i\i\\\\\\\\\\\\\\\\\\\\\\\\\\\\i\\\\\\\\\\\\\\\\\\\\i affix; 3129 0062 , University "dram This is to certify that the thesis entitled STATE FORMULATION OF LARGE-SCALE LINEAR TIME-INVARIANT BOND GRAPH MODELS presented by Benjamin Moultrie has been accepted towards fulfillment of the requirements for Ph . D . Jggee in Mechanical Engineering firfi/ J? Major professor (f. Date 5/3/77 0-7 639 ovmnur FINES ARE 25¢ PER DAY _ PER ITEM to remove Return to book drop r record. this checkout from you ‘ I STATE FORMULATION OF LARGE-SCALE LINEAR TIME-INVARIANT BOND GRAPH MODELS BY Benjamin Moultrie A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1979 ABSTRACT STATE FORMULATION OF LARGE-SCALE LINEAR TIME-INVARIANT BOND GRAPH MODELS by Benjamin Moultrie In this dissertation, topology-based equations are developed which give the effort-flow basis order for the juncture structure transformation of an arbitrary weighted junction structure. These equations 3e used to develop three upper bounds for the numberof distinct sets of port variables which can be used to specify weighted junction structure input-output relations. Each successive bound is shown to be numerically smaller and computationally more expensive than its predecessor. Examples are given which use the established bounds. Also, a causal assignment procedure is specified which simplifies the state model formulation process for linear time-invariant bond graphs. This result is used to develop an efficient computer implemented state model formulator for linear time-invariant bond graphs. The key storage features and novel matrix manipulation procedures of this state model formulator are explored, and key computer subroutines are given. The enhanced performance characteristics of this formulator are validated by computer test results. ACKNOWLEDGMENTS With apologies to Buffon: For in those few men whose head is steady, whose heart is compassionate, and whose sense is exquisite - there is substance, thought and reason; there is the art of speaking to the mind. Thanks Doc. ii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . vii NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . ix Chapter I. INTRODUCTION. 1 1.1 Background. . . . l 1.2 The Status Of Bond Graph Theory And Practice. . . . . . 3 1.3 Research Highlights And Dissertation Organization. . . . . . . . . . . . . . . . 5 II. BOND GRAPH JUNCTION STRUCTURES. . . . . . . . . 8 2.1 Junction Structure Terminology And Notation. . . . . . . . . . . . . . . . 8 2.2 Power Orientation . . . . . . . . 10 2.3 Analytic Properties Of Junction Structure Nodes . . . . . . . . . . . . . . 11 2.4 Basis Order Rules . . . . . . .‘. . . . . . 12 2.5 Causal Concepts . . . . . . . . . . . . . 15 2.5.1 Causal Assignment . . . . . . . . . . 16 2.5.2 Causal Completeness . . . . . . . . . 16 2.5.3 Causal Consistency. . . . . . . . . . 16 2.5.4 Causal Extension. . . . . . . . . . . 17 2.6 Causal Complexes. . . . . . . . . . . . . . 20 III. STATE MODEL FORMULATION FOR LINEAR TIME- INVARIANT BOND GRAPHS. . . . . . . . . . . 28 3.1 Field Multiports. . . . . . . . 29 3.2 The Standard Sequential Causality Assignment Procedure. . . . . . 31 3. 3 The State Model Formulation Algerithm . . . 32 iii Chapter IV. V. BIBLIOG APPENDI Appendi A. A COMPUTER IMPLEMENTED STATE MODEL FORMULATOR OF INCREASED EFFICIENCY 4.1 Design Features. . 4.1.1 Data Structures. 4.1.2 Causal Assignment. 4.1.3 Determination of Junction Structure Reducibility 4.1.4 Junction Matrix Formulation. 4.1.5 Matrix Inversion . 4.2 State Model Formulator Computer Test Results . . . . . . SUMMARY. RAPHY. CES. x DERIVATION OF THE BASIS ORDER RULES. A.l Basis Order And Simple Junction Structures . . A. 1. 1 Basis Order For Proper Simple Junction Structure Forests A.1.2 Basis Order For Proper Simple Junction Structures. A.1.3 Basis Order For Standard Simple. Junction Structures. A.1.4 Port Basis Order For Standard Simple Junction Structures A.2 Basis Order And Weighted Junction Structures . . . A. 2.1 Basis Order For Standard Weighted Junction Structures A.2.2 Port Basis Order For Standard Weighted Junction Structures AN APPLICATION OF THE BASIS ORDER RULES. B.l An Upper Bond For C As A Direct Application Of The Basis Order Rules 8.2 A Refined Upper Bound For C. THE RESOLUTION OF A CONFLICT RESULTING FROM A PORT BOND CAUSAL ORIENTATION. THE IMPACT OF THE STANDARD SEQUENTIAL CAUSALITY ASSIGNMENT PROCEDURE (SSCAP) ON THE REDUCED JUNCTION MATRIX . iv Page 40 41 41 42 42 43 44 46 53 55 S8 58 S8 58 66 69 71 72 73 74 8O 81 84 92 94 Appendix Page E. PRUNING AND SOME ASPECTS OF JUNCTION STRUCTURES. . . . . . . . . . . . . . . . . . . 101 F. COMPUTER SUBROUTINES. . . . . . . . . . . . . . 107 Table 10 LIST OF TABLES Analytic Properties Of Junction Structure Multiports Analytic Properties Of Field Multiports (with exactly one incident bond) Junction Matrix Storage For Test Examples Processing Time For State Model Formulation. Subroutine For Causal Complex Identification Subroutine For The Determination Of Causal Complex Solvability Subroutine For The Computation Of The Reduced Junction Matrix. . . Subroutine For The Construction Of The Junction Matrix. Subroutine For The Determination Of Junction Matrix Entry Position Subroutine For Matrix Inversion. vi Page 25 35 52 52 107 110 115 125 129 130 Figure 2. 1 LIST OF FIGURES Generic categories of bond graph multiports Junction structure examples. Examples of power orientation. A11 consistent causal forms for junction structure multiports. Examples of junction structure multiports with inconsistent causal forms Example of a causal complex (shown in solid lines) Consistent causal forms for field multiports Diagram of the standard sequential causal assignment procedure. The reduced junction matrix equation The junction matrix equation Derivation of the reduced junction matrix. Example of a list pair with the corresponding matrix Test example 1 Test example 2 Test example 3 vii ’Page 22 23 24 26 26 27 36 37 38 39 39 49 50 50 51 Figure A A. .1 2 A l-junction proper SJS. A O-junction proper SJS. Example of Lemma A.l construction procedure Transformation for converting a standard SJS to a proper SJS Transformation for converting a WJS to a SJS Example of a weighted junction structure. . . . . . . Causally augmented weighted junction structure . Unlabelled causal orientations of a weighted junction structure Symbolic bond graph representation with fields identified Example of subgraph generation by pruning . Symbolic bond graph representation with source field pruned . Junction structure with labelled nodes Example of the pruning procedure viii Page 76 76 77 79 79 89 90 91 98 99 100 105 106 Note number number sum of sum of number number number number number number number number number number number number that of of NOMENCLATURE external 1-junctions external O-junctions the degrees of the l-junctions the degrees of the O-junctions of of of of of of of of of of of of A

N + Nf<(N +B -N F 0 1 1 B f)max=“o 1 1’ since NfiNB° Therefore, Nf<0 implies that C is not an n-port. Suppose NF=O where Nf=(Nf)max' Then N =0 + O=NB-(NO+B F -N =NB-(J++NI) 1 l) =(P1+PO)-J+=NP-J+ + J+=Np + J_=O Therefore, NF=O implies that C is not an n-port (this con- clusion represents an exclusion of a case where C acts like a source of flow to its environment [26]). Similarly, N510 implies that C is not an n-port. Thus, in order for C to be an n-port, it is necessary that 0\'1/<: m n+m:l l I Dissipation: gR/fi n+m:l 3 Figure 3.1. Consistent causal forms for field multiports. ohsonOHQ ucoscuwmmw Hmmsmu Hmwucozcom ppmwcmpm ocu mo Empmmflm .N.m whamwm ‘ . 22.2%on __ 00‘ 5 mm» .5325 I :5ng MSG ‘ mzom 322 fig... 3?... E. I as... j - 7 amass n28 T , 021w 4 mg zoEESmE €832 - . 53.28 .1 m>Z<>Ema mac . L1 _ gomEH mac - , - _ cz Ease 928 _ .M w muéofi gag , . oz ? 1.85.28 Emazou mam . w # - 02— , 3.558 :28 mm» 85% .522 LS VGCtOT VCCCOT vector VGCtOT vector VGCtOF VGCtOT VGCtOT 38 _ . - — 1 - u —1 S11 S12 S13 S14 41 3d = 821 S22 S23 S24 Ed Bin S31 832 83:5 334 —Dout LE. .5418“ 543 544-191 of inputs to independent storage elements of inputs to dependent storage elements of inputs to the dissipation elements of inputs to the source elements of outputs from the independent storage elements of outputs from the dependent storage elements of outputs from the dissipation elements of outputs from the source elements Figure 3.3. The reduced junction matrix equation. 39 gout S1 S2 2in Kant 83 S4 Kant Kin - vector of all inputs to the junction structure Kant - vector of all junction structure internal variables yout - vector of all outputs from the junction structure Figure 3.4. The junction matrix equation Xout = SIZ-in + SZX-int Y-‘mt = 531m + S4y-int (a) v = (I - S )-IS v —int 4 3—in (b) _ _ -1 Xout ' [31 + Sz(I 54) Sijin (C) Figure 3.5. Derivation of the reduced junction matrix. (a) The expanded junction matrix equation. (b) Internal variables in terms of inputs. (c) The unpartitioned reduced junction matrix equation. IV. A COMPUTER IMPLEMENTED STATE MODEL FORMULATOR OF INCREASED EFFICIENCY The ENPORT—4 program is a powerful tool for the modeling, analysis, and simulation of multiport systems. When given a bond graph description of a system, ENPORT-4 selects physi- cally-meaningful state variables and derives the system state model, eigenvalues, and time response. The many addi- tional features, available options and outputs, and the struc- ture of ENPORT-4 are discussed in the program's documenta- tion [8]. Although ENPORT—4 provides the system analyst with a broad array of system information, it has significant in- adequacies which have a profound affect on program perfor- mance. Most of these inadequacies are revealed in the graph reduction and state model formulation procedures. In the following sections, deficiencies are identified in the ENPORT-4 graph reduction and state model formulation pro- cedures, and modifications are discussed which increase overall program efficiency. These modifications have been implemented in the ENPORT-5 program which is currently under development. Key ENPORT-S graph reduction and state model formulation subroutines are listed in Appendix F. 40 41 4.1 Design Features 4.1.1 Data Structures At various stages in the bond graph processing procedure assorted graph parameter and structural information must be retained or manipulated. In general, when interpreted in matrix form, this information results in a sparse matrix analogous to a graph incidence or adjacency matrix [27]. A major deficiency of ENPORT-4 is its use of full storage (storage which includes all matrix zero entries) in multi- dimensional arrays for the retention and manipulation of bond graph information. A major improvement in efficiency is realized in ENPORT-5 by minimizing data storage requirements through the use of a sparse-matrix-based storage format. This is achieved by using push-down stacks and linked data structures.[28-30]. In particular, the ENPORT-5 graph reduction and state model formulation procedures employ simple lists for the retention and manipulation of data. These lists are grouped in pairs, where the entries of each list are ordered. In each list pair, one list contains the nonzero entries of an implied matrix of known dimensions, and the other list contains the coordinates of the matrix entries where each coordinate pair is converted to a unique number. The conversion of a coor- dinate pair is accomplished by representing the position of a matrix entry as the entry's column coordinate added to the product of the matrix column dimension and one less than the 42 entry's row coordinate. An example of a list pair is given in Figure 4.1. 4.1.2 Causal Assignment The assignment of causality is an important stage in the processing of a bond graph. The ENPORT-4 program uses a causal assignment scheme which is a modification of SSCAP (the standard sequential causality assignment procedure) in that the scheme gives priority to user specified causal orien- tations. Although this feature provides the knowledgeable user with a great deal of flexibility, the unwary user may specify causal orientations which may violate system con- straints (such as constraints imposed by sources), give a false indication of system order, or create uncertainty in the state model formulation procedure. The causal assignment scheme employed by the ENPORT-5 program is a direct implementation of SSCAP in which the user cannot specify causalorientatiOnSLHHjl.after all source bonds and storage bonds have been causally oriented. This scheme not only eliminates the difficulties discussed above, but also guarantees that if a reduced junction matrix exists, then it has the simplified form identified in section 3.2. 4.1.3 Determination Of Junction Structure Reducibility The ENPORT-4 procedure for determining the reducibility of the junction matrix represents an additional area of inefficiency. The junction matrix equation was given in 43 Figure 3.4. As illustrated in Figure 3.5, the junction matrix is reducible if the matrix (I-S4) is nonsingular. In ENPORT-4, junction matrix reducibility only can be deter- mined during the process of attempting to invert the matrix (I-S4). Based on previous work, the reducibility of the junc- tion matrix depends on the solvability of causal complexes [12-14,l9,20]. Stating the case more explicitly, the junction matrix is reducible if and Only if each causal complex is solvable. As an explicit step in the ENPORT-5 causal assignment procedure, causal complexes are identi- fied and their graph locations are communicated to the user. Prior to the formulation of the junction matrix, each causal complex is tested for solvability in order to determine junction matrix reducibility. If any causal complex is determined to be unsolvable, then bond graph processing aborts and the user is notified of all unsolvable causal complexes. 4.1.4 Junction Matrix Formulation As an intermediate step in the formulation of the junction matrix in ENPORT-4, a matrix equation is explicitly formed for each junction structure node. The entries of each node matrix are then placed in the junction matrix in accordance with bond classifications and orderings. The ENPORT-5 program does not explicitly form a matrix equation for each junction structure node. Instead, the 44 junction matrix is constructed directly by using node causal forms, bond power orientations, and graph model parameters to obtain the coefficients of the summation, identity, and proportionality output equations for each junction struc- ture node. Specifically, causal, power, and parameter in- formation is used to identify the flow output variable and the coefficients of the corresponding flow input variables for each O-junction, the effort output variable and the coefficients of the corresponding effort input variables for each l-junction, the identity relations for each 0- junction and l-junction, and the pr0portionality relation- ships for each transformer and gyrator. Once determined, each of the above coefficients (except zeros) is directly stored in a compact junction matrix where each entry position is determined by bond classifications and order- ings, and the dimensions of the implied full storage junc- tion matrix. 4.1.5 Matrix Inversion As illustrated in section 3.3, several calculations in the state model formulation algorithm require the computa- tion of a matrix inverse. The matrix inversion routine used by ENPORT-4 is a Gauss-Jordan procedure which selects a matrix entry of greatest magnitude for the pivot at each 45 stage of the deflation process. In ENPORT-4, the selection of a pivot requires a row and column scan of mostly zero entries since each matrix is generally sparse and in a full storage format. A result of this research was the development of the sparse matrix inversion subroutine which is employed by the ENPORT-5 program. This subroutine is called "INPRD" (INverse-PRoDuct). Its development was motivated by the lack of a matrix inversion routine which can take advantage of the special features of the equations in the state model formulation algorithm. A very important consideration in the development of any sparse matrix inversion routine is the possible increase in the storage requirements for the inverse of a sparse matrix [31]. The INPRD subroutine effectively eliminates the problem of storage growth by taking advantage of the features of the matrix calculations in the graph reduction and state model formulation procedures. In these procedures, matrix inverses in calculations appear in the form A'lB where the matrix product A-18 relates sets of junction structure variables. INPRD is a Gauss-Jordan type pro- cedure which controls storage requirements by accepting the generally sparse matrices A and B as inputs and return- ing the generally sparse matrix A-1B as output. Note that A.1 is not explicitly computed unless B is the compatible identity matrix. INPRD applies directly to B a set of 46 transformations which represent elementary row operations for the reduction of A to the identity matrix. In order to mini- mize round-off errors, a matrix entry of greatest magnitude in A is selected as the pivot at each stage in the process of deflating A. The benefits of INPRD are evidenced by the performance characteristics of the ENPORT-5 program. 4.2 State Model Formulator Computer Test Results In this section, some performance aspects of the ENPORT- 4 and ENPORT-5 state model formulators will be considered. In particular, processing times and junction matrix storage requirements will be assessed for three test examples inter- actively processed on the CDC 6500 computer. The processing time will be interpreted as the CP (central processor) execution time consumed from the point of parameter input tothe pointcfi?state model output. Storage considerations are limited to the junction matrix, since it is the largest system matrix in the formulation process. For each example considered, the processing time and junction matrix storage space requirements of ENPORT-4 will be used as benchmarks. The first test example is a structural model of a lever mechanism with inertia load (see Figure 4.2). The second test example is a structural model of a beam-block trans- ducer system (see Figure 4.3). The final test example is a structural model of a radar pedestal position control system (see Figure 4.4). Each of the above examples may be 47 found in the user's manual for the ENPORT-4 program, where each is studied in detail [8]. For each test example, the computational results are contained in Table 3 and Table 4 for storage requirements and processing time respectively. From Table 3, it is observed that the ENPORT-S formulator requires significantly less storage (as typified by the junction matrix) than does the ENPORT-4 formulator for a given bond graph model. .The ENPORT-4 storage requirement for the junction matrix is given by (Np+2NI)2. The ENPORT-5 storage requirement for the junction matrix is given by 4(Np+2NI-1). Thus, the difference between the bond graph model storage demands of the ENPORT-4 and ENPORT-5 formulators becomes increasingly dramatic as the number of bonds in a graph model increases. From Table 4, it is observed that the ENPORT-5 formu- lator provides a significant savings in processing time for the given test examples. In general, the size (and sign) of this savings is a function of several variables, e.g., the number of causal complexes, and the density and dimen- sions of matrices to be manipulated. As an explicit case, consider the multiplication of an (nxm) matrix by an (mxp) matrix, neither of which contains a zero entry. In a full storage format, this matrix multiplication requires nmp scalar multiplications and n(m-l)p scalar additions. In ENPORT-5, this matrix multiplication requires nmp scalar multiplications, nmp scalar additions, and 2nm(p+l) element 48 comparisons. Note that full storage matrix multiplication requires the same number of scalar operations irrespective of the sparsity of either matrix factor. In general, the above matrix multiplication in ENPORT-5 requires n(m-r)(p-s) scalar multiplications, n(m-r)(p-s) scalar additions, and 2nm(p+l)-rn(p+2)-2ns(m-r/2) element comparisons where r is the average number of zeros per row in the (nxm) matrix and s is the average number of zeros per row in the (mxp) matrix. Note that the worst case is given for the number of element comparisons. It is seen that as r and 5 increase, matrix multiplication in a full storage format rapidly becomes com- putationally more demanding than matrix multiplication in ENPORT-5. Thus, although it is possible for the processing time required by ENPORT-S to exceed the processing time required by ENPORT-4, this possibility is minimized by the general sparsity of system matrices and the "inverse-multi- plication" feature of the INPRD subroutine. In conclusion, the combined results of Table 3 and Table 4 suggest that the ENPORT-5 state model formulator not only enhances the processing performance and capabilities of the ENPORT-5 program, but also contributes to a reduced dollar cost for the operation of a linear time-invariant bond graph simulation program. 49 '2‘ [1‘ -l 3 5 6 3 8 :4. 1.12.. Entry list Position list Figure 4.1. Example of a list pair with the corresponding matrix. 50 SE C 1 4 1 6 TF 7 1 5 Z ‘ 3 I R Figure 4.2. Test example 1 2 C 1 12 TE 11 O 10 1 9 GY Figure 4.3. Test example 2 51 I R 13 14 12 1 15 TF CY 11 16 0 0—-1-—C 1—17—1 9 ~\4L\\\ 18 / 1 R 6 5 2 19 GY 4 1 1 GY 3 1 I SE Figure 4.4. Text example 3 52 Table 3. Junction Matrix Storage For Test Examples Example (1)W4 (2)ws WS/W4 1 81 32 .395 2 289 64 .221 3 841 112 .133 (l) W4ENumber of storage words for junction matrix in ENPORT-4. (2) WSENumber of storage words for junction matrix in ENPORT-5. Table 4. Processing Time for State Model Formulation Example I (1)PT4 (Z)PTS PTS/PT4 1 0.604 ’ 0.138 0.228 2 0.905 0.346 01382 3 1.420 0.927 0.653 (1) PT4EProcessing time (in seconds) for ENPORT-4. (2) PTSEProcessing time (in seconds) for ENPORT-5. V. SUMMARY The basis order rules are among the most significant results achieved in this investigation. For weighted junc- tion structures, these topology-based formulations provide the bond graph analyst with the composition of a basis for the junction structure transformation. In addition, when used with Theorem 3, the basis order rules provide a "good" estimate of the number of distinct basis variable sets for the junction structure transformation. Herein, it was shown that the standard sequential causality assignment procedure assures that the S S 23’ 32’ and 833 blocks of the reduced junction matrix are zero for °a reducible junction structure. This resulted in'a major simpfijicationijlRosenberg's state model formulation algo- rithm which serves as a model for the ENPORT-S state model formulator [1fl. As indicated by the computer results in Chapter IV, the ENPORT-5 state model formulator provides heretofore unreal- ized efficiency and speed in the automated processing of linear time-invariant bond graphs. The key features of this formulator are (l) the use of sparse-matrix-based storage, (2) the determination of junction structure reduci- bility prior to the formation of the junction matrix, (3) the direct construction of the junction matrix, and (4) the 54 versatile sparse-matrix-based INPRD subroutine. As a result of the storage and processing efficiency of the ENPORT-5 state model formulator, the ENPORT-S program has a greatly enhanced capacity for the processing of large bonds graphs. Future advances in graph processing efficiency can be achieved by the development of a general technique which does not require matrix inversions for the determination of junction structure reducibility. A step in this direction can be made by the development of ”basis order rules" which are applicable to any junction structure containing a gyrator. Such formulations would offer necessary conditions for the reducibility of an arbitrary junction structure, as well as serve as aids for the determination of a basis for the junc- tion structure transformation. BIBLIOGRAPHY [11 [2] [31 ['41 [51 [61 [71 [8] [9] BIBLIOGRAPHY Chorafas, D.N., Systemsand.Simulation, Academic Press, 1965. Malmberg, A.F., "NET-2 Network Analysis Program-User's Manual Release 9", HDL-OSO-l, Braddock, Dunn, and McDonald, Inc., El Paso, Texas, 1973. Nagel, L. W. , and Pederson, D. O. "SPICE (Simulation Pro- gram with Integrated Circuit Emphasis)", ERL- M382, Electronics Research Laboratory, College of Engineering, Univ. of California, Berkeley, Ca. 94720, April, 1973. Chace, M. A. , and Angell, J. C. "Users Guide to DRAM (Dynamic Response of Articulated Machinery)", Design Engineering Computer Aids Laboratory, Department of Mechanical Engineering, University of Michigan, Ann Arbor, Mich., Feb., 1976. Dix, R. C. "A Users Manual for MEDUSA (MEchanism- Dynamics- -Universal System Analyzer)", Mechanics, MechanicaT and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, ILL. , Jan., 1975. Pilkey, W., and Pilkey, B., eds, Shock and Vibration Computer Programs - Reviews and Summaries, SVM-lO, The Shock and Vibfation Information Center, Naval Research Laboratory, Washington, D.C., 1975. Bowers, J.C., O'Reilly, J.E.,znu1Shaw, G.A., "SUPER* SCEPTRE - User's Manual", DAAA-Zl-73-C-0655, AMC, Picatinny Arsenal, Dover, N.J., 1975. Rosenberg, R.C., "A User' 5 Guide to ENPORT- 4", John Wiley, Inc., 1974. Paynter, H.M., Analysis and Design of Engineeripg Systems, M.I.T. Press, 1960. 55 [101 [ll] [12] [13] [14] [15] [161 [17] [18] [191 [20] [211 56 Karnopp, D.C., and Rosenberg, R.C., Analysis and Simu- lation of Multiport Systems - The Bond Graph Approach to Physical System Dynamics, M.I.T. Press, 1968 (68A35012). Gebben, V.D., "Bond Graph Bibliography for 1961-1976", Trans. ASME, J. Dyn. Sys., Meas., and Control, 99(1977) pp. 143-145. Nobuhide, 8., "Bond Graphs: Structural Properties and Augmentation Algorithm", Presented at the Conference on System Control Theory and its Applications to Dynamic Economic Models, Nagoya City University, Mar. 27-29, 1976. Ort, J.R., and Marten, H.R., "The Properties of Bond Graph Junction Structure Matrices", Trans. ASME, J. Dyn. Sys., Meas., and Control, 95(1973) pp. 362-367. Perelson, A.S., "Bond Graph Junction Structures", Trans. ASME, J. Dyn. Sys., Meas., and Control, 97(1975) pp. 189-195. Rosenberg, R.C., "State-Space Formulation for Bond Graph Models of Multiport Systems”, J. Dyn. Sys., Meas., and Control, Trans. ASME, 93(1971) pp. 35-40. Dixhoorn, J.J. van, "Simulation of Bond Graphs on Mini- computers", J. Dyn. Sys., Meas., and Control, Trans. ASME, 99(1977) pp. 9-14. Busacker, R.G., and Saaty, T.L., Finite Graphs and Net- works - An Introduction with Applications, McGraw-Hill, 1965. Harary, F., Graph Theory, Addison-Wesley, 1972. Rosenberg, R.C., and Andry, A.N., Jr., "Solvability of Bond Graph Junction Structures with Loops", IEEE Trans. on Circuits and Systems, 26(1979) pp. 130-137. Rosenberg, R.C., and Andry, A.N., Jr., "Solvability of Certain Classes of Bond Graph Junction Structures", Pro- ceedings of the Second International Symposium on Large Engineering Systems, Univ. of Waterloo, Waterloo, Ontario, Canada, May, 1978, pp. 537-541. Ort, J.R., and Martens, H.R., "A Topological Procedure for Converting a Bond Graph to a Linear Graph", Trans. ASME, J. Dyn. Sys., Meas., and Control, 96(1974) pp. 307-314. \\I I\) . J (Erapns'fi .I. r—v Di Perglson, A.S., Discussion re [13], Trans. Asme, J. ' 'ys., Meas., and Control, 98(1976) pp. 209-210 Peix)!son, L S. :1nd (Ester. (3.F., "Bond.(3rarflus and.I.1nea1‘ I ranklin Ilh stit1te 302(1976)159-185 Karnopp, 9. C., and Rosenberg, R.C., System Dynamics \ “Unilie u \pproach, John Wiley Inc., 1374. Roscntm org, R.C.. ”Essential Gyrators and Reciprocity in 111 ticn Structures”, (accepted for publication) ,.J C J. Franklin Institute. Karnopp, D.C., ”Some Bond Graph Identities Involving Junction Structures”, 7Trans. ASME, J. Dyn. Sys., Meas., and Control, - (197 5) pp. 439-440. rste Computational Structures, Korfhagc, R.R., )1 jcau mic Press. 19: Berztiss, A.T., Data Structures - Theory and Practice, Academic Press, 1971. Knuth, I).E., The jnn:(1f (omzute 1f Progranmruig, Vol. I, Chap. 3, \ddison—Wesley, I973. Stone, H.S., Introduction to Computer Organization and Data Structures, McGraw—hill, 1973 Iowarson, R.P., Sparse Matrices, Academic Press, 1973. Riordan, J., Combinatorial Identities, Wiley, 1968. APPENDIX A DERIVATION OF THE BASIS ORDER RULES A;l_ Basis Order and Simple Junction Structures In this section we derive a pair of general compu- tational rules for predicting the order and variable-type composition of an N-port SJS basis. An N-port JS has exactly N EN-nodes. It is common usage to refer to bonds (0,EN) or (1,EN) as port bonds. Motivation for these rules is derived by consider- ing the number of free variables which remain following the imposition of a set of independent constraint equations on a set of system variables. Several types of proper SJS's are studied first; then the results are extended to standard SJS's. In passing, alternate forms of the order rules for proper SJS's are given. The order rules are presented here as Theorem 1. Theorem 1: Every standard SJS satisfies the relations (1)E = NB + NO - B0 - N1 and (11)F = NB + N1 - B1 - N0. A.1.l Basis Order for Proper Simple Junction Structure Forests Initially we establish Theorem 1 for an arbitrary proper SJS tree G by demonstrating that G can be obtained 58 59 from a forest of separate 1-junctions and O-junctions by a series of subgraph concatenations. It is observed that junctions satisfy the order rules. Following the assumption that G contains more than a single junction, a O-junction in G is identified as a base node to which is added an appropriate number of l-junctions and O-junctions, of specified degrees, which yields a SJS equivalent to G. Additionally, it is noted that the concatenation of a junction to a proper SJS tree yields a proper SJS tree which satisfies the order rules, thus yielding the results for G. Finally, by considering the order rules for each component, the results are extended to an arbitrary proper SJS forest. Lemma A.1: Every proper SJS forest satisfies the relations E = NB + N0 - B0 - N1 and F = NB + N1 - B1 - No. Prior to proving Lemma A.l, two definitions are needed. First we state that two distinct SJS's are conformable if one contains a 0 and the other contains a 1. We now define the graph concatenation operator C where C(G,H) = K is a binary operation performed on two proper SJS's (G and H) which are conformable to yield a third proper SJS (K). Let G be a connected proper SJS and H be a connected proper SJS where VG n VH = 0. Also, let u be a l-junction and uEN be an EN-node 1n VG, EN-node in VH’ where (u,uBN) EKG and (v,vEN) eXH. Then C[G(u), and v be a O-junction and vEN be an H(v)] will denote the connected proper SJS K where 60 VK [V u V { } and G H] VEN’uEN X K [X G u XH u {(v,u)}] - {(v,vEN),(u,uEN)}. Note that if K = C[G(u), H(v)] and K' = C[G(u), H(v)] (different EN-nodes are removed) then K and K' are isomorphic. For C[G(u), H(v)], it will be said that G and H are "concaten- ated". Note that C[H(v), G(u)] = C[G(u), H(v)]. We now pro- ceed with the proof of Lemma A.l. Proof: Let G (m) denote a connected proper SJS such that 1 VG (m) consists of exactly one l-junction and exactly 1 m EN-nodes, where m22. Thus C (m) has the form shown 1 in Figure A.l. Observe that the definition of a l-junction applied to 61(m) yields the results E = N + N - B - N B 0 0 1 U“) + (0) " (0) ' (1) In - l and ’1'] ll 2 + Z I w I 2 II II H B 1 1 0 U“) + (1) " 0n) ‘ (0) These results agree with Lemma A.l. Let G0(n) denote a connected proper SJS such that VG (n) consists of exactly one O-junction and exactly n EN- 0 nodes, where n22. Thus, G0(n) has the form shown in Figure A.2. G (n) Observe that the definition of a O-junction applied to 0 yields the results E B+NO-BO-N1 (n)+(1)-(n)-(O) ll 2 l and '11 ll 2 + Z I w I 2 II II :5 a H B 1 1 0 (n)+(0)-(01-(1) 61' These results agree with Lemma A.1. Note that Lemma A.1 applies to a forest with an arbitrary number of components, each of which is a 0- or l-junction together with a set of EN-nodes. Now consider a concatenation involving G1(m) for mZZ. Let G be an arbitrary connected pr0per SJS where VG contains at least one 0-junction, say v, and let EG and FG be given. Observe that K = C[G(v),Gl(m)], where m22, contains one less effort variable and one less flow variable than GLJGl(m) < V 11V (m), XGIJXG (m)>. since 0(X (m)) - l. c; c K) = 00(19ch 1 l l (”0" denotes "order of".) Also, K and GLJGl(m) yield the same number of independent flow constraint equations, since the number of junctions and junction degrees are unchanged. Then, clearly EK = EG + EG UN)‘ 1 and FK = FG + FG UB)- l. 1 l (m): _ (m): , Let AE1 _ EK EG and AFl - FK FG. Then AE(m)=E(m)-l=m-2andAF(m)=F(m)-l=0. 1 G1 1 G1 where m22. That is, as a result of a concatenation involving Gl(m) the incremental changes in E and F are known. Now consider a concatenation involving 60(n) for n22. Let G be an arbitrary connected proper SJS where VG contains at least one l-junction, say u, and let EG and FG be given. Observe that K = C[G(u), 60(n)], where n22 62 contains one less effort variable and one less flow variable than GlJGO(n). Also, K and GIJG (n) yield the same number 0 of independent effort constraint equations and the same num- ber of independent flow constraint equations. Therefore, EK = EG + EG(n) - l and F FG + FG(n) - l. K (n) , (n) - Let AEO _ EK EG and F0 _ FK FG' Then 1E (n) E (n) — 1 = 0 and 1F (n) = F (n) - 1 = n - 2, 0 GO 0 G0 where n22. We now establish lemma A.1 for an arbitrary proper SJS tree. Let G be an arbitrary proper SJS tree. Suppose G contains and N are N l-junctions and N O-junctions; not both N 1 0 1 0 zero, since G is proper. If NO=0, G is a l-junction com- ponent; if N1=O, G is a O-junction component. In either case, we are done. Therefore, assume N1>0 and NO>0. Enumerate the O-junctions in V by v ,v ,...,vJ , G l 2 N0 and the l-junctions 1n VG by vN +1,vN +2,...,vN +N . Let 0 0 0 l (0) . - (1) . VG {v1,v2,...,vNO} and VG {VN0+1’VNO+2"'”VNO+N1}' Then VG(1)r1VG(O) = g and VG(1)lJVG(O) contains all junctions in G. The junctions in G will now be partitioned according to their distances from v1. Let S_l = 0 and S1 = {veVG(1)11VG(0)|d(v1,v) = i}, where i = 0,1,2,... Note that Siswsj = 0 if i f j. The order of VG(1)LJVG(O) is finite and G is a tree imply that there exists a smallest 63 positive kiN + N -1 such that d(v1,v):k for all veVG(1) VG(O). l 0 Without loss of generality, assume k is odd. Then k-l k-l I—z ) (0) (T) (1) .3 521 ‘ V0 and .3 S21-1 7 VG ° 1-0 1-0 Relabel the elements in Si so that vi j is the jth element in 81’ where Oiiik and l:j:o(Si). Let ai,j = deg(vi,j) for in G. Also, let G = G (deg VI), and let G. . be the 0,0 0 1.) proper SJS tree corresponding to vi j where Gi j is either GICm) or G0(n) V.. 1,1 if i is odd or even, respectively, and m or 11:61. .. 1,3 Now we will reconstruct G from a proper SJS forest of NO O-junction and N1 l-junctions by using the internal bonds of G as a directory. Note that the concatenation of two con- formable proper SJS trees yields a proper SJS tree. Let G = 00,0, and let 01,1 = G1,0(S0)be the proper 1,0 SJS tree obtained from the series of concatenations of 01 0 ’ 1,j)€xG; i.e., 133:0(81). Let GZ,O=Gl,o(SO)’ and let 62,1 be the proper SJS with Gl,j at v0,1 for each 3 such that (v0’1,v tree obtained from the series of concatenations of 02 0 with 9 G at v1,1 for each j such that (vl,l,v2 .)eXG. Let G 2,1 ,J 2,2 be the proper SJS tree obtained from the series of concatena- tions of G2 with G 1,2 for each j such that (v1,2’ ’1 2,j V2,j)€xG‘ In general, let Gi n be the proper SJS tree obtained from the series of concatenations of G, 1 1,n- for each j such that (vi-lnfvi,j)€XG’ where Gi with G1 . at v. ,0: i-1,0(S.- liiikand l:n:o(Si_ Then G =0. 1)' k,o(Sk_1) 64 An example of the construction procedure is given in Figure A.3. The construction procedure involves NO-l concaten- ations of proper SJS trees having the form 60(n), and N1 (m) concatenations of proper SJS trees having the form G1 Therefore, N II+N EG = E6 = EG + _20 AE0(deg vi) + .20 1 AE1(deg Vi) k,o(Sk_1) 0,0 i=2 1=N0+l NO+N1 NO+Nl = (l) + (0) + 2 (deg v.-2) = 2 deg v.-2N1+1 and i=N +1 1 i=N +1 1 0 0 N LI+N PG = FG = PC + .20 1P0(deg V1) + .20 1 AFl(deg Vi) = k,o(Sk_1) 0,0 1=2 1=N0+1 N0 N0 (deg vl-l) + 2 (deg v.-2) + 0 = 2 deg vi-ZNO + 1. i=2 1 i=1 N0 Observe that (i) B = Z deg v., 0 . 1 1=l N +N (11) B = 20 1 deg v , l . 1 1=N0+l (iii) N = N + N - l (Euler's Rule), (iv) N = B + B - N (v) P = B - N (vi) P = B - N 65 Therefore, if G is a proper SJS tree, then N0+N1 E r .8 deg vi -2N1 + l = NB + N0 - BO - N1 1—N +1 0 N0 F = 121 deg vi = 2N0 + 1 = NB + N1 - Bl - NO' Note that deg via 2, Is isN implies that E2 1 and F2 1. 0, We now extend the results to an arbitrary proper SJS forest G. Let G1, G2,..., Gn be the components of G. Then each component G1 is a proper SJS tree. Therefore, BG. 7 NBC. + N00. 7 N10. and 1 1 PG. = NBG. + NlG. - BIG. - NOG.’ where ls ISIL 1 1 l 1 1 Then 11 n E = 2: E = z (N 1+ N - B - N ) G i=1 oi i=1 B0i 001 00 10i n n n n = z N + z N - z B G. - E N = N N B N , i=1 BGi i=1 00i i=1 0 1 1:1 10 BC 00 00 1G and n 11 Fe 7 .E F0. ' E (NBC. + N10. BIG. ‘ Nos.) 1-1 1 1-1 1 1 1 1 n n n 7 .E NBG. + E N10. ' .E ”00 7 NBG NlG B10 NOG’ 1—1 1 1-1 1 1—1 where EG 2 n and F(3 2 11. Hence, if G is a proper SJS forest, then G satisfies the rela- tionsE=NB+NO-B -N andF=N +N -B -N 0 1 B 1 1 0' 66 Recall that B0 is the total number of (external and internal) bonds incident to O-junctions. Observe that in a proper SJS every internal bond is incident to exactly one l-junction and exactly one O-junction; thus, in a proper SJS, NI is included in BO. Therefore, for a proper SJS, NB = P1 + BO' Similarly, for a proper SJS, NB = P0.+ B1. These results and Lemma A.1 validate the following corollary. Corollary 1.1: Every proper SJS satisfies the relations (1) E N + P - N 0 l 1 (ii) F N1 + PO - NO. A.1.2 Basis Order for Proper Simple Junction Structures Now Theorem 1 will be established for an arbitrary connected proper SJS G. This will be accomplished by removing cycle bonds from G until a spanning tree is obtained, and then demonstrating that the previous relations remain valid when these bonds are replaced. These results will then be extended to an arbitrary proper SJS. Lemma A.2: Every proper SJS satisfies the relations NB + N0 - B0 - N1 and [T1 II F = NB + N1 - B1 - NO. Prior to proving Lemma A.2 we define a transformation T which removes cycles from a connected proper SJS, and define a transformation S which creates a cycle in a connected proper SJS. 67 To obtain T, let G be an arbitrary connected proper SJS. Assume G contains at least one cycle, say C. Let b be an arbitrary bond in C, and let vEN and uEN be EN-nodes. Then there exists a unique l-junction veVG and a unique O-junction ueVG such that b = (v, u). Let T [G(v, u)] H 7 V0 U {VEN’UEN EN)’ (u,uEN)}]-{(v,u)}. Then, clearly, H is a connected proper SJS denote the SJS H where V } and XH£E[XGu{(v,v which contains one less cycle than G. Observe that N1H = NlG’ ”on 7 ”00’ p111 7 p10 + 1' and Pon 7 P00 + 1' To obtain S, let G be an arbitrary connected proper SJS. Suppose there exist l-junction veVG and O-junction ueVG such that (v,u)¢XG. Also, assume there exist EN-nodes vEN and uEN in VG such that (v,vEN) and (u,uEN) are in X6. Then } and let S (G,v,u) denote the SJS H where VHEEV {v G ' EN’ uEN XH Etxcu{(V,u)}] - {(v, VEN),(u,uEN)}. Then H is unique (to an isomorphism), and H is a connected proper SJS which contains one more cycle than G. Observe that S(G,v,u) contains One less flow variable and one less effort variable than C, where S(G,v,u) and G yield the same number of independent flow constraint equations and the same number of independent effort constraint equations. There- fore, if H = S(G,v,u), then B - 1 and F = F - 1. EH 7 G H G Let ABS 5 EH - EG and AFS s FH - FG Then AES = -l and.AFS = -1. We now proceed with the proof of Lemma A.2. 68 Proof: Since for a proper SJS we have NB = P1 + B0 and NB = P + B it is sufficient to show that every proper 0 1’ SJS satisfies the relations F = N1 + P0 - N0 and E = NO + P1 - N1. Consider an arbitrary proper SJS G. If G is a forest, then, done, by Lemma A.1. Therefore, assume G con- tains at least one cycle. Let G be connected and let GOEEG. Also, let vi and ui denote l-junctions and O-junctions respectively, for all 1. Then G contains some cycle C1. 0 Let b1 = (v1, ul) be an arbitrary bond in C1, and set G1 = TIGO,(v1,u1)]. In general, 1f Gi-l contains some cycle Ci’ let bi=(vi,ui) be an arbitrary bond in 2., and set Gi=T[ G. 1 1-1’ Wi,ui)]. Then the order ofXG finite implies that G contains a finite number of cycles, which implies that there exists a smallest positive integer k such that Gk is a spanning tree. Note that N = N 7 N16, P06 = p06 + k’ and PIG = 06k k k k P16 + k. G is a proper SJS tree. Therefore, N 0G’ 16 k E = N + P - N ,and F = N + P - N Gk 0Gk 1Gk le GK 1Gk Observe that SIGi’Vi’ui] = Gi-l’ 1=k,k-l,...,l; i.e., k appli- cations of S to Gk yields GO. Therefore, + k (AE ) = (N + P - N ) + k(-l) S 06k le le B7 .. _ = .. (”00 I pic I k N10) k N00 I P10 N10 and F=F +k(AF)=(N +P -N )+k(-1) G Gk S 1Gk 0Gk OGk = (N16 I Poc I k ' N00) 7 k 7 N10 I poo 7 N00' E = E G Gk 69 Thus, if G is a connected proper SJS, then G satisfies E = NO + P1 - N1 and F = N1 + P0 - NO. Assume G 15 not connected. Let 61’ GZ"°°’Gn be the components of G. Then each eomponent of G is a connected proper SJS which implies that BC. I NOG. I PlG. ' N16. and FG. I NlG. I POG. ' N0c; ’ 1 1 1 1 1 1 1 where lsisn. Therefore, n n n n E == 2 .F = 23 N + Z: P - Z N' "N P N , G i=1 Gi i=1 0Gi 1:1 161 1:1 1G 0G 16 16 n n n n and F = 23 F E IN + Z I’ - 22 N = N P N . G i=1 Gi 1:1 lGi 1:1 OGi i=1 0Gi 1G CG CG Hence, if G is a proper SJS, then G satisfies the relations E = NO + P1 - N1 and F = N1 + P0 - NO, the relations E = NB + NO - B0 - N1 and F = NB + N1 - B1 - N0. and thus, G satisfies A.1.3 Basis Order for Standard Simple Junction Structures In this section we prove Theorem A.1. Theorem 1: Every standard SJS satisfies the relations E=NB+NO-BO-Nl and F = NB + N1 - Bl - NO. The proof of Theorem 1 is preceeded by the definition of the transformation T which reduces the number of bonds formed by nodes of the same type in a standard SJS. 70 Let G be an arbitrary standard SJS. Assume G is not proper, i.e., G contains a bond of the form (v1,v2) where v1 and v2 are nodes of the same type. Then for (vl,v2)eXG, where v1 and v2 are nodes of the same type, let u be a junctionof a node-type distinct v1 and v2. Without loss of generality, if v1 and v2 are EN— nodes, then let u be a l-junction. Then T(G,v1,v2) will denote the SJS H where 'HEVGU{u} and XHEEXG - {(v1,v2)}]u {(vl,u),(u,v2)}. Observe that H is a standard SJS which con- tains one less bond of the form (1,1), (0,0) or (EN,EN) than G (see Figure A.4). Note that H = T(G,v1,v2) contains one more effort variable and yields one more independent effort constraint equation than G. Therefore, EH=FG' Similarly, H contains one more flow variable and yields one more independent flow constraint equation than G. Thus, FH=FG. Observe that if v1 and v2 are N l-junctions, then N 1,,N N N N BH= BGI ‘1H= lG’ 0H: 06 B0H=BOG+2. Also, if v1 and v2 are O-junctions or EN-nodes, +1, BlH=BlG’ and 1H=N16I1’ 0H= OG’ 1H= icI 0H= We are now prepared to establish the order rules for an then NBH=NBG+1, N N N B B 2, and B B 06' arbitrary standard SJS. Proof: Let G be an arbitrary standard SJS. If G is a proper SJS, then done, by Lemma A.2. Therefore, assume G is not proper. Let k0,k1, and k2 be the number of bonds in G of the forms (0,0),(1,l) and (EN,EN) respectively. Let k=kn+k1+k2 nil, mil EN 1 EN n>0, m>0 n>l, m>l Transformation for converting a standard SJS to a proper SJS Transformation for converting a WJS to a SJS. EN -——————-D' EN Figure A.4. Figure A.S. APPENDIX B AN APPLICATION OF THE BASIS ORDER RULES Definition: For a WJS (weighted junction structure), a causal form is feasible if it does not violate any l-junction, O-junction, or TF node constraint, and every port bond is causally oriented. The effort-flow variable composition of a basis can be determined from the topological properties of a WJS. The composition is given by the basis order rules, the general froms of which are NE = NB + NO - BO - Nl - NT and (1) NF = NB + N1 - B1 - NO - NT Example 1 The WJS in Figure 3.1 has the following topological pro- perties: NB = 12, N0 = 2, N1 = 3, N = 2, B0 = 5, B = T 1 The basis order rules yield NE=12+2-5-3-2=4 and 80 9. NF=12+3-9-2-2=2 This input pattern is illustrated by the causally augmented WJS in Figure 3.2. Henceforth, all weighted junction structures will be con- sidered as n-port structures. In the process of obtaining a "good" upper bound for the number of unlabelled feasible port bond causal orienta- tions, three expressions for upper bounds will be derived where each successive expression requires greater knowledge of the junction structure and yields a smaller upper bound. B.1 An Upper Bound For C As A Direct Application Of The Basis Order Rules. A question of particular interest concerns the number of distinct port-variable bases which an n-port possesses. The basis order rules yield an upper bound for the number of such bases. From the fact that every port bond can accept exactly two causal orientations it is easily seen that ( c_% (m where C is the total number of unlabelled feasible port bond causa1 orientation and U0 = 2 NP. 82 However, an improvement over UO can be obtained by a direct application of the basis order rules. Given NP port bond with NE efforts as inputs, a (gen- erally) smaller upper bound can be expressed as CiUi (n where NP U=< > M) 1 NE and m%mn fi 01mim (9= (3 0 otherwise, for integers n and m. Notice that Ul represents a significant reduction from the coarse upper bound U0. Noting that NP = NE + NF, (6) one obtains the related form N NP 1 N P P ()= . =(). (7) NE NEINF! NF As would be expected, the results are symmetric with respect to effort and flow variables. 83 The inequality U1 3 U0 can be demonstrated by expressing ZNP as a binomial expansion. I Recalling the Binomial Expansion Theory, n D (a + b)r1 = z (bamkbk, k=O I let a = 1, b = 1, and n = NP. Then N N NP NP 2P=(l+1)P=Z(k). (8) N N Then for O 5 NE E NP and 0 1 NF i NP, (NE) and (NP) F are merely symmetrical terms in (8). Thus U1 1 U0. In particular, if NP 3 1, then U1 < UO . Example 2 Referring to Figure B.1, N = 6, N = 4, and N = 2. P E P Then U0 = 26 = 64 and U1I(2)IZ§IB_II15 Thus, C i 15 < 64. 84 8.2 A Refined Upper Bound for C In general, U1 can be improved upon considering the constraint equations associated with WJS elements. It will be assumed that the WJS of interest contains no external TF elements. This assumption results in no loss of generality, due to the causal properties of TF element [25]. Let e be the number of port effort inputs to external O-junctions. Similarly, let f be the number of port flow inputs to external 1-junctions. For a given e, if "e" port efforts are inputs to the A0 external O-junctions, then P0 - e port flows are inputs to the remaining PO - e port bonds which are incident to O-junctions. Thus, NF - (PO - e) port flows are inputs to the Al external l-junctions, and NE - e port efforts are inputs to the remaining Pl — (NF - PO + e) port bonds which are incident to l-junctions. Note that - P + e (9) and 85 Then (10) - ME A0 IJ2 I E ( e) (N e-LE A1 ) (ll) F-P0+e for some ME and LE. If NE i A0, then ME = NE‘ If NE > A0, then ME = A0. Thus, ME = min(NE,AO) . (12) Suppose Pl 2 NE' Then there are at least NE - Pl port effort inputs to the A0 external O-junctions. Then L = N - Pl' Observe that f = 0 when e = NE - Pl’ E E Suppose Pl > NE' Then NF > PO, and there can be a minimum of zero port efforts inputs to the AO external O-junctions. Then LE = 0. Observe that f = NF - PO > 0 when e = 0. Thus, (13) I" ll max (0, NE - P1) Equation (11) is symmetric in e and f. Thus, U2 can be formulated in terms of specified port flow inputs, if desired. 86 By (9) we have e NE - Pl + f, since N. + N = P + P E F 0 1' Then, by (5), ME A A NE A A Pi A A Z (e0)(N -Pl+e) I Z (e0)(N -% +e) I Z (N -3 +f)(fl) e=LE ‘F 0 e=LE F o f=LF E 1 $1 A1 ) A1 gr AO ) A1 = ( _ ( ) = ( _ ( ) szF NE Pl+f f fILF NE P1+f f where M? = min (NF, A1) and LF = max (0, NF - PO). It will now be shown that U2 i U1. Suppose e port efforts are inputs to the A0 external (A1) NF-P0+e consistent input assignments of NE - e port efforts to O-junctions. Then there are at most causally the Pl port bonds incident to l-junctions, given causally consistent input assignments of NF - PO + e port flows to the P1 port bonds incident to l-junctions. P l ) Observe that (N -e is the number of unrestricted input E assignments of NE - e port efforts to the Pl port bonds incident to l-junctions. Therefore, Ai Pi (NF-P0+e) i (NE-e) (14) Therefore, by (5) and (14). 87 MI (A0>< I1 > NI (AO>< A1 > egLE e NF-P0+e egO e NF-P0+e N N . EA P EP P P+P N o 1 o l_Ol_P < ego (e )(NeIe) : ego (e )(NEIe) - ( NE ) - (NE) Therefore, U2 1 U1. In particular, whenever Al < P1 or A0 < PO, U2 is a significant improvement over U1. Example 3 Again referring to Figure B.1, A0 = 0, A1 = 3, P0 = 0, P1 = 6, NE = 4, and NF = 2. Therefore, M and LE = max (0, 4-6) = 0. Thus, E _ 0 3 _ which is much lower than the upper bound of 15 obtained in example 2. In addition, the junction structure in Figure B.l has exactly three unlabelled causal orienta- tions which are given in Figure 3.3, hence, C = U2 in this example. The significance of U2 is captured by the following theorem. Theorem 3: The number of distinct basis variable sets for a WJS transformation is bounded above by MEA A M? A A u2 = z < 0 < 8 +f) e=LE NE- 1 l )( ) = = min(4, 0) = O 88 where M = min(N = max(O, N E E’ E EIPL)’ MF = min(NF, Al), and LF = max(O, NF-PO). A0), L 89 EN.__.1___EN EN EN \ l—TF—O —TF—o—-1 EN \EN Figure B.l. Example of a weighted junction structure. 90 EN _m___EN EN EN \IP—Tn—O—ATF—jo—h/ .N/ N Figure B.2. Causally augmented weighted junction structure. 91 EN -Ill—EN EIII/\il— TFl—_l_—| TF—-|O ——1i/\EN M /\ EN IN’N EN I—II—EN /EN 1|— TFl—— OF—TFF—OF-l EX “\EN EN }-—1|—— EN EN EN -l—-1TF ——|(|)—1TF——{o ——-|1/\ EN EN Figure 3.3. Unlabelled causal orientations of a weighted junction structure. APPENDIX C THE RESOLUTION OF A CONFLICT RESULTING FROM A PORT BOND CAUSAL ORIENTATION Property'ZJJ Let G be a junction structure with a port bond b. Suppose a causal orientation (and subsequent causal extension) of b results in a causal con- flict, then the opposite causal orientation (and subsequent causal extension) of b will not yield a causal conflict. Proof: It will be assumed that each causal orientation of b is followed by the causal extension process, and that G is acausal (since the pruning process of Appendix E can be initially applied to remove all causally oriented bonds and causally completed nodes). G contains a node of degree greater than two since a causal orientation of b results in a causal conflict. From a graph theoretic perspective, a causal orientation of b defines a "walk” of G [18]. Let ENO be the field-node incident to b. Then there exists a shortest path in G joining ENO and a JS-node, V0, of degree greater than two such that (1) every JS-node (exclusively) between ENO and V0 has degree two, and (2) there exists no shorter path in G joining ENO and a JS- node of degree greater than two. 92 93 Let d denote the path bond incident to V0. A causal orientation of b resulting in a causal conflict implies that the resulting causal orientation of d gives d strong causal implication with tespect to V0. The reversal of the causal orientation of b results in a reversal of d's causal orien- tation, since ENO and V0 are joined by a sequence of JS-nodes of degree two. This gives d weak causa1 implication with respect to V0, which leaves VO causally incomplete; conse- quently, no causal information propagates beyond V0 and no causal conflict can result. Hence, if a causal orientation of b yields a causal conflict, then the opposite causal orientation of b will not yield a causal conflict. APPENDIX D THE IMPACT OF THE STANDARD SEQUENTIAL CAUSALITY ASSIGNMENT PROCEDURE (SSCAP) ON THE REDUCED JUNCTION MATRIX Theorem D.l: Let G be a bond graph with an algebraically reducible junction structure. Assume that G can be completely and consistently causally oriented by SSCAP. Then dependent storage field inputs are determined by source field and independent storage field outputs. More- over, no dissipation field input is determined by a dependent storage field output. Proof: It will be assumed that (l) a causal assignment is always followed by causal extension, (2) each field multiport is a one—port, and (3) each field multiport is adjacent to a O-junction or l-junction. It has been shown that a linear time-invariant field multiport with n ports can be replaced by n one-port field multiports [25]. Thus, for this reason and due to the causal characteristics of junction structure nodes, the above assumptions result in no loss of generality. Consider the acasual representation of G = G given in 1 1 is the number of sources, m1 is the number of storage multiports, and p1 is the number of dis- Figure D.l where n sipation multiports. Without loss of generality, assume nlzl. 94 95 In the following discussion only consistent causal orienta- tions of source bonds will be considered. Causally orient the bond incident to an arbitrary source in G1. Let u1 denote the output from this source. Prune (see Appendix E) from G1 all resulting causally oriented bonds and causally completed nodes. (Note that no causal conflicts can result from source bond orientations due to the theorem's hypothesis). Let G2 be the bond graph obtained from G1 by the pruning procedure (see Figure D.2). If any dependent storage field bond or any dissipation field bond is pruned from G then each corresponding dependent storage field input or 1’ dissipation field input is determined by 111 together with prior information. If G2 # Q, then done. Assume G2 f B. Then G has the acausal form given in Figure D.l where n1, 2 m1, and p1 are replaced by n2, m2, and p2 respectively. Observe that if nzzl, then the source bond causal orienta- tion process together with the bond graph pruning procedure can be repeated (at most n1 times) until all source bonds of G have been causally oriented. Then there exists a smallest positive integer ksn1+l such that Gk contains no sources. (If n1=0, then k=l). nlzl a. k22. Then each dependent storage input and each dissipation input specified in Gh is determined by u uz, . . ., uh where lshsk-l 1’ and uh is defined similar to ul. 96 Consider Gk' Gk has the acausal form given in Figure D.3. If mk=0, then done. Assume mk>0. Assign integral causality to an arbitrary storage bond, b, in Gk' If a causal conflict results, then restore Gk to its acausal form and select a different storage bond to causally orient. In this case, property 2.2 of the causal extension process indicates that sufficient input information was available to determine the variables associated with bond b, i.e., the storage element incident to b is dependent and its in- put is determined by ul, uz, . . ., uk_1. Suppose the inte- gral causal orientation of b does not result in a causal conflict in Gk' Let x1 be the resulting output of the storage node incident to the oriented b. Let G be the k+l bond graph obtained from Gk by the pruning procedure. If any dependent storage bond or any dissipation bond is pruned from Gk’ then each corresponding dependent storage input or dissipation input isdetermined by x1 in Gk’ and therefore, is determined by ul, uz, . .I., uk_1, x1 in G. Gk+1 has the acasual form given in Figure D.3 with k replaced by k+l. If mk+l=0’ then done. If mk+1f0, then the above storage bond integral orientation and graph pruning process can be repeated (at most mk times) until the integral orientation of any remaining storage bond yields a causal conflict, i.e., there is a smallest integer 130 such that either the integral causal orientation of each storage bond in Gk+2 results in a causal inconsistency or mk+2=0 where limk, As noted above, if the integral causa1 orientation of 97 a storage bond results in a causal conflict, then the bond's associated variables are determined by previously specified source and (independent) storage outputs. Then eacl depen— dent storage input and each dissipation input specified in Gh is determined by u., . . ., 111 if l2 iHZOL-J HO C-l—«ZI "V-H 'm¢UFCUW 'HDU mmz C CHWXQM a 12 yucca). ‘1. «Hmm tun XA i U20 KO ' i—¢ OWD04 o A 03" Coo—noucuu D Z” *ZZPQOI£ 0 2H '0 onuh H 04 me» 2» U 00 l um '02“. x 00 COJ2H iZUP UOU NJPQO OX i<~fl— U): I» .AA.“<.V1 nmp-4P!- X r-O—OOJZODVH—HJ Z 3. .1me 0 Oi cum: mun. 2 OZVHQMCUQ- ~u¢zzzeu PjHéu. U \42U owl—oz: 2O KH\ O\:‘.. O‘NL—CL oun— uzxnamooa-«uamomm (murmurs-u. IF‘IHUZ“ --o \m OGXUC£P XOLU F'z '~a\ “new 0mm u-l o DUZZWZZZOQWUJ OWN UL‘JUUVUX 'C-"L.JZ'~LILK40 murxxxxr-r-a ¢HZO Q U 2 U .4 U H R _l ‘I N d o x U) a- , D H 0'" (A. 3‘ U-u-«I H H 3 15cm ' UNLAJ A ’m j VIXZ. W v um.) Lu U) zoz X I-0 n. O\ Lul- H .J a: mom .12 I m a :0 mm a- H Lu LAO-m: I? o-IHv u-o 13¢ L2 04.. oo- . PNL‘J UHt—n-oou c: .- "V8230 O .4 GO XL: .JHP'F-DQL'J ‘DCF‘UNO ‘NC < ZZv-‘IND mmmomn sew (DOG-.0. ZZlI—OIIHH 27-bi— «HHHhr-v-Dacozomr-sz: u u u HH‘F‘L‘PL‘FKL—Pk‘zzo (Jamxxouuumommmuuuz d 7-. uuuuo HUG a: Nine-nun 000 EL HHHHH uuu “N” U Of Causal Complex Solvability l' Fhe Determination Subroutine For Table 6. IESIPLX I o X“ 0 ha' 0 A (I)- n . 0 203 .H A O IN .070) o 3 "‘ 0.! OULLUA :0 3:: Co O‘lqflm v 2 n2 mite-Q c: H it 9 up . .gjv y— (n ILA QUW‘7H O. 4 I: ongP P O U“ Zuzacx H A P!— DJUC): .- O 22 (I) 0.13:: A W O“. (A cue-1 o O V «a: ~ma~~< ID! & CH 0 mtm— All ( ufl. ‘~VOHL< ‘0< 0.: an qmwr-Q v—< c4 rm mhmox oz Nu HH Umhn< No UN 2" XU'LQZ 1'" c: O 0U mgr-om; o as <~ GO (VJ Our-£0) 'CO‘ UGAOO QZH JXA~L 99.4.4 o- t— I- O vanaCw ' Q!- AJAA (\AO‘) moo anfiV u GHQ 02!: .1): O O xu-c I H r-UAAVLPD- < I “IA umouxucé :uxm uJI—UQiZI o 0.4 CAN! 0 >CL ou031AA9r-r-IIU "U4 I Ao-u-n-UUAu Q'UVIIA—I-JJ “LI- 0 :1 OGAOUUQI-NDQ. FZU~UUr—v-u-~2 .JVIIXXVU-IU'IKW" UHJUUF¢HHHWHH < D O. .3 (continued) Table 6. 113 U! 99 NI" 2 NM AH U wag-I HF'I .I o o + CO 2 CO 2 A I-I- hJ I-I- U I 00 .I GO .1 + (3‘9 ‘9 I 00 9 02 AA ‘A AA A Z": GIN qu- Nu) P- 0.1.] o 0 4;.) o o :J .1. 00 4'0. GO 0. #A mu; NH mm H HH 0 o O z- o o O V O :1 AA 1* om AA A (I) h- #N «Ho-4 H zo. «Iv-1 H (L A z): 9 f LLJH ? 'I' H O W” HH U .1" AH O NH N O .4" .I-J I- . A .J-I l- :A A I- ! A Us. 0 Al— UV Q 0!— O A)- xx 9 HD xx o 23-0 0 a: 1:3“. A.1 l o. 2.2. A um. I— A I o. UU <<2v o o .— it! u {— 2U DD (\mm QC <_IAV) A 441.510) Z‘ L) 040.. 44 a<—m. u: “‘40. << XH+H <4 X>IH 0 X>+~ 00 (Iv-0A lo (Iv-4A U (INA com IIIZO" 60‘ tuxm U ZlIIO‘ 0 01-0 OAU o 0 OH oAv 0 01-4 ans-t o GU§PPIIU UU’I—I—IIU A‘IN-PIIU “UFO-JAM wa-UJAU Ar-UUAUJ U o 0.: .mp— no“ a 0;.) «1,:— 0mg; or..v- cu DmmthDQNDfl-QQPHJQDUQPHDQD Z<-u.< all—.1032! fiWI—ZX'IO< "D 0:! 6—32 O 0 ImU 0C 933%.“: «u. 3x0. CMOO. Z CZI-Z OH C3‘I Z ((0‘3 mm z-I-¥ :UC (IOU 12‘ a: ( F“. 012.] can. <0 )I-CI C NU<- I“ afia>=una Unjuozu Atiiiifiififii Ififififififiiifiifittttiittiiitttiiiit 0 U N H c- m c 0‘ < I- O < P O o W U A I— < m I- £ < H D < x U < 4 A13: 9 0 Lu (III-- N 3U H z c U) "A (DU) m '37 a: .‘z D A I II- wx U OLAJ -' um c: 62 I- 'v U) 4A I tau-0N 4 4%? U ”-4 H HWH I- ".103 U no.3: 20 M1 Stunt-04' ‘)V) as! 1(3le 22:35-04 U m In I U UUUUUUUUUUUUUUUJU x U .1 D. Z 0 Lu U .1 a: o .4 H' o q U m m D D D O < qu I- 0 C21 U D '9 N I-V) A .1 UH I: m =4 H Am (III-0N U 23¢ <1an .Jx o: I-IZH D.N.ut- 01-413 UCL‘U I-Z‘ JI- O.) tat-I? aswuxz om-x Huan- XZU I-Kfl mum a: u deD OJHZ U22" II LLUVO Paw—04° club-c HLLmur— XUHI‘ .JHdEJ—q U U < u 4 H J: D 2 U ‘7 H UU UUU (continued) Table 7. (I) p— D l I- A D I U I'- 3 I- 0 Z LL L‘ 14.! IL A IL on U G AI- :0 U _I 0‘ (JO 4 ALL. 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A x I‘- ‘J U D U a; C I- 2 G! v LL I-I _’ U 0 z D A L9 2 A z < 1 H ‘H XXV x ¢3~+ Iuwx: U in 'H man: 0 #:UX? 22 0 I " "UII 0 QC! 0 AH 'QA HAM II II AZ-4 II H O QUHV‘Jme—OfiA .lIl no" N: H Udan—F‘OZVG’ V "VSIUNJZKZHIDWJ EUNGJHUHHNNX ' vgvg v uzouudzooom “HQUF‘HZUQQA _ G I- O t In a) In I N u: I U U \J U" Ao-ImII AAv-I chnazzu 7'21: 2 HH" UVII IOJJJOWWA II I~ II II X‘JQLL '2 ch-INuNI-u-nw Luv-IQ mm" an 1131 on HOHHHUu-NQH ZXAQJJ)-h—IH2;H¥ dS-IRANSFCRHATION 0 GO sm- U C I {‘4 “I I U (continued) LL I H II A XL” MI ' 33H (II ’A HI-I vv :4 2 xx U AA H CO F- . O < I-I- LX _I_.l «um 00 cm: AA “2". AA 0) 0“ HA ‘g-IA UV (II XLLZZ GHUIXZ F- m\4~v I OZIII-I- W0 9 AQJ} jnF-IHHO-I ”Hwy POI-“LAI- ZOXKHA H a: o. 0. O I n I U 121 IE 64509KHIKBEX*IK)QKHIIKIIIASUHIJI9J31’IADEX1) A H S C' A H ’3 II Emu J Ao-Iq o Inno— “ < 7 000 X” I-I—m MI OQII 4 gx won 0 Z . AAo-g H In IIA H A049 h O «"3 x x cam—I < x 22-. m u Amp-I33 I: U A..I CDXX C3 D-JJH m m xx A 2mm 2 LL 0 0.1 0 £ we 9 xomm H HFPI LL. 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