homotopia de punto fijo.pdf

222IEICE TRANS. FUNDAMENTALS, VOL.E85A, NO.1 JANUARY 2002

PAPER

A Practical Approach for the Fixed-Point Homotopy

Method Using a Solution-Tracing Circuit

Yasuaki INOUEa), Saeko KUSANOBU, and Kiyotaka YAMAMURA, Regular Members

SUMMARY Finding DC operating-points of nonlinear cir-cuits is an important and dicult task. The Newton-Raphsonmethod employed in the SPICE-like simulators often fails to con-verge to a solution. To overcome this convergence problem, ho-motopy methods have been studied from various viewpoints. Thexed-point homotopy method is one of the excellent methods.However, from the viewpoint of implementation, it is importantto study it further so that the method can be easily and widelyused by many circuit designers. This paper presents a practicalmethod to implement the xed-point homotopy method. A spe-cial circuit called the solution-tracing circuit for the xed-pointhomotopy method is proposed. By using this circuit, the solu-tion curves of homotopy equations can be traced by performingthe SPICE transient analysis. Therefore, no modication to theexisting programs is necessary. Moreover, it is proved that theproposed method is globally convergent. Numerical examplesshow that the proposed technique is eective and can be easilyimplemented. By the proposed technique, many SPICE users caneasily implement the xed-point homotopy method.key words: homotopy method, xed-point homotopy, DCoperating-point, circuit simulation, SPICE

1. Introduction

In circuit simulation, nding DC operating-points ofnonlinear circuits is an important and dicult task.The SPICE-like circuit simulators [1], widely utilizedfor designing LSIs, employ the Newton-Raphson (NR)method for solving modied nodal (MN) equations [2].However, the NR method or its variants often fail toconverge to a solution unless the initial estimation pointis suciently close to the solution [3].

To overcome this convergence problem, homotopymethods have been studied from various viewpoints [3][26]. There are many papers from the theoretical view-point. However, it is important to study further fromthe practical viewpoint so that the methods can be eas-ily and widely handled with the SPICE-like simulators.

In the SPICE-like simulators, many practical built-in models for semiconductor devices are contained, forinstance, the Gummel-Poon model for bipolar junctiontransistors (BJTs) [1], [25], [27]. In order to apply the

Manuscript received March 26, 2001.Manuscript revised July 26, 2001.Final manuscript received September 25, 2001.

The authors are with the Faculty of Integrated Culturesand Humanities, University of East Asia, Shimonoseki-shi,751-8503 Japan.

The author is with the Faculty of Science and Engi-neering, Chuo University, Tokyo, 112-8551 Japan.a) E-mail: [email protected]

homotopy methods to practical circuit simulation, it isvery important to develop a practical method which canbe implemented on the existing simulators containingsuch built-in models. Recently, such practical studieshave been remarkably progressed mainly by the authors[12], [15], [16], [18], [21][26]. By these studies, the ho-motopy methods can now solve bipolar analog circuitswith more than 16000 elements, which belong to a classof the largest-scale circuits available with the currentbipolar analog LSI thechnology. These studies havegreatly contributed to reducing the LSI developmentperiod and increasing the performance or functionalityof consumer products while reducing the cost.

In the homotopy methods, the Newton homotopymethod and the xed-point (FP) homotopy method(or its variants) are well-known. These methods areproved to be globally convergent for MN equations[16], [18], [22]. In particular, the FP homotopy methodhas an excellent property that a random choice of ini-tial point gives a bifurcation free homotopy path withprobability-one [9].

There are several approaches to the implementionof the homotopy methods using the existing simulators.Regarding the Newton homotopy method, it can be im-plemented on the SPICE-like simulators with a smallmodication (about 1000 steps of statements) to the ex-isting programs [15], [16], [18]. By the approach, modi-ed nodal equations with more than 39000 variables aresolved eciently with global convergence [16], [18], [24][26]. The approach using solution-tracing circuits withSPICE is the easiest way [12], [23], [26]. This approachrequires no modication to the existing programs sothat it can be widely used by SPICE users. However,with regard to the FP homotopy method or its variants,although it is an excellent method, the implementationhas been considered to be not so easy as that of theNewton homotopy method.

In this paper, an implementation technique for theFP homotopy method is presented. Our technique canbe implemented as easily as that for the Newton homo-topy method. Therefore, by using our technique, theFP homotopy method can be easily and widely han-dled by many circuit designers. In Sect. 2 as prelimi-

The algorithms proposed in [15], [16], [18] were imple-mented on SPICE in SANYO Electric Co., Ltd. and theyare working very well in circuit simulation.

INOUE et al.: A PRACTICAL APPROACH FOR THE FIXED-POINT HOMOTOPY METHOD223

naries, the FP homotopy method for MN equations andthe predictor-corrector curve-tracing algorithm are re-viewed. In Sect. 3, a solution-tracing circuit for the FPhomotopy method is proposed. Moreover, an extendedversion of the FP homotopy method is proposed. InSect. 4, the proposed method is proved to be globallyconvergent. Numerical examples are shown in Sect. 5.It is further shown in Sect. 6 that the FP homotopymethod can be easily implemented directly on the ex-isting simulators on the basis of our technique.

2. FP Homotopy Method for MN Equations

In this section, we review the FP homotopy method forsolving systems of equations of the form

f(x) = 0,f() : Rn Rn. (1)In the MN equation, Eq. (1) is rewritten as follows [16],[18], [22]:

fg(v, i) , Dgg(DgTv) +DEi+ J = 0 (2a)

fE(v) , DETv E = 0 (2b)

where f = (fg,fE)T , fg : Rn RN , fE : RN

RM , x = (v, i) Rn, and n = N +M . The variablevector vRN denotes the node voltages to the datumnode and the variable vector iRM denotes the branchcurrents of the independent voltage sources. Also, thecontinuous function g : RK RK is a VCCS (voltage-controlled current source) type. In addition, Dg is anNK reduced incidence matrix for the g branches andDE is an NM reduced incidence matrix for the inde-pendent voltage source branches. Moreover, J RN isthe current vector of the independent current sourcesand E RM is the voltage vector of the independentvoltage sources.

Consider solving the MN equation (1) or (2) bythe FP homotopy method [4], [19], [21], [22]. The FPhomotopy equation is written as follows:

h(x, t) , tf(x) + (1 t)A(x x0) = 0 (3)where t is the homotopy parameter, A is some nnmatrix, and h is called the FP homotopy. Equation (3)is constructed so that it has a trivial solution x0 whent = 0 and that it is equal to Eq. (1) when t = 1. The setof solutions to Eq. (3) is called the solution curves (orthe homotopy paths) of the homotopy equation. In thispaper, we assume that the solution set consists of one-dimensional dierentiable curves only. It is known thatthis assumption holds almost always under mild con-ditions if the FP homotopy is used [4], [9], [14]. Thenthe solution curve of Eq. (3) is traced starting from theknown solution (x0, 0) at t = 0. If the solution curvereaches the t = 1 hyperplane at (x, 1), then a solutionx is obtained. Note that the rst term tf(x) in Eq. (3)

is a function of t. That is the main reason why the FPhomotopy method cannot be easily implemented to theexisting programs. If we try to do it, large modica-tions have to be made to the existing subroutines whichconstruct the Jacobian matrix and the excitation vec-tor in the existing programs. It is almost impossible ina practical situation.

It is shown that the FP homotopy method is glob-ally convergent for MN equations if we use, as A,

A =[GFP1N 00 RFP1M

](4)

where 1N and 1M are N N and M M identity ma-trices, respectively [22]. Also, GFP and RFP are scalarconstants [21]. Note thatA is a regular diagonal matrixin this case. It can be modied for various applications.However, the global convergence property is not alwaysguaranteed.

The solution curve of the homotopy equation canbe traced by the BDF (Backward Dierentiation For-mula) curve-tracing algorithm [7]. In the BDF al-gorithm, the solution curve is parameterized by itsarc-length s, and the following system of dierential-algebraic equations

h(x(s), t(s)) = 0 (5a)

iI

(xi(s))2 + t(s)2 = 1 (5b)

is solved, where I {1, 2, , n}, (|I| = m, m n),denotes a subset of indices of the components of x,x = dx/ds and t = dt/ds. Equation (5a) is the ho-motopy equation and Eq. (5b) describes the relation-ship between the arc-length and the components of thesolution curve projected into an (m + 1)-dimensionalEuclidean space [12], [16]. Equation (5) is solved byapplying the backward dierentiation formula with thepredictor-corrector algorithm [28]. Note that this anal-ysis method is similar to the Gear (or BDF) methodfor the transient analysis in circuit simulators [1], [28].

3. Solution-Tracing Circuit

In this section, we propose a solution-tracing circuit forthe FP homotopy method and some extension of theFP homotopy method. In our approach proposed inthis paper, Eq. (5) is solved by the following two steps.

Step 1: Transform some part of Eq. (5) into some cir-cuit, i.e., the circuit describes the part of Eq. (5).

Step 2: Execute the transient analysis, by using aSPICE-like simulator, for the circuit (to be solved)together with the circuit in Step 1.(1)If necessary (as discussed later), compute theinitial transient solution xI at t = t0, 0 < t0 1,before the transient analysis.


Fig. 1 The circuit describing Eq. (5b).

Fig. 2 The circuit describing part of homotopy equation (6).

Otherwise, set xI = x0.(2) Execute the transient analysis using xI and t0

as the initial values.

Note that the arc-length s is regarded as the time inthe transient analysis. Also, by utilizing some com-mand provided by the SPICE-like simulators, the tran-sient analysis automatically stops when t reaches 1.In Step 2, we begin to trace the solution curve ofEq. (5) starting from (xI , t0), where xI is the solutionto h(x, t0) = 0 and close to x0, and t0 is a positive valuesuciently close to zero. By utilizing t0 > 0, we canapply the solution-tracing circuit technique to the FPhomotopy method. Since the rst term in Eq. (3) doesnot completely disappear at t = t0, the initial solutionxI at t = t0 is not generally equal to x0. However, itis very easy to compute the initial transient solution,since, in Eq. (3), the second term becomes major att = t0 (t0 0) and A(x x0) is linear.

If GFP and RFP are suciently large and t0 is closeenough to 0, then xI becomes very close to x0, i.e.,xI x0. For instance, if x0 = 0 in practical circuits,then we may regard xI as virtually equal to 0. Thusthe computation of the initial transient solution is notalways necessary. In this way, the solution curves ofhomotopy equations can be traced by performing theSPICE transient analysis. Therefore, no modicationsto the existing programs are necessary.

Consider transforming Eq. (5) into circuits includ-ing some special circuit. Dividing Eq. (3) by t > 0 [19]and considering Eqs. (2) and (4), we obtain

Dgg(DgTv)+DEi+J+(1t)t

GFP (vv0)=0 (6a)

DETv E (1 t)

tRFP (i i0) = 0. (6b)

Note that the last terms in the left-hand side (LHS)of Eq. (6) correspond to the FP homotopy method andthe other terms remain in the MN equation (2) of theoriginal circuit to be solved. Therefore, in order totransform Eq. (5) into circuits, we need an additionalcircuit which describes Eq. (5b) and the last terms inthe LHS of Eq. (6). The circuit is called the solution-tracing circuit for the FP homotopy method.

The solution-tracing circuit we propose is shownin Figs. 1 and 2. Equation (5b) can be transformedinto the circuit shown in Fig. 1, and the last terms inthe LHS of Eq. (6) can be transformed into the circuitshown in Fig. 2.

The circuit in Fig. 1 is constructed in a mannersimilar to that for the Newton homotopy method [12].In the gure, vi and ij are obtained as node voltagesin the dierential circuit, where vi and ij are the volt-age at some node i I and the current through somevoltage source j I in the original circuit to be solved,respectively. Applying KCL to the node in the square-and-sum circuit, we can see that it describes Eq. (5b),from which t is obtained as the node voltage in thesquare-and-sum circuit. Then t is integrated in the in-tegration circuit to obtain t as the node voltage. Theconductance G is required only to avoid the topologicalrestriction of the circuit simulator used. A negligiblesmall value G 0 is used so that it is regarded asvirtually open.

In Fig. 2, is obtained as the node voltage andas the current through the voltage source in the -generation circuit, where = (1 t)/t. Using this ,we make, as shown in the gure, the VCCS describingthe last term in the LHS of Eq. (6a) and the CCVS(current-controlled voltage source) describing the lastterm in the LHS of Eq. (6b). The terminals of theVCCS are connected to node i and the datum node


in the original circuit, respectively. The CCVS is con-nected in series with voltage source j in the originalcircuit. In this way, we can make VCCSs and CCVSsfor all i {1, 2, , N} and j {1, 2, ,M}, respec-tively.

In our approach proposed in this paper, A is ex-tended in a more general form

A =

[DgGFP Dg

T0

0 RFP1M

](7)

where Dg is some N K reduced incidence matrixfor appropriate K braches in the original circuit to besolved. Furthermore, Eq. (5b) is also extended as

(Bx)T (Bx) + t2 = 1 (8)

where B is some (K +M)n matrix represented as

B =

[Dg

T0

0 1M

]. (9)

It is apparent that DgTv denotes the branch volt-

age vector for the K branches. Choosing a branch inthis way is called setting a probe to the branch [12],[16]. It is natural to set probes to appropriate nonlinearbranches, for instance, the base-collector (BC) and/orthe base-emitter (BE) branch(es) of each BJT in theoriginal circuit [16]. In this case, A is not always regu-lar. When Eqs. (4) and (5b) with m = n are used, it iscalled the diagonal probes. In this case, A is regular.It is trivial to modify the solution-tracing circuit intothat for Eqs. (7)(9). Notice that A has to be regularif we start at t = 0. In our approach, since we start att > 0, A need not be regular as shown in Sect. 4. Noticealso that the BJTs are almost regarded as diodes whent = t0 if we utilize the BC probes or the BE probes forall the BJTs. Namely, if a conductance 0GFP largeenough is connected in parallel with the base-collectoror the base-emitter terminals of some BJT in practicalcircuits, then the connections can be almost regardedas the diode connections [16], where 0 = (1 t0)/t0.This implies that it is also very easy to compute theinitial transient solution. Moreover, the diodes satisfya favorable property discussed in the next section.

4. Global Convergence of the FP HomotopyMethod

In this section, we show a theorem which guarantees theglobal convergence of the FP homotopy method pro-posed in the previous section, where we utilize t0 > 0andA not necessarily regular. In order to make the dis-cussion simple and more general, we use the fundamen-tal modied cut-set (MC) equation [16], [22] instead ofthe MN equation. The reason for considering the fun-damental MC equation is that it is equivalent to the

MN equation, but has a more favorable form for theproof. The MC equation is an extension of the cut-setequation and a more general formulation including theMN equation. Therefore, to discuss the global conver-gence of the FP homotopy method for h(x, t) denedby the MN equation, it is sucient to discuss that de-ned by the MC equation [16], [22].

Consider a tree which consists of all independentvoltage source branches and some of the g branches.Let the voltage vector for the tree branches be u RNand the variable vector be

x =[ui

]. (10)

Then we can formulate the MC equation of the formsimilar to Eq. (2). By considering Dg and DE as thefundamental cut-set matrices and J as the cut-set cur-rent sources, Eq. (2) can be regarded as the fundamen-tal MC equation. Furthermore, by considering Dg asthe fundamental cut-set matrix for the K branches, thesame discussion applies to Eqs. (7)(9). In the funda-mental MC equation, DE is represented by the follow-ing simple form:

DE =[1M0

]. (11)

Also, by separating the rst M rows and the remainingrows and by separating the rst N M columns andthe remaining columns, Dg is represented as

Dg =[DgEDgg

]=[

0 DglE1NM Dglg

]. (12)

In addition, u and J are represented as

u =[uEug

](13)

J =[JEJg

](14)

where uE RM , ug RNM , JE RM , and Jg RNM .

Moreover, in order to make the discussion moregeneral, we introduce a positive semi-denite diagonalmatrix GFP = diag(GFPj), j = 1, 2, ...,K, whose K di-agonal elements are GFP s > 0 and the others are zeros.Then we may rewrite the rst diagonal block in Eq. (7)as DgGFP Dg

T=DgGFPDTg , and Eq. (7) becomes

A =[DgGFPD

Tg 0

0 RFP1M

]. (15)

In this case, GFPDgTu = GFPvb represents the currentvector of the K branches of GFP whose branch voltagevector is vb.

When we use the fundamental MC equation and


Eq. (15) as A, the FP homotopy h = (hg,hE)T ex-tended in Sect. 3 is written as follows:

hg(x, t) = tDgg(DTg u) + tDEi+ tJ

+ (1t)DgGFPDTg (uu0) (16a)

hE(uE , i, t) = tuE tE (1t)RFP (ii0) (16b)where hg : Rn+1 RN , hE : R2M+1 RM , and x0 =(u0, i0). Furthermore, from Eqs. (10)(14), hg(x, t) inEq. (16a) is rewritten as follows:

hgE(x, t) = tDgEg(DTg u) + ti+ tJE

+ (1 t)DgEGFPDTg (u u0) (17a)

hgg(u, t) = tDggg(DTg u) + tJg

+ (1 t)DggGFPDTg (u u0) (17b)where hgE : Rn+1 RM and hgg : RN+1 RNM .Note that, in this section, Dg and DE are the funda-mental cut-set matrices, and J is the cut-set currentsources.

As the property that a fairy general class of resis-tive elements satises, we dene the following terminol-ogy [6].

Denition: A continuous function g : RK RK issaid to be uniformly passive on vpb if there exists a > 0such that (vb vpb )T

(g(vb) g(vpb )

) vb vpb 2 forall vb RK . Resistive elements such as BJTs, diodes (includingBJTs in the diode connections), tunnel diodes, andpositive linear resistors are known to be uniformly pas-sive on certain points [6]. In particular, the diodes andthe positive linear resistors satisfy the above propertyon any point. Thus the uniform passivity is a very mildcondition. In this paper, we assume that g is uniformlypassive on certain points and that

(g(vb) + 0GFPvb

)is uniformly passive on vIb = D

Tg u

I , where 0 =(1 t0)/t0 and xI = (uI , iI). If we connect a su-ciently large conductance 0GFP in parallel with eachbranch of appropriate nonlinear branches of g, we mayassume the uniform passivity on vIb . Then the followingtheorem holds.

Theorem 1: Consider the FP homotopy given byEq. (16). Assume that g is Lipschitz continuous [3] andthere exists a vpb such that g is uniformly passive onvpb . Assume also that

(g(vb) + 0GFPvb

)is uniformly

passive on vIb . Then for any initial point xI Rn

the solution curve of h(x, t) = 0 starting from (xI , t0)reaches t = 1.

Since the proof of this theorem is very long, it is shownin the Appendix. As previously pointed out, the MNequation is equivalent to the fundamental MC equa-tion. Therefore, Theorem 1 can be applied also to the

FP homotopy dened by the MN equation in Sect. 3.By this theorem, the global convergence of the FP ho-motopy method in Sect. 3, starting at t > 0 and usingthe homotopy given by Eqs. (3), (4), (6) and (7), is the-oretically guaranteed.

5. Numerical Examples

We show here three examples. In these examples, theinitial conditions used are t0 = 102, 0 = 99 andx0 = (v0, i0) = 0 unless otherwise specied.Example 1: Consider the negative resistance (NRES)circuit shown in Fig. 3 [5]. The circuit has threeoperating-points. In this circuit, SPICE default param-eters are used for the BJT model [1]. Figure 4 shows thesolution curves traced by the proposed technique, wherethe DC operating-point(s) is (are) found at t = 1. Inthe gure, v(i) represents the voltage of node i. Marksare plotted on the curves at every two time steps, whichindicate the number of steps and the step-length usedfor tracing the solution curve by the transient analy-sis. Figure 4(a) shows the solution curves of the NREScircuit with the diagonal probes. Figures 4(b), (c) and(d) show the curves with the BC probes, with the BEprobes using the initial condition v0BE=0.7, and withthe BC and BE probes using v0BE=0.7, respectively.Figures 4(a) and (c) show dierent paths and reachthe identical DC operating-point. Figures 4(b) and (d)show similar paths and reach the three DC operating-points in the same order. The BE probes shown inFig. 4(c) are most ecient in this example, where 37steps are used for nding the DC operating-point. Asshown in this example, our technique can be easily im-plemented by using the existing simulator.Example 2: We next consider the diode-transistor(DITR) circuit shown in Fig. 5 [22]. The circuit isknown to have a very sti solution curve and three DCoperating-points. Figure 6 shows the curves of the cir-cuit with the diagonal probes. Our technique nds allthe three DC operating-points successfully.Example 3: We nally consider UA741 OP AMP [1].The circuit is a high gain operational amplier and con-sists of 29 elements including 22 BJTs. Part of this cir-cuit is used as a dicult example in the reference [14].As was done in the reference the model parameter VA(Early voltage) is set to zero for all the BJTs. Thus

Fig. 3 Negative resistance circuit.


Fig. 4 Solution curves of NRES circuit.

the open-loop gain of the circuit becomes very high.Figure 7 shows the curves of the circuit with the BEprobes using v0BE=0.7. The number of steps used fornding the DC operating-point is 39. From our experi-ence in applying our technique to many other circuits,we found that the BE probes are more eective thanthe other probes for a class of practical circuits (such asUA741) having the topological loop composed of volt-age source branches and BJT branches only. When weuse the diagonal probes the number of steps is 587, theBC probes 187, and the BC and BE probes 179, re-spectively. Furthermore, the BE probes are also moreeective than the Newton homotopy method for that

Fig. 5 Diode-transistor circuit.

Fig. 6 Solution curves of DITR circuit.

Fig. 7 Solution curves of UA741.

class of circuits. When we apply the Newton homotopymethod to UA741, the number of steps is 111.

6. Direct Implementation on SPICE-Like Sim-ulators

As shown above, by utilizing the solution-tracing cir-cuit technique, the FP homotopy method can be im-plemented with no modications to the existing sim-ulators. In this section, we further show that the FPhomotopy method can be easily implemented directlyon the existing simulators.

As discussed in Sect. 3, Eq. (6) is composed of theoriginal circuit to be solved and the other additionalterms. Hence, if we replace Eq. (5a) with Eq. (6), thelinearized equation of Eq. (5), used in the BDF curve-tracing algorithm, can be written as


[(Jo 00 0

)+ JH

][xt

]=[bo0

]+ bH (18)

where Jo and bo are the Jacobian matrix and the ex-citation vector of the original circuit, respectively. JHand bH are the Jacobian matrix and the excitation vec-tor of Eq. (5b) and the last terms in the LHS of Eq. (6),respectively.

Equation (18) shows that the Jacobian matrix andthe excitation vector of the linearized equation canbe constructed by only adding JH and bH to thoseof original circuit to be solved. This implies that weneed no modications to the existing programs butonly an additional subroutine which constructs JH andbH . Therefore, by using our technique, the FP homo-topy method can be easily implemented directly on theSPICE-like simulators through a small modication tothe existing programs.

7. Conclusions

In this paper, a practical implementation technique forthe FP homotopy method has been presented for nd-ing DC operating-points of nonlinear circuits. We haveproposed a solution-tracing circuit for the FP homo-topy method and an extended version of the FP ho-motopy method. Moreover, we have proved that theproposed method is globally convergent. Furthermore,we have shown numerical examples to demonstrate theeectiveness of the proposed technique. We have alsoshown that the proposed FP homotopy method can beeasily implemented directly on the existing simulators.

Acknowledgment

The authors would like to thank Professor HiroakiYamada of University of East Asia for his encourage-ment. They are also grateful to Emeritus ProfessorKazuo Horiuch of Waseda University, also Professorof University of East Asia, and Professor Akio Ushidaof Tokushima University for their fruitful discussions.Moreover, they wish to thank Professor Eigo Kaji ofUniversity of East Asia for his assistance and helpfuldiscussions.

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[23] A. Ushida, Y. Yamagami, I. Kinouchi, Y. Nishio, and Y.Inoue, An ecient algorithm for nding multiple DC solu-tions based on SPICE oriented Newton homotopy method,Proc. 2001 IEEE Int. Symp. Circuits & Syst., pp.V-447V-450, May 2001.

[24] K. Yamamura, Pathways from theory to practice, J.IEICE, vol.81, no.1, pp.3336, Jan. 1998.

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[26] Y. Inoue, Large-scale circuit analysis using homotopy con-tinuation methodsA pathway from theory to practice inKaruizawa, Proc. 13th Workshop on Circuits & Syst. inKaruizawa, pp.173180, April 2000.

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[28] R.K. Brayton, F.G. Gustavson, and G.H. Hachtel, A newecient algorithm for solving dierential-algebraic systemsusing implicit backward dierentiation formulas, Proc.IEEE, vol.60, no.1, pp.98108, Jan. 1972.

Appendix: Proof of Theorem 1

To prove Theorem 1, it is sucient to prove that[4],[16],[22]

1. for all x {x Rn|x = xI}, h(x, t0) = 0 holds,and

2. there exists an r > 0 such that for all x {x Rn| x r} and t [t0, 1), h(x, t) = 0 holds.

We rst prove the above item 1 in the following lemma.

Lemma 1: For all x {x Rn|x = xI}, h(x, t0) =0.

Proof: Let 0 = (1 t0)/t0 and xI = (uI , iI).Since h(xI , t0) = 0, h(x, t0) = h(x, t0) h(xI , t0)holds. Multiplying h(x, t) =

(hg(x, t),hE(uE , i, t)

)Tin Eq. (16) at t = t0 by[

(u uI)(i iI)

]T/t0

and considering uTDEi = uTE i = iTuE , we obtain[

(u uI)(i iI)

]Th(x, t0)/t0

=[(u uI)(i iI)

]T{h(x, t0) h(xI , t0)}/t0

= (u uI)T {Dgg(DTg u) +DEi+ J+ 0DgGFPDTg (u u0)Dgg(DTg uI)DEiI J 0DgGFPDTg (uI u0)} (i iI)T {uE E 0RFP (i i0) uIE +E + 0RFP (iI i0)}

= {DTg (u uI)}T[{g(DTg u) + 0GFPDTg u}

{g(DTg uI) + 0GFPDTg uI}]

+ (i iI)T 0RFP (i iI). (A 1)

Since(g(vb) + 0GFPvb

)is uniformly passive on vIb =

DTg uI , there exists a gG > 0 such that

{DTg (u uI)}T[{g(DTg u) + 0GFPDTg u}

{g(DTg uI) + 0GFPDTg uI}]

gGDTg (u uI)2.From Eqs. (12) and (13), DTg u ug holds. There-fore, Eq. (A 1) becomes[

(u uI)(i iI)

]Th(x, t0)/t0

gGDTg (u uI)2 + 0RFP i iI2 gGug uIg 2 + 0RFP i iI2.

Hence, for all ug and i such that ug = uIg or i = iI ,[(u uI)(i iI)

]Th(x, t0)/t0 > 0

holds, which implies h(x, t0) = 0. So far, we have dis-cussed the components ug and i of the variable vectorx. Consider also the other component uE . If i = iI ,then clearly h(x, t0) = 0 holds. Therefore, we assumethat i = iI . Then from Eq. (16b),

hE(uE , i, t0) = hE(uE , i, t0) hE(uIE , iI, t0)= t0uE t0E (1 t0)RFP (iI i0)

t0uIE + t0E + (1 t0)RFP (iI i0)= t0(uE uIE).

Therefore, for all (uE , i) {(uE , i) R2M |uE =uIE , i = i

I}, hE(uE , i, t0) = 0 holds, which impliesh(x, t0) = 0. Hence, for all ug, i, and uE suchthat ug = uIg or i = iI or uE = uIE , h(x, t0) = 0holds, which implies for all x {x Rn|x = xI},h(x, t0) = 0. Next, we prove the above item 2 in the following lem-mas. First of all, we have the following lemma as pre-liminaries for the subsequent lemmas.

Lemma 2: There exist a > 0, a 1 > 0 and a 2 > 0such that for all x Rn and t [t0, 1),

(u u0)Thg(x, t)/t (DTg u vpb 1)2 2

+ (uE u0E)T (i+ JE). (A 2)

Proof: Let = (1 t)/t. Multiplying hg(x, t) inEq. (16a) by (u u0)T /t, we obtain

(u u0)Thg(x, t)/t= {DTg (u u0)}Tg(DTg u)+ (u u0)TDEi+ (u u0)TJ+ {DTg (u u0)}TGFPDTg (u u0).


Since (0, 0], GFP is positive semi-denite,uTDE = uTE , and u

TJ = uTEJE + uTg Jg, we obtain

(uu0)Thg(x, t)/t {DTg (uu0)}Tg(DTg u)+(uEu0E)T (i+JE)

+ (ug u0g )TJg. (A 3)From the assumption that g is Lipschitz continuous anduniformly passive on vpb , there exist an L > 0 and a > 0 such that for all u RN

g(DTg u) g(vpb ) LDTg u vpb

(DTg u vpb )T {g(DTg u) g(vpb )} DTg u vpb 2.Also, DTg u ug holds. Let 3 = g(vpb ) and4 = DTg u0 vpb . Hence, we obtain

{DTg (u u0)}Tg(DTg u)= (DTg uvpb DTg u0+vpb )T

{g(DTg u)g(vpb )+g(vpb )}= (DTg u vpb )T {g(DTg u) g(vpb )}

+ (DTg u vpb )Tg(vpb ) (DTg u0 vpb )T {g(DTg u) g(vpb )} (DTg u0 vpb )Tg(vpb )

DTg u vpb 2 (3 + 4L)DTg u vpb 34. (A 4)

Substituting Eq. (A 4) into Eq. (A 3) and consideringug u0g DTg uDTg u0 + vpb vpb

DTg u vpb + 4we obtain

(u u0)Thg(x, t)/t DTg u vpb 2

(3 + 4L+ Jg)DTg u vpb 34 4Jg+ (uE u0E)T (i+ JE)

= {DTg u vpb (3 + 4L+ Jg)/(2)}2 (3 + 4L+ Jg)2/(4) 4(3 + Jg)+ (uE u0E)T (i+ JE).

Let 1 = (3 + 4L + Jg)/(2) and 2 = (3 +4L+ Jg)2/(4) + 4(3 + Jg). Therefore, we ob-tain Eq. (A 2). In order to prove the above item 2, we divide the dis-cussion into three steps by dividing the variable vectorx into the three components, uE , ug, and i. In the rststep, regarding uE , we obtain the following lemma.

Lemma 3: There exists an r1 > 0 such that for alluE {uE RM |uE r1} and t [t0, 1), h(x, t) =0.

Proof: If hE(uE , i, t) = 0, then clearly h(x, t) = 0holds. Therefore, we assume that hE(uE , i, t) = 0.Then from Eq. (16b) we obtain

i = 1R1FP (uE E) + i0. (A 5)Substituting Eq. (A 5) into Eq. (A 2), we obtain

(u u0)Thg(x, t)/t {DTg u vpb 1}2 2

+ (uE u0E)T {1R1FP (uE E)+ i0 + JE}. (A 6)

The last term in Eq. (A 6) becomes(uE u0E)T {1R1FP (uE E) + i0 + JE}

= 1R1FP {uTEuE (u0E +E)TuE + u0ETE}

+ uTE (i0 + JE) u0ET (i0 + JE)

1R1FP {uE2 (u0E +E+ RFP i0 + JE)uE (E+ RFP i0 + JE)u0E}. (A 7)

Substituting Eq. (A 7) into Eq. (A 6) and consideringits rst term being positive, we obtain

(u u0)Thg(x, t)/t 2 + 1R1FP {uE2

(u0E +E+ RFP i0 + JE)uE (E+ RFP i0 + JE)u0E}

= 1R1FP {uE2 (u0E +E+ RFP i0 + JE)uE (E+ RFP i0 + JE)u0E RFP2}. (A 8)

Since t [t0, 1) and (0, 0] (for instance, t0 = 102and 0 = 99 ), there exist a 5 > 0 and a 6 > 0 suchthat

u0E +E+ RFP i0 + JE 5

(E+ RFP i0 + JE)u0E+ RFP2 6.Therefore, Eq. (A 8) becomes

(u u0)Thg(x, t)/t 1R1FP (uE2 5uE 6).

Since t > 0 and RFP > 0, there exists an r1 > 0 suchthat uE r1 implies (uu0)Thg(x, t)/t > 0. Hence,for all uE {uE RM |uE r1} and t [t0, 1),hg(x, t) = 0 holds, which implies h(x, t) = 0. In the second step, with regard to ug, we have the fol-lowing lemma.

Lemma 4: There exists an r2 > 0 such that for allu {(uE ,ug) RN |uE < r1, ug r2} and t [t0, 1), h(x, t) = 0.


Proof: Multiplying hgg(u, t) in Eq. (17b) by (ug u0g )

T /t, we obtain

(ug u0g )Thgg(u, t)/t= {DTgg(ug u0g )}Tg(DTg u) + (ug u0g )TJg

+ {DTgg(ugu0g )}TGFPDTg (uu0).

From the relation DTggug =DTg uDTgEuE

(ug u0g )Thgg(u, t)/t= {DTg (u u0)}Tg(DTg u) + (ug u0g )TJg

{DTgE(uE u0E)}Tg(DTg u)+ {DTg (u u0)}TGFPDTg (u u0) {DTgE(uE u0E)}TGFPDTg (u u0)

= {DTg (u u0)}Tg(DTg u) + (ug u0g )TJg {DTgE(uE u0E)}T {g(DTg u) g(0)} {DTgE(uE u0E)}Tg(0)+ {DTg (u u0)}TGFPDTg (u u0) {DTgE(uEu0E)}TGFPDTg (uu0).

Since GFP is positive semi-denite and GFPvb GFP vb

(ug u0g )Thgg(u, t)/t {DTg (u u0)}Tg(DTg u) + (ug u0g )TJg

{DTgE(uE u0E)}T {g(DTg u) g(0)} {DTgE(uE u0E)}Tg(0) {DTgE(uE u0E)}TGFPDTg (u u0)

{DTg (u u0)}Tg(DTg u) ugJg u0g Jg DTgE(uE u0E)g(DTg u) g(0) DTgE(uE u0E)g(0) DTgE(uEu0E)GFP DTg (uu0).

Since g is Lipschizt continuous

(ug u0g )Thgg(u, t)/t {DTg (u u0)}Tg(DTg u)

ugJg u0g Jg DTgE(uE u0E) L DTg u DTgE(uE u0E)g(0) DTgE(uEu0E)GFP DTg (uu0).

From Eq. (A 4),

(ug u0g )Thgg(u, t)/t DTg u vpb 2

(3 + 4L)DTg u vpb 34 ugJg u0g Jg DTgE(uE u0E) L DTg u DTgE(uE u0E)g(0) DTgE(uEu0E)GFP DTg (uu0).

(A 9)Since DTg u ug, we obtain

DTg u vpb 2 ug2 2DTg uvpb + vpb 2. (A 10)

Substituting Eq. (A 10) into Eq. (A 9) and arrangingit, we obtain

(ug u0g )Thgg(u, t)/t ug2 {3 + 4L+ DTgE(uE u0E)(L+ GFP )+ 2vpb } Dgu ugJg 34 u0g Jg (3 + 4L)vpb DTgE(uE u0E)GFP DTg u0 DTgE(uE u0E)g(0)+ vpb 2. (A 11)

Since (0, 0] and uE < r1, there exist a 7 > 0, a8 > 0 and a 9 > 0, such that

{3 + 4L+ DTgE(uE u0E)(L+ GFP )+ 2vpb } Dg 7

Jg = 8

34 + u0g Jg+ (3 + 4L)vpb + DTgE(uE u0E)GFP DTg u0+ DTgE(uE u0E)g(0) vpb 2 9.

Hence, Eq. (A 11) becomes(ug u0g )Thgg(u, t)/t

ug2 7u 8ug 9.From u ug+ uE < ug+ r1, we obtain

(ug u0g )Thgg(u, t)/t ug2 (7 + 8)ug (r17 + 9).

Therefore, there exists an r2 > 0 such that ug r2implies (ug u0g )Thgg(u, t)/t > 0. Hence, for all u {(uE ,ug) RN |uE < r1, ug r2} and t [t0, 1),hgg(u, t) = 0 holds, which implies h(x, t) = 0. From Lemmas 3 and 4, we obtain the following lemma.


Lemma 5: There exists an r3 > 0 such that for allu {u RN |u r3} and t [t0, 1), h(x, t) = 0.

Proof: Set r3 = r1 + r2. If u r3, then eitheruE r1 holds or uE < r1 and ug r2 hold.Hence, from Lemmas 3 and 4 the conclusion follows.

In the third step, with respect to i, we prove the fol-lowing lemma.

Lemma 6: There exists an r4 > 0 such that for allx {(u, i) Rn|u < r3, i r4} and t [t0, 1),h(x, t) = 0. Proof: Multiplying hgE(x, t) in Eq. (17a) by iT /t, weobtain

iThgE(x, t)/t

= i2 + iT {DgEg(DTg u) + JE+ DgEGFPDTg (u u0)}

i2 iDgEg(DTg u) + JE+ DgEGFPDTg (u u0).

Since u < r3 and (0, 0], there exists a 10 > 0such that

DgEg(DTg u) + JE+ DgEGFPDTg (u u0) 10.

Hence, we obtain

iThgE(x, t)/t i2 10i.Therefore, there exists an r4 > 0 such that i r4implies iThgE(x, t)/t > 0. Hence, for all x {(u, i) Rn|u < r3, i r4} and t [t0, 1), hgE(x, t) = 0holds, which implies h(x, t) = 0. From Lemmas 5 and 6, we have the following lemma.

Lemma 7: There exists an r > 0 such that for allx {x Rn|x r} and t [t0, 1), h(x, t) = 0. Proof: Set r = r3+r4. If x r, then either u r3 holds or u < r3 and i r4 hold. Hence, fromLemmas 5 and 6, the conclusion follows.

Therefore, from Lemmas 1 and 7, we have proved The-orem 1.

Yasuaki Inoue was born in Niigata,Japan, on September 6, 1945. He re-ceived a diploma from the Departmentof Electronics, Nagaoka Technical HighSchool, Niigata, Japan, in 1964 and theD.E. degree in electronics and communi-cation engineering from Waseda Univer-sity, Tokyo, Japan, in 1996. From 1964to 2000, he was with Sanyo Electric Co.,Ltd., Gunma, Japan, where he was en-gaged in research and development in ana-

log integrated circuits and analog/digital CAD systems. In SanyoSemiconductor Company, he was General Manager of the CADEngineering Department from 1993 to 1998 and the Memory De-velopment Department from 1998 to 2000. He holds over fortypatents. Since 2000, he has been a Professor with the Depart-ment of Integrated Cultures and Humanities, also a Professorwith the Graduate School of Integrated Science and Art (Mul-timedia Masters Course), University of East Asia, Shimonoseki,Japan. His research interests include numerical analysis of non-linear circuits and systems, analog circuits, and CAD systems.He was an Associate Editor of the IEEE Transactions on Cir-cuits and Systems Part II from 1997 to 1999. He received theIshikawa Award from the Union of Japanese Scientists and En-gineers in 1988 and the Distinguished Service Award from theScience and Technology Agency, the Japanese Government, in1999. Dr. Inoue is a member of IEEE.

Saeko Kusanobu was born inOkayama, Japan. She received the B.S.degree in mathematics from Shimane Uni-versity, Matsue, Japan, in 1994 and theM.S. and D.E. degrees in information sci-ence from Hiroshima University, Hiro-shima, Japan, in 1996 and 1999, respec-tively. Since 2000, she has been an As-sistant Professor with the Department ofIntegrated Cultures and Humanities, Uni-versity of East Asia, Shimonoseki, Japan.

Her research interests include numerical analysis of nonlinear cir-cuits and systems, and the applications of the statistical tech-nique to image analysis. Dr. Kusanobu is a member of theJapanese Society of Computational Statistics and the Japan Sta-tistical Society.


Kiyotaka Yamamura was born inTokyo, Japan, on December 19, 1959. Hereceived the B.E., M.E., and D.E. degrees,all in electronics and communication engi-neering, from Waseda University, Tokyo,Japan, in 1982, 1984, and 1987, respec-tively. From 1985 to 1987, he was a Re-search Assistant with the School of Sci-ence and Engineering, Waseda University.From 1988 to 1998, he was an AssociateProfessor with the Department of Com-

puter Science, Gunma University, Kiryu, Japan. Since 1999, hehas been a Professor with the Department of Electrical, Elec-tronic, and Communication Engineering, Chuo University, To-kyo, Japan. His research interests include numerical analysis ofnonlinear circuits and systems. From 1994 to 1998, he was anAssociate Editor of the IEICE Transactions on Fundamentals.He received the Niwa Memorial Award from the Niwa Memo-rial Foundation in 1986, the Shinohara Memorial Young En-gineer Award from IEICE in 1986, the Inoue Research Awardfor Young Scientists from the Inoue Foundation in 1989, theTelecom-System Technology Awards from the Telecommunica-tions Advancement Foundation in 1990 and 1999, the Best PaperAward from IEICE in 1999, the IBM Japan Science Prize in 1999,and the Ohm Technology Award from the Promotion Foundationfor Electrical Science and Engineering in 2000. Dr. Yamamura isa Senior Member of IEEE.

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