Power Conserving Multiport Systems, Junction Structures and Bond Graphs

Copyright © IF AC Control Science a nd T ec hno logy (8th Triennial W orld Congress) Kyo to , J a pa n , 1981

POWER CONSERVING MULTIPORT SYSTEMS, JUNCTION STRUCTURES AND BOND GRAPHS N. Suda and A. Enoki Department of Control Engineering, Faculty of Engineen'ng Science, Osaka University, 1· 1 Machikaneyama , Toyonaka, Osaka 560, japan

Abstract. The present paper establishes the theoretical basis of junction structures of bond graphs. A JUNCTION and a BOND are re-defined based on the power conserving multiport system studied in the foregoing work. The junction structures of bond graphs are reduced to JUNCTION STRUCTUREs, that is, the set of JUNCTIONs and BONDs. Keywords. theory

Bond graphs; Multiport systems; Models; System analysis; System

INTRODUCTION Bond graphs are graphical representations of phisical systems (Karnopp and Rosenberg, 1968, 1975; Dixhoorn and Evans, 1974). They employ the concept of power flow between interconnected parts of systems to model it. The interconnection features of the modeled system are represented by collections of four junction elements, namely , 0- and l-junctions, transformers and gyrators. They are called junction structures of bond graphs and many properties of junction structures have been studied and discovered by many authors. As Rosenberg (1979) mentioned, there are three major classes of junction structures, namely, simple, weighted and general junction structures. Ort and Martens (1973), Perelson (1975) and Perelson and Oster (1976) discussed the propertie s of the simple and weighted junction structures. The studies of Rosenberg and Andry (1979) and Rosenberg (1978, 1979) are concerned with general junction structures. The general theory to the three classes is, however, less comprehensive. The present authors also investigated simple junction structures (Suda and Enoki, 1977a, 1977b). We have been considering such questions as, "What a re the essential conditions o f junc tions and bonds ?", "Are there any other junctions ?", "What kind of classes of equations can be represented by junction structures of bond gr aphs ?" and "Is there a unified approa ch to the three classes of junction structure s ?".

PCMP's, and their properties were studied. From these properties it was proved that the class of equations which are representable by junction structures of bond graphs, is exactly the same as that of PCMP's. The aim of this paper is to extend the foregoing studies and establish the unified methodology of the analyses of the three classes of junction structures. Based on the PCMP a JUNCTION and a BOND are re-defined. The JUNCTION is defined as the special case of the PCMP and the BOND is defined as a two-port JUNCTION. From the properties of the JUNCTION a ZERO-JUNCTION and a ONE-JUNCTION can also be re-defined as t wo kinds of reciprocal JUNCTIONs. It is shown that a PCMP is repre sented a s a JUNCTION STRUCTURE which is a collection of JUNCTIONs and BONDs. The relationship amon g PCMP's, JUNCTION STRUCTUREs and junction structures of bond graphs, is described. POWER CONSERVING MULTIPORT SYSTEMS The system under consideration is supposed to have n ports, and f or port k a pair of variables, (Vk , ik )' is defined such that the product of them is power, i.e., Pk =i~XVk (k=1,2,···,n). Let V~ [ V 1 V2 ··· vn lT and i =[ i 1 i 2 ... i n l T . Suppose v and i satisfy the following equations:

Fv + Gi

= 0

(1 )

where F and G are nxn real matrices. Two conditions are defined for this s ystem.

For the purpose of answering above questions, we have defined a power conserving multiport s ystem (PCMP) with two fundamental conditions, that is, causality and power conservation" in the foregoing work (Suda and Enoki, 1977c). Necessary and suff ic ient conditions of the PCMP were discovered. Some concepts with the PCMP were defined, such as a gy rator index, reciprocity , reducibilit y and connections of

Condition 1 There exist an integer p (O~p ~n ) and an nxn permutation matrix P=[ P1 Pz l such that rank [FPl GP2 l = rank [F Gl

p /if

=

n

(2)

Condition 2 Let (Vk ' ik ) be an arbitrar y solution of Eq. (1), then

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N. Suda and A. Enoki

v T 1: i = 0 (3) where 1:=diag.(ol,02,···,On) and +1 if power flows out of the system at port k ok = { -1 if power flows into it at port k

A system (F,G) which satisfies above two conditions, is called a power conserving multiport system (PCMP). Condition 1 means that a set of n variables, one at each port, can be freely chosen and once the values of them are fixed then those of the remaining n variables are determined uniquely. The meaning of Condition 2 is obvious.

o~

s

~

n

The index s has the following properties. Theorem 2 Suppose the pair (F,G) is a PCMP, then the following three propositions are equivalent. (i) rank F + rank G = n + s (ii) rank All + rank A2 2 = s (iii) rank Fl:GT = s Since the rank of a skew-symmetric matrix is even, Corollary 2 The index s is even.

Necessary and sufficient conditions for the PCMP are given as follows.

Let y~s/2 and y is called a gyrator index . The gyrator index is, as will be stated later, the minimum number of gyrators which is necessary for representing the PCMP by a bond graph.

Theorem 1 The following four propositions are equivalent. (i) The pair (F,G) is a PCMP. (ii) There exist an nxn nonsingular matrix To and an nxn permutation matrix Po such that

For an n-port PCIW, as mentioned in the definition of the PCMP, i is an input at p of the n ports and v is an input at the other (n-p) ports. The number p is given in the following theorem.

TOFPo=[Ip

o

p

-AI2J}P -An

n-.:p

}

J}

o

ToGPo

(4a)

n-p p

(4b)

In_p } n-p ----v-n-p and (5)

where

[~:: ~::].

n-l i f n is odd

v

Fl:GT + Gl:FT =

0

(8)

Theorem 1 (ii) shows the input-output relation of a PCMP and (iii) means that a weakened version of Tellegen's theorem holds for a PCMP. Theorem 1 (iv) gives an easy way to decide whether the pair (F,G) is a PCMP or not. The following result is obtained from the proof of Theorem 1, which provides us how to find the matrix P, as well as the integer p of Condition 1. Corollary 1 Suppose the pair (F,G) is a PCMP, then rank [FPI GP2l = rank [F Gl = n holds for any permutation matrix P=[PI P2l ......,............,..... which satisfies rank FPI = rank F ~

The minimum value of s is zero and the maximum value, smax' is given as follows:

n if n is even

1:0 = PoT 1: Po (6) (iii) The matrix [F Gl has full rank, n, and two solutions of Eq. (1), say (v,i) and (v,~), satisfy the following relationship: vT 1: i + T 1: i = 0 (7) (iv) The matrix [F Gl has full rank, n, and Fl:GT is skew-symmetric, i.e., A

Theorem 3 Let y be the gyrator index of the PCMP (F,G). Then the number of ports where i is an input to the PCMP is one of the following: rank F, rank F 2, rank F 4, ••••• , rank F - 2y (= n - rank G)

f

n-f

f

Now some properties of the PCMP are studied. From the definition of the PCMP n ~ rank F + rank G ~ 2n Let s ~ rank F + rank G - n, then s satisfies

Let us study the case in which s i s zero. Corollary 3 The following six propositions are equivalent. (i) s = 0 (ii) All = 0 and A22 = 0 (Hi)

nGT

=

0

(iv) The number of ports where i is an input is rank F and the one where V is an input is rank G, that is, both of them are fixed. (v) There exist an integer d and an nxn nonsingular matrix To such that

To[F Gl

FO-

=

[

O] }d G } n- d

(9)

(vi) A strengthened version of Eq. (7) holds, 1. e . , v T 1: = T 1: i = 0 (10)

i

v

A PCMP is called r eciprocal if y=O and antireciprocal if ylO. Equation (10) shows that Tellegen's theorem holds for the reciprocal PCMP. Now the irreducibility of a PCMP is defined. Definition 1 Suppose the pair (F,G) is an n-port PCMP. If there exist an integer l (O
.......,....,......,.-l

n-l

Power Conserving Multiport Systems a PCMP, then the PCMP (F, G) is said to be

reducible . I t is noted that if the pair (FP1, GP1) is a PCMP then the pair (FP 2 , GP 2 ) is also a PCMP. Lemma 1

G2 )

,

Suppose the pairs, (Fl, Gl ) and (F

are PCMP' s, then a pair

([01:2J, [GOl

)

~22J

is a reducible PCMP. The followin g theorem gives necessary and sufficient conditions of the reducibility. Theorem 4 Suppose the pair (F,G) is a PCMP, then the following three propositions are equivalent. (i ) The PCMP (F,G) is reducible. (ii) There exist an nxn nonsingular matrix To and an nxn permutation matrix P o such that

ToFPo

=

[FOl :2]~ ~-l

(lla)

GO1

O] }l (lIb) } n- l hold for an integer l (O
=

[

G2

This theorem shows that any complex PCMP can be made by connecting simple PCMP's. It is easy to connect two irreducible PCMP's by a pair of ports, but this theorem also considers the case to connect more than two pairs of ports in an irreducible PCMP and the case to connect two irreducible PCMP's by several pairs of ports. In these two cases, the system made by connection may not preserve the properties of the original PCMP, while in the first case they are preserved as they are. It is an important point that Theorem 5 also considers those cases. The relationship between the connection of PCMP's and the gyrator index is described. Theorem 6 The gyrator index dose not increase by the connection of PCMP's. The following corollary is concerned with the connection of reciprocal PCMP's. Corollary 7 The system made by connecting pairs of ports in a reciprocal PCMP, is another reciprocal PCMP. JUNCTION STRUCTURES

trix where A is the matrix defined in Eq. (6). Lemma 1 shows that a direct sum of two or more PCMP's always becomes a reducible PCMP and it is shown from Theorem 4 that if a PCMP is reduc ible then it can be decomposed into a direct sum of two or more irreducible PCMP's. The property (iii) of Theorem 4 provides us with an easy wa y to check whether a PCMP is reducible or not. Some more properties with the reducibility are given. Corollary 4 If a PCMP (F, G) is irreducible, then the matrix [F Gl does not have zero columns. Corollary 5 If a PCMP is irreducible, then at any port k neither vk nor i k is determined of itself regardless of the other variables

Vj , ij (j =1, 2,···,n , j l k ). Co rollar y 6 The gyrator index of a reducible PCMP is equal to the sum of those of the irreducible PCMP's whose direct sum is the ori ginal one . At the end o f this section, a connection of PCMP' s is defined. We select q (q
In this section a JUNCTION, a BOND and a JUNCTION STRUCTURE are re-defined based on the PCMP discussed in the previous section. We use capital letters to express the junction, the bond and the junction structure redefined in this section. First a JUNCTION is defined as follows. Definition 3 Suppose the pair (F,G) is an irreducible n-port PCMP. If there exist an integer q (O~q~n ) and an nxn permutation matrix Q=[ Ql Q21 such that

q

n:.cj

rank [FQl

GQ 21 1 (12a) FQ2 1 n-l (12b) then the pair (F,G) is called a JUNCTION. rank [GQ l

A JUNCTION is the special case of the PCMP that satisfies Eqs.(12a) and (12b). Before the meaning of Eqs.(12a) and (12b) are clarified, necessary and sufficient conditions of the JUNCTION are given. Theorem 7 Suppose the pair (F,G) is a PCMP, then the following three propositions are equivalent. (i) The pair (F,G) is a JUNCTION. (ii) There exists an nxn nonsingular matrix T such that

T[ FQ l GQ21

i01Wl °2 W2

=

L

Definition 2 To connect q pairs of ports in a PCMP is to put Vak=Vbk and iak=ibk for each

0

Theorem 5 A s ystem made by connecting pairs of ports in a PCMP is another PCMP.

Z2

[

(13a)

0···

Zl

k (k=1,2, ... ,q ). The intuitive meaning of the connection is obvious. The connection o f more than two PCMP's is e quivalent t o the connection of one PCiviP t hat i s a di rect sum of them.

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o

0

(l3b)

where QTLQ=diag.(OI,02,···,On) and zi-l/Wi and wi's are nonzero constants (i=1,2,···,n). (iii) Let A be the matrix defined in Eq.(6), then

N. Suda and A. Enoki

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rank A = 2 Since it is easy to find the matrix A from Corollary 1, Theorem 7 (iii) is used to identify if the PCMP is a JUNCTION. The following corollary provides us a direct way to identify whether the pair (F,G) is a JUNCTION or not. Corollary 8 Suppose the pair (F,G) is a PCMP, then the pair (F,G) is a JUNCTION if and only if

rank [GP l FP 2 1 = 2 holds for the permutation matrix P=[P l P 2 1 mentioned in Corollary 1. Using the permutation matrix Q=[ Ql Q2 1 mentioned in Definition 3, we can rewrite Eq.(l) in the form:

[FQ l

GQ2 1IQ 1T~J LQ2T~

In Eq.(14)

+ [GQ l FQ2 1IQ1Ti ] = 0

x~[~::~J

LQ2 Tv and

Y~ [~::~J

(14)

are called

strong and weak variables of the JUNCTION (F,G), respectively. Let xk and Yk be the strong and weak variables of a JUNCTION J at port k, respectively (k=1,2,··.,n). For each port k, port k gives the JUNCTION J a strong causal determination if xk is an input and Yk is an output, and a weak causal determination if Yk is an input and xk is an output. From Theorem 7 (ii) Corollary 9 The pair (F,G) is a JUNCTION if and only if the following equations hold: ~ = X2 = .•... = xn (lSa) Wl W2 wn G1W 1Yl + G2W2Y2 + ... + GnwnYn = 0 (lSb) where xk and Yk are strong and weak variables at port k (k=1,2, ... ,n), respectively. A JUNCTION is an irreducible PCMP characterized as follows: Exactly one of the strong variables can be freely chosen and once the value of this strong variable is fixed then all the others are determined uniquely. All the weak variables except one can be freely chosen and once the values of these weak variables are fixed the remaining one is uniquely determined as their linear combination. If the strong variable is chosen at some port, then the weak variable should be chosen at the other ports, since either the strong variable or the weak variable is freely chosen at every port. In view of the above a JUNCTION should satisfy the following causality condition. Cl Exactly one of the ports gives the strong causal determination, and all the others give the weak causal determination to the JUNCTION. Some properties about the gyrator index and the reciprocity of a JUNCTION are given. Theorem 8 The gyrator index of a JUNCTION is either zero or one.

A JUNCTION is called reciprocal if the gyrator index is zero and antireciprocal if it is one. Theorem 9 (a) For a reciprocal n-port JUNCTION, the number of ports where i is the strong variable is either zero or n . (b) For an antireciprocal n -port JUNCTION (F, G), the number of ports where i is the strong variable is rank G - 1 (= n - rank F + 1). From the results of Theorem 9 (a), reciprocal JUNCtIONs can be classified into two kinds. Definition 4 A ZERO-JUNCTION is a reciprocal JUNCTION at every port of which v is the strong variable and a ONE-JUNCTION is a reciprocal JUNCTION at every port of which i is the strong variable. The following corollary is derived from Theorem 3. Corollary 10 (a) The number of ports where i is an input is n-l for an n-port ZERO-JUNCTION, and one for an n-port ONE-JUNCTION. (b) The number of ports where i is an input is either rank F or rank F - 2 (= n - rank G) for an antireciprocal n-port JUNCTION (F,G). Here we define a BOND. Definition S A BOND is a two-port JUNCTION that satisfies Gk l +ak 2=0 where kl and k2 are two ports of the JUNCTION. According to Definition 3, if the pair (F,G) is a BOND then Eqs.(12a) and (12b) become rank [FQl GQ 2 1 = rank [GQ l FQ 2 1 = 1 For BONDs, therefore, we cannot make distinction of the strong and weak variables, nor the strong and weak causal determinations. A BOND only connects one JUNCTION to another JUNCTION. Suppose a BOND K is incident with two JUNCTIONs, J l and J 2 • Let Xl and Yl be the strong and weak variables of J l at K, respectively, and let X2 and Y2 be the s trong and weak variables of J 2 at K, respectively. Lemma 2

Either (a) or (b) holds.

(a) {Xl = aY2

(b) { Xl = CJ.X2

ay 1

ay 1

=

X2

=

Y2

where a is a nonzero constant. We call a the modulus of a BOND. Two kinds of BONDs are defined from Lemma 2. Definition 6 A BOND is called even causal if (a) of Lemma 2 holds, and odd causal if (b) holds. It is noted that Lemma 2 holds for the triple (J l ,K,J 2 ) and therefore whether a BOND is even causal or odd causal is determined by the triple of J l ,J 2 and K. Obviously there are n kinds of causal patterns for an n-port JUNCTION and there are two for a BOND. If a BOND is even causal, then it gives the strong causal determination

Power Conserving Multiport Systems to both JUNCTIONs connected to it in one causal pattern and the weak causal determination in another causal pattern. If a BOND is odd causal, then it gives the strong causal determination to one JUNCTION and the weak causal determination to another. Some useful properties and equivalence rules about BONDs and JUNCTIONs are given. Lemma 3 Suppose a BOND K is incident with two JUNCTIONs, J 1 and J 2 • If K is odd causal, then the triple (J 1 ,K,J 2) is a JUNCTION. Lemma 4 Suppose a self-loop-BOND K is incident with a JUNCTION J. If K is odd causal, then the strong variables of J are zero if and only if Eqs.(15a) and (15b) of J hold for an arbitrary value of the modulus of K. Lemma 5 Suppose a self-loop-BOND K is incident with a JUNCTION J. If K is even causal, then J is equivalent to the JUNCTION with deletion of K from J for an arbitrary value of the modulus of K. Lemma 6 Suppose two parallel-BONDs, Kl and K2 , are incident with two JUNCTIONs. If both Kl and K2 are even causal, then they can be replaced by one BOND which is equivalent to them for arbitrary values of the moduli of Kl and K2 . A set which consists of JUNCTIONs and BONDs that join two JUNCTIONs, is called a J UNCTION S2'RUCTURE .

Lemma 3 shows that an odd causal BOND is included in a JUNCTION and it is an even causal BOND that connects two JUNCTIONs in a JUNCTION STRUCTURE. It follows from Lemmas 4 through 6 that a JUNCTION STRUCTURE can include neither a self-loop-BOND nor parallelBONDs. The relationship between PCMP's and JUNCTION STRUCTUREs is given in the following. Theorem 10 A JUNCTION STRUCTURE is a PCMP. Conversely any PCMP can be represented as a JUNCTION STRUCTURE. At the end of this section, one more causality condition is given which is concerned with the JUNCTION STRUCTURE, in addition to

Cl. Some terminologies are defined. A path is an alternating sequence of JUNCTIONs and BONDs, JO,Kl,Jl,K2,"',Jn-l,Kn,Jn, in which all the JUNCTIONs and all the BONDs are distinct and each BOND is incident with the two JUNCTIONs immediately preceding and following it. A path is called an causaZ path if every JUNCTION in the path has a strong causal determination given to it by a BOND in a path. A closed path is called a bond ring . A closed causal path is called a causa Z bond ring . We give the causality condition as follows. C2

There should be no causal bond ring.

The JUNCTION STRUCTURE can escape from "the vicious circle" of input-output relation by

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the condition C2. CORRESPONDENCE WITH BOND GRAPHS The relationship among PCMP's, JUNCTION STRUCTUREs and junction structures of bond graphs is discussed in this section. In the previous section it was proved that a PCMP can be represented as a JUNCTION STRUCTURE. We will show that the class of the constitutive equations of junction structures is exactly the same as that of PC~W's. We begin with the following lemma. Lemma 7 A O-junction of bond graphs is a ZERO-JUNCTION and a I-junction is a ONE-JUNCTION. A bond, a two-port junction, a twoport transformer and a two-port gyrator with power "flowing through it" are BONDs. Corollary 11 A O-junction, a I-junction, a two-port transformer and a two-port gyrator are PCMP's. Hereafter a two-port transformer and a twoport gyrator are simply called a transformer and a gyrator, resprctively. Lemma 8 A sub graph of a bond graph is a BOND if it consists of the series connection of transformers, gyrators and two-port junctions and power flows into one port and out of the other. Conversely any BOND can be represented as above. From Lemmas 7 and 8 the junction structure of a bond graph can be easily coverted to a JUNCTION STRUCTURE whose BONDs are even causal or odd causal. Here we discuss the adequateness of causality of junction structures. The junction structures of bond graphs consist of four elements, that is, O-junctions, I-junctions, transformers and gyrators. The causality of such junction structures should observe not only the input-output relation of Condition 1 of the PCMP, but also two conditions, Cl and C2, of the JUNCTION STRUCTUREs. It is well-known that the two conditions are a sufficient condition of Condition 1. The important thing is that the causality is one of the fundamental structural properties of systems. Therefore once the causal pattern of a junction structure is fixed, it is desiable that it is preserved regardless of the moduli of transformers and gyrators. In view of the above the causality of a junction structure should be assigned. From Lemma 4 in the previous section, if there exists a JUNCTION J with which an odd causal self-loop-BOND is incident, then all the BONDs connected to J, as well as J, can be deleted, since the strong variables of J are all zero. Lemma 5 shows that an even causal self-loop-BOND can be deleted by itself. Even causal parallel-BONDs can be replaced by an even causal BOND from Lemma 6. A reciprocal BOND is odd causal if and only if it is incident with similar JUNCTIONs.

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N. Suda and A. Enoki

An antireciprocal BOND is odd causal if and

only if it is incident with two different kinds of JUNCTIONs, namely, a ZERO-JUNCTION and a ONE-JUNCTION. We can assume, without loss of generality, that there are no self-loop-BOND and no parallel-BONDs. The following result is obtained from Lemma 3. Theorem 11 A subset of a JUNCTION STRUCTURE is a JUNCTION if and only if there exists a path between any two JUNCTIONs of the subset that consists of only odd causal BONDs. From this theorem a JUNCTION STRUCTURE can be a set of JUNCTIONs and even causal BONDs that join two JUNCTIONs. Therefore any junction structure of bond graphs can be easily converted to a JUNCTION STRUCTURE whose BONDs are even causal. Now we consider the junction structures of bond graphs. Lemma 9 A O-junction, a I-junction and a transformer are reciprocal PCMP's and a gyrator is an antireciprocal PCMP. The gyrator index of a gyrator is one. The relationship between PCMP's and junction structures is stated as follows. Theorem 12 The constitutive equations of a general junction structure are a PCMP. Conversely any PCMP can be represented as a general junction structure. This theorem shows that the class of the constitutive equations which are representable by junction structures of bond graphs, is exactly the same as that of PCMP's. Corollary 12 The constitutive equations of a weighted junction structure is a reciprocal PCMP. Conversely any reciprocal PCMP can be represented as a weighted junction structure. The following result is a generalized version of Corollary 12. Theorem 13 The gyrator index of the PCMP represented as a general junction structure with A gyrators, is not greater than A. Any PCMP with gyrator index y can be represented as a general junction structure with y gyrators. Corollary 13 An n-port PCMP can be represented as a general junction structure which n includes at most [2] gyrators. CONCLUSION In the present paper the theoretical basis of the junction structures of bond graphs was established. Power conserving multiport systems (PCMP's) were defined, which satisfy two conditions --- Condition 1 is from causality and Condition 2 is from power conservation, both of them are quite reasonable. Some

properties about the PCMP are given. Based on the PCMP, a JUNCTION and a BOND are defined, which are the special cases of the PCMP. The leading principle in defining the JUNCTION is that the strong and weak variables were used as a pair of variables which defines power. The JUNCTION and the BOND defined in this paper are, so to speak, the generalized junction and bond of bond graphs, respectively. Using them the junction structures of bond graphs can be analyzed in a unified way. The junction structure of a bond graph is converted to a JUNCTION STRUCTURE which is a collection of JUNCTIONs and BONDs that connect two JUNCTIONs. The JUNCTION STRUCTURE is a PCMP and any PCMP can be represented as a JUNCTION STRUCTURE. Thus the relationship among PCMP's, JUNCTION STRUCTUREs and junction structures has become clear. REFERENCES Dixhoorn, J.J., F.J. Evans, (Ed.) (1974). Physical Structure in Systems Theory. Academic Press, London. Karnopp, D., R.C. Rosenberg (1968). Analysis and Simulation of Multiport Systems. The M.I.T.Press, Cambridge, Mass. Karnopp, D., R.C. Rosenberg (1975). System Dynamics: A Unified Approach. John Wiley and Sons, N.Y. Ort, J.R., H.R. Martens (1973). The properties of bond graph junction structure matrices. Trans. ASME Ser. G, 95, 362-367. Perelson, A.S. (1975). Bond graph junction structures. Trans. ASME Ser. G, 22, 189-195. Perelson, A.S., G.F. Oster (1976). Bond graphs and linear graphs. J. Franklin Inst., 302, 159-185. Rosenberg, R.C. (1978). On gyrobondgraphs and their uses. Trans. ASME Ser. G, lOO, 76-82. Rosenberg, R.C. (1979). Essential gyrators and reciprocity in junction structures. J. Franklin Inst., 308, 343-352. Rosenberg, R.C., A.N. Andry (1979). Solvability of bond graph junction structures with loops. IEEE Trans. Circuits & ~, ~, 130-137. Suda, N., A. Enoki (1977a). A new matrix representation of bond graphs and equivalence condition. Trans. Soc. Instrum. & Control Eng., 13, 324-329. Suda, N., A. Enoki (1977b). On the adequate augmentation of bond graphs. Trans. Soc. Instrum. & Control Eng., 13, 445-450. Suda, N., A. Enoki (1977c). A general theory of multiport systems. Preprints for Symposium on Systems Theory, Soc. Instrum. & Control Eng., 19-24.