An Assessment of Current Qualitative Simulation Techniques
Contents
1. and Kuipers amp Chiu 87 proposed solutions The state of the art is presented in Kuipers et al 89 Principle Roughly speaking a variable may exhibit chatter if its derivative is unconstrained Chatter doesn t occur for the simple spring because the acceleration has a behavior similar to the position whose derivative is explicitly represented Basically if at some time point a variable transitions to a critical point that is its derivative becomes zero then its qualitative value in the next open interval of time is determined by its second derivative If no information is provided about this second derivative then QSIM will branch on each possible future Multiple occurrence of this phenomenon leads to intractable branching The method proposed in Kuipers et al 89 is performed in three steps Identify equivalence classes of variables in the QDE likely to chatter Derive an expression called the curvature constraint for the second derivative of one variable among each class Use the sign of the second derivative to constrain simulation l Derivation can be performed manually or automatically by an integrated algebraic manipulator Curvature constraints provide useful information only when they are not ambiguous that is when the sign of the second derivative is positive or negative When it is zero then the same problem occurs and the next qualitative value of a chattering variable is given by the sign of its third derivative For2. D G Qualitative Reasoning with Higher Order Derivatives Proceedings of AAAI 84 86 91 1984 Dalle Molle 90 Dalle Molle D Qualitative Simulation of Dynamic Chemical Processes Technical Report Al TR 89 107 The University of Texas at Austin 1989 Doyle amp Sacks 89 Doyle J amp Sacks E Stochastic Analysis of Qualitative Dynamics Proceedings of IJCAI 89 1187 1192 1989 Farquhar amp Kuipers 90 Farquhar A amp Kuipers B J QSIM User s Manual Technical Report Al TR 90 123 The University of Texas at Austin 1990 Forbus 84 Forbus K D Qualitative Process Theory Artificial Intelligence 24 5 85 168 1984 Fouch amp Kuipers 90a Fouch P amp Kuipers B J Reasoning about Energy in Qualitative Simulation an Report Al TR 90 xx forthcoming The University of Texas at Austin I Fouch amp Kuipers 90b Fouch P amp Kuipers B J Introducing Energy into Qualitative Simulation Submitted to AAAI 90 Hobbs 85 Hobbs J R Granularity Proceedings of IJCAI 85 432 436 1985 Ishida 89 Ishida Y Using Global Properties for Qualitative Reasoning A Qualitative System Theory Proceedings of IJCAI 89 1174 1179 89 Kuipers 86 Kuipers B J Qualitative Simulation Artificial Intelligence 29 289 338 1986 Kuipers amp Chiu 87 Kuipers B J amp Chiu C Taming Intractable Branching in Qualitative Simulation Proceedings of IJCAI 87 1078 1085 1987 209 Fouch amp Kuipers An Assessment
3. and non conservative ones It consists of computing the sign of the work of conservative and non conservative forces and checking that this is compatible with the sign of the variation of kinetic energy The Energy Constraint like the Non Intersection Constraint takes advantage of two main features of QSIM it creates new landmarks during simulation thus providing enough information to compute the sign of the quantity defined above and it describes behaviors of a system directly in terms of its variable histories which makes non loca reasoning very natural Results Currently QSIM has to be provided with the name of variables representing conservative and non conservative terms C and N For both the simple and the damped springs the decomposition of the acceleration is trivial For the simple spring C X t A t and N t 0 For the damped spring C X t Fs t and N t F p t With the Energy Constraint QSIM is able to determine that the two first behaviors of figure 1 5 are spurious with the following justifications e a Inconsistent between tO and t4 Ke var 0 C work NC work 0 b Inconsistent between tO and t4 Ke var C work 0 NC work 0 and thus we come up with only one genuine behavior for the simple spring Applying the Energy Constraint to the damped spring also allows to get the correct behavioral description as shown in figures 3 4 and 3 5 We come up with one infinite pseudo cyclic b
4. shows that reasoning with higher order derivatives and introducing curvature constraints allows to eliminate this phenomenon but that the real problem for the damped spring is that the level of description is not appropriate the system is not constrained enough for QSIM to predict a total ordering of the relative occurrence of events and behaviors keep proliferating This phenomenon is referred to as Occurrence Branching An alternate way to get rid of chattering which turned out to solve in part the occurrence branching problem is to shift to a higher level of description by ignoring irrelevant distinctions This is described in section 2 2 All the preceding techniques still do not provide a good behavioral description of the damped spring QSIM still derives behaviors which are genuinely incompatible with any actual system that abstracts to the model This problem of incompleteness mainly stems from a combination of the loss of quantitative precision with the local character of qualitative inferences A way to get a global view of a system behavior is to use the Phase Space representation as described in section 3 1 This allows QSIM to derive that some properties of a system cannot change through time The final step to get a correct behavioral description of the spring system is to introduce energy considerations as shown in section 3 2 A summary is provided in section 4 With the help of all these techniques QSIM is now able to derive importa
5. the damped spring Zoo Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 5 E 5 Figure 2 1 Tree of behaviors of the damped spring using curvature constraints Deriving curvature constraints in the presence of monotonic M or M constraints requires additional assumptions about them the semantics of M Y X is that there exists a function f such that V z Y t f X 1 with f x gt 0 This implies that dY X but oX and oY are not related The assumption that X oY is called the sign equality assumption and is applied by the algebraic manipulator This assumption is obviously satisfied when f is linear but more generally when f s shape is relatively smooth Results Figure 2 1 shows the tree of behaviors for the spring with friction when simulation is allowed to reach time point t7 Instead of 56 behaviors at time tz only 4 are produced But the tree is still growing exponentially Figure 2 2 shows the three possible behaviors at time tz Time tz is defined either by X crossing zero A reaching a critical value or both at the same time In the linear case the relative occurrence of these events is a system property Lee et al 87 that is a property that depends on the parameters of the system but that that does not vary during simulation In the general case it depends on the functions Fr and Fs and thus any order on the occurrence of events is possible The point is t
6. value ign then branching will no longer occur since it is impossible to detect critical values Ignoring directions of change has three main advantages over using information about higher order derivatives No assumption is made about the nature of M or M instances and thus soundness is preserved e No algebraic manipulation has to be made to derive higher order derivative properties e It eliminates occurrence branching caused by a variable reaching a critical value However in some cases Information about higher order derivatives is required to filter out genuinely spurious behaviors Dalle Molle 89 Kuipers er al 89 that the ignore qdirs method does not rule out Higher order derivative constraints usually perform better than ignore qdirs on non oscillatory systems e It may be very interesting to know explicitly the direction of change of some variable l Short for Ignoring the qualitative direction of change 2 02 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques d Results Figure 2 3 shows the tree of behavior of the damped spring using the ignore qdirs method for the acceleration Instead of 60 behaviors at time t7 only 7 behaviors are produced at the same time and 50 if simulation is allowed to run until time t79 but we still have too many behaviors j Ld m a Figure 2 3 Tree of behaviors of the damped spring ignoring the direction of change of the a
7. without crossing it if the friction force is strong enough This is an over damped behavior Otherwise the system will exhibit decreasing oscillations and what we really get Figure 1 3 shows the behavior tree of the simple and damped springs when behaviors are allowed to reach time point fg and t Clearly this is not as simple as expected l Farquhar amp Kuipers 90 Throop et al 90 describe in detail how to use the QSIM is implementation 2 A filled circle represents a state at some time point an empty circle a state at some time interval a filled circle surrounded by a larger circle a quiescent stale and an empty circle surrounded by a larger one a cyclic state identical to a prior state in the same behavior States followed by dashed lines are states whose successors have not been computed yet due to a resource cut off Time increases from left to right LI Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 3 a b Figure 1 3 Behavior trees produced by the QSIM kernel a simple spring time limit tg 26 behaviors b damped spring time limit t4 56 behaviors Problems with the damped spring Figure 1 4 shows a particular behavior of the damped spring Analyzing the distinctions among all the behaviors reveals that only the acceleration has distinct behaviors The explanation is that the acceleration is constrained only by continuity it is the sum of a continuously increasing function th
8. 19s An Assessment of Current Qualitative Simulation Techniquest Pierre Fouch amp Benjamin Kuipers Department of Computer Sciences The University of Texas at Austin Austin TX 78712 USA Email fouche cs utexas edu kuipers cs utexas edu Abstract QSIM is a powerful Qualitative Simulation algorithm which now includes many features that have proven to be necessary in Qualitative Simulation These features are reasoning with Higher Order Derivatives having Multiple Levels of Abstraction reasoning in the Phase Space representation and reasoning about Energy The aim of this paper is to provide a comprehensive view of all these techniques by explaining their rationale showing the problems they address and how they interact Remaining problems in Qualitative Simulation are also discussed Main Topic Qualitative Simulation This work has taken place in the Qualitative Reasoning Group at the Artificial Intelligence Laboratory The University of Texas at Austin Research of the Qualitative Reasoning Group is supported in part by NSF grants IRI 8602665 IRI 8905494 and IRI 8904454 by NASA grants NAG 2 507 and NAG 9 200 and by the Texas Advanced Research Program under grant no 003658175 Pierre Fouch holds a grant from Rhone Poulenc 196 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques l Introduction Qualitative Physics has experienced a rapid growth since its birth generally dated to the spe
9. cceleration Browsing the tree reveals that many of them are intuitively spurious and figure 2 4 shows one of them the block seems to oscillate with increasing amplitude when X is positive and constant amplitudes when X is negative But reasoning about oscillations requires a global point of view i rl i i i a i 31 A 2 N no P 7 Figure 2 4 An intuitively spurious behavior 3 From a Local to a Global Point of View The filtering techniques we have discussed so far have been purely local Determining the validity of state transitions has been made by considering only two successive states and the basic validity criterion has been continuity of variables But the first thing that we are thinking of when we are looking at a spring is oscillation And dealing with oscillation requires a global view of a system behavior it involves comparing the state of a system at 203 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 8 a given time with its state at some time before and these two time points are not successive in our qualitative description For instance expressing that the amplitude of oscillations starts increasing in figure 2 4 requires comparing the qualitative magnitude of the block position at time points t7 and fg One way to reason globally about a system is to use a phase space representation 3 1 Changing of Representation the Phase Plane View The Phase Space for a system is th
10. cial issue of the Artificial Intelligence Journal of December 1984 Among all the formalisms that have been developed QSIM originally designed by Kuipers 86 has been greatly improved by many researchers since It now includes several features that have proven to be necessary in Qualitative Simulation like reasoning with higher order derivatives de Kleer amp Bobrow 84 Kuipers amp Chiu 87 having multiple levels of description Hobbs 85 Kuipers amp Chiu 87 reasoning in the phase space representation Sacks 87 Struss 88 Lee amp Kuipers 88 Doyle amp Sacks 89 and reasoning about energy Fouch amp Kuipers 90a 90b So far these techniques have been described separately and have not been compared to each other Consequently it was not easy to decide even for someone very familiar with Qualitative Simulation which one to apply to solve a practical problem This paper is an attempt to provide a comprehensive view of these techniques for each of them we describe their rationale and intuitive appeal and we compare their relative efficiency on two simple examples widely used in the Qualitative Physics literature a block spring system with or without friction The first part of the paper presents the models and the result of their simulation with the QSIM kernel It is shown that simulation is intractable mainly because of a phenomenon known as Chatter The second part is devoted to local reasoning techniques Section 2 1
11. e any linearity assumption to demonstrate that Qualitative Simulation applies to non linear systems For the simple spring we can directly model that the acceleration is inversely related to the position of the block For the damped spring the friction force Fp is inversely proportional to the speed of the block and again we shall not assume that this relation is linear Figure 1 2 shows the models as they are given to QSIM define QDE simple spring define QDE damped spring quantity spaces quantity spaces X minf 0 inf X minf O inf V minf O inf V minf O inf A minf O inf A dminf O inf constraints FF minf 0 inf d dt X V FS minf 0 inf d dt V A constraints M X A minf inf d dt X V 0 0 d dt V A inf minf M FS X minf inf 0 0 inf minf M FF V minf inf 0 0 inf minf add FS FF A Figure 1 2 QSIM models of the simple and damped springs Simulation with the QSIM kernel what we expect We start the simulation with the spring stretched and the block immobile The beginning of the expected behavior is the block moving towards its rest position What it will do next depends on the friction force If the motion is frictionless then the block will move across its rest position reach another extreme and move back to its original position One can describe this behavior as a stable oscillatory behavior If friction occurs then it can move towards the rest position
12. e Cartesian product of a set of independent variables that fully describe the system In practice it provides another view of system behavior a point in the phase space represents a system state and a trajectory a behavior Principle of the Non Intersection Constraint A major theorem about the existence and uniqueness of the solution of an autonomous first order ordinary differential equation system has a direct equivalent in the phase space representation A trajectory which passes through at least one point that is not a critical point cannot cross itself unless it is a closed curve In this case the trajectory corresponds to a periodic solution of the system Lee amp Kuipers 88 and Struss 88 discovered that this property can be conveniently translated into qualitative terms for second order systems In this case the phase space is a plane and a general intersection criterion can be established even if trajectories are described qualitatively This criterion can then be used to rule out trajectories that intersect themselves Figure 3 1 shows the Non Intersection Constraint at work the behavior corresponds to figure 2 4 In the phase plane V A the trajectory is a closed curve and thus the behavior should be cyclic and trajectories in other phase planes be closed curves too But this is not the case and the behavior is labeled as spurious x at the end of each trajectory VveA X vs V Figure 3 1 The Non intersection Constraint a
13. e friction force and a continuously decreasing function the spring force Since we have no more information about these two functions their sum can be a decreasing steady or increasing function of time While all these behaviors are real behaviors they only occur if the functions relating speed and friction force and position and spring force have uncommon shapes If one can make additional assumptions about their second derivatives which are valid for actual spring systems these behaviors are no longer possible This is developed in section 2 1 Velocity Figure 1 4 One chattering behavior of the damped spring igo Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 4 Problems with the simple spring The first tree of figure 1 3 starts branching at time ty Figure 1 5 displays the three possible behaviors computed until that time In behavior a the block stops before its initial position In behavior b the block goes beyond and in behavior c the system comes back to its initial state Of course only the last behavior is genuine Kuipers 86 In fact solving the problem requires a global point of view on the system behavior Figure 1 5 Three way branching of the simple spring 2 Local Reasoning Techniques 2 1 Reasoning with Higher Order Derivatives The phenomenon that occurs for the damped spring is a particular instance of a more general problem called Chatter de Kleer amp Bobrow 84
14. e more deeply the differences Qsim does not allow the simple spring to reach quiescence e The reason for the time difference between the first branching occurrences t4 for the simple spring and ts for the damped spring is that the variables X and A are in phase for the simple spring but there is a phase difference for the damped spring This phase difference implies the creation of distinct time points for the events A crosses 0 and X crosses 0 Both trees exhibit more branching on the branch corresponding to increasing oscillations The explanation is given in section 5 But of course only damped oscillations are valid for the damped spring and only stable oscillations for the simple spring Determining unambiguously the character of oscillations damped or stable requires energy considerations 3 2 Reasoning about Energy Intuitively increasing and decreasing oscillations for the simple spring figure 1 5 a and b are spurious because the system is conservative the mechanical energy is conserved 20S Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 10 and the potential energy of the block depends only on its position Fouch amp Kuipers 90a 90b developed a method implemented as a global filter called the Energy Constraint that automates this reasoning It is based on a qualitative interpretation of the Law of Conservation of Energy and on the decomposition of processes into conservative
15. ed spring is linear then it cannot become over damped after the first oscillation We have to extend QSIM so that it can automatically determine whether a behavioral property is a system property But at the moment these words are written we do not know how Conclusion We have seen that with the help of several filtering techniques using Curvature Constraints Ignoring the Direction of Change of a variable using the Non Intersection or the Energy Constraint QSIM is now able to simulate systems that were previously intractable We hope this paper will help QSIM users choose the appropriate methods to simulate their models However another possible improvement and direction of research is to automate the process of choosing these methods Acknowledgments Pierre Fouch is particularly indebted to his advisor Jean Paul Barth s from the University of Compi gne France for his constant support and useful advice This paper benefited from discussions with many members of the Qualitative Reasoning Group at the University of Texas at Austin especially Wood W Lee Thanks to Adam Farquhar for a careful reading of a previous draft of this paper 208 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques a References de Kleer amp Brown 84 de Kleer J amp Brown J S A Qualitative Physics Based on Confluences Artificial Intelligence 24 7 83 1984 de Kleer amp Bobrow 84 de Kleer J amp Bobrow
16. ehavior exhibiting decreasing oscillations and an infinite set of quiescent behaviors for the system can become over damped each time the block is moving toward its rest position If the system was linear it could not become over damped after the first oscillation but without a linearity assumption these are genuine behaviors Figure 3 4 Tree of behaviors of the damped spring H ATu Hh NM FT oh MT Mif Acc ier Lise Figure 3 5 Decreasing oscillations of the damped spring Velocity 4 Summary Here are two tables that summarize the results of applying these different techniques to the simple and damped springs Figures in bold type face indicate that the number of behavior is correct lThis method is actually not restricted to mechanical systems and is applicable to any second or higher order system 2 06 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 11 1 a ee Or at Danana p a gnore Qdir and Nom Intersection Constraint S5 Open Problems Apparently we are now able to correctly simulate any reasonably complex system without any problem Of course this is not true and some problems still remain Constraint Checking VS Limit Analysis Currently QSIM uses landmarks to both check constraints by the means of corresponding values and perform limit analysis Here are two examples to illustrate that this can cause problems sometimes apa for some reason we have to kn
17. er J ed Readings in Qualitative Reasoning about Physical Systems Morgan Kaufmann Publishers 1989
18. hat even if these distinctions are real we do curvature constraints are always not ambiguous See Kuipers ef al 89 for a complete discussion of the use of Higher Order Derivatives in Qualitative Simulation 13X is a notation used by de Kleer amp Bobrow 84 and represents the sign of X s derivative By extension J X represents the sign of X s n derivative and PX the sign of X 2See Kuipers et al 89 for a discussion of physical situations in which these assumptions may be violated 20l Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 6 b Figure 2 2 Three way branch at time tz not care about them When we describe the behavior of a spring informally we are interested in the position of the block its speed and also whether it is accelerating or decelerating In other words we are just interested in the sign of the acceleration but not its derivative the level of description is not appropriate and we are getting lost in too detailed a behavioral description 2 2 Switching to a Higher Level of Description Ignoring irrelevant distinctions In the preceding section simulation was branching according to the ordering of the following events A reaches a critical value and X crosses zero The former event is not of particular interest and Kuipers amp Chiu 87 developed a method to ignore this irrelevant distinction If the directions of change inc dec and std are collapsed into a single
19. it if the sequence X has a limit when n tends to infinity From the simulation we know that X is a decreasing sequence and that zero is a lower bound Thus X has a limit Let X be this limit Suppose we choose Xlo equal to XI From the definition of a limit we must have X XI for all n The only value Xlo for which X is constant is zero and thus X 0 We plan to generalize and incorporate this kind of reasoning into QSIM Asymptotic behaviors can they be reached in finite time Determining whether this limit can be reached in finite time can be done using the phase plane representation So far the non intersection constraint has been used to check that a trajectory does not intersect itself but it also prohibits intersection of any two trajectories in the same phase portrait if they correspond to different initial conditions Since we know that the asymptotic behavior of the damped spring corresponds to immobility at the rest position which is itself a possible trajectory actually a point we can conclude that the asymptot can never be reached in finite time Doyle and Sacks 88 89 developed a general methodology to interpret trajectories in the phase space representation and certainly some of their techniques could be used in the QSIM framework Is a behavioral property a system property Currently QSIM is not able to determine that some behavioral properties of a system do not change through time For instance if the damp
20. nt properties of industrially significant systems Fouch amp Kuipers 90a 90b However some problems still remain The way QSIM handle correspondences between qualitative values and creates new landmarks is not satisfactory QSIM provides no result about asymptotic behaviors Finally QSIM cannot always determine whether a behavioral property is a system property This is described in section 4 1 Basic Qualitative Simulation The Spring Block system figure 1 1 consists of a block connected to a spring laying on a horizontal table The block position is referenced by a variable X the origin being the rest position The frictionless system will often be referred to as the simple spring and the other as the damped spring Though extremely simple from a structural point of view deriving their behaviors qualitatively has turned out to be challenging We know that the lFor a detailed description of QSIM see Kuipers 86 see Kuipers 89 for a tutorial view 2For instance de Kleer amp Brown 84 Forbus 84 Kuipers 86 Weld 87 Trav amp Dormoy 88 Struss 88 Lee amp Kuipers 88 Ishida 89 L37 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 2 TTT j NAN Id dS eer 3 O x Figure 1 1 The Spring Block System force Fs exerted by the spring on the block is inversely proportional to its elongation X While the relation between Fs and X happens to be linear Hook s law we shall not mak
21. of Current Qualitative Simulation Techniques b Kuipers 89 Kuipers B J Qualitative Reasoning Modeling and Simulation with Incomplete Knowledge Automatica Vol 25 No 4 pp 571 585 1989 Kuipers et al 89 Kuipers B J Chiu C Dalle Molle D Throop D Higher Order Derivative Constraints in Qualitative Simulation Technical Report Al TR 90 116 The University of Texas at Austin 1989 Kuipers et al 87 Lee W W Chiu C Kuipers B J Developments Towards Constraining Qualitative Simulation Technical Report Al TR 87 44 The University of Texas at Austin 1987 Lee amp Kuipers 88 Lee W W amp Kuipers B J l Non Intersection of Trajectories in Qualitative Phase Space A Global Constraint for Qualitative Simulation Proceedings of AAAI 88 286 291 1988 Sacks 87 Sacks E Piecewise Linear Reasonin Proceedings of AAAI 87 655 659 1987 Struss 88 Struss P Global Filters for Qualitative Behaviors Proceedings of AAAI 88 275 27 1988 Throop et al 90 Throop D amp the Qualitative Reasoning Group QSIM Maintainer s Guide Technical Report Al TR 90 124 The University of Texas at Austin 1990 Trav amp Dormoy 88 Trav L amp Dormoy J L Qualitative Calculus and Applications Proceedings of the 12t World Congress on Scientific Computation Paris 1988 Weld 87 Weld D S Comparative Analysis Proceedings of IJCAI 87 959 965 1987 Weld amp de Kleer 89 Weld D S amp de Kle
22. ow the direction of change of the acceleration for the damped spring Simulating it taking into account higher order derivatives and applying energy filtering produces 103 behaviors at time f79 many of them being spurious Figure 5 1 shows one of them where the variables Fp Fs and A only are plotted se ep ee 8 F amp Ac here Lie Figure 5 1 One spurious behavior of the damped spring In this behavior the acceleration A is greater at time tjo than at time t3 But between t3 and tzo Fs has decreased and Fp t3 F r t 9 the add constraint fails to recognize an inconsistency Because at time t3 A s qmag is a not a landmark but an interval no corresponding values were created that would have allowed to detect the inconsistency One way to deal with this is to propagate landmarks across constraints allowing corresponding values to be created and consequently spurious behaviors to be eliminated However these landmarks will be inserted into quantity spaces and QSIM will then take them into account to perform limit analysis As they do not represent real qualitative distinctions this will lead to intractable occurrence branching One could also allow correspondences between intervals This would solve the above problem but may fail in other cases if an interval is refined later in the simulation The appropriate solution is to distinguish the two tasks constraint checking and limit analysis landmarks necessary to check constraints do no
23. t always have to appear in a quantity space As a second example consider a spring for which energy is provided instead of being dissipated Simulation of such a system produces a tree similar to figure 4 2 except that lFor instance consider the following episode qval A 9 lt Aj A2 inc gt qval A t9 t7 lt Ay A2 inc gt qval A t lt A3 std gt with A3 Aj A Att we know that qmag A t 0 Aj A3 207 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 12 branches corresponding to steady or decreasing oscillations are pruned The tree exhibits a three way pattern of branching at each oscillation once QSIM has determined that the block goes further away than the previous maximal position Xp then when the block comes back the ordering of events X crosses Xp and A crosses zero is undetermined leading to occurrence branching But after Xp is crossed a first time Xp is not of particular interest any longer and it should be withdrawn from the quantity space We are currently i P e with new more flexible methods to manage creation and withdrawal of andmarks Asymptotic behaviors what are they Another issue is to determine the asymptotic behavior of oscillatory systems Let us see on the damped spring example how this can be formulated Let X be the maximal value of the position when X is positive in the n cycle Intuitively we can say that the oscillations tend to a lim
24. t work Results Figure 3 2 shows the relative efficiency of the Non Intersection Constraint on the damped spring example Instead of 50 behaviors only 15 are created at time to Note that only one cyclic behavior is now detected One can see that the tree is divided into three main branches at time fg The upper branch corresponds to increasing oscillations the middle to damped oscillations and the lower to stable oscillations It is interesting to compare this tree with the one obtained for the simple spring using the Non Intersection Constraint figure 3 3 The two trees are very similar For the simple spring branching 1One must notice that in the quantitative case only one among all possible couples of independent variables is necessary to check the non intersection property However since reasoning qualitatively implies losing a certain amount of information all possible phase planes must be taken into account to fully capture the Non Intersection Constraint one phase plane may contain information that is not present in another one 204 Fouch amp Kuipers An Assessment of Current Qualitative Simulation Techniques 9 Figure 3 2 Tree of behaviors for the damped spring with NIC until time t o occurs at time fz instead of time ts and the branch corresponding to decreasing oscillations is placed at the top of the tree instead of at the middle Figure 3 3 Tree of behaviors for the simple spring with NIC until time to Let us analyz
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