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Determination of Parameter Values in Linear Electrical and Electronic Circuits
Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan, 1981
DETERMINATION OF PARAMETER VALUES IN LINEAR ELECTRICAL AND ELECTRONIC CIRCUITS T. Ozawa* and S. Shinoda** *Department of Electrical Engineering, Kyoto University, Sakyo, Kyoto, japan **Department of Electrical Engineering, Chuo University, Bunkyo, Tokyo 112, japan
Abstract. The problem of determining parameter values in a linear electrical or electronic circuit from measurements of voltages and/or currents , is studied. First , equivalent circuit transformations are introduced to eliminate an inaccessible node where neither measurements nor applications of voltages and/or currents are possible. Next , if there are more than one inaccessible node , a sequence of transformations ar~ performed, and a sequence of circuits NO ' Nl' .. , Nf-l and Nf are obtained, where NO is the original circuit and Nf is the final circuit. Then a procedure and conditions for determining the parameter values in these circuits are given . The procedure is efficient , because it is based on a graph theoretic approach and involves solving sets of linear equations only, whereas conventional methods require solving nonlinear equations. Keywords. Parameter estimation; fault detect i on; linear systems; integrated c ircuits; graph theory; equiva lent circuit transformations.
Solving a set of nonlinear equations is time consuming. Several subsequent papers have considered this problem (Bedrosian and Berkowitz , 1962; Hayashi, Hattori and Sasaki , 1 967) . Recently Navid and Wilson (1979) gave a sufficient condition for the solvabi lity of the nonlinear equations. The condition is applicable to an active(elec tronic) circuit as well as a passive circuit.
INTRODUCTION with modern integrated circuit technology it is possible to manufacture electrical and electronic circuits with great complexity . These integrated cir cuits are embedded on semiconductor chips, and are not assemblies of elements of known values . Their circuit e lements cannot be taken ou t individually for measurements, even if they are faulty o r if t hey are stray elements of unknown values. Such faults must be detected without disassembling the circuits . Presently , therefore, there is a great need for systematic method to determine element values or parameter values(functions of element values including element values themselves) in them.
Another formulation for determining para meter values in a passive circuit was given by Shinoda (1970). His measurements consist of those of vo1tages and/or currents at test elements(branches) at a single circuit state (Applied voltages and/or currents are not varied.). This formulation was generalized by Ozawa and Kajitani (1979) for an active circuit which contains both elements of known values and unknown values . Their results give a base for various approaches to parameter-value determination, including that given in this paper. One of their theorems clarifies the voltages and currents which cannot be determined from the measurements at a single circuit state. Then these unknown variables are to be determined by changing the circuit state.
The problem of de terming parameter values in a linear passive(electrical) circuit from measurements at test terminals(nodes) was first considered by Berkowitz (1962). His measurements consist of those of currents due to the applied voltages of known values at the test terminals . The short- circuit transfer admittances are first obtained from the measurements at multiple circuits states (Applied voltages are varied to change the voltages and currents in the circuit . ) . Then t he equations relating the element values (conductances, etc.) to the short- circuit transfer admittances are solved to determine the element values . The equations are nonlinear, even if the circuit is linear. A few necessary conditions for the solvability of the equations are derived . This kind of measurements are , in general, difficult to perform , especially for electronic circuits.
Calculation of parameter values from node voltages is considered by Trick , Mayeda and Sakla (1979). They derived linear equations for element values by use of Tellegen ' s theorem, and gave solvability conditions for two special types of circuits . Their equa tions are very complex. Simpler equations can be derived from node voltages (El - Turky,
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T. Ozawa and S. Shinoda
and Vlach, 1980). The circuit considered in this paper may contain transistors and other active elements as well as passive two-terminal elements(resistors, capacitors and inductors). A transistor can be represented by a model consisting of resistors and voltage - controlled current sources. Likewise most active elements have their controlled source models. Therefore, we assume the elements in the circuit are passive two-terminal elements and controlled sources, especially voltage-controlled current sources (A voltaa,e controlled current source consists of a dependent current source and a voltage sensor which picks up the voltage controlling the dependent source. Thus this element has two branches.)
current sources are shown in Fig. l(b), but a two-terminal element should be connected between every pair of the peripheral nodes. The parameter value of the two-terminal element is denoted by gjk' if it connected between peripheral nodes j and k. It is called a g-parameter. The parameter of a voltage-controlled current source is called an a-parameter or an a-parameter respectively. The sensor branches are omitted in the figures.
It is assumed that circuit topology(how elements are connected) is known, and that all the element values are unknown. If some of the element values are known, some voltages and/or currents can be calculated from the measurements (Ozawa and Kajitani, 1979). In the equations for determining parameter values the calculated voltages and/or currents need not be distinguished from the measured voltages and/or currents. All of them can be considered as known variables, and the following discussions can be applied with a slight modification. The element values and the parameter values are all admittances, unless specified. We assume that they are nonzero, and that there is no special relation among the element values.
n
(a)
~~--~----~D n-l
1
an - ll (vn_l-v l )
ELIMINATION AND REVIVAL OF INACCESSIBLE NODES
a12 (v l -v 2 )
Equivalent Circuit Transformations n
'ie consider first the elimination of a single inaccessible node by equivalent circuit transformations. The node to be eliminated and the elements connected to it form a star. The transformations introduced here are extensions of the famous y-fi transformation. They can be classified into several types depending on the positions of the sensors and dependent sources in the star. Type A transformations. In case the star has dependent current sources controlled by the voltages between the peripheral nodes and the center, or in other words , in case both the dependent sources and their sensors are incident to the node to be eliminated, we introduce type A transformations. There can be complex transformations, if the star has many dependent sources, but to see the nature of the transformation of this type it would be best to consider that illustrated in Fig. 1, where the star has two dependent sources. The star in Fig. l(a) is equivalent to the circuit in Fig. l(b). To avoid a complex drawing only the dependent
(b)
Fig. 1 Type A transformation
The relation between these parameters and the element values gj(j=1,2, .. ,n), al and a 2 in the star can be derived as follows. gjk=gjgk/ O
(j=1,2, .. ,n-l;k=2,3, .. ,n: k>j)
~jl=gjal/o
(j=2,3, .. n-l:j~1,n)
aj2=gja2/o
(j=1,3, ..
(1)
Q12=a l a 2 /o
,n- 2 ,n;j~2,n-l)
(2) (3) (4)
where
(5) The short-circuit transfer admittances of the transformed circuit are (gjk=gkj if j>k in the following):
Determination of Parameter Values
ylk=glk'
y 2k =g2k (k=3, .. ,n) yjk=gjk (j=3,4, .. ,n-2;k=3,4, .. ,n:k;oij)
(7)
(6)
Yjl:gjl+ajl (j=2,3, .. ,n-2)
(8)
yj2=gj2+aj2 (j=1,3, .. ,n-2)
(9)
Y =q +a +a +a (10) n-l 1 n-l 1 n-l 1 12 12 Yn - l 2=gn-l 2 -a 12 -a 32 -· .-a n _ 2 2 -a n2 -Cl 12 (11) (12) Yn - l k =gn-l k +a (k=3,4, .. ,n-2,n) k2 a (13) Ynl =§nl- 21 - 31-' . n_l 1 12 Y =g +a +a +a (14) n2 n2 n2 21 12 (15) ynk=gnk+akl (k=3,4, .. n-l).
a
-a
-a
Type B Transformations. In case the star has dependent current sources controlled by the voltages between peripheral nodes, or in other words, in case only the dependent sources but not the sensors are incident to the node to be eliminated, we introduce type B transformations. One of the transformations of this type is depicted in Fig. 2. Again only dependent sources are shown in Fig. 2(b), and two-terminal elements 9jk (j=1,2, .. ,n-l;k=2,3, .. ,n:k>j) between pairs of peripheral nodes should be added to the figure. For each of the dependent current bk(vk-v ) in the star, dependent l
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currents bjk(Vl-Vk) (j=1,2, .. ,n:j;oik) flow from nodes j(j =1, 2, .. ,n:j;oik) to node k. The parameter bjk is called ab-parameter. The relation between the parameter values and the element values in this transformation is as follows. (j=1,2, .. ,n-l; k=2,3, .. ,n:k >j) (16)
(j=1,2, .. ,n;k=m,m+l, .. ,n:k;oij) (17) where (18) The short-circuit transfer admittances are (§jk=gkj if j >k): yjk=gjk (j=1,2, .. ,n;k=2,3, .. ,m-l:k;oij)
(19)
Y'k=§'k+b'k (j=1,2, .. ,n;k=m,m+l, . . ,n:k;oij) J J J (20) Y'l=§'l-b, -b, l-··-b, (j=2,3, .. ,m-l) J J Jm Jm+ In (21) Yjl=gjl-bjm-bjm+l-··-bjj_l-bjj+l-··-bjn
+b l ,+b 2 ,+ .. +1:), J
J
l,+b, 1,+··4 , . J- J J+ J nJ
(j=m,m+l, .. ,n)
(22)
Type C Transformations. In case only the sensors but not the dependent current sources are incident to the node to be eliminated, we define type C transformations. This type of transformations is similar to type B. Therefore the details of the transformations are omitted.
2
There can be a mixture of the above types.
computation of Element Values. Suppose that an inaccessible node is eliminated by a transformation. We will show conditions for computing the element values in the original circuit from the parameter values in the transformed circuit.
n (a)
Type A Transformation. If logarithms of eqs. (1)-(4) are taken, linear equations for variables xgj=log(gj/s) (j=1,2, .. ,n) and xaj=log(aj/s) (j=1,2) are obtained. They are:
b 2m (v 1 -vm) 2
X +X =log g12 gl g2 x +x =log g13 gl g3 m x x n (b)
Fig. 2
Type B transformation
gl al
+x +x
a2 a2
=log
a12
(23)
=log 0'12
There are n(n-l)/2+2(n-2)+1 equations in (23), but not all of them are necessary. Let M be the coefficient matrix of (23). M has two l's on each row and can be regarded as the edge-to-node incidence matrix of a
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T. Ozawa and S. Shinoda
graph . We use this graph as an auxiliary graph to give a solvability condition for (23). Let GA be the graph . GA has n+2 vertices corresponding to gj(j=1,2, . . , n), al and a2(They are identified also by these simbols.), and edges between gj and gk(j=l , 2, .. ,n-l;k=2,3, .. ,n:k > j), between al and gj (j= 2,3, . . ,n- l), between a2 and gj(j=1 , 3 , . . , n- 2,n) , and between al and a2 ' The edges in GA correspond to the equations in (23). The number of unknown variables in (23) is n+2. We will show which n+2 equations in (23) are necessary and sufficient to determine the x variables . The coefficient matrix of the n+2 equations which are chosen from (23) is denoted by Ms' Hs is a submatrix of M, and the edges corresponding to the rows of Ms together with their end vertices form a subgraph of G]\.. Let this subgraph be denoted by Gs ' Obviously we have: Lemma 1. Hs is singular if Gs contains an isolated vertex. Suppose Gs contains a cycle with c edges. Let Mc be the submatrix of Ms which corresponds to this cycle. We can easily show: Lemma 2. Mc is singular if c is even , and is nonsingular if c is odd(c~3) . Lemma 3. Ms is nonsingular if and only if Gs contains no isolated vertex and each connected component of Gs contains exactly one cycle with odd number~3 of edges. Proof. Necessity: From Lemma 1 Gs cannot have an isolated vertex . We can assume Ms is a block diagonal matrix , each block corresponding to a connected component of Gs . If there is a connected component containing no cycle , there must be a connected component containing more than one cycle and vice versa, since the number of edges of Gs is equal to the number of vertices. The blocks corresponding to these components are rectangular(not square). Thus if there are such components , Ms is singular . From Lemma 2 a connected component contajns no eve n cycle. Sufficiency: We need to consider a connected component, which corresponds to a square block of Ms' Suppose a nonseparable part of the component contains no cycle. The submatrix corresponding to it can be made triangular by proper permutation of rows and columns , if the column corresponding to its cut-vertex is removed . From this fact together with Lemma 2 we see the block corresponding to the component is nonsingular. In view of Lemma 3 we define a particular subgraph of GA . A dendroid in any graph G is defined to be a subgraph of G such that it contains all the vertices of G and each of its connected components contains exactly one cycle with odd number~3 of ed0es. Theorem 1.
Among the equations in (23)
the
equations corresponding to the edges of a dendroid in GA are necessary and sufficient to determine the unknown variables in (23) . Proof. From Lemma 3 we can determine Xgj (j=1,2, .. ,n), xal and x a 2' Then s=(gl+g2+" +gn+al+a2)/s can be calculated. Thus we can determine gj(j=1 , 2, . . ,n) , a l and a 2 . If the element values gj(j=1,2, .. ,n), al and a2 are to be computed from the short-circuit admittances, the following auxiliary graph GAY is used. GAY has n vertices correspond ing to gj(j=1 , 2, .. ,n) and edges between vertices gl and gk(k=3,4, .. ,n-l,n), between vertices g2 and gk(k=3,4, .. ,n - l,n) and between vertices gj and gk(j=3 , 4 , . . , n - 2;k=3 , 4, .. ,n - l , n:k~j). The edges of GAY correspond to the short- circuit admittances each of which has only one term in the right hand side of the equation in eqs . (6) and (7). Theorem 2 . The unknown variables x g j(j=1 , 2 , .. ,n) can be determined from the admittances corresponding to a dendroid in GAY ' Then al/s can be determined from one of Yjl(j=2 , 3, .. ,n-2) and Ynk(k=3 , 4, .. , n - l), and a2/s, from one of Yj2(j=1,3, .. ,n-2) and Yn - lk(k=3, 4 , .. ,n- 2,n). From these values gj(j=1,2 , .. , n), al and a2 can be determined. The right hand side of the equation for Ynk ' Yj2 or Yn- lk in Theorem 2 consists two terms , as seen in eqs . (8) (15) (9) or Type B Transformations . The auxiliary graph for determining gj(j-l,2, .. ,n) and bK(k=m, m+l , .. , n) from g- and b-parameters is denoted by GB' GB has 2n-m+l vertices corresponding to gj(j=1,2 , .. ,n) and bk(k=m,m+l, .. ,n), and edges between vertices gj and gk(j=1 , 2 , .. , n - l; k=2 , 3 , .. , n:k>j) and between vertices g. and bk(j=1 , 2 , .. , n;k=m , m+l , . . , n:k~j) . The eages correspond to the equations in (16) and (17) . Theorem 3 . The element values gj(j=1,2, . . , n) and bk(k=m , m+l, .. , n) can be determined from the equations, among those in eqs. (16) (17) , which correspond to the edges of a dendroid in GB ' Again, by taking the logarithms of the equations , we need to solve a set of linear equa tions for unknown variables x g j(j=1,2, .. , n) and xbk(k=m,m+l , . . ,n) , where xbk=log(bk/s). If we want to determine the element values from the short - circuit admittances , we form an auxiliary graph GBY' which has n vertices g . (j=1,2, .. ,n) and edges between vertices gj a~d gk(j=1,2, .. ,n;k= 2 , 3, .. ,m- l:k~j). The edges of GBy correspond to the short - circuit admittances, each of which has only one term in the right hand side of the equation in eq. (19). Theorem 4. The element values gj(j=1 , 2, .. , n) can be computed from the short - c~rcuit admit tances corresponding to the edges of a dendroid in GBy . Then bk(k=m , m+l, . . n) can be
Determination of Parameter Values
determined from these element values and one of Yjk(j=1,2, .. ,n:jik). The short-circuit admittance Yjk in the theorem has only two terms in the right hand side of the equatuion in (20). The admittances given by eqs. (21) and (22) may be used, if available. Modification of Auxiliary Graphs. For the simplicity of the discussion we have ignored the element which may exist between the peripheral nodes of the star. If an element exists, for example, between nodes j and k of the star, it is parallel to 9jk in the transformed circuit. The values of parallel elements cannot, in general, be obtained separately by voltage measurements only. Thue gjk is unknown and the equation 9j9k/a=gjk cannot be used to determine the element values of the star. Therefore the edge in the auxiliary graphs which corresponds to this equation should be deleted. Modification of the auxiliary graphs and the theorems is summarized as follows. (i) If there is a two-terminal element between nodes j and k of the star, then delete the edge between vertices gj and 9k in any of GA, GAY' GB and GBy . (ii) For type A transformation: If there is a dependent current source between node j(j=2,3, .. , or n-l) and node n of the star which is controlled by voltage Vj-Vl' then delete the edge between vertices gj and al in GA. Omit Yjl in Theorem 2. If there is a dependent current source between node j(j= 1,3, .. ,n-l or n) and node n-l of the star which is controlled by voltage Vj-V2' then delete the edge between gj and a2 in GA. Omit Yj2 in Theorem 2. If there is a dependent current source between nodes n-l and n which is controlled by voltage vl-v2' then delete the edge between al and a2 in GA· (iii) For type B transformation: If there is a dependent current source between nodes j and k (j=1,2, .. ,n;k=m,m+l, .. ,n:kij) controlled by voltage vl-vk' then delete the edge between vertices gj and gk in GB. Omit Yjk in Theorem 4. (iv) If any of the parameter values or short-circuit admittances in the transformed circuit cannot be determined from the measurements by some reason or other, delete the edge corresponding to it from the auxiliary graphs or omit it from the theorems.
Eliminations and Revivals of Inaccessible Nodes. If there are more than one inaccessible node, we repeat the transformations introduced in the previous section, and eliminate them one by one. Let NO be the original circuit and let N+, N2 , .. , and Nf be the resultant circu~ts obtained by the sequence of transformations, where f is the number of inac-
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cessible nodes. At each elimination an auxiliary graph is constructed. An elimination of a node, in general, gives rise to parallel elements and the auxiliary graph should be modified accordingly. The final circuit Nf usually has many parallel elements. These parallel elements cannot be separated by the voltage measurements, but the sum of their parameter values (admittances) can be obtained. As for the short-circuit admittances of the transformed network, they can be derived by adding the admittances given by eqs. (6)(15) or (19)-(22) to the admittances originally existing at the relevant node pair. In this way the short-circuit admittances of Nf can be derived and expressed in terms of the parameter values of the elements in Nf. From the parameter values or the shortcircuit admittances of Nf, we are to determine the element values in NO. The computations of the element values or parameter values are performed in the reverse order of the elimination. The eliminated nodes are revived one by one, and we get Nf-l from Nf, Nf-2 from Nf-l' and so forth. The steps to obtain Nk-l from Nk by reviving node k, which is the center of the star in the transformation, are as follows. 1° Find whether a dendroid exists in the auxiliary graph. If ther is one, compute the parameter values in the star in Nk-l from the parameter values in NK which correspond to the edges of the dendroid. 2° Compute, using necessary equations in eqs. (1)-(4) or (16) (17), the §-, ~- &- or b-parameters which are in parallel with the other elements in Nk' but disappear in Nk-l· Subtract the values from the sum of the parameter values or the total short-circuit admittances of Nk' to obtain the parameter values of Nk-l. Nl , N2 , .. , and Nf depend on the order of eliminated nodes, and thus the possibility of existence of dendroids in the auxiliary graphs varies depending on the order. Theorem 5. If step 1° is possible for each k=f, f-l, .. and 1, the element values in NO can be determined from the parameter values in Nf by solving sets of linear equations only.
DETERMINATION OF PARAMETER VALUES FROM NODE VOLTAGE MEASUP£MENTS Suppose we have eliminated all the inaccessible nodes and have Nf. Suppose also Nf has m accessible nodes where both appications of cuurents and measurements of voltages are possible, and p partly accessible nodes where only measurements of voltages but not applications of currents are possible. Let vl' v2'··' vm and jl' j2'··' jm be the node voltages and the applied currents at accessible nodes 1, 2, .. , m
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T. Ozawa and S. Shinoda
respectively, and let v m+ l ' v m+2"" vp be the node voltages at partly accessible nodes m+l, m+2, .. , p respectively. They are related by open-circuit impedances as Zll Z12
Zlm
Z21 Z22
Z2m
v jl
..............
Z Z ml m2
..
2
j2
Z mm
v
..............
Z Z pl p2
v
l
(24) m
jm
Z pm)
v
From the node voltages of Nf' we can compute the branch voltages by use of Kirchhoff's voltage law. Let i and j be the vectors of the branch currents and the applied currents respectively. Then from Kirchhoff's current law we have Qi=Qjj, where [Q -Qjl is the fundamental cutset matrix or the incident matrix of the current graph derived from Nf . The branch currents and the branch voltages are related by i=Vg, where V is the diagonal matrix with the branch voltages on the diagonal, and 9 is the vector of the parameter values . Substituting this equation into the relation above, we get (25) The number of equations in (25) is usually less than the number of unknown parameter values. In order to qet an enough number of equations for unknown variables, we vary the applied currents and produce different circuit states. We distinguish the variables at the k-th circuit state by superscript (k) . Then from m independent applied currents we usually get m circuit states, and we have:
QV(2) g= QV(m)
Q.j (1) J Q.j (2) J
Q,j
The p roposed method can be applied to o ther physical systems which have electrical or electronic circuit analogues. Since controlled sources can be used, varieties of systems can be modeled by electronic circuits. For example, input-output relations
x1 =
p
Thus Zik (k=1,2, .. ,p) can be directly obtained from the voltages vl' v2"" vp due to the application of a unit current at node i only. If there is no partly accessible nodes, the short-circuit admittances can be obtained by inverting the opencircuit impedance matrix in eq. (24).
QV (1)
and electronic circuits from measurements. The procedures are efficient b ecause they are graph theoretic and invo l ve solving sets of linear equatio ns only.
(26)
(m)
J
Some conditions for determining parameter values of 9 from eq. (26) are given by Ozawa and Yamada (1981).
CONCLUDING REMARKS We have given conditions and procedures for determining parameter values in electrical
(27)
x2 = can be expressed by a circuit containing two resistors and two controlled sources. One method to obtain an electrical or electronic model of a system is to use a bond graph(vanDixhoorn and Evans (Ed.) 1874). Acknowledgements. This work was p artly supported by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan under Grant Cooperative Research (A) 435013(1979-1980) .
REFERENCES Bedrosian, S.R. and Berkowitz, R.S. (1962). Solution procedure for single-element-kind networks. IRE Int. Conv. Rec. Part 2, 16-24. Berkowitz, R.S. (1962). Conditions for network - element-value solvability. IRE Trans. Circuit Theory, CT-9, 24-29. El-Turky, F.M. and Vlach , J. (1980). Calculation of element values from node voltage measurements. 1980 Int. Symp. on Circuits & Syst. Proc . 170-172. Hayashi, S., Hattori, Y. and Sasaki, T. (1967). Conditions on network-elementvalue evaluation. Electronics and Communications in Japan, 50, 118-127 Navid, N. and Wilson Jr. A.N. (1979). A theory and an algorithm for anolog circuit fault diagnosis. IEEE Trans. Circuits & Syst. CAS-26, 440-457. Ozawa, T. and Kajitani, Y. (1979). Diagnosability of linear active networks. IEEE Trans. Circuits & Syst. CAS-26, 485-489. Ozawa , T. and Yamada, H. (1981). Conditions for determining parameter values in a linear active network from node voltage measurements. 1981 Int. Symp. on Circuits & Syst. Proc. Shinoda, S. (1970). A minimization problem in network diagnosis. Trans. Inst. Elec. Comm. Eng. Japan, 53-A, 569-570 . Trick, T.N., Mayeda, W. and Sakla, A.A. (1979) . Calculation of parameter values from node voltage measurements. IEEE Trans. Circuits & Syst., CAS-26, 466-474. vanDixhoorn, J.J. and Evans, F . J. (Ed.) (1974). Physical Structure in Systems Theory, Academic Press, London.
Determination of Parameter Values Discussion to Paper 59 . 1 R. Konakovsky (Federal Republic of Germany) : First, does the quality of the control depend on the number of subdivisions of the system in subsystems? Next, how do you decide between two possibilities of a subdivision? And finally, is the subdivision given by the system structure itself and does the reliability in the figure mean for a given period of time? M. Funabashi (Japan): In answer to your first question, it very much depends on the structure of the original problem, and so is very difficult to answer in general. I think we must study that case by case. The period of time is 1 hour. H. Wedde (Federal Republic of Germany) : Your concept of reliability does not yet take into consideration the communication between the (decentralised) computers. Do you not think that this is a practically important aspect? M. Funabashi (Japan): Consideration of the communication is very important. However, our study has excluded it because it does not so much effect the control algorithm, and our approach is that from the control scheme. We will probably include it in the future. D.R. Powell (France) : Could you please justify your assumed relationship between reliability and computer power? At what mission time was the reliability evaluated (reliability is a time-dependent function)? M. Funabashi (Japan): The relationship I talked about is the empirical one, which coincides with the generally accepted tendency that reliability of microprocessors is higher than that of large-scale processors. The time was 1 hour. Discussion to Paper 59.2 R . Konakovsky (Federal Republic of Germany) : There are also methods for fault diagnosis in electronic systems. Can you explain the difference between the fault diagnosis of mechanical and electronic systems? S. Hirai (Japan): Usually in the case of diagonal methods for electronic circuits, check points for diagnosis are prepared beforehand. Therefore, it is very difficult to detect unexpected failure. Furthermore, causality in an electric system is much simpler than in a mechanical one. Our diagonal system puts emphasis on describing mechanical objects which have very complicated causality in a general and simple manner. D.R . Powell (France) : Although you do state that your failure analysis technique assumes only 1 fault, do you think that it will be possible to extend the technique to multiple failure situations which, I believe, are frequent in mechanical systems due to chain reaction effects. S. Hirai (Japan) : It is an important and interesting problem that we extend our method for multiple failure situations . We are now studying the method. A simple method in the present stage is to detect and repair each abnormal part one by one using this diagonal system. eST 3 - H'
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Y. Nozaka (Japan): I would like to ask two questions froD the practical standpoint. First, how much time does it take in the case of diagnosis of air-conditioner and second, what is the main feature of this new method compared to conventional methods such as vibration analysis? S. Hirai (Japan): The mean response time for each data input is one or two seconds. The main feature of our system in the present state is to describe the function of machines and its deviation in a simple and general manner and to perform the diagnosis on the basis of this description . The conventional methods you pointed out will be incorporated in our system in future as the extended sensing teChnique . Discussion to Paper 59.4 T. Soeda (Japan) : Stochastic signals are not always stationary. What do you think of the accuracy of the model when we apply this method to the practical complex problems? T. Fukuda (Japan): I assumed here that the process is stationary , because many plants operate in a steady state. If not, we cannot apply this method directly, because we cannot determine the order of the model. If it is known a priori , we can model the time varying system accurately. R. Konakovsky (Federal Republic of Germany): What represents the top of your fault tree diagram and what are the primary events at the bottom? T. Fukuda (Japan): In this special case, I employed the event that the heating section will burn out . In general, we can employ some events as a top event, which we try to prevent. The list of basic events employed have already been discussed in the paper. Discussion to Paper 59.5 R. Konakovsky (Federal Republic of Germany) : In your method you transform the network until the final network. You then determine the parameters of the final network and from here you trace back to the original network to get its parameters? Do you simplify the equation after the transformation and what is the relation between the number of accessible and inaccessible points? How many measurements are necessary to determine the parameters? T. Ozawa (Japan): That is correct . By taking the logarithms , the set of nonlinear equations are converted to a set of linear equations whose coefficients are alII. There is no particular relation between the numbers of accessible nodes and inaccessible nodes . In general, however, the number of unknown elements increases as the number of inaccessible nodes increases . The number of possible measurements for transfer admittances is limited to n(n - l) where n is the number of accessible nodes . The number of unknown ele ments must obviously be less than or equal to this number n(n-l). The number of measurements necessary is equal to the number of unknown elements .
DETERMINATION OF PARAMETER VALUES IN LINEAR ELECTRICAL AND ELECTRONIC CIRCUITS T. Ozawa* and S. Shinoda** *Department of Electrical Engineering, Kyoto University, Sakyo, Kyoto, japan **Department of Electrical Engineering, Chuo University, Bunkyo, Tokyo 112, japan
Abstract. The problem of determining parameter values in a linear electrical or electronic circuit from measurements of voltages and/or currents , is studied. First , equivalent circuit transformations are introduced to eliminate an inaccessible node where neither measurements nor applications of voltages and/or currents are possible. Next , if there are more than one inaccessible node , a sequence of transformations ar~ performed, and a sequence of circuits NO ' Nl' .. , Nf-l and Nf are obtained, where NO is the original circuit and Nf is the final circuit. Then a procedure and conditions for determining the parameter values in these circuits are given . The procedure is efficient , because it is based on a graph theoretic approach and involves solving sets of linear equations only, whereas conventional methods require solving nonlinear equations. Keywords. Parameter estimation; fault detect i on; linear systems; integrated c ircuits; graph theory; equiva lent circuit transformations.
Solving a set of nonlinear equations is time consuming. Several subsequent papers have considered this problem (Bedrosian and Berkowitz , 1962; Hayashi, Hattori and Sasaki , 1 967) . Recently Navid and Wilson (1979) gave a sufficient condition for the solvabi lity of the nonlinear equations. The condition is applicable to an active(elec tronic) circuit as well as a passive circuit.
INTRODUCTION with modern integrated circuit technology it is possible to manufacture electrical and electronic circuits with great complexity . These integrated cir cuits are embedded on semiconductor chips, and are not assemblies of elements of known values . Their circuit e lements cannot be taken ou t individually for measurements, even if they are faulty o r if t hey are stray elements of unknown values. Such faults must be detected without disassembling the circuits . Presently , therefore, there is a great need for systematic method to determine element values or parameter values(functions of element values including element values themselves) in them.
Another formulation for determining para meter values in a passive circuit was given by Shinoda (1970). His measurements consist of those of vo1tages and/or currents at test elements(branches) at a single circuit state (Applied voltages and/or currents are not varied.). This formulation was generalized by Ozawa and Kajitani (1979) for an active circuit which contains both elements of known values and unknown values . Their results give a base for various approaches to parameter-value determination, including that given in this paper. One of their theorems clarifies the voltages and currents which cannot be determined from the measurements at a single circuit state. Then these unknown variables are to be determined by changing the circuit state.
The problem of de terming parameter values in a linear passive(electrical) circuit from measurements at test terminals(nodes) was first considered by Berkowitz (1962). His measurements consist of those of currents due to the applied voltages of known values at the test terminals . The short- circuit transfer admittances are first obtained from the measurements at multiple circuits states (Applied voltages are varied to change the voltages and currents in the circuit . ) . Then t he equations relating the element values (conductances, etc.) to the short- circuit transfer admittances are solved to determine the element values . The equations are nonlinear, even if the circuit is linear. A few necessary conditions for the solvability of the equations are derived . This kind of measurements are , in general, difficult to perform , especially for electronic circuits.
Calculation of parameter values from node voltages is considered by Trick , Mayeda and Sakla (1979). They derived linear equations for element values by use of Tellegen ' s theorem, and gave solvability conditions for two special types of circuits . Their equa tions are very complex. Simpler equations can be derived from node voltages (El - Turky,
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T. Ozawa and S. Shinoda
and Vlach, 1980). The circuit considered in this paper may contain transistors and other active elements as well as passive two-terminal elements(resistors, capacitors and inductors). A transistor can be represented by a model consisting of resistors and voltage - controlled current sources. Likewise most active elements have their controlled source models. Therefore, we assume the elements in the circuit are passive two-terminal elements and controlled sources, especially voltage-controlled current sources (A voltaa,e controlled current source consists of a dependent current source and a voltage sensor which picks up the voltage controlling the dependent source. Thus this element has two branches.)
current sources are shown in Fig. l(b), but a two-terminal element should be connected between every pair of the peripheral nodes. The parameter value of the two-terminal element is denoted by gjk' if it connected between peripheral nodes j and k. It is called a g-parameter. The parameter of a voltage-controlled current source is called an a-parameter or an a-parameter respectively. The sensor branches are omitted in the figures.
It is assumed that circuit topology(how elements are connected) is known, and that all the element values are unknown. If some of the element values are known, some voltages and/or currents can be calculated from the measurements (Ozawa and Kajitani, 1979). In the equations for determining parameter values the calculated voltages and/or currents need not be distinguished from the measured voltages and/or currents. All of them can be considered as known variables, and the following discussions can be applied with a slight modification. The element values and the parameter values are all admittances, unless specified. We assume that they are nonzero, and that there is no special relation among the element values.
n
(a)
~~--~----~D n-l
1
an - ll (vn_l-v l )
ELIMINATION AND REVIVAL OF INACCESSIBLE NODES
a12 (v l -v 2 )
Equivalent Circuit Transformations n
'ie consider first the elimination of a single inaccessible node by equivalent circuit transformations. The node to be eliminated and the elements connected to it form a star. The transformations introduced here are extensions of the famous y-fi transformation. They can be classified into several types depending on the positions of the sensors and dependent sources in the star. Type A transformations. In case the star has dependent current sources controlled by the voltages between the peripheral nodes and the center, or in other words , in case both the dependent sources and their sensors are incident to the node to be eliminated, we introduce type A transformations. There can be complex transformations, if the star has many dependent sources, but to see the nature of the transformation of this type it would be best to consider that illustrated in Fig. 1, where the star has two dependent sources. The star in Fig. l(a) is equivalent to the circuit in Fig. l(b). To avoid a complex drawing only the dependent
(b)
Fig. 1 Type A transformation
The relation between these parameters and the element values gj(j=1,2, .. ,n), al and a 2 in the star can be derived as follows. gjk=gjgk/ O
(j=1,2, .. ,n-l;k=2,3, .. ,n: k>j)
~jl=gjal/o
(j=2,3, .. n-l:j~1,n)
aj2=gja2/o
(j=1,3, ..
(1)
Q12=a l a 2 /o
,n- 2 ,n;j~2,n-l)
(2) (3) (4)
where
(5) The short-circuit transfer admittances of the transformed circuit are (gjk=gkj if j>k in the following):
Determination of Parameter Values
ylk=glk'
y 2k =g2k (k=3, .. ,n) yjk=gjk (j=3,4, .. ,n-2;k=3,4, .. ,n:k;oij)
(7)
(6)
Yjl:gjl+ajl (j=2,3, .. ,n-2)
(8)
yj2=gj2+aj2 (j=1,3, .. ,n-2)
(9)
Y =q +a +a +a (10) n-l 1 n-l 1 n-l 1 12 12 Yn - l 2=gn-l 2 -a 12 -a 32 -· .-a n _ 2 2 -a n2 -Cl 12 (11) (12) Yn - l k =gn-l k +a (k=3,4, .. ,n-2,n) k2 a (13) Ynl =§nl- 21 - 31-' . n_l 1 12 Y =g +a +a +a (14) n2 n2 n2 21 12 (15) ynk=gnk+akl (k=3,4, .. n-l).
a
-a
-a
Type B Transformations. In case the star has dependent current sources controlled by the voltages between peripheral nodes, or in other words, in case only the dependent sources but not the sensors are incident to the node to be eliminated, we introduce type B transformations. One of the transformations of this type is depicted in Fig. 2. Again only dependent sources are shown in Fig. 2(b), and two-terminal elements 9jk (j=1,2, .. ,n-l;k=2,3, .. ,n:k>j) between pairs of peripheral nodes should be added to the figure. For each of the dependent current bk(vk-v ) in the star, dependent l
1807
currents bjk(Vl-Vk) (j=1,2, .. ,n:j;oik) flow from nodes j(j =1, 2, .. ,n:j;oik) to node k. The parameter bjk is called ab-parameter. The relation between the parameter values and the element values in this transformation is as follows. (j=1,2, .. ,n-l; k=2,3, .. ,n:k >j) (16)
(j=1,2, .. ,n;k=m,m+l, .. ,n:k;oij) (17) where (18) The short-circuit transfer admittances are (§jk=gkj if j >k): yjk=gjk (j=1,2, .. ,n;k=2,3, .. ,m-l:k;oij)
(19)
Y'k=§'k+b'k (j=1,2, .. ,n;k=m,m+l, . . ,n:k;oij) J J J (20) Y'l=§'l-b, -b, l-··-b, (j=2,3, .. ,m-l) J J Jm Jm+ In (21) Yjl=gjl-bjm-bjm+l-··-bjj_l-bjj+l-··-bjn
+b l ,+b 2 ,+ .. +1:), J
J
l,+b, 1,+··4 , . J- J J+ J nJ
(j=m,m+l, .. ,n)
(22)
Type C Transformations. In case only the sensors but not the dependent current sources are incident to the node to be eliminated, we define type C transformations. This type of transformations is similar to type B. Therefore the details of the transformations are omitted.
2
There can be a mixture of the above types.
computation of Element Values. Suppose that an inaccessible node is eliminated by a transformation. We will show conditions for computing the element values in the original circuit from the parameter values in the transformed circuit.
n (a)
Type A Transformation. If logarithms of eqs. (1)-(4) are taken, linear equations for variables xgj=log(gj/s) (j=1,2, .. ,n) and xaj=log(aj/s) (j=1,2) are obtained. They are:
b 2m (v 1 -vm) 2
X +X =log g12 gl g2 x +x =log g13 gl g3 m x x n (b)
Fig. 2
Type B transformation
gl al
+x +x
a2 a2
=log
a12
(23)
=log 0'12
There are n(n-l)/2+2(n-2)+1 equations in (23), but not all of them are necessary. Let M be the coefficient matrix of (23). M has two l's on each row and can be regarded as the edge-to-node incidence matrix of a
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T. Ozawa and S. Shinoda
graph . We use this graph as an auxiliary graph to give a solvability condition for (23). Let GA be the graph . GA has n+2 vertices corresponding to gj(j=1,2, . . , n), al and a2(They are identified also by these simbols.), and edges between gj and gk(j=l , 2, .. ,n-l;k=2,3, .. ,n:k > j), between al and gj (j= 2,3, . . ,n- l), between a2 and gj(j=1 , 3 , . . , n- 2,n) , and between al and a2 ' The edges in GA correspond to the equations in (23). The number of unknown variables in (23) is n+2. We will show which n+2 equations in (23) are necessary and sufficient to determine the x variables . The coefficient matrix of the n+2 equations which are chosen from (23) is denoted by Ms' Hs is a submatrix of M, and the edges corresponding to the rows of Ms together with their end vertices form a subgraph of G]\.. Let this subgraph be denoted by Gs ' Obviously we have: Lemma 1. Hs is singular if Gs contains an isolated vertex. Suppose Gs contains a cycle with c edges. Let Mc be the submatrix of Ms which corresponds to this cycle. We can easily show: Lemma 2. Mc is singular if c is even , and is nonsingular if c is odd(c~3) . Lemma 3. Ms is nonsingular if and only if Gs contains no isolated vertex and each connected component of Gs contains exactly one cycle with odd number~3 of edges. Proof. Necessity: From Lemma 1 Gs cannot have an isolated vertex . We can assume Ms is a block diagonal matrix , each block corresponding to a connected component of Gs . If there is a connected component containing no cycle , there must be a connected component containing more than one cycle and vice versa, since the number of edges of Gs is equal to the number of vertices. The blocks corresponding to these components are rectangular(not square). Thus if there are such components , Ms is singular . From Lemma 2 a connected component contajns no eve n cycle. Sufficiency: We need to consider a connected component, which corresponds to a square block of Ms' Suppose a nonseparable part of the component contains no cycle. The submatrix corresponding to it can be made triangular by proper permutation of rows and columns , if the column corresponding to its cut-vertex is removed . From this fact together with Lemma 2 we see the block corresponding to the component is nonsingular. In view of Lemma 3 we define a particular subgraph of GA . A dendroid in any graph G is defined to be a subgraph of G such that it contains all the vertices of G and each of its connected components contains exactly one cycle with odd number~3 of ed0es. Theorem 1.
Among the equations in (23)
the
equations corresponding to the edges of a dendroid in GA are necessary and sufficient to determine the unknown variables in (23) . Proof. From Lemma 3 we can determine Xgj (j=1,2, .. ,n), xal and x a 2' Then s=(gl+g2+" +gn+al+a2)/s can be calculated. Thus we can determine gj(j=1 , 2, . . ,n) , a l and a 2 . If the element values gj(j=1,2, .. ,n), al and a2 are to be computed from the short-circuit admittances, the following auxiliary graph GAY is used. GAY has n vertices correspond ing to gj(j=1 , 2, .. ,n) and edges between vertices gl and gk(k=3,4, .. ,n-l,n), between vertices g2 and gk(k=3,4, .. ,n - l,n) and between vertices gj and gk(j=3 , 4 , . . , n - 2;k=3 , 4, .. ,n - l , n:k~j). The edges of GAY correspond to the short- circuit admittances each of which has only one term in the right hand side of the equation in eqs . (6) and (7). Theorem 2 . The unknown variables x g j(j=1 , 2 , .. ,n) can be determined from the admittances corresponding to a dendroid in GAY ' Then al/s can be determined from one of Yjl(j=2 , 3, .. ,n-2) and Ynk(k=3 , 4, .. , n - l), and a2/s, from one of Yj2(j=1,3, .. ,n-2) and Yn - lk(k=3, 4 , .. ,n- 2,n). From these values gj(j=1,2 , .. , n), al and a2 can be determined. The right hand side of the equation for Ynk ' Yj2 or Yn- lk in Theorem 2 consists two terms , as seen in eqs . (8) (15) (9) or Type B Transformations . The auxiliary graph for determining gj(j-l,2, .. ,n) and bK(k=m, m+l , .. , n) from g- and b-parameters is denoted by GB' GB has 2n-m+l vertices corresponding to gj(j=1,2 , .. ,n) and bk(k=m,m+l, .. ,n), and edges between vertices gj and gk(j=1 , 2 , .. , n - l; k=2 , 3 , .. , n:k>j) and between vertices g. and bk(j=1 , 2 , .. , n;k=m , m+l , . . , n:k~j) . The eages correspond to the equations in (16) and (17) . Theorem 3 . The element values gj(j=1,2, . . , n) and bk(k=m , m+l, .. , n) can be determined from the equations, among those in eqs. (16) (17) , which correspond to the edges of a dendroid in GB ' Again, by taking the logarithms of the equations , we need to solve a set of linear equa tions for unknown variables x g j(j=1,2, .. , n) and xbk(k=m,m+l , . . ,n) , where xbk=log(bk/s). If we want to determine the element values from the short - circuit admittances , we form an auxiliary graph GBY' which has n vertices g . (j=1,2, .. ,n) and edges between vertices gj a~d gk(j=1,2, .. ,n;k= 2 , 3, .. ,m- l:k~j). The edges of GBy correspond to the short - circuit admittances, each of which has only one term in the right hand side of the equation in eq. (19). Theorem 4. The element values gj(j=1 , 2, .. , n) can be computed from the short - c~rcuit admit tances corresponding to the edges of a dendroid in GBy . Then bk(k=m , m+l, . . n) can be
Determination of Parameter Values
determined from these element values and one of Yjk(j=1,2, .. ,n:jik). The short-circuit admittance Yjk in the theorem has only two terms in the right hand side of the equatuion in (20). The admittances given by eqs. (21) and (22) may be used, if available. Modification of Auxiliary Graphs. For the simplicity of the discussion we have ignored the element which may exist between the peripheral nodes of the star. If an element exists, for example, between nodes j and k of the star, it is parallel to 9jk in the transformed circuit. The values of parallel elements cannot, in general, be obtained separately by voltage measurements only. Thue gjk is unknown and the equation 9j9k/a=gjk cannot be used to determine the element values of the star. Therefore the edge in the auxiliary graphs which corresponds to this equation should be deleted. Modification of the auxiliary graphs and the theorems is summarized as follows. (i) If there is a two-terminal element between nodes j and k of the star, then delete the edge between vertices gj and 9k in any of GA, GAY' GB and GBy . (ii) For type A transformation: If there is a dependent current source between node j(j=2,3, .. , or n-l) and node n of the star which is controlled by voltage Vj-Vl' then delete the edge between vertices gj and al in GA. Omit Yjl in Theorem 2. If there is a dependent current source between node j(j= 1,3, .. ,n-l or n) and node n-l of the star which is controlled by voltage Vj-V2' then delete the edge between gj and a2 in GA. Omit Yj2 in Theorem 2. If there is a dependent current source between nodes n-l and n which is controlled by voltage vl-v2' then delete the edge between al and a2 in GA· (iii) For type B transformation: If there is a dependent current source between nodes j and k (j=1,2, .. ,n;k=m,m+l, .. ,n:kij) controlled by voltage vl-vk' then delete the edge between vertices gj and gk in GB. Omit Yjk in Theorem 4. (iv) If any of the parameter values or short-circuit admittances in the transformed circuit cannot be determined from the measurements by some reason or other, delete the edge corresponding to it from the auxiliary graphs or omit it from the theorems.
Eliminations and Revivals of Inaccessible Nodes. If there are more than one inaccessible node, we repeat the transformations introduced in the previous section, and eliminate them one by one. Let NO be the original circuit and let N+, N2 , .. , and Nf be the resultant circu~ts obtained by the sequence of transformations, where f is the number of inac-
1809
cessible nodes. At each elimination an auxiliary graph is constructed. An elimination of a node, in general, gives rise to parallel elements and the auxiliary graph should be modified accordingly. The final circuit Nf usually has many parallel elements. These parallel elements cannot be separated by the voltage measurements, but the sum of their parameter values (admittances) can be obtained. As for the short-circuit admittances of the transformed network, they can be derived by adding the admittances given by eqs. (6)(15) or (19)-(22) to the admittances originally existing at the relevant node pair. In this way the short-circuit admittances of Nf can be derived and expressed in terms of the parameter values of the elements in Nf. From the parameter values or the shortcircuit admittances of Nf, we are to determine the element values in NO. The computations of the element values or parameter values are performed in the reverse order of the elimination. The eliminated nodes are revived one by one, and we get Nf-l from Nf, Nf-2 from Nf-l' and so forth. The steps to obtain Nk-l from Nk by reviving node k, which is the center of the star in the transformation, are as follows. 1° Find whether a dendroid exists in the auxiliary graph. If ther is one, compute the parameter values in the star in Nk-l from the parameter values in NK which correspond to the edges of the dendroid. 2° Compute, using necessary equations in eqs. (1)-(4) or (16) (17), the §-, ~- &- or b-parameters which are in parallel with the other elements in Nk' but disappear in Nk-l· Subtract the values from the sum of the parameter values or the total short-circuit admittances of Nk' to obtain the parameter values of Nk-l. Nl , N2 , .. , and Nf depend on the order of eliminated nodes, and thus the possibility of existence of dendroids in the auxiliary graphs varies depending on the order. Theorem 5. If step 1° is possible for each k=f, f-l, .. and 1, the element values in NO can be determined from the parameter values in Nf by solving sets of linear equations only.
DETERMINATION OF PARAMETER VALUES FROM NODE VOLTAGE MEASUP£MENTS Suppose we have eliminated all the inaccessible nodes and have Nf. Suppose also Nf has m accessible nodes where both appications of cuurents and measurements of voltages are possible, and p partly accessible nodes where only measurements of voltages but not applications of currents are possible. Let vl' v2'··' vm and jl' j2'··' jm be the node voltages and the applied currents at accessible nodes 1, 2, .. , m
1810
T. Ozawa and S. Shinoda
respectively, and let v m+ l ' v m+2"" vp be the node voltages at partly accessible nodes m+l, m+2, .. , p respectively. They are related by open-circuit impedances as Zll Z12
Zlm
Z21 Z22
Z2m
v jl
..............
Z Z ml m2
..
2
j2
Z mm
v
..............
Z Z pl p2
v
l
(24) m
jm
Z pm)
v
From the node voltages of Nf' we can compute the branch voltages by use of Kirchhoff's voltage law. Let i and j be the vectors of the branch currents and the applied currents respectively. Then from Kirchhoff's current law we have Qi=Qjj, where [Q -Qjl is the fundamental cutset matrix or the incident matrix of the current graph derived from Nf . The branch currents and the branch voltages are related by i=Vg, where V is the diagonal matrix with the branch voltages on the diagonal, and 9 is the vector of the parameter values . Substituting this equation into the relation above, we get (25) The number of equations in (25) is usually less than the number of unknown parameter values. In order to qet an enough number of equations for unknown variables, we vary the applied currents and produce different circuit states. We distinguish the variables at the k-th circuit state by superscript (k) . Then from m independent applied currents we usually get m circuit states, and we have:
QV(2) g= QV(m)
Q.j (1) J Q.j (2) J
Q,j
The p roposed method can be applied to o ther physical systems which have electrical or electronic circuit analogues. Since controlled sources can be used, varieties of systems can be modeled by electronic circuits. For example, input-output relations
x1 =
p
Thus Zik (k=1,2, .. ,p) can be directly obtained from the voltages vl' v2"" vp due to the application of a unit current at node i only. If there is no partly accessible nodes, the short-circuit admittances can be obtained by inverting the opencircuit impedance matrix in eq. (24).
QV (1)
and electronic circuits from measurements. The procedures are efficient b ecause they are graph theoretic and invo l ve solving sets of linear equatio ns only.
(26)
(m)
J
Some conditions for determining parameter values of 9 from eq. (26) are given by Ozawa and Yamada (1981).
CONCLUDING REMARKS We have given conditions and procedures for determining parameter values in electrical
(27)
x2 = can be expressed by a circuit containing two resistors and two controlled sources. One method to obtain an electrical or electronic model of a system is to use a bond graph(vanDixhoorn and Evans (Ed.) 1874). Acknowledgements. This work was p artly supported by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan under Grant Cooperative Research (A) 435013(1979-1980) .
REFERENCES Bedrosian, S.R. and Berkowitz, R.S. (1962). Solution procedure for single-element-kind networks. IRE Int. Conv. Rec. Part 2, 16-24. Berkowitz, R.S. (1962). Conditions for network - element-value solvability. IRE Trans. Circuit Theory, CT-9, 24-29. El-Turky, F.M. and Vlach , J. (1980). Calculation of element values from node voltage measurements. 1980 Int. Symp. on Circuits & Syst. Proc . 170-172. Hayashi, S., Hattori, Y. and Sasaki, T. (1967). Conditions on network-elementvalue evaluation. Electronics and Communications in Japan, 50, 118-127 Navid, N. and Wilson Jr. A.N. (1979). A theory and an algorithm for anolog circuit fault diagnosis. IEEE Trans. Circuits & Syst. CAS-26, 440-457. Ozawa, T. and Kajitani, Y. (1979). Diagnosability of linear active networks. IEEE Trans. Circuits & Syst. CAS-26, 485-489. Ozawa , T. and Yamada, H. (1981). Conditions for determining parameter values in a linear active network from node voltage measurements. 1981 Int. Symp. on Circuits & Syst. Proc. Shinoda, S. (1970). A minimization problem in network diagnosis. Trans. Inst. Elec. Comm. Eng. Japan, 53-A, 569-570 . Trick, T.N., Mayeda, W. and Sakla, A.A. (1979) . Calculation of parameter values from node voltage measurements. IEEE Trans. Circuits & Syst., CAS-26, 466-474. vanDixhoorn, J.J. and Evans, F . J. (Ed.) (1974). Physical Structure in Systems Theory, Academic Press, London.
Determination of Parameter Values Discussion to Paper 59 . 1 R. Konakovsky (Federal Republic of Germany) : First, does the quality of the control depend on the number of subdivisions of the system in subsystems? Next, how do you decide between two possibilities of a subdivision? And finally, is the subdivision given by the system structure itself and does the reliability in the figure mean for a given period of time? M. Funabashi (Japan): In answer to your first question, it very much depends on the structure of the original problem, and so is very difficult to answer in general. I think we must study that case by case. The period of time is 1 hour. H. Wedde (Federal Republic of Germany) : Your concept of reliability does not yet take into consideration the communication between the (decentralised) computers. Do you not think that this is a practically important aspect? M. Funabashi (Japan): Consideration of the communication is very important. However, our study has excluded it because it does not so much effect the control algorithm, and our approach is that from the control scheme. We will probably include it in the future. D.R. Powell (France) : Could you please justify your assumed relationship between reliability and computer power? At what mission time was the reliability evaluated (reliability is a time-dependent function)? M. Funabashi (Japan): The relationship I talked about is the empirical one, which coincides with the generally accepted tendency that reliability of microprocessors is higher than that of large-scale processors. The time was 1 hour. Discussion to Paper 59.2 R . Konakovsky (Federal Republic of Germany) : There are also methods for fault diagnosis in electronic systems. Can you explain the difference between the fault diagnosis of mechanical and electronic systems? S. Hirai (Japan): Usually in the case of diagonal methods for electronic circuits, check points for diagnosis are prepared beforehand. Therefore, it is very difficult to detect unexpected failure. Furthermore, causality in an electric system is much simpler than in a mechanical one. Our diagonal system puts emphasis on describing mechanical objects which have very complicated causality in a general and simple manner. D.R . Powell (France) : Although you do state that your failure analysis technique assumes only 1 fault, do you think that it will be possible to extend the technique to multiple failure situations which, I believe, are frequent in mechanical systems due to chain reaction effects. S. Hirai (Japan) : It is an important and interesting problem that we extend our method for multiple failure situations . We are now studying the method. A simple method in the present stage is to detect and repair each abnormal part one by one using this diagonal system. eST 3 - H'
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Y. Nozaka (Japan): I would like to ask two questions froD the practical standpoint. First, how much time does it take in the case of diagnosis of air-conditioner and second, what is the main feature of this new method compared to conventional methods such as vibration analysis? S. Hirai (Japan): The mean response time for each data input is one or two seconds. The main feature of our system in the present state is to describe the function of machines and its deviation in a simple and general manner and to perform the diagnosis on the basis of this description . The conventional methods you pointed out will be incorporated in our system in future as the extended sensing teChnique . Discussion to Paper 59.4 T. Soeda (Japan) : Stochastic signals are not always stationary. What do you think of the accuracy of the model when we apply this method to the practical complex problems? T. Fukuda (Japan): I assumed here that the process is stationary , because many plants operate in a steady state. If not, we cannot apply this method directly, because we cannot determine the order of the model. If it is known a priori , we can model the time varying system accurately. R. Konakovsky (Federal Republic of Germany): What represents the top of your fault tree diagram and what are the primary events at the bottom? T. Fukuda (Japan): In this special case, I employed the event that the heating section will burn out . In general, we can employ some events as a top event, which we try to prevent. The list of basic events employed have already been discussed in the paper. Discussion to Paper 59.5 R. Konakovsky (Federal Republic of Germany) : In your method you transform the network until the final network. You then determine the parameters of the final network and from here you trace back to the original network to get its parameters? Do you simplify the equation after the transformation and what is the relation between the number of accessible and inaccessible points? How many measurements are necessary to determine the parameters? T. Ozawa (Japan): That is correct . By taking the logarithms , the set of nonlinear equations are converted to a set of linear equations whose coefficients are alII. There is no particular relation between the numbers of accessible nodes and inaccessible nodes . In general, however, the number of unknown elements increases as the number of inaccessible nodes increases . The number of possible measurements for transfer admittances is limited to n(n - l) where n is the number of accessible nodes . The number of unknown ele ments must obviously be less than or equal to this number n(n-l). The number of measurements necessary is equal to the number of unknown elements .