Il

]P1,w := col(I1, V1) and

observe that y1 = H1w and u1 = K1w. Eq. (18) can thus be writtenas

dx1dt= A1x1 + B1K1w (19)

H1w = C1x1 + D1K1w.

Now consider anothermulti-port linear electrical network con-sisting of a finite number of capacitors, inductors, transformers andgyrators for which V2 = col(V21, V22, . . . , V2l) denotes the vec-tor of port voltages and I2 = col(I21, I22, . . . , I2l) denotes thevector of respective port-currents. Consider partitions of V2 andI2 given by V2 = col(V21, V22) and I2 = col(I21, I22) such that


Fig. 2. Interconnection of multi-port electrical networks.

dim(V21) ∈ {0, 1, . . . , l}, dim(I21) = dim(V21) and the transferfunction from u2 := col(V21, I22) to y2 := col(I21, V22) exists andis a hybrid transfer function for the network. Let G2 ∈ Rl×l(ξ) de-note this transfer function matrix. Then G2 is lossless positive real.Let B2 denote the space of all admissible trajectories col(y2, u2).Define l2 := dim(V21). Define

P2 :=

Il2 0l2×(l−l2) 0l2×l2 0l2×(l−l2)0(l−l2)×l2 0(l−l2)×(l−l2) 0(l−l2)×l2 Il−l20l2×l2 0l2×(l−l2) Il2 0l2×(l−l2)0(l−l2)×l2 Il−l2 0(l−l2)×l2 0(l−l2)×(l−l2)

.Define H2 :=

[Il 0l

]P2and K2 :=

[0l Il

]P2and observe that

y2 = H1col(I2, V2) and u2 = K2col(I2, V2). Let a minimal state-space representation for the network be given by

dx2dt= A2x2 + B2u2 (20)

y2 = C2x2 + D2u2.

Now consider the interconnection of the two networks as depictedin Fig. 2. From Kirchhoff’s voltage and current laws, we obtainV1 = V2 and I1 = −I2. Eq. (20) can then be written as

dx2dt= A2x2 + B2K2Σw (21)

H2Σw = C2x2 + D2K2Σw.

From KYP lemma, it follows that there exists P2 ∈ Rx2×x2 such thatP2 = P>2 > 0 and

ddt(x>2 P2x2) = u

>

2 y2 + y>

2 u2

= V>21I21 + I>21V21 + V>

22I22 + I>22V22= V>2 I2 + I>2 V2 = −V

>

1 I1 − I>1 V1.

Define

Bx := {col(x1, x2) ∈ C∞(R,Rx1+x2) | ∃w

such that (19) and (21) hold}.

Define E := diag(P1, P2) and observe that QEBx> 0 since P1 and P2

are positive definite. Also observe that ddtQE(x) = 0 for all x ∈ Bx.Hence from Theorem 11, it follows thatBx is oscillatory, i.e the in-terconnection of two multi-port lossless networks leads to an os-cillatory behaviour.Now assume that B1 and B2 in Eqs. (18) and (20) respectively

have full column rank. From Lemma 20, it follows thatB1 andB2are J-lossless. If B denotes the space of all admissible trajectoriescol(I1, V1) corresponding to the interconnected network, then it iseasy to see that B =

(P>1 B1

)∩(ΣP>2 B2

). Define B′1 := P

>

1 B1

and B′2 := P>

2 B2. It can be verified that P>1 = P−11 = P1, P

>

2 =

P−12 = P2 and P>

1 JP1 = P>

2 JP2 = J . Consequently from Lemma 15,it follows that B′1 and B′2 are J-lossless. Since B = B′1 ∩ (ΣB′2),from Theorem 16, it now follows thatB is oscillatory.

Remark 21. The significance of Corollary 18 is now explained.Consider amulti-port linear electrical network consisting of a finitenumber of ideal capacitors, inductors, transformers and gyratorswith a given scattering description (see pp. 30–31 of [4]). Let V1 andI1 denote the vectors of port voltages and currents respectivelywith equal dimension l. In a scattering description of the network,the external variables are the incident voltage V i1 :=

12 (V1+I1) and

the reflected voltage V r1 =12 (V1− I1) and the transfer function S ∈

Rl×l(ξ) from V i1 to Vr1 is called the scattering matrix of the network,

i.e if the governing law of the network is P( ddt )Vi1 = Q ( ddt )V

r1 ,

then the scattering matrix is S := PQ−1. LetB1 be the space of allpossible trajectories col(

√2V r1 ,√2V i1) corresponding to the given

network. Define

Ξ :=1√2

[−Il IlIl Il

]and B1 := ΞB1. It is easy to see that B1 consists of all admissibletrajectories col(I1, V1) corresponding to the network. Assume thatthe given network is such that B1 is J-lossless. Observe that Ξ =Ξ−1 = Ξ> and Ξ JΞ = Σ . From Lemma 15, it follows thatB1 isΣ-lossless.Now consider another network for which V r2 and V

i2 denote

the reflected and incident voltages. Assume that like in the caseof the earlier network, the space B2 of all admissible trajectoriescol(√2V r2 ,√2V i2) isΣ-lossless. Now consider the interconnection

of the two networks as depicted in Fig. 2, i.e interconnection insuch a way that V r1 = V

i2 and V

i1 = V

r2 . If B denotes the space of

all admissible trajectories col(√2V r1 ,√2V i1) corresponding to the

interconnected network, then it is easy to see thatB = B1∩(JB2).From Corollary 18, it follows thatB is oscillatory.

4.3. Interconnection of one-port lossless electrical networks

Consider a lossless one-port electrical network, for which thesystem equation is

n(ddt

)V = d

(ddt

)I (22)

where V and I denote the voltage across the port and the currentthrough the network respectively. Define B as the set of alladmissible trajectories col(I, V ) : R → R2 that obey Eq. (22).From the theory of electrical networks, it follows that Z defined byZ := n

d is lossless positive real. Assume thatB is controllable. KYPlemma can nowbe used in order to prove thatB is J-losslesswhere

J :=[0 11 0

].

Consider first the case, where deg(n) < deg(d). Since nd is losslesspositive real, observe that a minimal state-space representation ofB given by

dxdt= Ax+ BV

I = Cx+ DV

will be such that D = 0, B ∈ Rx and B 6= 0, because from KYPLemma, it follows that [B = 0] H⇒ [C = 0] H⇒ [I = 0], which isnot true. From Lemma 20, it follows thatB is J-lossless. In a similarway, it can be proved thatB is J-lossless also if deg(n) > deg(d).J-losslessness ofB can also be proved by anothermethodwhich

uses properties of Hurwitz polynomials and of positive QDFs. Thismethod has the advantage that it does not refer to state-spacerealizations. This method is now given. From the controllability ofB, it follows that n and d are co-prime. Since nd is lossless positive


Fig. 3. Interconnection of one-port electrical networks.

real, from the material in Appendix C, it follows that both d and nare oscillatory, one of them is even and the other is odd and thepurely imaginary roots of one are interlaced between those of theother. From Theorem 24, Appendix, it follows that n+d is Hurwitz.It is easy to see thatB = Im

(M( ddt )

), whereM := col(n, d). Define

Φ(ζ , η) :=n(ζ )d(η)+ d(ζ )n(η)

ζ + η.

From Theorem 23, Appendix, it follows that QΦ > 0. Consequentlyfrom Lemma 14, it follows thatB is J-lossless.Now consider the interconnection of two lossless one-port

electrical networks as depicted in Fig. 3. Let the system equationsfor the two networks be given by

n1

(ddt

)V1 = d1

(ddt

)I1

n2

(ddt

)V2 = d2

(ddt

)I2.

Assume that each of the sets {n1, d1} and {n2, d2} consists of co-prime polynomials. When the two networks are interconnected asdepicted in the figure, from Kirchhoff’s voltage and current laws,we obtain V1 = V2 and I1 = −I2. From the discussion of Sec-tion 4.2, it follows that the behaviourB consisting of all admissibletrajectories col(I1, V1) corresponding to the interconnected net-work is oscillatory. This result can also be proved by making use ofproperties of Hurwitz polynomials and of autonomous lossless be-haviours as follows. Observe that the characteristic equation for theresulting autonomous system is r := n1d2+ n2d1. It is now provedthat r is oscillatory which implies that the resulting autonomoussystem is lossless.It is known that (n1+d1) and (n2+d2) are both Hurwitz. Hence

their product

p = (n1 + d1)(n2 + d2)= (n1n2 + d1d2)+ (n1d2 + n2d1)

is also Hurwitz. Consider four cases.

• Case 1: n1 and n2 are even and d1 and d2 are odd. In this case, ris the odd part of p and hence from Theorem 24, it is oscillatory.• Case 2: n1 and d2 are even and n2 and d1 are odd. In this case, r isthe even part of p and hence from Theorem 24, it is oscillatory.• Case 3: d1 and n2 are even and n1 and d2 are odd. In this case, r isthe even part of p and hence from Theorem 24, it is oscillatory.• Case 4: d1 and d2 are even and n1 and n2 are odd. In this case, ris the odd part of p and hence from Theorem 24, it is oscillatory.

This proves that the interconnection of two lossless one-portnetworks of the type depicted in Fig. 3 always results in a losslessautonomous system.The problem of decomposition of an oscillatory behaviour with

a given characteristic polynomial as an interconnection of twoSISO behaviours, such that one has a lossless positive real transferfunction and the other has a lossless negative real transfer functionis now considered and an algorithm is provided for the same. Notethat this problem can be considered as an inverse problem to theone where an autonomous behaviour which is an interconnectionof two SISO behaviours such that one has a positive real transfer

function and the other has a negative real transfer function isanalysed. The solution to this problem also provides ways ofdecomposing an autonomous electrical lossless circuitwith a givencharacteristic polynomial as an interconnection of two one-portlossless electrical circuits.

Algorithm 22. Data: An oscillatory even polynomial r ∈ R[ξ ] ofdegree 2m.Output: Two J-lossless behaviours B1 and B2, such that B =

B1 ∩B′2 has its characteristic polynomial equal to r , where

J :=[0 11 0

],

B′2 := ΣB2, with

Σ :=

[−1 00 1

].

Step 1 Either choose an odd polynomial r1 ∈ R[ξ ] of degree2m + 1 in such way that the roots of r are interlacedbetween those of r1 or choose an odd polynomial ofdegree 2m − 1 in such a way that the roots of r1 areinterlaced between those of r .

Step 2 Factorize the polynomial r + r1 into two factors p, q ∈R[ξ ], i.e r + r1 = pq. Let pe and po be the even and oddparts of p and let qe and qo be the even and odd parts of q.

Step 3 Output:

B1 = ker[pe

(ddt

)−po

(ddt

)]B2 = ker

[−qo

(ddt

)qe

(ddt

)].

With reference to the above algorithm, observe that

r + r1 = (peqe + poqo)+ (peqo + poqe).

Define s1 := peqe + poqo and s2 := peqo + poqe. Then it is easyto see that s1 is even and s2 is odd, and hence s1 = r which is thecharacteristic polynomial ofB1 ∩B′2. Since r and r1 obey interlac-ing property, from Theorem 24 it follows that r+r1 is Hurwitz, andconsequently both p and q are Hurwitz. Conclude from Theorem24that both the pairs (pe, po) and (qe, qo) obey interlacing property.DefineM1 := col(po, pe) andM2 := col(qe, qo). It is easy to see thatB1 = Im

(M1( ddt )

), andB2 = Im

(M2( ddt )

). Define

Φ1(ζ , η) :=pe(ζ )po(η)+ po(ζ )pe(η)

ζ + η

Φ2(ζ , η) :=qe(ζ )qo(η)+ qo(ζ )qe(η)

ζ + η.

From Theorem 23, it follows that QΦ1 and QΦ2 are both positive.Consequently by definition, B1 and B2 are J-lossless. This provesthe correctness of Algorithm 22. Observe also that the transferfunctions po/pe and −qe/qo corresponding to the behaviours B1andB′2 are lossless positive real and lossless negative real respec-tively.Observe that in step 1 of Algorithm 22, there are infinite

number of ways of choosing r1 such that rr1 is lossless positivereal. For each of these ways, in step 2, there are a finite numberof ways of factorizing (r + r1). Hence, there are infinite numberof ways of choosing two behaviours with lossless positive realand lossless negative real transfer functions respectively, such thattheir intersection has for its characteristic polynomial, a givenoscillatory polynomial.


5. Conclusion

The main result of this paper is Theorem 16 where it is provedthat the interconnection of two J-lossless behaviours leads to anoscillatory behaviour. The properties J-lossless behaviours and ofautonomous lossless behaviours have been used to prove thistheorem in the multi-variable case. In the case of interconnectionof SISO lossless behaviours, it has been shown in Section 4.3 thatTheorem 16 can also be proved by making use of properties ofHurwitz polynomials and of positive QDFs. The relationship ofTheorem 16with KYP lemma has also been described in this paper.The algebra of two-variable polynomial matrices has been usedthroughout as a tool in proving many of the results in this paper.

Appendix A. Interlacing property and positive quadratic differ-ential forms

Given below is a theorem that relates positivity of a quadraticdifferential form to an interesting property relevant to this paper,known as the interlacing property.

Theorem 23. Let r1 ∈ R[ξ ] be given by r1(ξ) = (ξ 2 + ω20)(ξ2+

ω21) . . . (ξ2+ ω2n−1), where ω0 < ω1 · · · < ωn−1 ∈ R+ and n is a

positive integer. Define r ′(ξ) := (ξ + ω20)(ξ + ω21) . . . (ξ + ω

2n−1);

r2(ξ) := ξ r1(ξ) and r(ξ) := ξ r ′(ξ). Then the following hold:

1. Let f1(ξ) be a polynomial of degree less than or equal to n − 1.Define

φ1(ζ , η) :=ηr ′(ζ 2)f1(η2)+ ζ r ′(η2)f1(ζ 2)

ζ + η.

Then Qφ1 > 0 if and only if f1(−ω20) > 0 and the roots of f1 are

interlaced between those of r ′, i.e along the real axis, exactly oneroot of f1 occurs between any two consecutive roots of r ′.

2. Let f2(ξ) be a polynomial of degree less than or equal to n. Define

φ2(ζ , η) :=ζ r ′(ζ 2)f2(η2)+ ηr ′(η2)f2(ζ 2)

ζ + η.

Then Qφ2 > 0 if and only if f2(0) > 0 and the roots of f2 areinterlaced between those of r .

Proof. See proof of Theorem 2, p. 1527, [8]. �

Appendix B. Hurwitz polynomial

A Hurwitz polynomial is a polynomial with all its roots in theopen left half of the complex plane. Given below is an interestingproperty of a Hurwitz polynomial which is used in this paper.

Theorem 24. Consider a polynomial p(ξ) = p′(ξ 2)+ξp′′(ξ 2), wherep′, p′′ ∈ R[ξ ]. Assume that the leading coefficient of p is positive.Define

ω :=

{the root of p′ with the smallest absolute value if deg(p′) ≥ 1,0 if deg(p′) = 0.

Define p1(ξ) := ξp′′(ξ). p is Hurwitz iff either one of the followingholds:

• deg(p′) > deg(p′′) ≥ 1; p′ and p′′ have distinct roots on thenegative real axis; the roots of p′′ are interlaced between those ofp′ and p′′(ω) > 0.• deg(p′) = deg(p′′) ≥ 1; p′ and p′′ have distinct roots on thenegative real axis; the roots of p′ are interlaced between those ofp1 and p′(0) > 0.• deg(p′′) = 0, deg(p′) = 1, p′ has a root on the negative real axisand p′′(ω) > 0.• deg(p′′) = deg(p′) = 0 and p′(0) > 0.

Proof. See proof of Theorem 1, p. 107 of [11]. �

Given below are the definitions for the even and odd parts of agiven polynomial.

Definition 25. The even part pe ∈ R[ξ ] and odd part po ∈ R[ξ ] ofa given polynomial p ∈ R[ξ ] are defined as

pe(ξ) :=p(ξ)+ p(−ξ)

2

po(ξ) :=p(ξ)− p(−ξ)

2.

If pe, po ∈ R[ξ ] respectively denote the even and odd partsof a given Hurwitz polynomial p ∈ R[ξ ], then with reference toTheorem 24, observe that

pe(ξ) = p′(ξ 2)po(ξ) = ξp′′(ξ 2).

From Theorem 24, it follows that Z := pe/po has one of the follow-ing two forms:

Z(ξ) =H(ξ 2 + ω21)(ξ

2+ ω23) . . . (ξ

2+ ω22m−1)

ξ(ξ 2 + ω22)(ξ2 + ω24) . . . (ξ

2 + ω22m−2)(23)

or

Z(ξ) =H(ξ 2 + ω21)(ξ

2+ ω23) . . . (ξ

2+ ω22m−1)

ξ(ξ 2 + ω22)(ξ2 + ω24) . . . (ξ

2 + ω22m)(24)

where H ∈ R+,m ∈ N and

0 < ω1 < ω2 < ω3 < ω4 < · · · .

Appendix C. Positive real transfer functions

Definition 26. A rational matrix B ∈ Ru×u(ξ) is called positive realif the following conditions hold

1. All elements of B are analytic in the open right half plane.2. B∗(λ)+ B(λ) ≥ 0 for Re(λ) > 0.

Definition 27. A rational matrix B ∈ Ru×u(ξ) is called losslesspositive real if the following conditions hold

1. B is positive real.2. B∗(jω) + B(jω) = 0 for all ω ∈ R, with jω not a pole of anyelement of B.

A matrix A ∈ Ru×u(ξ) is called lossless negative real if−A is losslesspositive real. If A ∈ R(ξ) is lossless positive real, then it is shownin [12], pp. 49–50 that A has one of the four forms of which two areshown in the right hand side of Eqs. (23) and (24) and two otherforms are their reciprocals.

Appendix D. Hybrid transfer function matrix of a multi-portelectrical network

The following material is from [4]. Consider a multi-port elec-trical network consisting of a finite number of ideal resistors, ca-pacitors, inductors, transformers and gyrators for which V denotesthe vector of port voltages and I denotes the vector of respectiveport-currents, both having dimension equal to l. Consider parti-tions of V and I given by V = col(V1, V2) and I = col(I1, I2)such that dim(V1) ∈ {0, 1, . . . , l} and dim(I1) = dim(V1). Defineu := col(V1, I2) and y := col(I1, V2). Assume that the transferfunction G from u to y exists. G is called a hybrid transfer functionmatrix of the network if limξ→∞ G(ξ) < ∞. In [13], it has beenproved that there exists at least one hybrid transfer function ma-trix for a given multi-port network.


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interconnection of -lossless behaviours

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