NME DOMAIN MODELS OF PHYSJCAL SYSTEMS AND EXISTENCE OF SOLUTIONS YEN“ £50.. the D99?“ of 53%. D. MICBEGAN 5mm Mm Jam Lyle Wirth 1962 This is to certify that the thesis entitled ”TIME DOMAIN MODELS OF PHYSICAL SYSTEMS AND EXISTENCE OF SOLUTIONS" presented by John Lyle Wirth has been accepted towards fulfillment of the requirements for PhoDo degree in Elect. Eng. .)‘ ' /‘l' _._J- I 4127.74.¢-—v)/~’ xv Major professor v . Date May 2]., 1962 0-169 LIBRARY Michigan State University TIME DOMAIN MODELS OF PHYSICAL SYSTEMS AND EXISTENCE OF SOLUTIONS by John Lyle Wirth AN ABSTRACT OF A THESIS Submitted to the School of Advanced Graduate Studies of Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1962 V '0?! 'w- anr 1 w m "D ‘31 “‘— v» .... 3*: "s a “up U. 66“. O 1". CA,“ ., .ko ~"-Lu (F ”(lav-al.- ~Cyh_ . v‘ P"‘.‘rp rcvv" 0‘ Q ”‘4 a u , _' hv“““ g I ‘1‘ \k V“e & ‘ a ABSTRACT TIME DOMAIN MODELS OF PHYSICAL SYSTEMS AND EXISTENCE OF SOLUTIONS by Jehn Lyle Wirth For a number of years, attempts have been made to develOp analyti- cal techniques for the study of what are known as nonlinear systems. Such subjects as Cauchy-Taylor transforms, describing functions, etc., have been develOped and used with some success. Advancements in the area of computer technology and in the study of numerical analysis, have produced very practical methods for obtain- ing solutions of nonlinear equations. In addition to being able to gener- ate solutions of a given mathematical model of a system, it is also possi- ble to use the computer to formulate the system model starting from the mathematical models of the individual components and their interconnection pattern. Considerable progress has been realized in programming existing modeling techniques for computer execution. In order to fully utilize the potentiality of the digital computer in the analysis of systems, however, the modeling procedures should pro- duce a system model in a form directly suited for solution by numerical methods. These models may differ substantially from models produced by Alf“ 5'er!- .-.~. .. lop ".“$'* - ,“a-* I P 4 3.0 u. G‘- . .. _--“ _l _ b‘hQU V-“{ ~ rav- .J“ ,1 M. u-‘ . -v- L m... L.“ a...) ' .4: “box,” 0‘, . w,, . 0' V, v.“ “~‘e . 5:“. Abstract John Lyle Wirth existing techniques. Almost all known numerical methods for solving simultaneous, ordi- nary differential equations require that the differential equations are in normal form, i.e., in the form pig. d O = fi(xlooo Xn,t) , i = l) 2’ 000 n. This thesis presents a formulation procedure for formulating a normal- form model of linear and nonlinear systems. To put the formulation pro- cedure on a rigorous basis, a.number of existence theorem pertaining to linear and nonlinear systems are presented. In the develOpment, the basic tools are matrix algebra and linear graph theory. Such Special cases as ideal transformers, ideal levers, etc. are considered and treated in detail. Examples include such systems as compound pendulums, rocket trajectory, saturable core reactors, and systems containing ideal transformers. TIME DOMAIN MODELS OF PHYSICAL SYSTEMS AND EXISTENCE OF SOLUTIONS by John Lyle Wirth A THESIS Submitted to the School of Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1962 ACKNOWLEDGEMENT The author wishes to thank Dr. H. E. Kbenig for his guidance and encouragement during the preparation of this thesis and during the period of research preceeding its completion. He is also appreciative of the com- ments and direction offered by Dr. G. P. Weeg. The author also wishes to thank the National Science Foundation and the United States Steel Cor- poration who contributed to the author's support during the period of re- search. -11... SECTION I . II . III . IV. VI. INTRODUCTION . . LINEAR AIGEBRAIC SYSTEMS TABLE OF CONTENTS NONLINEAR AIEEBRAIC SYSTEMS . . . . SYSTEMS DESCRIBED BY AISEHRAIC AND DIFFERENTIAL EQUATIONS . . . . A PROCEDURE FOR FORMULATING NONLINEAR MATI‘WATICAL MO DEIS C O O O O O O O 0 CONCLUSION . . . . -iii- 16 22 36 71 LIST OF FIGURES Figure Page . 3.1 Tunnel Diode Characteristics . . . . . . . . . . . . . . 19 5.1 A Typical R-L-C System . . . . . . . . . . . . . . . . . 37 5.2 A.Mechanical System.. . . . . . . . . . . . . . . . . . #5 5.3 An Electrical System . . . . . . . . . . . . . . . . . . h9 5.h Compound Pendulum . . . . . . . . . . . . . . . . . . . 5h 5.5 An Electrical System . . . . . . . . . . . . . . . . . . 59 5.6 Orbital System . . . . . . . . . . . . . . . . . . . . . 6% 5.7 Saturable Reactor . . . . . . . . . . . . . . . . . . . 68 -iv- LIST OF APPENDICES APPENDIX Page A. Some Theorems and Definitions . . . . . . . . . . . . . 73 -v... I. INTRODUCTION Historically, much of the analysis of systems of electrical com- ponents has been carried out by means of transform techniques. These analysis techniques have been developed to a level of sophistication not (1.2.3.10 paralled by other branches of engineering. (1) Blackwell and Koenig , in their concepts of systems, have ex- tended these orderly and precise methods of formulation to linear systems of all types of discrete components. The only prerequisite is that there exist two fundamental variables - an "across" variable which sums zero around circuits, and a "through" variable which sums zero at the vertices and suitable linear equations relating the "through" and "across" vari- ables of each component. Attempts have been made to extend transform techniques to include the nonlinear cases(5). At best, these efforts have met with limited success in regard to both generality and ease of application. The objective of this thesis is to set forth precise and orderly techniques for formulation mathematical models of nonlinear systems in general. These models are to be in a form amenable to solution by known techniques. In the pursuit of this objective, many of the concepts and develOpments of Reed(3), Kbenig and Blackwell(l) have been used and ex- tended. -2- Numerical methods of solution of systems of algebraic and/or differential 6.7.12), equations are also considered fundamental to this development( Sections II and III of the thesis deal entirely with systems com- posed of components described by algebraic equations. Here, such special "(8) component types as "perfect couplers" and "gyrators are treated in both the linear and nonlinear setting. Necessary conditions for the existence of a unique solution for a linear system are also present in terms of the system topology. Section IV deals with the more general system.described by both algebraic and differential equations. Theorems pertaining to the exist- ence of solutions to nonlinear systems are presented with the linear system.considered as a special case. The proof of the existence theorems, in most cases, present practi- cal methods for constructing the solutions to the classes of problems covered by the theorems. Section V is therefore devoted to developing algorithms for generating a.mathematical model of the system in a form suitable for solution by numerical methods. These methods, in general, extend to a.more general class of systems than those covered explicitly by the existence theorems. Even though the solution is not known to exist, a solution can often be constructed by these methods. II. LINEAR AISEBRAIC SYSTEMS Consider first the class of systems in which the mathematical models (component equations) of all the components are algebraic. When the component models are all linear, sufficient conditions for the exist- ence of a solution are given by the following theorem: Theorem 2.1: Let the component equations for a system having a.graph G be of the form X=Z(t)Y+F(t) (2.1) where the matrix Z(t) = [ziJ(t)] is positive definite* for all t on 1 constants, and where the vector valued function F(t) = [fi(t)] is defined for all t on the interval I defined by I = [t:to _<_ t 5 t1] , to and t I. Then there exists a unique solution for the variables X and Y of the system graph for all t on I. Proof: For any forest T of the system graph G, the fundamental circuit and cut-set equations can be written as [B U] = 0 [U at] Y = o l :b l b (2.2) C YC where the subscripts b and c refer to branch and chord variables, respectively. Let the components equations in (2.1) be rearranged \— * For the definitions of terminology and notation used throughout, see Appendix A. -4- and partitioned so as to show the branch and chord variables explicitly as separate vectors, thus Xb le(t) Zl2(t) Yt F%b(t) (2.3) xc-= Z21(t) 222(t) Ye + Fc(t) Substituting (2.3) into (2.2), we have [Bl U] -Ell(t) 212(t)] ‘irb + [131 U] *Eb(t) = o 221(t) Z22(t) . Yc Fc(t) (2.A) Substituting the cut-set equations from (2.2) into (2.A) then gives “C. [Bl U] le(t) 212(t) Bl + [Bl U] Fb(t) = 0(2.5) 221(t) 222(t) U [Ye] Fc(t) Since Z(t) is positive definite, the rearranged coefficient matrix shown in (2.3) is also positive definite. Therefore, by Theorem A.A, the triple product shown in (2.5) is none singular for all t on I and we have the unique solution -1 [Ye] = ['Bl U] le(t) Z12(t) B: > [Bl U] Fb(t) (2.6) Z2100) 222“: U Fc(t) The variables Yb, Xb’ and XC can then be found uniquely from (2.2) and (2.3). The above theorem and proof are very similar to results obtained by Reed in some unpublished works. The only difference is that Reed also ..5- included the requirement that the matrix Z(t) be symetric as well as posi- tive definite. In one sense, Theorem 2.1 is quite general in that it places no restrictions upon the topology of the system, i.e., the components may have any mode of inter-connection. Hewever, the requirement that Z(t) be positive definite rules out cases where some of the component variables are known functions of the independent variable t. This latter case is covered by the following theorem: Theorem 2.2: Let the component equations for a system with graph G be of the form x1 = z(t)yi + Fl(t) x0 = Fo(t) (2.7) Y: = F2(t) where l) Z(t) and Fl(t) satisfy the hypothesis of Theorem.l.l, 2) Fo(t) and F2(t) are defined for all t on I, and 3) there exists a forest T of G such that the elements corresponding to the entries of X0 and Yé are branches and chords, respectively, of T. Then there exists a unique solution for all variables of the system graph for all t on I. Proof: As in the proof of the previous theorem, let the first set of equations in (2.7) be rearranged and partitioned accord- ing to the branch-chord classification of the elements for the forest T. .— xlb zllct) 2120.) :11: Flb(t) (2 a) = + 0 xlc Z2l(t) Z22(t) ch Flb(t) The fundamental circuit and cut-set equations can be written as __ .— — . - w t t B11 1312 U 0 x0 = 0 U o -Bll 4321 YO . _ o B B O U X t t Y — 21 22 le O U -B12 -B22_ ylb2 (2.9) lc lc§ X2 ) Y2 Substituting (2.8) into the circuit equations in (2.9) and then substituting the cut-set equations in (2.9) into this result gives [BllJ[XO] + [B12 u] le(t) 212(t) B12 + Z21(t) Z22(t)‘ U [ch] (2.10) " -'t [312 U] Flb(t) + [312 U] le(t) 212(t) B22 z o Flc(t) 221(t) 222(t) 0 [Y2] By Theorem.A.A, (2.10) can be solved uniquely for ch in terms of X0, Y2, and t. However, since X0 and Yé are given by (2.7), ch is then known as a unique function of t alone. The remain- ing system variables can then be computed uniquely from (2.9) and (2.8). All systems of linear algebraic two-terminal components are covered by Theorems 2.1 and/or 2.2. Since symmetry is not assumed in the matrix c.7- Z(t), many systems of multiterminal components are also covered. There are, however, many systems of multiterminal components which do not fall into the framework of the hypothesis of these theorems. In particular, equations of the form x1 = o 212(t) Yi (2 11) x2 Z21(t) o Yé occur quite often in mixed mechanical-hydraulic systems. Mechanical and electrical systems may also have component equations of the form X1 I O Hl2(t) 31 Y2 H21(t) 0 x2 (2.12) Since the first equation in (2.12) is of the same form as the circuit equations, as might be suspected, certain topological configurations lead to inconsistencies in the system equations. These inconsistencies are avoided by the hypothesis of Theorem 2.3. Theorem 2.3: Let the component equations for a system having a graph G be of the form given in (2.7) and (2.13). _x o H (t) Y 3 _ l2 3 Yb — -Hi2t(t) o xh (2.13) ‘where l) (2.7) satisfies the hypothesis of Theorem 2.2, 2) the elements corresponding to X and YA are contained 3 in the branch set and chord set, reSpectively, of the forest T of Theorem 2.2, and with the variable X* representing all across variables except X -8... 3) det [U -le(t) B2] ,é o for all t on I where B2 is the coefficient matrix of the fundamental circuit equations defined by the elements corresponding to Xh’ namely [B1 B2 U] xet = 0 x3 X1. 3 and X“. a Then there exists a unique solution for all the system variables for all t on Proof: I. The essential parts of this proof have been considered by Blackwell(u). In particular, Blackwell has shown that under the conditions of the hypothesis, the combined rank of the circuit, cut-set and terminal equations of the form shown in (2.13) is maximums From this result the first equation in (2.13) can be com- bined with the fundamental circuit equations and the second equation in (2.13) can be combined fundamental cut-set equations to yield reduced sets of equations which are independent of the variables X3, Y3, Xh’ or Yfi. The resulting equations are B B U0 bx? UO-t -Bt FY— —11 -12 o = 0 E11 21 o = o x t t Y E21 E22 0 U lb 0 U “13-12 ’B22 lb (2.11.) x Y lc lc x2 Y2 _ _J [— _. The entries in the submatrices B13, however, are no longer simply +1, -1, or 0, but are now function of t depending upon the entries -9- of H12(t). The proof is completed by noting that (2.1%) is in the same form.as that shown in (2.9). Hence, by the proof of Theorem 2.2, the solution for ch exists and is unique. The remaining variables can be computed uniquely using (2.1M), (2.13), (2.7) and the fundamental circuit and cut-set equations. Although not as obvious as in the case of equations of the form shown in (2.12) and (2.13), Systems having component models of the form shown in (2.11) may also be interconnected in such a way that a solution is not possible. TOpological conditions sufficient for the existence of a solution are given by the following theorem, Theorem 2.h: Let the component equations for a system having a graph G be of the form shown in (2.7) and (2.15), x3 = Z3(t) Y3 (2.15) where 1) Z3(t) is a square skew-symmetric matrix for all t on I, 2) (2.7) satisfies Theorem 2.2, 3) Z(t) is a non-null matrix and A) there exists a forest T satisfying 3) of Theorem 2.2 such that the elements corresponding to X are contained in the 3 branch set. Then there exists a unique solution for all the variables of the system graph for all t on I. -10- For the forest T, the fundamental circuit and cut-set equations are "' “I —' t t-_' — ‘1 B11 1312 B13 U 0 x0 U o o Bll B21 YO X = O t t Y = 0 B21 B22 B23 0 U lb 0 U 0 -B12 B22 lb x Y (2.16) 3 t t 3 x o 0 U B13 B23 Y 10 ‘__ __ 1c X2 . _?2__ Substituting (2.7) and (2.15) into the circuit equations and then substituting the cut—set equations into this result, we have .—-v —— "" _[ [B11] [x0] + [Bl2 U] zll (t) 212 (t) Hie + Lfgl (t) 222 (t) U [Yic] [B12 U] zll (t) 212 (t) B52 [Y2] + [B13] [Z3(t)] [Bi3] [ch] + 221 (t) Z22 (t) o t [B13] [23(t)] [B23] [Yé] = o In order to be able to show that there exists a unique solution for ch’ it must be shown that — _. JL—— __._ A = [B 2 U] zll(t) 212(t) B t 1 + [B13] [Z3(t)] [B13] 221(t) Z22(t) U — _ d— —— is nonsingular. Consider the quadratic form XtAX -11- where X is a nonzero vector. If it can be shown that this quadratic form is always greater than zero, then A is positive definite and, be Theorem A.2, nonsingular. By the definition of A, we have XtAX=Xt [B12 U] -le(t) 212(t) B32 + xt [313] [Z3(t)] [Bi3] x -Z2l(t) 222(t). U X (2.18) Since the coefficient matrix .E11(t) Z12(£3— 32100 222(t) is non-null and positive definite, the first term in (2.18) is al- ways greater than zero. The second term, on the other hand, is always exactly zero since Z3(t) is skew-symmetric (Theorem A.8). Therefore, A is positive definite and, by Theorem A.2, nonsingular for all t on I. It has been established that there exists a unique solution for Y c' The remaining system variables can be 1 obtained uniquely from (2.16), (2.7), and (2.15). For systems containing components with mathematical models of both forms (2.11) and (2.12), the hypothesis of Theorems 2.3 and 2.h can be combined to yield sufficient conditions for the existence of a solution. It should also be noted that although the above theorems are pre- sented in terms of the open circuit parameters, i.e., the component equations -12- are explicit in the across variable X, these same theorems can be stated and proved using the short circuit parameters. The proofs are almost identical to those considered above when the roles of X and Y are inter- changed. A review of the hypothesis of Theorems 2.1 through 2.4 shows that each have one point in common. In each case, if the hypothesis of the theorem is satisfied, it is possible to define a forest T of the system graph such that the component equations can be arranged in a form showing the branch across variables and the chord through variables as an ex- plicit function of the remaining component variables. In fact, this is a necessary condition for the existence of a unique solution for all the variables of the system graph. Theorem 2.5: For a system composed of components described by linear algebraic equations, a unique solution exists for all the system variables for any t on an interval I only if there exists some forest T of the system graph such that the component models can be written in the form . 390 All“) A12“) Yb ~Fb(t) = + (2.19) Ye A21(t) A22(t) Xe Fc(t) where the subscripts b and c denote branch and chord variables, respec- tively. Proof: Consider the set of equations consisting of an arbitrary set of component equations and the circuit and cut-set equations of an arbitrary graph G, i.e., -13- __ __ i. 7 B1 .Be 0 0 X1 _ —7 o o o A A x l 2 2 = o (2.20) Gl(t) G2(t) G3(t) Gu(t) Y3 F(t) where the elements of the system graph have been arbitrarily divided into sets identified by the subscripts 1, 2, 3, and A. Note that the union of sets 1 and 2 constitutes the complete system graph as does the union of sets 3 and 4. Further, B1 con- tains exactly e - v + p rows and A1 contains exactly v - p rows, where e is the number of elements, v is the number of vertices, and p is the number of parts in the graph G. The proof is ex- tablished by showing that the coefficient matrix of (2.20) is singular unless the hypothesis of the theorem is satisfied. First, consider expanding the coefficient matrix in (2.20) by Laplace's Expansion about the first e rows. Selecting an arbitrary set of e columns in these two rows, the corresponding determinant,after some rearrangement, is B o (:l)det [o A] (2.21) This determinant is zero unless B and A are square, i.e., B must be of order (e-v+p)x(e-v+p) and A must be of order (v-p)x(v-p). Further, by Theorem A.9, detB and detA are non-zero if, and only if, the elements corresponding to the columns of A and B constitute a forest and compliment of a forest, respectively, of the system -lh- graph. Therefore, the determinant of the coefficient matrix in (2.20) can be written as g (:1) det Bi det AJ det [G1,GJ] (2.22) 13 where AJ has columns corresponding to some forest Tj’ Bi has columns corresponding to the compliment of some forest Ti’ and [G1,GJ] represents the co-factor of the columns of Bi and Aj’ The determinant of the coefficient matrix in (2.20) is nonzero only if at least one term of the sum shown in (2.22) is nonzero. Thus, there must exist at least two forest Ti and T (i may equal J j) such that det [Gi.GJ] # 0 Therefore, the last equation in (2.20) can be solved for Xbi and ch to yield xa Hi1“) Hem Y... F. + Y. = H.21(t) H22(t) x. Fc(t) CJ C1 (223) If i=3, the theorem is proved. If i#j, then by means of a tree transformation we have Y . c1 and ci ci -15- where Cl and C2 are nonsingular. Therefore we have xbi U 0 Huh) H12“) 02 o Ybi = -l Yci 0 cl H2l(t) Hé2(t) 0 U xci ’U o Fb(t) + -1 (2.2%) 0 cl Fc(t) and the proof is complete On the basis of Theorem 2.5, many of the sufficiency requirements incorporated in previous theorems are also necessary. For example, Theorem 2.5 implies that for the existence of a unique solution it is necessary that there exist a tree which contains all of the elements having Specified across variables and containing no elements having specified through variables. This result, stated earlier by Blackwell(u), is, therefore, a Special case of the more general result given by Theorem 2.5. This theorem also states the restrictions on the topology of systems containing perfect couplers and gyrators. III. NONLINEAR ALGEBRAIC SYSTEMS The basic difficulty encountered in the analysis of nonlinear algebraic systems is that very little is known about the existence and uniqueness of a solution to a set of arbitrary nonlinear algebraic equations. Therefore, Theorem A.6 is essential to the following develop- ment. Through the use of this theorem, four existence theorems exactly parallel to Theorems 2.1 through 2.h are presented concerning nonlinear systems. No attempt is made, however, to arrive at any necessary con- ditions for the existence of solutions. Theorem 3.1: Let the component equations for a system with graph G be of the form x = F (Y,t) (3.1) where l) the Jacobian matrix. 2E. exists and is strictly positive ar definite for all t on an interval I defined by I = [tzto 5 t 5 t1], t and t constants, and O l , 8f 2) all partial derivatives a—y—i- are bounded and satisfy a J Idpschitz condition with respect to Y. Then there exists a unique solution for all the variables of G for all t on I. Proof: Let T be an arbitrary forest of the system graph. Then the component equations can be rewritten as -16- -17- xb _ F1 (Yt’ Yc’ t) xc F2 (Yb, Yé, t) F' (Y', t) (3.2) —. .—.— Then F' (Y', t) has the same properties as F(Y, t) in (3.1). Sub- situting (3.2) into the circuit equations, (2.2), gives [Bl U] Fl (Yb, YC, t) o (3 3) F2 (Yb. Yc’ t) By substituting the cut-set equations from (2.2) into (3.3) we then have [B U] F (Bt Y Y t) l l l C’ c’ O (3 h) t 0 F2 (Bl Y5, Y5, t) Since the Jacobian matrix J of (3.4) is given by 'BFl aFl] F t7 [B U] ———- ———- B1 1 3Y5 3Y5 3F2 arz ‘3 J (3‘5) ——— ———- U 3Y5 aYc J is strictly nonsingular for all t on I by Theorems A.5 and A.3. Therefore (3.h) satisfies the hypothesis of Theorem.An6 and.con- sequently possesses a unique solution for YE. All other system variables can be computed uniquely from (2.2) and (3.1). -18- In the above proof, the method of attach is directly parallel to that used in the linear case. The only significant change is the use of Theorems A.3, A.5, and A.6 instead of their linear counterparts. There- fore, the following theorems are stated without proof. Theorem 3.2: Let the component equations for a system having graph G be of the form x1 = Fl (Y1, t) x0 = F0(t) (3-6) Y2 = F2(t) where 1) Fl (Y1, t) satisfies 1) and 2) of Theorem 3.1, 2) Fo(t) and F2(t) are defined for all t on I, and 3) the topology of the system is such that there exists a forest T of the system graph containing the elements corresponding to X and not containing the elements corres- O ponding to Y2. Then there exists a unique solution for all the variables of the system for all t on I. Before proceeding to the cases involving perfect couplers and gyrators, a closer examination of the hypothesis of Theorems 3.1 and 3.2 is in order. For most two terminal components, the equations relating the terminal variables of the various components are such that all partial derivatives exist and are bounded. Therefore, networks of diodes, non- linear springs, etc., are known to have unique solutions. However, -19- such components as the tunnel diode have characteristics such as shown in Figure 3.1. v Figure 3.1 Tunnel Diode Characteristics For such a characteristic, the partial derivative of v with resPect to i is seen to vanish at two points and the hypothesis of the theorems are not satisfied. Systems involving such components may, or may not, have a unique solution - a result observed experimentally. Turning now to perfect couplers and gyrators in nonlinear systems we have the following theorems. Theorem 3.3: Let the component equations for a system.with graph G be on the form shown in (3.6) and (3.7): F t x3 = 309.. ) (3.7) where, in addition to the hypothesis of Theorem 3.2, 3F3 3F), 1) the entries of -——- and -———- are bounded and satisfy a aXu 3Y3 Lipschitz condition for all t on I, 2) the tOpology of the system is such that there exists a forest T of the graph G which satisfies Theorem 3.2 and also contains the _20- elements of G corresponding to X and contains no elements of G 3 corresponding to Yh’ 3) the fundamental circuit equations defined by the elements of G corresponding to Yfi, i.e., the equations [B B U] rxfl = 0 X3 are such that 3F3 det [ - 5-1- B2] 74 0 hr and det[--——liBt];éo for all (Xh’ Y3) and all t on I. Then there exists a unique solution for all the system variables for all t on I. Theorem.3.h: Let the component equations be of the form shown in (3.6) and (3-8)) X3 = F3(Y3) 13) (3'8) where, in addition to the hypothesis of Theorem 3.2, 5F 1) 3?; is skew-symmetric for all Y3 and all t on I, 3 BF3 2) the entries of 572' are bounded and satisfy a lipschitz L») -21- condition for all t on I, 3) the tOpology of the system is such that there exists a forest T which satisfies 3) of Theorem 3.2 and also contains the elements corresponding to X and, 3 h) the vector valued function Fl(Yl’ t) of 3.6 is non-null. Then there exists a unique solution for all the system variables for all t on I. Again, as in the case of linear systems, the nonlinear theorems can be stated in terms of the short-circuit parameters. The implicit form F (x, Y, t) = 0 can also be used. waever, this form is not particularly amenable to theoretical discussion unless conditions sufficient for obtaining one of the explicit forms are imposed. It does have considerable merit from the standpoint of formulation and will be used to advantage in Section V. IV. SYSTEMS DESCRIBED BY ALGEERAIC AND DIFFERENTIAL EQUATIONS In order to investigate the existence of solutions for systems con- taining components described by ordinary differential equations, the exist- ence of solutions for systems of differential equations must be investi- (ll) (13) gated. Bellman have established such existence , and others theorems for systems of ordinary differential equations in normal form, i.e., in the form EiI'E-i = fi (X1, X2, 000 Xn) t), i = l) 2’ 00' n0 Abian and Brown(15) have also established an existence theorem.based upon the more general form dx f(a—_E,X,t)=0 However, in this latter theorem, sufficient conditions are imposed to facilitate the Solution for the derivative term. Therefore, for the dis- cussion to follow, only theorems pertinent to the normal form are exploited. Aside from the problems of existence and uniqueness of a solution, there is also the very practical problem of actually obtaining the solution for a set of simultaneous ordinary differential equations. Here again the normal form seems to be the most convenient. This form is not only necessary for solution by means of analog integration, but is also -22- -23- in a form easily integrated by any one of a number of numerical integra- (6.12). tion procedures Therefore, by formulating a normal form model of the system, both the existence and the actual solution itself can be es- tablished. Systems Described by Differential Equations: First, consider the class of all systems made up of components which are described by ordinary differential equations only. Theorem h.l: let the component equations for a.system having graph G be of the form x F (x , x , Y , Y , t) g? 1 = l 1 2 1 2 2 NZ) (1“) Y2 F2 (X1. x2. Y1. Y. t) where l) F(Z) satisfies a Lipschitz condition and 2) the system topology is such that there exists a forest T of the system.graph G such that the elements corresponding to Xl are branches and the elements corresponding to Yé are chords. Then for any given set of initial conditions xl (to) Yé (to) I H there exists a unique solution for all the variables in the system.for all t such thatt Etst O + c, c > O. 0 Proof: For the forest indicated in 2) of the hypothesis, the fundamental circuit and cut-set equations are [B U] x = 0 [U -B ] Y = o 1 1 1 1 (L2) X2 Y2 Substituting (4.2) into (4.1) we have X1 Fl(Xl, -lel, BtY2, Y2. 13) 1.. = t 5 F'(Z') (1+3) dt Y2 32(xl, -lel, B Y2, Y2, t) where z 3 (X1, Y2). By Theorem A.10, F' satisfies a Lipschitz condition. Therefore, by Theorem A.ll, (4.3) has a unique solution for the given set of initial conditions I. The remaining variables can be calculated uniquely from (4.2). The class of systems covered by Theorem 4.1 is indeed quite small. For example, Systems containing circuits of capacitors, cut-sets of in- ductors, etc., are not within the framework set forth in the above theorem. In an effort to extend Theorem 4.1 to a more general situation, consider the following: Theorem 4.2: Let the component equations for a system with graph G be of the forms shown in (4.1) where 1) part 1) of the hypothesis of Theorem 4.1 is satisfied, 2) the matrix of functions 22 2F. ) is defined everywhere, is strictly positive definite, and contains -25- bounded entries which satisfy a. Lipschitz condition. Define S:L to be the subgraph containing all elements of G having component equations which are explicit in the derivative of the across variable. Let Tl be a forest of 81 and let T be a forest of the complete system 1’ i.e., every element of T1 is also contained in T. Then for every set of initial conditions of the form graph such that TDT le (to) i 1: Y2c (to) where the subscripts b and c denote branch and chord variables, re- spectively, there exists a. unique solution for all the variables of the + c, c 2 0. system for t S t _<_ t O 0 Proof: The proof of this theorem is considered as a special case of Theorem 4 . 3 following . Systems Described by Algebraic and Differential Equations: Up to this point, the component equations have been taken as either a set of al- gebraic equations or as a set of differential equations. However, in most problems presented by modern technology, neither of these two situations is likely to arise. Rather, the system under consideration is more likely to contain components which are described by both algebraic and differential equations. A number of existence theorems pertaining to systems described by mixed algebraic-differential equations can be established by combining the results of the previous theorems. Specifically, by combining Theorem2.2 and Theorem 4.2 we have the following: Theorem 4.3: of the form ~26- Let the component equation for a system having graph G be X17 = Fl(Xl, x2, x3, Y1, Y2, Y3, t) = F(Z) (4.5) 32-4 _F2(Xl, x2, x3, Y , Y2, Y3, t) r _. x31 F3(Y3, tn x0 = FO(t) (l..6) _YL-— —Fu(t) _ Define the following subgraphs of the system graph G: Let T let 81 be 1) 2) The subgraph consisting of all the elements corre8ponding to XO. The subgraph consisting of all the elements in S1 and all elements having component equations explicit in the de- rivative of the X . 1 The subgraph containing all the elements in 82 and all the elements correSponding to X3. The subgraph consisting of all the elements in S and 3 all the elements having component equations explicit in the derivative of Y2. The complete system graph G. forest of Si such that Tiz’Ti-l and let T5 = T*. Further, F(Z) satisfy a Lipschitz condition, (4.6) satisfy the hypothesis of Theorem 3.2 for the forest T*, .27- 3) the vector function Fo(t) and F4(t) be such that Ego and %%1+ exist and satisfy a Lipschitz condition, and 4) the matrix QF DF 1 is defined and strictly positive definite with entries which are bounded and satisfy a.lipschitz condition. Then for any given set of initial conditions le(to) Y2c(t0) l H 'where the subscripts b and c denote branch and chord variables, respec- tively, there exists a unique solution for all the system variables for t :5 t;5 t + c, c >-o. O 0 .Proof: For the forest T* it can be shown that the fundamental circuit and cut-set equations are of the form _B;i B12 0 0 U o o '5—[ ‘io" B21 B22 B23 32h 0 U O O x1b = [B U] Xb = 0 B31 B32 0 B3u o 0 U o x2b x __41 B42 343 BAA O O O U_i X3b C x1c (4.7) x2e x3c in... -28- and Y (4.8) .-U- For the forest T*, the first equation in (4.6) can be parti- tioned and rearranged in the form x , F (Y , Y , t) 3b 3b 3b 3c (4 9) x3c F3C(Y3b, Y3C, t) Substituting (4.9) into the third equation in (4.7) gives B B X + U F t = [31 3%) 0 [B34 ] 3b 0. The remaining systems variables 0 O can be calculated uniquely from(4.20), (4.25), (4.6), (4.7), (4.8) and (4.22). Other theorems may also be generated by taking various combina- tions of the theorems of Sections II, III, and IV. Further, additional generalizations can be included so that the theorems presented are more generally applicable. For example, in Theorem 4.3, a.more general form of (4.6) might be considered, or the Iipschitz and differentiability properties might be considered over some defined region rather than every closed region. However, these added generalizations add considerably more detail to the already complicated proof and might serve only to further obscure a very important consideration - the formulation of a systematic procedure for obtaining a normal-form.model of a nonlinear system, a ,i. n. .0, V... in!“ V V. A PROCEDURE FOR FORMUIATING NOW MATHEMATICAL MODELS In the previous sections, several existence theorems are presented and proved. From a practical standpoint, these theorems as such might well be considered to be of academic interest only. In practice, the solution is usually assumed to exist and the only remaining problem is that of actually obtaining it. However, the proof of Theorem 4.3 also serves as a basis for defining a formal procedure for obtaining mathe- matical models of nonlinear systems in normal form. Once such a model has been established, there are a number of existing numerical and analog techniques for actually constructing the solution. Systems 9: First Order Components: First, consider the general class of systems with components described by algebraic or first order ordinary differential equations, i.e., systems having components described by equations of the form dx. - E1 = fi(xl,x2,ooe,xn,yl,eoo,yn,t) , l = l,2,oeo,I (501) d‘. 5%.] = fj(xl’X2’...’xn)yl’...)yn,t) ) J = I+l,...,J (5'2) 0 = fk(xl,X2,.o.,Xn,yl,...,yn,t) , k = J+l)ooo,n (5.3) In order to obtain the normal-form model for a.representative system.of this class, (5.1), (5.2), (5.3), and the circuit and cut-set -36- -37- equations from the system graph are combined as indicated in Section IV. To select the variable to appear in the final system model, select a forest by defining the following subgraphs: Sl - The subgraph composed of all elements having across variables specified as a function of t by their respective component equations. S - The subgraph composed of all elements in S1 and all elements having component equations of the form Shown in (5.1). S — The subgraph composed of all elements in $2 and all elements having component equations of the form shown in (5.3). S4 - The subgraph composed of all elements in S3 and all elements having component equations of the form shown in (5.2). SS - The complete system graph. .A forest of the system graph is then choosen by defining T to be a forest of the subgraph Si such that Ti:)Ti-l (every element contained in Ti-l is also contained in Ti)° Then T5 is the desired forest. As an simple illustration, consider the example shown in Figure 5.1a. —-/vv\/\/\. 5 I O l 37“ 4 3| I lI—P (a) (b) Figure 5.1 A Typical R-L-C System -38- The component equations are of the form -x1- -( l/Cl )Yl _ x2 (1/02)Y2 x (l/C )Y Yu (l/Lh)xh 0 Rsy5 - xS M0_J _xO - f(t) The defined subgraphs then consist of the following elements: S - element 0. l 82 - elements 0,1,2,3. S3 - elements 0,1,2,3,5. S4 - all elements S - all elements Thus, T contains only element 0 and T contains elements 0 and any two 1 2 of the elements 1, 2, or 3. Since T2 is a tree of the complete system graph, trees T3, T4’ and T5 are identical to T2. One of the three possible trees T is shown by heavy lines in Figure 5.1b. 5 Writing the fundamental circuit and cut-set equations for the graph of Figure 5.1b we have -39 - 1 O O O O O l O -l O O l -l O -l -l -l Because of the unit matrices appearing in the fundamental circuit and cut- set equations, the chord across and branch through variables can be ex- pressed as a function of the branch across and chord through variables. Thus, (5.5) can be rewritten in the form rYO o o -1 yl = - -1 -l *1 _y2_ :1 -l 0 a y4 -y5. (5-6) This result can now be substituted into the component equations to yield, in the case of the present example, lOOl—‘CDOO F(l/Cl)(Y3 + Ya - Y5) (l/Ce)(y3 + Ya) (1/03) (Y3) ("l/L1,) (-xl - x2) R x + x 5%" 1 o [ xO - f(t) (5-7) -hO- Notice that the coefficient matrix multiplying the derivative vector in (5.7) has maximum rank. This prOperty is guaranteed by the specific choice of forest outlined earlier. shown in (5.8). By elementary row operations, (5.7) can be reduced to the form "(l/cl) (,3 + yu - ys) - O (l/Ce) (y3 + ya) (l/Lh.) ( ”X1 -X2) (1/Cl + 1/02 + 1/c3) y3 + (1/01 + 1/02) yu - (1/cl) y5 R x + x sys ‘ 1 o x - f(t) J (5-8) The last three equations in (5.8) can now be solved for the variables not appearing in the derivative vector on the left. By substituting this re- sult into the top three differential equations we have the desired form, namely where a b c') -xi] '3. d e f x2 + k (5.9) La h i pr 1 [f(t)] _ _ L J l" A. shows that -41- = .(C1 + C2)/(R5) (0102 + c2 0 + c 301) 3 '= o = -(c2)/(clc2 + 0203 + c 301) = (03)/(R5) (ch2 + 02c3 + c3cl) = o = (cl)/(clc2 + 02c3 + 0301) = ’(l/Lh) = a = 0 review of the steps demonstrated by the example of Figure 5.1 there are five basic steps leading to the normal form: 1). Select an apprOpriate forest of the system graph. 2). write the fundamental circuit and cut-set equations in a form explicit in the chord across and branch through variables. 2 3). Substitute the fundamental circuit and cut-set equations 4) 5)- into the component equations. . By elementary operations, eliminate the coefficient matrix multiplying the derivative vector. Solve the algebraic equations for the variables not appear- ing in the derivative vector and substitute this result into the differential equations. -L2- Each of the five steps listed above is well defined and is, at least in principle, easy to carry out. Further, in most situations that arise in practice, step 4 is not required. In these cases, the first four steps can often be carried out by ”inspection". The example used to demonstrate each of the above steps was linear for simplicity only. In fact, by defining C1 = Ci(xi) and LJ = Lj(yj)’ the notation used in the above example also suffices for the nonlinear situation. Notice, however, that if R5 is considered as a function of y5, it is quite likely that the fifth equation in (5.8) cannot be solved explicitly for yS. If such is the case, the normal form cannot be ob- tained explicitly. Rather, the nonlinear algebraic equation must be carried along and solved simultaneously with the differential equations. Before considering the subject of systems of components described by higher order differential equations, it should be pointed out that the formal procedure used above for selecting a forest of the system graph is simply a.precise way of stating a.very simple concept. That is, the in- dicated method of choosing a forest simply implies that 1) all the Specified across drivers are made branches, 2) as many as possible of the elements having equations explicit in the derivative of the across variable are branches, 3) as many as possible of the elements having equations explicit in the derivative of the through variable are chords, and 4) all the through drivers are chords. -43- This specific choice of forest is made to insure that the co- efficient matrix of the derivatives vector in (5.7) has maximum rank. It is therefore possible, by elementary operations, to reduce (5.7) to normal form. Higher Order Systems: The procedure given for components modeled by first order differential equations is directly extendable to some of the higher order forms. The only necessary changes occur in the choice of the forest and in the form of the component equations. Consider a component equation of the form n n-l d x dx d x n = f(x: _dt’ °°° —n-_l ) Y) 15) (5'10) dt dt This equation can be reduced to first-order form by simply defining new variables to replace the lower order derivatives. For example, the equation 3 2 d x dX d X ”'3 = f (X: EE': "EV Y: t) (5°11) dt dt can be written in the form A...“ — . O. x f (X, X) x ) Y) t) .51. . .. dt x = X (5.12) x x -44- The last two equations in (5.12) Simply define the variables used to replace the lower order derivatives. The resulting set of first order equations can then be treated as indicated in the previous example. The extension to higher order systems also requires a slight al- teration in the choice of forest. In order to obtain the desired prOperties, the subgraphs S and S4 must be further subdivided according to the order 2 of the derivative of the variable associated with each element. Thus, de- fine the following subgraphs: S - The subgraph consisting of the elements having com- 2.3 ponent equations explicit in the 3th derivative of the across variable and all elements which are contained in 82,J+1’ Jfimh If J = n, where n is tne order of the highest order derivatives of an across variable, then 82 n also includes all elements of S ) 1' S4,k - The subgraph consisting of elements having component equations explicit in the kth derivative of a through variable and all elements which are contained in S4,k-l’ k:#l. If k=l, 84,1 also contains all elements of S3. Now define T1 to be a forest of 31’ T . to be a forest of S ., GLCu ) ) 2g 23 such that Tlc'fe’nc T2,n-1C CT2,1C T3C Tim ...CTu’mCT5 -h5- where n and m.represent the highest order derivative of an across and through variable, respectively, that appears in the component equations. As before, T5 is the desired forest. It should be emphasized once again that the definition of sub- graphs and their associated forests is simply a.precise way of writing down a simple intuitive notion. The scheme for selecting these subgraphs is such that the elements modeled by the higher order differential equations are given priority in as far as their being selected as branches or chords, depending upon whether the component equations are explicit in the derivative of an across or a through variable, respectively. To illustrate the procedure for component models involving higher order equations, consider the simple mechanical system shown in Figure 5.2a. 0\ (a) (b) Figure 5.2 A Mechanical SyStem The component equations for this system are of the form -h6- Pd2x'_ — _ l (l/Mlhrl 2 dt 2 d x 2 (l/M2)Y2 2 dt “ = .13) 3 (l/B3)Y3 dt 1 3’4 ‘ Km. 3’5 ' st5 Y6 ’ f(t) The defined subgraphs and corresponding forests are: Sl - null Tl - null 52,2 - elements 1, 2. T2,2 - elements 1, 2. S - elements 1, 2, 3. T ,1 - elements 1, 2. 2,1 2 S3 - elements 1, 2, 3, 4, 5. T3 - elements 1, 2. S4 - elements 1, 2, 3, 4, 5. T4 - elements 1, 2. S5 - elements 1, 2, 3, 4, 5, 6. T5 - elements 1, 2. Writing the component equations in first order form we have x1 (l/Mlb'l x1 x1 x2 (1/M2)y2 d x i 'd—t: 2 = 2 (5.14) x 1 B 3 ( / 3)Y3 O yllr " KuXu ys ' st5 - _ 3’6 " f(t)- The fundamental By elementary row Operations, (5.16) reduces to -h7- circuit and cut-set equations for the forest T are 15) into (5.14) we have *1 (l/M2) (Y3 + YA - Y5 - Y6) i2 (l/B3) y3 - £1 + 22 y4 ' K4 (*1 ' x2) ys ' K5 (X2) .Ye ' f(t) 1 -(l/M1) (-Y3 ”yu) i1 (l/M2) (Y3 + Ya - Y5 - Y6) - *2 Uilq (l/B3)Y3 Xl y)‘ " KAI- (x1 "' X2) d . '3? x2 y5 ' K5 (‘2) - Lx2] -Y6 ‘ f(t) (5.15) (5-16) (5.17) -48- The normal form model of the system is obtained when the last four algre- braic equations in (5.17) are solved for y3, yu, y5, and y6 and the re- sult substituted into the remaining differential equations. Thus we have “21' r(i/Ml) (133m2 — 5.1)) + K4(X2 - x1) ‘ 3? ”fl = l . . (5.18) x2 (l/M2) (B3(xl - x2)) + K4(X1 - x2) - K4X2 - f(t) x2 5‘2 It can be seen by this example that the formulation procedure for the higher order component equations is much the same as in the first order case. It differs only in that additional stepsare necessary to first reduce the higher order forms to an equivalent first order form and that a slightly different procedure is used for choosing the forest. It should also be noted that when substituting the fundamental circuit and cut-set equations into the component equations, the derivative of one or more of the circuit or cut-set equations may be required to effect the substitution. Special Forms g£.Algebraic Equations: In the two previous parts of this section, a formulation procedure is outlined for systems described by first and higher order differential equations. With one exception, all of the indicated steps can be carried out in general. However, certain situations exist where the required solution of the algebraic equations -49- either does not exist, or is nonunique. Consequently, special procedures _must be employed to deal with these cases. To demonstrate the nature of the problem, consider the system shown schematically in Figure 5.3a. LIJV A l (a) (b) Figure 5.3 An Electrical System The component equations for this system are 7x1- F(l/C1)Yl- X2 (1/02)Y2 gr 0 = R3y3 - x3 (5.19) 0 x0 - f(t) 0 ya + Ky5 _0 _ -x5 ' Kx4.. IProceeding as indicated in the earlier part of this section, we have -50.. ”x1‘ "(l/cl) <-yy) x2 (1/02) (-Y3 - Y5) d o R3Y3 4 (X2 - X0) 5% c) = x0 - f(t) (5'20) 0 U4 + KY5 -0- [_x2 'le _ Normally, the last four equations in (5.20) would be solved for y3, yh, ys, and x as a function of x1, x , and t. But since the last equation 0 in (5.20) is independent of y3, yu, yS, and x0, no such unique solution exists. In an attempt to circumvent this difficulty, consider the de- rivative of the last equation of (5.20) with reSpect to t. dx dx 5752 “K atl-O (5.21) If xl(t) and x2(t) are any two functions which satisfy (5.21), then x2(t) -le(t) = c for some constant c. Therefore, if it is also required that x2(to) -le(t0) = o -5 1- then any two functions which satisfy (5.21) and the above boundry con- dition also satisfy the last equation in (5.20) and conversely. There- fore, the last equations in (5.20) can be replaced by (5.21) without loss of precision giving [xl‘ [(1/01) ('Yu) ‘ EL O = R3y3 - (x2 - x0) (5.22) at 0 x0 - f(t) 0 fig - del _ __ _dt dt - where x2020) - le(to) = 0. At first glance it would seem that such a substitution defeats the purpose of the formulation procedure because it introduces derivatives in the right-hand side of the equations. Note, however, that these derivatives are also contained on the left-hand side of (5.22). The tOp two differential equations in (5.22) can there- fore be substituted into the last equation to yield le- F(l/Cl) (‘31,) 3 X2 (l/Ce) (’Y3 ‘y5) 9}.- O = R3y3 ' (X2 " X0 (523) at 0 x0 _ f(t) _0 _ _(K/Clhru - (l/Cl) (y3 + y5)_ -52- By solving for y3, yh, yS, and x0 in the last four equations in (5.23) and substiuting this result into the first two equations, we have the normal form 7le "(K/(133(02K2 + Cl)))(f(t) - x2)- g}; = (5.24) “x2“ (Kg/(33(C2K2 + Cl)))(f(t) - x2) where x2(to) - le(t0) = 0. Note that the condition imposed upon the initial conditions must be carried along to assure that the solution to (5.24) also satisfies (5.20). The general situation typlified by this example is one in which, are performing the first four steps of the formulation procedure, the equations take on the form at o - F (z z t) . 2 l’ 2’ where for some k, - ' _ i.e., one of the algebraic equations is a function of the entries in Zl -53.. alone. Since F2 contains the same number of equations as there are entries in Z2, it follows that from F2, no unique solution can be found for 22 as a function of 21 and t. Taking the derivative of (5.26) with respect to t we have afék dz -.._ -——1J = O (5.27) d . azlJ t J provided that the indicated parital derivatives exist. If the added condition I _- f2k (zl(to), to) .. o (5.28) is imposed, then any set of functions [zlj(t)] which satisfy (5.27) also satisfy (5.26) and conversely. Substituting from (5.25) for the derivatives in (5.27) yields c)f _ Z 2k flJ (21, 22, t) _ o (5.29) 213 J Usually, at least for first order systems, (5.29) is dependent upon 22. Therefore, if (5.26) is replaced by an equivalent expression, namely (5.29) and the constraint (5.28), the difficulty is eliminated. -51,- To illustrate further, consider the compound pendulum shown in Figure 5.4a. 03‘ 13 .I‘ 03 04 T (a) (b) Figure 5.4 Compound Pendulum For this example, the component equations are -iolq F(l/Ml)yol U xo1 iol £11 (l/M1)yvl + g x11 U11 d U02 (l/M2)yo2 EU. xo2 = ice (5°30 £12 (1/M2)y12 + g x12 £12 0 x§3 + x§3 - R3 0 x03 5'13 " x13 ”'03 0 x34 + ”$4 ‘ RE _0 _‘ _Xo4y'r4 ' £43454 j where the subscripts o and 1 denote the two co-ordinates of motion. -55 - For the forest of the system.graph (Figure 5.4b) which contains elements 01, 11, U2, and t2, the fundamental circuit equations are 1 r- -i l O O 0 x01 0 1 O 0 x02 = 0 0 l 0 xTl 0 0 0 1- xT2 X o3 X04 X 13 x TU- B (5.31) Substituting (5.31) and the associated cut-set equations into (5.30) gives F- i1] 0 x01 x11 x11 xo2 x02 x12 x12 0 ”(l/Ml) (YO), - 3'03) 5‘01 (l/Ml) (YT)L - 5’13) + s 5‘11 (l/ME) ('You) U02 (l/Mg) (-Y,r)+) + s 1212 2 2 x0]- + xTl - R3 x01 y13 - x11 yo3 2 2 2 (x02 - x01) + (x12 U x11) 3 R4 (X -X)Y -(X -X)Y L 02 01 14 T2 Tl 04‘ (5.32) -56- Note that the ninth and eleventh equations in (5.32) are functions of the variables in the derivative vector only. Taking the derivative of these equations with respect to t we have x EEE£~ + x del - 0 Cl dt 11 dt - (5 33) dx 02 01 T2 T1 - ____. - —___. - ____. - = 0 (x02 Xol) ( dt dt ) + (X12 X11) ( dt dt ) with the constraints x x + x x = 0 01 01 Tl Tl (5 34) a d>-x d>f+-x2 U4 12 11 02 01 3'1“- [O _ -5a By solving (5.38) for y03, yT3, you, and y14 and substituting the result into the first eight equations in (5.30), the normal—form model for the system is obtained. Note, however, that the initial conditions used to obtain any desired solution to the system model are not completely ar- bitrary. They are subject to the added conditions that they must satisfy (5.34) and (5.35) for t = to, the initial value of the independent vari- able. In this latter example, it was necessary to differentiate the alge- braic equations twice. In general, if some of the component equations are of nth order, n differentiations of the algebraic equations may be re- quired. In an attempt to put order into the problem presented above, note that the equations which lead to difficulty in the three previous examples are the perfect coupler models encountered earlier in Section II, i.e., a component equation which relates an across variable to another across variable. Also notice that when attempting to obtain a solution for the algebraic equations, the variables in the derivative vector are treated essentially as if they were known functions of t. Therefore, the question of whether or not a solution exists for the algebraic e- quations, at least in the linear case, can be answered by determining whether or not any forest of the class outlined in the first part of this section also satisfies the hypothesis of Theorem 2.5 in-so-far as the elements of the graph having algebraic component equations are concerned. -59.. For the example shown in Figure 5.3a, the only possible choice of forest includes elements 0, l, and 2. This automatically classifies elements 4 and 5 as chords. Consequently, since the component equations for these elements are the hypothesis of Theorem 2.5 is not satisfied andxu>unique solution exists for the algebraic equations. In this way, at least in the linear cases, the need for special formulation procedures can be anticipated from the form of the component equations and the manner in which the components are interconnected. As a final illustration of systems containing perfect coupler models, consider the system shown in Figure 5.5a. H 5‘ (a) (b) Figure 5.5 An Electrical System -60- Since the component equations are of the form [xl [(1/clhrl ' X2 (l/C2)y2 0 X5 " lell- 3.,- ° = x6 " "2"4 (5.39) 0 U4 + Kly5 + Kéy6 O R3Y3 ' X3 _Oj _xO-f(t) - only two forests can be selected consistent with the procedure outlined in the first part of this section. Neither of the two forests, (0, l, 2, 3) or (O, l, 2, 4), are acceptable from the standpoint of Theorem 2.5. It is evident, therefore, that the algebraic equations are such that the normal form cannot be obtained without special procedures. Writing the fundamental circuit and cut-set equations from Figure 5.5b and substitut- ing this result into (5.39) we have le- F(l/Cl) ('33) U X2 (l/Ce) ('Y6) 0 x1 - le4 g5. o = x2 - new,L (5.40) 0 -y3 + Kly5 + Kéyé O R3y3 - X4 + x0 _ o _ _xO - f(t) J - 61.. Since the third and fourth equations are functions of xl, x , and x1+ only, 2 the five algebraic equations in (5.40) have no unique solution for y3, y5, y6, xh, and x0 as a function of x1, x2, of the equations are functions of x1 and x2 alone. It would appear, therefore, that the system shown in Figure 5.5 is beyond the scope of and t. But in this example, none the develOpment presented thus far. Notice, however, that by solving the fourth equation in (5.39) for xu, (5.39) can be rewritten in the form leq _(l/Clh'l x2 (1/02)Y2 0 x5 - (Kl/1:2)x6 2% o = x,L - (l/K2)x6 (5.41) 0 Y6 + (Kl/K2 )Y5 + (l/thqL U R3Y3 " x3 _oj _xo - f(t) .- Combining (5.41) with the circuit and cut-set equations then gives the form Fxl‘ F(l/Cl) ('y5) " X2 (l/C2) ('y6) 0 x1 " (Kl/K2)x2 g? o = x2+ - (l/K2)x2 (5-42) 0 Y6 + (Kl/gm5 - (Ugh, O R3y3 - xu + x0 _0" _x0 - f(t) - -62- But now the third equation does relate x and x and the normal-form l 2 model of the system can be obtained as in the previous examples. Thus, it is Seen that for systems involving multiport perfect couplers, the component equations should be written in a certain form. Specifically, if one of the elements, say en, corresponding to a perfect coupler is a chord for every forest of the type indicated in the first part of this section, the perfect coupler equations Should be written in the form _ - u _ r _ el 0 o ... hl yel x82 0 o ... h2 ye2 s '2 s s s = 0 we) xen-l O 0 ... hn-l Yen_ :h1 -h2 O _ en_ Similarly, if one of the elements, say e3, j‘< n, corresponding to a.per- fect coupler is a branch for every forest T indicated earlier, the 5 perfect coupler equations should again be in the form shown in (5.43). In the case of the example of Figure 5.5, element 6 is a chord for every forest T5. Therefore, a prOper form.for the perfect coupler equations is x” ' O O l/Ké YA = 0 (5.43) -63- The form - - - x6 0 o Ka/K‘ ' " l y6 x1+ - o o l/Kl y1+ = o (5 .45) _YSJ _-K2/Kl -1/1-Bo> 3.;2 x, (:L/Me» (Tye-31 (x0. x1. is,» where M(t) is the mass of the rocket, Bo(xo’ x 1’ x0) and B1(xo’x1’xo) are functions which determine the frictional effects of the atmosphere and To(t) and T1(t) represent the thrust of the rocket engine. The effect of gravity upon the rocket can be considered as a com- ponent having equations of the form 3’ "_3_ x o = -KM(t) (x2 + x2) 2 o yT o 1 x T -64- The position of the moon in its orbit around the earth is given by x sin K2t = Kl x cos Két A schematic diagram of the system under consideration and the system graph are Shown in Figure 5.6. (a) 7‘ UV (b) Figure 5.6 Orbital System In the graph, element 1 pertains to the rocket - earth component, elements 2 and 3 pertain respectively to the gravitational effects of the moon and the earth upon the rocket and element 0 pertains to the relative position -65- of the moon with respect to the earth. Combining the various component equations into one set we have [ioi' "(l/M(t)) (To(t) - Bo(xol’ x11. igl) + yoli' xol £11 £11 (l/M(t)) (TT(t) ' B1(xol’ x11’ U11) + ytl) x11 U11 3 2 2 2. g3) O = yo2 + K1M (~Rhyp-RuNh5y3 + rl ‘5 55) -o_ h-RhNLLSyT - (35 + RnNiS) y3 - x1 -f(y3) + Nusflmsz Note that because of the nonlinear function f(y3), no explicit solution for y3 is possible in the last equation in (5.55). The three nonlinear equations shown in (5.55) must be solved simultaneously in order to ob- tain a solution. VI CONCLUSION In the first sections of this thesis, a number of existence theorems are presented and proved. The second section is devoted en- tirely to systems having components modeled by linear algebraic equations and extended existing theorems to cover the more general case where the coefficient matrix is not symmetric. A.particularly interesting result of this section is presented in Theorem 2.5. Here, necessary conditions are established for the existence of a solution for a completely general linear algebraic system. This theorem.covers, as special cases, the restrictions on the topology of regulated sources, perfect couplers, etc. To establish results similar to those of Section II for nonlinear systems, the mathematical structure presented in Appendix A is found use- ful. This development introduces the definitions of strictly positive definite matrices and strictly nonsingular matrices. These definitions are in turn used in the presentation of theorems and proofs which are patterned after the linear cases. Probably the most significant result of this development is given by Theorem A.6. Here, sufficient conditions are presented for the existence of a solution to the vector equation F(X) = O. The proof, in addition to establishing the validity of the theorem, also provides new numerical techniques for obtaining the solutions for such equations. The solution is shown to exist by demonstrating that -71- -72 .. a certain differential equation has a solution. This solution than pro- vided the solution to F(X) = 0. Therefore, all the techniques currently available for solving differential equations are new applicable to the problem of solving algebraic equations. In fact, it can be show. that if the first order Euler method witn step size equa to one is used, the re- sulting numerical method is identical to the famil’ar Newton-Raphson method. The important point is, however, that higher order methods, such as the fourth order Runge-Kutta method, can be used with saall step size thereby eliminating the problem of convergence. In Section IV, differential equation models of system components are considered. The major result, presented in Theorem n.3, pertains to mixed algebraic and 'ifferential equation models. The proof of this re- sult is used as basis for defining a.formal procedure for generating {‘3 normal-form models of systems. The formulation procedure, presented in Section V, is applicable to first and higher order systems a though more development is needed in the latter case. Systems containing perfect couplers, gyrators, etc., also fit into the franework presented. Here, however, special forms of the component equations are required for systematic treatment. By combining the results of Section V with existing numerical methods for solving differential equations, it is possible to write general purpose digital computer programs to obtain solutions to a broad class of system problems. In this way, any of the systems presented as examples can be solved automatically by a.single computer progr a. APPENDIX A THEOREMS AND DEFINITIONS In this deveIOpment, it is convenient to refer to large numbers of equations or symbols by simplified notation. Therefore, matrix notation and matrix algebra‘g) are used throughout. A.matrix, or a.vector, is de- noted by upper case letters such as Z and G, while subscripted lower case letters donote individual entries in these matrices, e.g., zi, gij’ etc. The symbols X and Y are used exclusively for vectors. Vector valued functions of real vector variables are also useful and are denoted in the customary manner. For example, the symbols F(X) serve to denote the vector function f2(xl,x2, ... xn) F(X) = H: m(xl,x2, ... x ) FUnctions of more than one vector variable are written in the form r 1 fl(xl, ... xn, yl, ... ym) f2(xl, ... xn, yl, ... ym) F(X,Y) = m- fk(xl’ ... xn, yl, ... y ) -73.. -74- Differentiation in matrix notation has the following meaning: Q; H) H Q1 H: H m W H Q; >4 P Q, X [D QL X e» re to fit eh A) Q, ><| [—1 ‘94 [D \. :1. UR) Q}? (X.Y) = a c, N Q, Jfk afk af 5.2; 53:; ° 5::- As illustrated above, the arguments of functions are occasionally drOpped when there is no danger of misunderstanding. Two measures of a vector, the norm and bound, are also convenient. The norm of an n-dimensional vector X is defined as N(X) = E: xi 1 2 and the bound of X is defined as "XII = Max‘x,l i 1 In order to establish some of the properties required in the proofs presented in the main body of the thesis, a.number of definitions and -75.. Theorems are needed. Definition A.l: A real, square, constant matrix B is said to be positive definite if for every nonzero vector Y‘we have Y BY >'O Definition A.2: A.matrix B(X) whose entries are bounded, continuous, real valued functions of the vector X is said to be strictly positive de- finite if, for all X, there exists a constant c, c >'O, such that for any vector Y, YtB(X)Y > cN2(Y) Theorem A.l: If and only if B is a.positive definite matrix, then B is strictly positive definite. 2329:; Sufficient: Consider the quadratic form Y BY (la) Let (la) be rewritten in the form YtBY = N2(Y) 2th (2a) where N(Z) = 1. It must now be shown that there exists a constant -76- c, c > 0, such that 2th _>_ c for every Z such that N(Z) = 1. Suppose no such c exists. Then for every element of a sequence [ci], C1 > 0, there exists a Zi with N(Zi) = 1 such that ZIBZ. < c. 1 1 1 Now, choose a.sequence [oi] having zero as a limit. The corres- ponding sequence [Z1] represents an infinite, bounded set and, therefore, has a limit point, say Z. New choose a subsequence of the sequence [Z1] , say [Zk] , which has Z as a limit. Then given 6 >'O, there exists a constant m such that for all k >im, ZkBZk <6 By the continuity of the quadratic form, we then have Z BZ = 0. Thus, since N(Z) = l, B is not positive definite and the con- tradiction is complete. -77- Necessitv: If B is strictly positive definite, YtBY 3 cN2(Y), c > o t Then, if Y ,14 o, N(Y) > o and Y BY > 0. Theorem A.2:* If a matrix A is positive definite, then A is nonsingular. Proof: Assume to the contrami.e., that A is singular. Then there exists a nonsingular matrix K such that where A is an rxn matrix with r equal to the rank of A. Then 1 . we have A = KB = x1305)"l Kt = KCKt where C = B(K‘t)-l Therefore t xt AX = (K x) 1‘c(1 0, such that det A(X) ‘2 n for every X. -79- Proof: For any value of X, say X=X , A(Xl) is a positive doJinite 1 matrix. Therefore, there exists a.matrix X such that KA(X = T 1) where T is upper triangular, tnn is equal to the determinant of A(Xl), K is lower triangular, and knn = 1. Now consider the quadratic form YtA(Xl)Y = (th'l) KA (xl) Kt (th'l)t Since KA(Xl) = T, the n,n entry in K'A(xl)Kt is equal to the determinant of A(Xl). Thus, if z = ( o o o ... o 1), then ZtKA(Xl)XtZ = det A(Xl) Now let 05°)"l Y = 2. Then t Y = K Z and, from the prOperties of K, yn = 1. Therefore, for any value of x, there exists a Y such that N(Y) 2'1 and YtA(X)Y = det A(X) -80- Now suppose, contrary to the conclusion of the theorem, that for every n >‘O there exists an X such that det A(X) < n Then for n = c we have YtA(X)Y < c 5 crfly) for some X and Y and any c, c > 0. But since A(X) is strictly positive definite, t Y A(X)Y 3 cN2(Y) for every X and Y and some c. Hence the contradiction, and the theorem is proved. Theorem A.h: If A is a.positive definite matrix of order mxm and B is a mxn matrix with rank n, n 51m, then B AB is positive definite and, consequently, nonsingular. (11+) Proof: After Tokad and Kesavan . Theorem A.5z If A(X) is an nxn strictly positive definite matrix and B -81.. is a constant matrix of the form B =[Bl U] then t BA(X)B is strictly positive definite. Proof: Consider the quadratic form t Y BA(X)BtY = ZtA(X)Z where Z = BtY. Since Z=B Y = B1 11 [Y] it £011ch that N(Z) _>_ N(Y). Further, since A(X) is strictly positive definite, we also have that ZtA(X)Z _>_ cN2(Z) 3 cN2(Y) for some c, c >»O. Therefore YtBA(X)BtY 3 cN2(Y) -82- and BA.(X)Bt is strictly positive definite. Corollary A.5a: If A(X) is an nxn matrix of the form O O o A22(X) = A(X) where A22(X) is strictly positive definite and of order mxm, and if B is a pxn matrix of the form with p S m, then BA(X)Bt is strictly positive definite. Proof: The matrix B can be partitioned in the form where B is a.pxm matrix of the form 3 Then ..ynlvfigflmlfl .4 .‘y d. -83- which, according to Theorem A.5, is strictly positive definite. Theorem A.5b: If A(X) is strictly positive definite, and if B(X) is a row-column rearrangement of A(X), i.e., t B(X) = DA(X)D where D is a matrix with one and only one nonzero entry, +1, in each column and row, then B(X) is also strictly positive definite. Proof: Since A(X) is strictly positive definite, we have that ZtA(X)Z 3 cN2(Z) From the structure of the matrix D, it can be seen that D is nonsingular and if DtY = Z then N(Y)=N(Z). Therefore, for every Y there exists a Z such that we have YtDA(X)DtY = ZtA(X)Z 3 cN2(Z) = cN2(Y) Therefore DA(X)Dt is strictly positive definite. Definition A.3: A vector valued function F(Z) is said to satisfy a Lip- schitz condition if for every closed region R defined by “Z - Cujs cl we have, for some c depending on R, 2 “F(Zl) - F(22)l < c2 "21 .. 22' -Bh- where ZleR and ZeeR. Note that the above definition refers to every closed region R. If, for any reason, it is desired to consider a specific region, this region is to be stated explicitly. Theorem A.6: Let F(X,Y) be an arbitrary q-dimensional vector function of the q-dimensional variable X and the p-dimensional parameter Y. For every Y such that l) the entries of §_F_ exist everywhere, are bounded, and satisfy X Lipschitz condition with respect to X and 2) det fl 2 k > O for all X, k a constant, (ax there exists a unique X such that F(X,Y) = 0. Proof: Existence: Let X_be an arbitrary q—dimensional constant vector and cosider H(X,Y,Z) : F(X)Y) " ZF(2(_)Y) (3a) Differentiating (3a) with reSpect to z we have dH dF‘ dX a"; = 53f ‘5' ‘ F(X,Y) (ha) Note that the above partial derivatives exist by hypothesis. It is now claimed that there exists an X(z) such that (ha) vanishes. If such an X(z) exists, it must satisfy -1 ~33;- = ‘33 F(E’Y) (5a) 0a -85- -l where the entries in 53—:— are again bounded and satisfy a Lip- schitz condition. By Theorem.A.ll, (5a) does have a solution X(z) on the interval 0 _<_ z 5 1 for any set of initial conditions x(1). Substituting this solution into (5a) we have d , , = o 3% (X(z)Yz) (6a) Integrating (6a) with respect to 2 then yields, by the Fundamental Theorem of Calculus, O 0 g—E— (X(z), Y, 2) dz 0 dz = O l 1 (7a) H(X(O)1Y) )) ‘ H(X(l),Y,l) =F(X(O),Y) ‘ F(X(l),Y) + F(Ejy) Therefore if the initial condition is X(l) = X, the solution to (5a) evaluated at z=0, namely X(O), is a solution to F(X,Y) = O. Uniqueness: Let X} be an arbitrary vector and let X'(z) be the re- sulting solution to (5a). Then it has been shown that X'(O) is a root of F(X,Y), i.e., F(X'(O),Y) = 0. Now assume, contrary to the conclusing of the theorem, that there exists a second root, say Q. QfXWO). Define L to be the "line segment" Joining X} and Q, i.e., L=[Z:Z=OQ(_' -(a-1)Q, 050151] It is now claimed that there exists a 6 such that if X1 6 L, X2 e L, and le - X2“ < 5, then the solutions to the equations fi‘ixhsdzvh \ . , ... fin. 1.. aflfla“! I. .Fnh.‘ . .. -86- -1 . - dip _ gklfl) F(Xl,Y) , xl(1)..§l (8a.) _. l - dz -1 dxg .5_F_(X2;Y) F(Y,.Y) ; X2(l)=§2 (9a) a? = ax2 . namely xl(z) and X2(z), are such that .xl(o) x2(o). To show this result, it is first shown that the solution to (8a) for arbitrary X l e L lies in some closed region. Since a bound on the solution (8a) is given by the product of the length of the interval of integration and the maximum value of the derivative, for 0.3 2‘3 l'we have “X1(z) ' -)£1' 5 gm "F(EPY)” -l where m is an absolute bound on the entries in 3% . From the fact that ‘§§-exists and satisfies a Lipschitz condition, F(X,Y) satisfies a Lipschitz condition over the region L. Therefore, there exists a constant ml such that llF(}-‘-1’Y)l Sml and we have "X1(z) ' 2S1| 5 qmml Thus, - '-.-. - .. . i |X1(z) El |X1(z) E1 5 +§1| lx1(z) ‘ £1“ +| ”11""3- " QI AIA El ' é m2 IA -87- Therefore, for an arbitrary vector X _1 e L, the solution to (8a) is such that for O E z E l, |X1(z) - X'I _<_ m2 (10a) Next, it is shown that given 6 >>O, there exists a 5 >'O such that if 331 e L, {2 e L and IE1 - Eel < 5, then lxl(o) - x2(o)| < e. Sub- stracting (9a) from (8a) we have ”g; (x -x 2)||= aF l'(xl.Y) F(Y,.Yw g (x2.Y)'l F(X2,Y)l 2 Eng: (xl.Y)'l| lurid) - F(YQ.Y)| +"dfl (15.10"l -d_§ (X2.Y) llF(§2.Y)" 6x2 qm "F(Y,.Y) — F(Y,.Yfl qc2 INK-WY)“ "X1 ' le IA + where c2 = Max c1 and c1 are the Idpschitz constants for the entries -1 i 6F . in‘§E—- over the closed region defined by (10a). Since F(X,Y) satisfies a.Iipschitz condition, given El >'O there ex1sts a 51 >10 such that if "Xl - Xel < 51, then “F(Y,.Y) - F(§2.Y)I < e1L Under these conditions, we have "dz(X1X2)I < 93” £1 + qcami "X1 " x2" (11a) for 0.5.2 S l. -88- Then, since the boundries of the region “Xl(z) - Xa<2>| S "221 - r2" exp (chmlu-z» me + C 1 (exp (qc2m1(l-Z)) - l ) '2m1 are such that if (Zl - Z2) is any function contained in this boundry, d - = _ lde (Z1 Z2)“ gm 61 + qc2m1 “21 22! all solutions to (Be) and (9a) subject to the conditions imposed in the earlier parts of the proof must be contained in this region. Thus, for z = O we have that III€ llxlm) - x2 0 there exists a 5 >'O, namely 5 = min (51,52), such that if pi e L, £2 6 L, andIIXl — x2" < s then "xl(o) - x2(o)“ 5 e (10) . . By Theorem 7.5 of Apostol , under the conditions of the hypothe- sis, there exists an e-neighborhood of a root of F(X,Y) which con- tains only that root. Therefore, since both Xl(O) and X2(O) are roots, it follows that xl(o) = x2(o) -89- To complete the proof, choose a finite number of points from L, say Ek’ k = 0, l, ... 0, such that Ila-ail“ _! io-ZS 22.=Q- Then we have that xO(o) = X'(O) xl(o) = xO(o) >4 Q A O V I - Xo-l( Thus, X'(O) = X0(O). But, since X0 = Q, and since Q is a root of F(X,Y), we have that X0(z) = x0(1) = $0 = Q,= x0(o) = X'(O). Therefore, Q = X'(O) and the two roots are identical. This com- * pletes the proof. Theorem A.7: If, in addition to the hypothesis of Theorem A.6, F(X,Y) satisfies a Lipschitz condition with respect to (X,Y), then for any closed region R of the form "Y - C".E cl there exists a c2 depending on R such that if Y: e R and Yé e R, then * This proof was suggested by Mr. J. T. Olsztyn of General Motors Research Laboratories. -90- “X1 ' X2“ 5 C2 ”Y1 ' Y2" where F(Xl, Y1) = o and F(X2, Y2) = 0. Proof: Let Y1 e R and Y2 e R and consider the method of solution used in the proof of Theorem A.6. The solution to -1 dX 3F (X,Y2) F(Xl,Y2), x(1) = X 9 dz 1 “>21 evaluated at z = O is equal to X2. However, since F(X,Y) satisfies a Lipschitz condition with respect to (X,Y) for some c >‘O we have "F(Xi’Ye) " F(Xi’Y1)"-<-- ° "Y2 " Y1" Therefore, since F(X1,Yl) = O, "F(X1’Y2)n -<- C "Y2 ' Yll Since the solution cannot exceed the product of the maximum value of the derivative times the length of the interval we have Ila-xi .<. us - Yil where, as in the proof of Theorem A.6, m is an absolute bound on at; UM)" the entries of <5X Definition AA: A matrix B(X) is said to be skew-53mmetric if, for all -91- X, bij(x) = -bji(X) for 1 # J, and bij = O for i = 3. Theorem A.8: If B(X) is a skew-symmetric matrix, then for every vector X and every vector Y, YtB(X)Y = 0 Proof: Consider the quadratic form YtB(X)Y Then YtB(X)Y= Z yibiJ(X)yJ 1:3 = Zbiibfihi + Z yibiJOUYJ i 1.3 #3 But since bij(x) = ~bji(X), Z yibiim 3’1 = O 1.3 #1 Further, since bii(X) = 0, Z bii(X)yi = O 1 Thus, YtB(X)Y = o for all x. -92- Theorem.A.9: Given a 8 matrix (the coefficient matrix of the cut-set equations) for a graph of p parts, any q columns, q_§ (v - p), are in- dependent if, and only if, the columns correspond to elements which form a subgraph of some forest. Given a B matrix (the coefficient matrix of the circuit equations) for a graph of p parts, any q columns, 9.5 (e - v + p), are independent if, and only if, the columns correspond to elements which form a.subgraph of the compliment of some forest. Proof: After Blackwell and Reed. Theorem A.lO: If a vector valued function F(Z) satisfies a.Iipschitz condition, and if and where Fl also satisfies a.1ipschitz condition, then F'(Zl) : F(zl. Flap) also satisfies a Lipschitz condition. Proof: Consider a closed region R defined by R = [21: "21 - on _<_ cl] for arbitrary C and c Since F satisfies a Lipschitz I. l -93- condition, if Zi and Zi belong to R, then I _ n t _ n kiwi) F1&1)“ 5°?— "Z1 z1" 52°1°2 for some c2. Therefore, since Zi & C belongs to R, i - "F1(Zl) 151(0)“ E2clc2 ”F1“? ’ 31(0)" 5 2c1‘22 Let Rl be the region defined by R1 = [2: "z - 03' 5 cu] where C3 = C 5(a) and Since F satisfies a Lipschitz condition, there exists a constant c5 such that "F(Z') - F(Z")I _<_ c5 '2' -z"" if 2’ and Z" belong to R Thus, if Z' and Z" belong to R, 1° 1 l 4.. -9 then Z’ = Zi and Z" = Zi' F1 (2i) Fl (Zf) belong to R1 and "PW-21> - Mi)" = "Ferries” -F>l 5. c5 Zi Zi F1(Zi) F1(Z§ 5 c5 Max(l,C2) "21' - zi'" This completes the proof. Theorem A.ll: we have "Z ' C“ Eel “F(X) - F( where c 3 solution to c where forogtgcl/2 Proof: After Bellman. If, for any two vectors X and Y in the region R defined by Y)l 5C3 "X " Y" is a constant depending only upon R, thenthere exists a unique -95 - Corollary Atll: If the function F'(Z',t) satisfies a Lipschitz condition with respect to (Z',t) then the differential equation dZ' dt F'(Z',t), Z(to) = c' (12a) has a unique solution for t _<_ t _<_ t + C", C" > O. O 0 Proof: If the change of variable t = t - t is made in (12a) we have 1 o n _ I I . I _ I g§_ (tO + tl) _ F (2 (t0 + tl), tO + t1) , 2 (t0 + o) _ c dt 1 or 5124.. (H) F"(Z"(t ), t ) ; z"(o) = c' (l3a) dtl 1 1 where Z (t1) E.Z (tO + t1). Since the equation d1 dtl 1 ; «(0) = 0 has the solution 1 [Z'trw , [”“°’]= m T' 1 1(0) 0 If z(tl) (Z"(tl), T (tl)), then by Theorem A.ll, (lua) has a tl, (l3a) can be written as unique solution for O S t f c, c > O. 1 Therefore, since Z’(t) = Z'(t0 + t1) = z"(tl), _96- a solution for Z'(t) is established for t .5 t‘f t + c, c >.0. O 0 To show that the solution is unique, assume that a second solution, Xf(t),exists. Then a second solution, say Xf(tl), must exist for (lha), contradicting the conclusion of Theorem A.ll. l. 10. ll. 12. 13. 1h. 15. -97- REFERENCES Koenig, H. E. and Blackwell, W. A., Electromechanical System Theory, McGraw-Hill, 1961 . Seshu, S. and Reed, M. B., Linear Graphs and Electrical Networks, Addison-Wesley, 1961. Reed, M. B., A Foundation for Electric Network Theory, Prentice- Hall, 1961. Blackwell, W. A., The Linear Graph in System Analysis, A Thesis, Michigan State University, 1958. Ku, Y. H., Wolf, A. A., and Dietz, J. H., Taylor-Cauchy Transforms for Analysis of a Class of Nonlinear Systems, Proc. IRE., Vol. 1118, 912-922, 1960. Levy, H. , and Baggot, E. A., Numerical Solutions of Differential E uations, Dover, 1950. Householder, A. 8., Principles of Numerical Analysis, McGraw-Hill, 1953- Trent, H. M., Isomorphism Between Oriented Linear Graphs and Lumped Mechanical Systems, J. Acoust. Soc. Am., 500-527, 1955. Hohn, F. E. , Elementary Matrix Algebra, MacMillan, 1958. Apostol, T. M., Mathematical Analysis, Addison-Wesley, 1957. Bellman, R., Stability Theory of Differential Equations, McGraw-Hill, 1953. Milne, W. E. , Numerical Solution of Differential Equations, Wiley, 1953. Moulton, F. R., Differential Equations, Dover, 1958. Kesavan, H. K., and Tokad, Y., Terminal Representations of Electro- Magnetic Systems and Problems of Inductance Networks, To be Published in AIEE. Abian, S. , and Brown, A. B., On the Solution of the Differential Equation fix,y,y') = 0, American Mathematical Monthly, Vol. 66, No. 3, March 1959, 192-199. . - ' g—fn ‘ W-- . ttrflizi‘rr‘:‘:~'w‘ ! "SE 8 ‘I \ -3, ‘iilll 5"!‘0‘ ‘ - 11M" gigs-E milurmm' 7 1 7 4 8 7 1 3 o 3 9 2 1 I" H H H " All) H i 3 Imm’t‘un