Solving Complexity and Ambiguity Problems within QualitativeSimulationbyDaniel

chattering region and successor states 4) (a) chatter{box algorithm ° . . . . . ↑ . . . . . ↑ . . . . . ↑ . . . . . ° INF A-1 0 T0 T1 T2 amount (B) ° . . . . . ↑ . . . . . ↑ . . . . . ↑ . . . . . ° INF A-2 0 T0 T1 T2 amount (C) ↑ . . . . . ↑↓ . . . . . ↓ . . . . . ↓ . . . . . ° INF 0 MINF T0 T1 T2 d amount (B) ° . . . . . ↑ . . . . . ° . . . . . ↓ . . . . . ° INF N-1 0 MINF T0 T1 T2 d amount (C) (b) W-tube behavior plots In this example, the potentially chattering variables include Netflow-AC, Crossflow-BC, Delta-BC, Netflow-B, Netflow-A, Crossflow-AB, and Delta-AB (see gure 4.7). Due to the application of HOD constraints during the focused envisionment, NetflowB is the only variable that actually exhibits chatter. In the focused envisionment graph, cycles are represented by dotted lines. The chattering region is abstracted into a single state in the main simulation, represented by a 2. The qdir in the chattering region is represented by a bi-directional arrow in the behavior plot. Figure 4.6: chatter{box abstraction algorithm applied to the W-tube. 96 current state. To handle chatter around landmarks, the algorithm expands the potentially chattering region to include QSIM introduced landmarks and landmarks identi ed as chatter landmarks within the QDE. Analysis of the QDE Kuipers et al. (Kuipers et al., 1991) present an algorithm for identifying potentially chattering variables based strictly upon the constraints within the model. This technique is used in the automatic derivation of HOD constraints. We have extended the algorithm to include qualitative state information when appropriate and to reason about the full set of constraints provided by QSIM. Previously, the algorithm only handled a sub{set of QSIM's constraints. There are two main steps to the algorithm: Step 1 Partition the variables into chatter equivalence classes. Step 2 Identify equivalence classes that cannot chatter. Variables that are not included within non-chattering equivalence classes are considered to be potentially chattering. De nition 4.5 (Chatter equivalence) Variables V1 and V2 are considered chatter equivalent within a region of the state space if and only if the sign of V 0 1 is uniquely determined by the sign of V 0 2 and vice-a-versa. The identi cation of chatter equivalence classes is based upon the observation that two variables are chatter equivalent if they are related by a monotonic function constraint (e.g. M+ or M ) or a re nement of a monotonic function constraint (e.g. ADD(x; y; z) if either x, y, or z are constant). Note that the S+ and S constraints4 only apply the monotonicity restriction within a particular region of the state space. Thus, the current qualitative state must be used to determine if two variables related by an S+ or an S constraint are chatter equivalent. Kuipers et al. (1991) gives a complete listing of the conditions under which two variables can be identi ed as chatter equivalent. A chatter equivalence class is identi ed as non-chattering if either a variable in the class is constant or its derivative is explicitly represented in the model. The latter restriction, as discussed below, is relaxed if a variable's derivative is identi ed as potentially chattering around zero. Figure 4.7 demonstrates the detection of potentially chattering variables for the W-tube example. 4The S constraints describe a saturation function in which two variables, x and y, are monotonically related whenever x is within a region identi ed by two landmarks speci ed in the constraint. When x is outside this region, y is constant. 97 Inflow−A Amount−A Delta−AB Netflow−A Pressure−A Crossflow−AB ADD ADD