Static and Dynamic Abstraction Solves the Problem of Chatter in Qualitative Simulation

chattering region and successor states Final Behavior Tree Focused Envisionment Graph Before chatter abstraction Behavior Tree Potentially chattering region The links in the gure represent processes and the objects represent data structures. Two states linked by a dotted line in the envisionment graph represent the same qualitative state. Using chatter box abstraction, QSIM generates a single behavior. The chatter box state is represented by a square in the behavior tree. HOD constraints are used to eliminate chatter in all of the variables except NetflowB. Figure 3: Chatter box abstraction algorithm applied to the W-tube. Theorem 2 The set of real{valued trajectories described by a chatter box abstracted behavioral description is equal to the set described by an unabstracted tree with respect to the non{chattering variables and a super{set with respect to the chattering variables. Dynamic Chatter Abstraction Chatter box abstraction explores the entire chattering region of the state space via simulation. The number of states within this region is exponential in the number of chattering variables. Thus, exploring this region can become intractable as the size of a model grows and the number of unconstrained variables increases. Dynamic chatter abstraction provides a scalable solution by avoiding the need to perform a focused envisionment. Instead, the chattering variables within a region of the state space are identi ed by a dynamic analysis of the model and the current state. For each time{interval state, dynamic chatter abstraction determines whether the state is contained within a chatter box and if so identi es the set of variables that chatter within this region. It uses an understanding of the restrictions that are asserted by each constraint within the model along with the current qualitative state to perform this task. For example, in the W-tube model if HOD constraints are not used, the direction of change for NetflowB is restricted only by the constraint NetflowB + Flow-BC = Flow-AB In the time{interval following the initial state, all three variables are increasing. Thus, in this state the qdir of NetflowB is unconstrained and NetflowB is free to chatter. Dynamic chatter abstraction, however, must reason not only about the qualitative values contained within the current state, but also about how these values change as variables begin to chatter. Once NetflowB chatters, its derivative can change sign and the above addition constraint no longer restricts the derivative of Flow-AB. If Flow-AB is not prevented from chattering by other constraints, it is free to chatter and must be identi ed as such within the current chatter box. A more detailed presentation of the algorithm is contained in (Clancy and Kuipers, 1997b). Identifying the set of chattering variables The algorithmmust determine if a variable v can chatter following the current time{interval state before a non{chattering distinction occurs. If such a variable exists, then the current state is contained within a chatter box. To address this issue, two questions must be answered: Consistency Is there a consistent state in which v is free to chatter? Reachability Can this state be reached from the current state only through changes occurring in other chattering variables? Such a state is called chatter{reachable. To answer these questions, dynamic chatter abstraction uses a chatter dependency graph to de ne the conditions under which a variable can chatter with respect to the current state. Two types of nodes exist within the chatter dependency graph (see gure 4). An equivalency node is created for each set of chatter equivalent variables within the model2. These nodes are connected through a directed AND-OR subgraph of intermediate dependency nodes. For an equivalency node EQsource, the leaves of the AND-OR subgraph correspond to equivalency nodes containing variables related to the variables within EQsource via a constraint within the model. The AND-OR subgraph can be viewed as a predicate that speci es the conditions under which the variables in EQsource are free to chatter. A consistent, chatter{reachable state satisfying this predicate exists if and only if the variables in EQsource chatter within the current region of the state space. Evaluating the dependency graph The dependency graph evaluation algorithm categorizes chatter equivalency classes as chattering, non-chattering, or chatter-unknown. The algorithm iterates through the equivalency classes moving them from the 2Prior to the simulation, variables are partitioned into chatter equivalency classes as in the chatter box abstraction algorithm. NetflowA (dec) Flow−AB (inc) Delta−AB (inc) EQ−1