Numerical Interval Simulation :Combined Qualitative and Quantitative Simulation to

Bound Behaviors of Non-Monotonic Systems*

1 Introduction

Marcos Vescovi, Adam Farquhar, and Yumi Iwasaki

Knowledge Systems Laboratory Stanford University701 Welch Road, Bldg . #C

CA94304 Palo Alto USAphone +1415 8519427 fax +1415 725 5850

vescovi, axf, iwasaki@ksl .stanford .edu

Abstract : Models of complex physical systemsoften cannot be defined precisely, either becauseof lack of knowledge or because the systemparameters change over time according tounknown phenomena . Such systems can berepresented by semi-quantitative models thatcombine both qualitative and quantitativeknowledge. This paper presents NumericalInterval Simulation, a method that can simulatesystems involving any continuous,nonmonotonic functions . We present asuccessful application of NIS to predict thebehavior of a complex process at a Brazilian-Japanese steel company. We claim that suchcapability of simulating nonmonotonic functionsis fundamental in order to handle real-worldcomplex industrial processes .

Mathematical models of complex physicalsystems cannot be precisely defined in manycases, either due to lack of knowledge orbecause parameters and functions change overtime according to unknown phenomena .Nevertheless, it is often possible to providereasonable bounds for the parameters andfunctions . Bounds on parameters are intervalranges . Bounds on incompletely knownfunctions take the form of a pair of functions,one to provide an upper bound and another toprovide a lower bound. Such models that canintegrate qualitative and quantitative knowledgeare called semi-quantitative models .

" This research is in part supported by the AdvancedResearch Projects Agency, ARPA Order 8607, monitoredby NASA Ames Research Center under grant NAG 2-581, and by NASA Ames Research Center under grantNCC 2-537 .

They can represent important classes of complexsystems such as chemical, electro-mechanical,nuclear, thermal, steel and other industrialprocesses [1] .

Many qualitative simulation methods are basedon constraint propagation . They generate allpossible states and use filtering techniques toeliminate impossible ones . Semi-quantitativemethods such as Q2 [2] and Fu-Sim [3] addnumerical information and take advantage of it inthe filtering process . The basic algorithmremains the same. Such semi-quantitativemethods do not fully exploit the quantitativeknowledge and produce only weak predictionsacross time intervals . This is primarily due tothe limitations of interval propagation acrossderivative constraints . The mean value theoremis used to constrain the ranges of a variable attwo adjacent time points to and to+1, where toand to+I are the temporal boundaries of aqualitative state . Those methods do not reasonabout the behavior of the system within aqualitative state.

This paper presents Numerical IntervalSimulation (NIS), a method that produces moreprecise simulations of semi-quantitative models.NIS performs numerical simulations usingmaximal and minimal values for the derivativesof the state variables . In order to calculate suchmaximal and minimal values, NIS uses intervalarithmetic to calculate intervals containing thepossible values of the derivatives and takes theirmaximum or minimum respectively . NIS iscomplete in the sense that its solutions bound allthe solutions of the semi-quantitative model.The method replaces the use of the mean valuetheorem with explicit integration .

NIS was originally presented at the 1992Qualitative Reasoning workshop [4] . At that

same workshop, Kay and Kuipers alsopresented their work on dynamic envelopes [5] .Both techniques share the same fundamentalinsights . Subsequent development has lead indifferent directions, although it should bepossible to unify the two lines of work. NIS hasbeen extended to handle non-monotonicfunctions and forcing functions, whereas thedynamic envelope method is more closelyintegrated with the QSIM [6] formalism . Thedynamic envelope method of Kay and Kuipers[7] derives and numerically simulates "extremalsystems" composed of "extremal equations" thatare bounds on the derivatives of the statevariables . Such "extremal systems" have beenautomatically generated only for systems ofmonotonic functions .

NIS is an extension of the fuzzy simulationmethods for linear systems proposed in [8] andfor piece-wise linear systems proposed in [9] .We claim that the extension presented in thispaper to handle nonmonotonic functions andforcing functions is fundamental to simulate real-world nonlinear complex systems . Wedemonstrate the usefulness of NIS by presentingits successful application to simulate a complexsintering process at Companhia Siderurgica deTubarao, a Brazilian-Japanese steel companylocated in Brazil .

This paper is organized as follows : We firstdescribe the Numerical Interval Simulationmethod . Second, we show two examples ofnumerical interval simulation of systems ofordinary differential equations composed ofnonlinear nonmonotonic functions . The firstsystem represents an electrical system withpower supplied by an AC generator, and thesecond represents an inverted pendulum attachedto an electric motor. We then discuss themathematical properties of the method . Wefinally present the application of NIS to acomplex industrial process . We close with adiscussion of related work and conclusions .

2 Numerical Interval Simulation

The Numerical Interval Simulation method takesas input a semi-quantitative model and generatesupper and lower bounds on the trajectories ofeach of the state variables in the model .The semi-quantitative model is composed of a setof equations, interval bounds on the constantparameters used in the equations, and intervalbounds on the initial values of the state variables .

The equations are (non-) linear first-orderordinary differential equations (ODES) of theform

The functions fk can contain the standardarithmetic operators such as addition,subtraction, multiplication, division,exponential, logarithm, sines, cosines, andconstant parameters . In addition, fk can alsocontain intervals in place of constant parametersand interval functions that bound the values ofthe actual parameters and functions . Intervalfunctions are functions that take intervals asarguments and return an interval . In otherwords, fk can contain, instead of a real-valuedfunction g(t, x1, . . . xn), an interval function h(t,il, . . . in ) such that g(t, x1, . . . xn) E h(t, il, . . .in ) where ij ^s are intervals and xj E ij forj = 1 ton .

NIS performs numerical simulation, usingextremal values for the derivatives of the statevariables . Such extremal values for thederivatives are calculated in the followingmanner:

x'k= fk(t, xi, x2, . . . xn) for k = 1 . . n

First, NIS uses interval arithmetic tocalculate the interval of possible values forthe derivatives of the state variables [10] .Since the functions fk can contain intervals(as parameters) and interval functions, thearithmetic operators in fk are actually theircorresponding interval arithmetic operators,and fk are interval functions as well . NISuses fk to calculate the intervals for thevalues of the derivatives .

Second, NIS takes the maximum orminimum of the resulting interval, dependingon whether the maximal or minimalderivative is required .

Later in the paper, we show that NIS is completein the sense that its result is guaranteed to boundall the possible solutions of the semi-quantitativemodel . The only restriction for the functions fkused in the proof of completeness is that suchfunctions must be continuous over the intervalsconsidered in the simulation . It means thatfunctions fk can be composed not only ofmonotonic functions but also of nonmonotonicones . NIS allows nonmonotonic functions overthe reals such as multiplication, sine, cosine,

exponential and logarithm, and the arbitraryinterval functions can also bound a set ofnonmonotonic arbitrary real functions .

The NIS method calculates maximal and minimalvalues for the derivatives at a given instant tn .Such extremal values are used by a numericalsimulator to determine the value of the statevariables at the next instant to+1- Severaldifferent methods could be used to perform thenumerical simulation . We present below NISusing both the Euler's method and the Runge-Kutta method.

2.1 Euler's Method NIS

Let xk(t) = [xkrnin(t), xkmax(t)] be the intervalsuch that its lower and upper bounds correspondto the minimum value xkm i n (t) and themaximum value xkmax(t) of state variable xk attime t .respectively . In the remainder of thispaper, we will use the notation [b] for the pointinterval [b, b] for brevity . The Euler's methodNIS for each simulation step can be describedas :

1) Calculate the maximum 8max(tn)i and theminimum &nin(tn)i values for the derivatives ofstate variables xi at instant tm in the followingmanner :

&nax(tm)i = max(fi(tm, xl, x2 . . .[ximax]- . .,xn))

&min(tm)i = max(f(tm, xl, x2. . . [ximin], . . .,xnA

where xj (j = 1 . . .n except i) are the interval[xjmin, xjmaxl-

Notice that, the functions fi are calculated takingas arguments the real maximal value ximax(tm)and the real minimal value ximin(tm) of statevariables xi at instant tm respectively, instead ofusing the interval [ximin(tm), ximax(tm)1- Thisis fundamental to provide accurate simulationsand will be discussed in more detail later in thepaper .

2) Calculate the maximum bound ximax(tm+1)and minimum bound ximin(tm+1) at instanttm+1 with the given step size of &:

where

ximax(tm+l) = ximax(tm) + St &nax(tm)i

ximin(tm+1) = ximin(tm) + St &nin(t n)i

2.2 Runge-Kutta NIS

We presented the Euler's method in two steps byfirst calculating the extremal values for thederivatives and second by calculating the valuesof the state variables for aims of clarity . Withthe Runge-Kutta NIS, the extremal values forderivatives are calculated several times insideeach iteration of the method. We present belowthe four-stage Runge-Kutta NIS method.We usethe symbols ® and ® to represent the intervalarithmetic relations addition and multiplicationrespectively . The equations are given for theupper bound, ximax. The equations for thelower bound are identical with min substitutedfor max throughout .

ximax(tm+l) = ximax(tm) + Wkilmax +2ki2max + 2ki3max + ki4max)

kil max = St max(fi(tm, xl, x2 . . ., [ximax]. . .xn)),

ki2max = St max(fi(tm + 28t, xl ® [211

[kllmin, kllmax], .--, [ximax + 2kilmaxl,

,1

xn ® [21 ®[knlmin, knlmax]))

ki3max = St max(fi(tm + ZS0, xl ® [2][kl2min, kl2max], .--, [ximax + Zki'maxl,

1xn e [21 ® [kn2min, kn2maxl))

ki4max = St max(fi(tm + St, xl ®[kl3min,kl3max], ---, [ximax + ki3max], - .-,xn®[kn3min, kn3max])) .

2.3 Two Examples

In this section, we show NIS simulations of twoelecrro-mechanical systems . Each physicalsystem is represented by ODES that includenonlinear nonmonotonic functions .

2.3 .1 Coupled two-stage electricalsystem

The first'-system is the coupled two-stageelectrical system shown in Figure 1 . The firststage is an RL series circuit with input DCvoltage Vin provided by a battery . The secondstage is an RC parallel circuit with input currentIin provided by an AC generator. The currentIin is a sinusoidal signal with amplitude andfrequency directly dependent on the first stagecircuit output current Il .

The following second-order system representsan instance of a such device:

V~ RIl' =

1 -11 11, where Vin = [4.028, 4.03]

for t _<20 and Vin = Ofor t > 20.

V '

Iin

Vc c, where 1

k

I sin k I tc = C -~

in= A l

(f l )

Figure 1 : The coupled two-stage electricalsystem .

Figure 2 shows a NIS simulation of suchsystem, given the following parameter and initialvalues :

RI = [408, 410] mfz,l=2H,Rc= [400, 5001 xac=10MF,kA = [16, 18110-8 , kf = [123, 1251 10-3,11(0) = 0 A, andVC(O) = 0 V.

Figure 2: NIS simulation of variable Vc of thecoupled electrical system .

2.3.2 Inverted pendulum

The second example is a well known problem innonlinear mechanics, known as the invertedpendulum. The pendulum is a stiff bar of lengthL which is supported at one end by a frictionlesspin . The support pin is given a rapid up-and-down motion s by means of an electric motor, s= A sin ax . Application of Newton's second lawof motion yields the equation L9"=(g - co2A sinwe) sin e, where 8 is the angular position of thebar (e = 0 when the bar is directly above thepin), and g is the acceleration due to gravity .

Following is the second-order systemrepresenting the inverted pendulum:

e' = 3e

Se' = L (g - w2 A sin mot) sin e

Figure 3 shows a NIS simulation of suchsystem, given the following parameter and initialvariables values :

g = 386.09 inlsec2,

L=10in,

A = [0.5, 0.55] in,C) = [5.3, 5.4] radlsec,

e (0) = 3tad, and30 (0) = 0 radlsec .

640

��

ILOW

EA ^ "Ail" , ~'---

.,�~M~Wl

"T"n"WXA~.~. i~AMM

t

Figure 3 : NIS simulation of variable 0 of theinverted pendulum.

2.4 Properties of NIS

We have analyzed some computational propertiesof NIS, namely completeness and soundness .NIS can be shown to be complete, but notsound .

2.4.1 About Completeness

NIS is complete in the sense that its result isguaranteed to bound all the possible solutions ofthe semi-quantitative model . The proof ofcompleteness follows the steps below :

" Consider the semi-quantitative model x'k =fk(t, xl, x2, . . . xn), where xk(t0) E IOk for k =1, . . .n .

" The function fmax = max(fk(t, xl, . . . [xk], . . .xn)) calculated by NIS is such thatb't, xk E R,Vx1, x2, . . . xn E I,b'x'k e R /x°k E fk(t, xl, . . . [xk], . . . xn) ,x'k :5 fmax(t, x1, . . . [xk], . . . xn) .

This is straightforward to prove since NIS usesinterval arithmetic operations that calculate the setof all possible values for the functions fk andonly then takes its maximum.

" For xmax such that x'max =fmax(t, x1, x2,. . . xn), we can prove that b't xk(t) _< xmax(t) .

The proof is based on the fact that, since thefunctions fk are continuous and xk(tO) _<xmax(tO), xk has to be equal to xmax first inorder to become greater than xmax . Since atany crossing point the derivative of xmax isgreater than the derivative of xk (proved in thestep above), xk will be only smaller or equal toxmax at an arbitrarily small time after thecrossing point and xk does not become greaterthan xmax.

" In a similar manner, considering anotherfunction fmin, we prove that b' t, xmin(t) _<xk(t) .

2.4.2 About Soundness

Soundness is the property of predictingbehaviors that are solution of at least oneinstance on the semi-quantitative model .Although NIS is not sound in general, themethod is very precise in most cases . In thissection we first discuss why NIS is usuallyprecise by comparing the method to our firstapproach for interval simulation [8] . We thendiscuss and show a particular case in which themethod diverges after a couple of interactions .

In order to point out why spurious behaviors areproduced, we will recall our first approach forinterval simulation [8] . The idea with the formermethod was to extend a numerical method bysubstituting its operators by the intervalarithmetic correspondents . The extended methodis complete but produces too much spuriousvalues . The main reason is that the arithmeticinterval operations are applied without takinginto account the interaction among variables .Consider the first order system x' = -kx, there isa strong relation between x and x', and bothvariables are operands in the extended Euler'smethod:

x(tn+1) =x(tn) ® (h ® x'(tn))

x(tn+1)= x(tn) ® (h ®(e (k ® x(tn)))

The possible values for x(tn+l) are only thosecalculated considering unique values for x(m) ineq.2 . The extended method, however, combinesall the possible values for x(tn) and x'(tn) .Imagine x1, x2 such that x1, x2 E x(tn), themethod considers the combination x(tn) =x1,

and x'(tn ) calculated for x(tn) = x2, whichwould probably produce a spurious value forx(tn+ l ) . Interactivity is partially overcome byNIS, since the functions fi take only themaximum ximax(tn) or minimum ximin(tn) ofstate variable xi, instead of the full intervallximin(tn), ximax(tn)] . For example, the NISEuler's method of the first order system x'=-ks,with k= lkmin, kmaxl, is

xmax(tn+1)=xmax(tn) +h(max (0 (klxmaxl)))

xmin(tn+1)=xmn(tn)+h(min (O (k ®1xminl)))-

We can easily prove that the interval 1xmin(tn+l ), xmax (tn+l )l produced by NIS isalways a sub-interval of x (tn+l) produced bythe extended method . Figure 4 illustrates this .The value xspurious(t2), calculated by theextended method, would surely be a spuriousvalue greater than xmax(t2), calculated by NIS.

t0

xmax(tl)=x(t0) +h(-kmin x(t0))

xmin(tl) =x(t0)+ h (-kmax x(t0))

xspurious(t2) = xmax(tl) + h(-kmin xmin(tl))

xmax(t2) = xmax(tl) + h(-kmin xmax(tl ))

Figure 4 : xspurious(t2) and xmax(t2) calculated bythe extended method and NIS respectively

The interval simulation showed in Figure 2 is avery precise simulation . The interval shrinkstowards the end of the prediction of anoscillatory behavior, what would rarely be thecase with other semi-quantitative methods.

However, other cases of interactivity betweenthe state variable and their derivatives can occur.For example, with systems of the form

x'k = fk(xk+1),x'k+l =fk+l(xk+2),

x'k+n = fk+n(xk),

the interactivity is among xi and other variablesappearing in the equation of x'i . NISsimulations of such systems tend to becomeunstable after several iterations . The invertedpendulum and other oscillatory systems are ofthis type . Figure 3 illustrates such an instability .

3 . The Complex Sintering Process

In this section, we present the successfulapplication of NIS to predict behavior of acomplex sintering process at CST CompanhiaSiderurgica de Tubarao, a Brazilian-Japanesesteel company located in Brazil . The sinteringprocess continuously produces sinter ore withvarious kinds of fine iron ore as the raw materialand lime stone as the binder. The process hastwo major goals : One is the stabilization ofoperation to produce strong sinter ore of uniformsize as the ferrous burden of the blast furnace .Another is optimization of the process tominimize the production cost under variousconditions and processing throughout the wholeiron works . Figure 5 illustrates the sinteringplant . The granulated raw material in the surgehopper is fed across the sinter bed width and isignited by the furnace. The material burns fromthe surface toward the bottom by the downwardair flow through the wind boxes . The material isshifted by the sinter bed towards the cooler.

Raw Material

CWind Boxes

Surge Hoper

Sinter Bed

ignition Furnace

Figure 5: The Sintering Plant

Cooler

The goal of the process operator is to control thesinter bed speed in order to maximize

productivity while maintaining safety . Too lowa speed causes low sinter production and quality ;high speed causes burning material to fall intothe cooler, damaging the equipment and possiblycausing a fire . An observable variable called theburn through point, Btp, is the primary variableto control . The Btp is supposed to be maintainedbetween 65 and 78%. Btp below the lowerboundary corresponds to low sinter qualitycondition and lost productivity, while Btp abovethe higher boundary corresponds to a dangerousoperation condition . The ideal is to keep the Btpin between 70 and 75%.

A precise model of the sintering process is notavailable due to the complexity and experimentalnature of the process . Indeed, the relationsamong variables change during the operation,influenced by unknown phenomena. Forexample, the coefficients significantly changewhenever there is a significant change in thequality of the raw material ; this is very difficultto monitor . However, it is possible to defineboundaries for the relations and parameters .Included in the semi-quantitative model of thesintering process used by NIS is the first orderrelation (the NIS simulated model actuallycontains a dozen of these first-order relations),

Btp'= P106(t - Z) - g(Btp),

where z is the constant delay between Btp andpressure P106, and g is a piece-wise linearfunction . Figure 6 shows the given boundariesfor g(Btp) . For more details, see [1] .

g(Btp)

0.7 0.75 Btp

Figure 6 : Part of the semi-quantitative model ofthe sintering process .

Actual productivity, safety and reliabilityrequirements demand optimum operation . In thecase of the sintering process, Btp is to be keptas close as possible to 75% . Small deviationscan have significant effects on the process . A

Btp increase of around 3% when its value is at75% correspond to a very dangerous operationregion that can cause fire and damage to thecoolers . Predictions that are too wide, either dueto lack of knowledge, or weaknesses in thesimulation method, are simply useless . Manyhours of operation during different periods of theyear were simulated, and the results obtained byNIS were very successful . With rareexceptions, the actual behavior of the processwas always bounded by NIS simulations (therare exceptions ocurred because of modelingerrors as opposed to NIS errors) . Furthermore,the predictions were considered tight enough tobe useful .

Figure 7 shows an example of a NIS predictedbehavior (the two outer thin lines) and theobserved behavior (the thick line) of the Btp .The NIS simulation failed in the interval [380,430] minutes, indicating that the boundarieswere not well chosen in this operating region .This is not of great concern since the system isnot supposed to operate in this region (Btpsmaller than 65%). This undesirable behavioroccurred in reality because of an operatormistake. Notice that the NIS simulationpredicted that the Btp would penetrate the regionbelow 65% at time 370 minutes . An on-linemonitoring system would have thus advised theoperator about such possibility, and it could havebeen avoided .

757065

BTP(%)

Min

Figure 7: NIS predicted boundaries and themeasured value of variable Btp.

Further work is in progress to run NIS on-lineas an adviser for the sintering process operators .The semi-quantitative model is based on the last6 months of operation . Since the quality of theraw material can significantly influencecoefficient values, more work is to be performedin order to increase the reliability of thesimulations .

I

I

I

I

I

I

i370

II380

I430

4. Related Work

Research in different areas such as qualitativephysics [2, 3, 15, 7], sensitivity analysis (12],tolerance banding [13] and interval analysis [10,14], address the problem of combiningqualitative and quantitative knowledge. We willfirst discuss the methods outside qualitativephysics.

Sensitivity analysis [12] is used to predict theeffect of small-scale perturbations to a system .Tolerance banding [13] is used to predict large-scale uncertainties . Both methods are normallyrestricted to linear models and allow uncertaintiesof parameters and initial values only . Accordingto Kay and Kuipers [7], bounding methods thatdo not rely on linearity assumptions have beendeveloped for VLSI simulation [18], but theyrely on domain-specific assumptions . IntervalAnalysis [10, 14] simulates semi-quantitativedifferential equations by recasting numericalsimulators to work with interval arithmetic. BothNIS and the dynamic envelope methodnumerically simulate extremal systems . Anadvantage of both approaches is that modelimprecision is separated from the errorintroduced by the simulator . Another advantageis that such methods can switch to morepowerful numerical simulators as they aredeveloped .

Inside the scope of qualitative physics, the firstsemi-quantitative methods such as Q2 [2] andFu-Situ [3] do not fully exploit the quantitativeknowledge available in the semi-quantitativemodel and produce only weak predictions acrosstime intervals . As model precision increases,NIS produces more precise simulations than Q2or Fu-Sim. The method replaces the use of themean value theorem with explicit integration overtime .

To establish the temporal correspondencebetween the observed values and the predictions,the sintering application uses synchronizedsampling as most industrial process monitoringsystems do. Tracking the process is thussignificantly facilitated . The imprecision of thesemi-quantitative model only affects variablevalues estimates . That provides a firmer groundfor comparing the results of the simulation withreal observations, which is crucial in real timemonitoring systems . In semi-quantitativemethods like Q2 or Fu-Sim, temporal durationsare calculated with the first order Taylor-

Lagrange formulae using quantity space valuesin the form of numeric or fuzzy intervals . It wasshown in [15] that the first order Taylor-Lagrange formulae is scarcely sufficient toprovide significant information . This is true,independent of the weakness directly related to aweak quantity space, at the neighborhood ofcritical points for which the derivative reacheszero . Indeed, zero derivative leads to oneinfinite boundary for the duration estimate . As aresult, time durations calculated for adjacentstates are often widely overlapped . It mayhappen that a given time instant belongs toseveral consecutive states, implying that variablevalues at this instant are very weaklyconstrained.

SIMGEN by Forbus and Falkehainer [16] alsouses qualitative and quantitative knowledge .SIMGEN generates a precise model from alibrary of predefined functions and performs anumerical simulation. The method does not usea semi-quantitative model or simulation .

Most closely related to NIS is the dynamicenvelope method [7] . Both methods numericallysimulate extremal systems . NIS uses intervalarithmmetic to calculate the extremal values forderivatives at each simulation step, whereas thedynamic envelope method generates the exremalsystems a priori . The dynamic envelope methodinherits certain limitations from the QSIMformalism upon which it is built . First, it doesnot allow forcing functions such as A sin on inthe example in Section 2.3 .2 . Forcing functionscannot be reasonably specified in a purelyqualitative framework, because the eventsgenerated by the forcing function must beexhaustively interleaved with the normalqualitative events . This results in a combinatoricexplosion . Second, the QSIM formalism doesnot allow for non-monotonic function inequations other than multiplication . Thedynamic envelope method also benefits from thequalitative representation . QSIM is able to splitdivergent behaviors, which allows the dynamicenvelope method to provide separate, tighter,bounds on each qualitative behavior, rather thana single, broad, bound that covers all of thebehaviors .

NIS can simulate systems involvingnonmonotonic functions and allows use offorcing functions . NIS is an extension of fuzzysimulation methods for linear systems proposedin [8] and in [4], and for piece-wise linear

systems proposed in [9] . Since a large class ofcomplex industrial processes are modeled bynonmonotonic functions and include forcefunctions, the capability of handling suchfunctions constitutes an important contribution tothe field of qualitative physics .

NIS simulations of certain oscillatory systemscan be unstable . The same limitation occurs withthe dynamic envelope method . An alternative,which consists in intersecting the results of thedynamic envelope method and of Q2, wasproposed in [7] in order to improve the results ofthe simulation in such cases . We are consideringthe integration of NIS with the methods underdevelopment at the University of Texas .

5 . Summary

The Numerical Interval Simulation methodproduces high precision simulations of semi-quantitative models. The method is complete inthe sense that its solutions bound all thesolutions of the semi-quantitative model . As theprecision of the semi-quantitative modelincreases, the method produces more precisesimulations than former semi-quantitativemethods like Q2 or Fu-Sim . Although NISproduces the same results as the dynamicenvelope method when simulating systemsinvolving only monotonic functions, the methodalso simulates systems containing arbitrarycontinuous, nonmonotonic functions. NIS alsoallows use of forcing functions . We claim thatsuch level of generality is fundamental tosimulate real-world nonlinear complex systems .

We demonstrated the usefulness of NIS bypresenting its successful application to simulate acomplex sintering process at CompanhiaSiderurgica de Tubarao, a Brazilian-Japanesesteel company located in Brazil . The resultshows that NIS is adequate to represent theavailable information and produces simulationswith the required level of precision . Theprototype has been tested and we are working torun NIS on-line as an adviser to the sinteringprocess operators .

References

1 . Vescovi, M., et al. Qualitative FuzzySimulation of Complex Processes : Applicationto an Iron Ore Sintering Plant. In Proceedings of

the Eighth Conference on Artificial Intelligencein Engineering . 1993 . Toulouse, France .

2 . Kuipers, B . and Berleant, D. UsingIncomplete Quantitative Knowledge inQualitative Reasoning. In Proceedings of theSeventh National Conference on ArtificialIntelligence . 1988 .

3 . Shen, Q. and Leitch, R. Integrating common-sense and qualitative simulation by the use offuzzy sets . In Proceedings of the FourthInternational Workshop on Qualitative Physics.1990 .

4 . Vescovi, M. and Trave-Massuyes, L. AConstructive Approach to Qualitative FuzzySimulation . In Proceedings of the SixthInternational Workshop on Qualitative Physics .1992 .

5 . Kay, H. and Kuipers, B . Numerical BehaviorEnvelopes for Qualitative Models . InProceedings ofthe Sixth International Workshopon Qualitative Reasoning about PhysicalSystems . 1992 . Edinburgh, Scotland.

6 . Kuipers, B., Qualitative Simulation . ArtificialIntelligence, 1986 . 29 .

7 . Kay, H . and Kuipers, B. Numerical BehaviorEnvelopes for Qualitative Models . InProceedings ofthe Eleventh National Conferenceon Artificial Intelligence . 1993 .

8 . Vescovi, M. La Representation desConnaissances et le Raisonnement sur lesSystemes Physiques . 1991, Ph. D. thesis,University de Savoie, in French .

9 . Vescovi, M ., et al . Numerical- BasedQualitative Fuzzy Simulation - An Extension forPiece-Wise Linear Systems. In Proceedings ofthe Nono Congresso Brasileiro de Automatica.1992 .

10 . Moore, R., Methods and Applications ofInterval Analysis . 1979, SIAM.

11 . Kay, H. Monitoring and Diagnosis ofMulti-Tank Flows using Qualitative Reasoning. 1991,Master's thesis, University of Texas at Austin,

12 . Deif, A., Sensitivity Analysis in Linearsystems. 1986, Springer-Verlag .

13. Lunze, J., Robust Multivariable FeedbackControl. 1989, Prentice Hall .

14 . Markov, S . and Angelov, R. An intervalmethod for systems of odes . In Lecture Notes inComputer Science #212 - Interval Mathematics1986 . Springer-Verlag, Berlin.

15 . Missier, A. Structures Mathematiques pourle Calcul Qualitatif - Contribution a la SimulationQualitative . 1991, Ph. D. thesis, I . N. S . A.,In French .

16 . Forbus, K. and Falkenhainer, B . Self-Explanatory Simulations : An Integration ofQualitative and Quantitative Knowledge . InProceedings of the Eighth National Conferenceon Artificial Intelligence . 1990 .

17 . Berleant, D. and Kuipers, B ., CombinedQualitative and Numerical Simulation with Q3,in Recent Advances in Qualitative Physics, B.Faltings and P. Struss, Editor. 1992, MIT Press :

18 . Zukowski, Charles A. The BoundingApproach to VLSI Circuit Simulation 1986 .Kluwer Academic Publishers, Boston .