Spatial Aggregation:Modeling and controlling physical fields

Christopher Bailey-Kellogg Feng ZhaoDepartment of Computer and Information Science

The Ohio State University2015 Nell Avenue

Columbus, OH 43210 U.S.A.{kellogg, fz}@cis, ohio-state, edu

Abstract

Many important physical phenomena, such as temper-ature distribution, air flow, and acoustic waves, aredescribed as continuous, distributed parameter fields.Controlling and optimizing these physical processesand systems are common design tasks in many sci-entific and engineering domains. However, the chal-lenges are multifold: distributed fields are concep-tually harder to reason about than lumped parame-ter models; computational methods are prohibitivelyexpensive for complex spatial domains; the underly-ing physics imposes severe constraints on observabilityand controllability.This paper develops an ontological abstraction andan aggregation-disaggregation mechani.qm, in a f~ame-work collectively known as spatial aggregation(SA), for reasoning about and synthesizing distributedcontrol schemes for physical fields. The ontologicalabstraction models physical fields as networks of spa-tial objects. The aggregation-disaggregation mecha-nism employs a set of data types and generic oper-ators to find a feasible control structure, specifyingcontrol placement and associated actions that satisfygiven constraints. SA abstracts common computa-tional patterns of control design and optimization ina small number of operators to support modular pro-gramming; it builds concise and articulable structuraldescriptions for physical fields. We illustrate the useof the SA ontological abstraction and operators in anexample of regulating a thermal field in industrial heattreatment.

Keywords. Qualitative reasoning; Spatial reason-ing; Ontologies; Decentralized control; Distributed AI;Programming languages.

IntroductionContinuous, distributed parameter fields are commonphysical phenomena: consider the temperature field ina building, the air flow around an airplane wing, or thenoise from a copy machine. There are enormous prac-tical benefits to reasoning about and controlling thesephysical processes and systems. For instance, the dragon an airplane can be reduced by analyzing and con-trolling the air flow around the wings. Temperature in

a "smart" building can be regulated to maximize oc-cupant comfort while minimizing energy consumption.Because of the rapid advance in micro-fabrication tech-nology that can integrate and produce micro-electro-mechanical system (MEMS) devices on a massive scale,we are becoming increasingly reliant on large networksof sensors, actuators, and computational elements toaugment our ability to interact with and control thephysical environment (Berlin 1994).

However, there is enormous challenge in controllingand optimizing physical fields using networks of sen-sors and controllers. The difficulties arise from threesources. First, a distributed parameter field is con-ceptuaUy harder to reason about and model than alumped parameter system such as a circuit. Spatialtopology, metric, material properties and physical lawsall come into play, in addition to the combinatorialstructures. The underlying physical processes mightbe nonlinear and defy analytic, closed form solution.Second, numerical methods developed for designingand controlling physical fields require solving large sys-tems of equations and hence are prohibitively expen-sive for large, irregular geometric domains and highlynon-uniform, nonlinear phenomena. Third, physicallaws constrain the ability of spatially distributed, localagents to sense and affect the environment (Seidman1996). Local sensors and control elements measureand interact with small neighborhoods around them.Macroscopic consequences are aggregated from localactions. Consequently, the design, programming, andcoordination of distributed computational agents im-mersed in physical media require abstraction mecha-nisms, inference methods, and programming languagesdifferent from those for reasoning about and control-ling centralized, lumped parameter models.

In this paper, we develop an ontological abstractionfor physical fields and an aggregation-disaggregationmechanism for reasoning about and synthesizing dis-tributed control schemes for physical fields. The onto-logical abstraction encodes neighborhood relations andequivalence classes and describes the structure and be-havior of a field in multiple layers of spatial objects.The aggregation-disaggregation mechanism finds a lea-

From: AAAI Technical Report WS-97-11. Compilation copyright © 1997, AAAI (www.aaai.org). All rights reserved.

sible control design for a physical field specifying con-trol placement and associated actions that meet givendesign objectives; it employs spatial object, field,and ngraph data types and generic operators such asaggregate, disaggregate, classify, interpolate,redescribe, and update that manipulate and trans-form objects. The ontological abstraction, genericoperators, and aggregation-disaggregation mechanismsare collectively called spatial aggregation (SA); have implemented the spatial aggregation language(SAL) to support modular programming in this frame-work. SA transforms higher-level control objectivesinto local control actions using divide-and-conquer, ex-ploiting physical constraints of fields manifested as lo-cality, continuity, and spatial and temporal scales. Weillustrate the use of the SA ontological abstraction andoperators in an example of temperature regulation of athermal field from an industrial heat treatment prob-lem. This work has significantly extended the pre-viously developed SA framework for data interpreta-tion (Yip & Zhao 1996; Bailey-Kellogg, Zhao, & Yip1996) to address new problems arising from control andoptimization of distributed parameter physical fields.

SA differs from existing numerical design and op-timization methods in several important ways. First,numerical methods build a discretization for a physicalfield. In contrast, SA constructs an explicit structuraldescription for the field from a numerical discretiza-tion, measurements, or other sources. This structuraldescription can be abstracted and articulated to sup-port a variety of inference, explanation, and tutoringtasks. Second, numerical methods optimize parame-ter values of distributed control problems, while SAsolves for structural design problems, specifying con-trol configurations including the number and locationsof controls. Third, SA performs local computation onnetworks of spatial objects, without the need to con-struct an explicit, global model. Numerical methodstypically require a global model for a field. Finally, SAencapsulates many pieces of domain-specific knowledgein its generic operators, while hiding low-level details.

A Spatial Aggregation Model ofPhysical Fields

Fields abstract a wide range of continuous, distributedphysical phenomena such as temperature, velocity, andimage data. Formally, a field maps one continuum toanother. A temperature field in a room maps a loca-tion to temperature, i.e., R3 -~ R1. Many importantreasoning and controlling tasks for physical fields re-quire compact structural and behavioral descriptionsof the fields.

Consider a problem from manufacturing: heat treat-ment of a metal sheet (Jaluria & Torrance 1986), shown in Figure 11. As part of the manufacturing pro-

1A similar problem arises from temperature control insemiconductor manufacturing, in which the oven temper-

/ "i "/Figure 1: Industrial heat treatment of a metal sheet.The control objective is to heat the material to adesired temperature at a small number of locations,shown as dark circles, so as to minimize temperaturefluctuation across the material.

cess, the temperature distribution over the sheet mustbe regulated at some desired profile over a period oftime to minimize damage to the material. To describethe thermal process in the metal sheet, we need tospecify physical laws, geometry, material properties,and boundary conditions. In addition, the control ob-jective specifies a desired thermal profile over the ma-terial.

Physical laws governing fields are described by par-tial differential equations. For instance, the physicalprocess of isotropic, steady-state heat diffusion is gov-erned by the Poisson equation:

kV2@ + ¢ = 0 (1)

where V2 is the Laplace operator, @ is the temperature,

k is material conduction coefficient, and Q is the sourcevalue representing heat per unit time and volume. ThePoisson equation also governs other physical phenom-ena such as gravity and electrostatics. In fluid dynam-ics, the motion of fluid is governed by the Navier-Stokesequation, another well-known partial differential equa-tion. Traditionally, detailed, labor-intensive numericalsimulations are employed to elaborate consequences ofthese models. In contrast, our objective is to constructa qualitative physics model for spatial physical fieldsso that behaviors of the fields can be inferred using asmall number of operations on a discrete representa-tion and explained in terms of object interaction andevolution.

A physical field exhibits multiple temporal and spa-tial scales due to physical properties of the domain,geometry, boundary conditions, and external forces.Mechanical vibrations travel at the speed of sound ina piece of material. Temperature decays with varyingrates along different directions, determined by materialproperty and geometric shape. Acoustic waves mix andreflect, according to wave lengths and boundary con-ditions. The scales and locality permit a continuous,distributed parameter field to be abstracted as a setof discrete objects by suppressing irrelevant details.Consequently, the field can be understood by inter-

ature over a surface of semiconductor wafer must be reg-ulated by a set of spatially distributed heating lamps toensure high yield. This is a challenging control problem inrapid thermal processing (RTP) of semiconductor wafersbecause temperature non-uniformity often leads to chip de-fects (Kailath & others 1996).

Figure 2: Temperature distribution in the dumbbell-shaped material, heated at the centers of heavy ends.The temperature gradient is described by isotherms.

preting a compact structural and behavioral represen-tation. For instance, after examining the thermal fieldin dumbbell-shaped material heated at the heavy ends(Figure 2), we discover that the isotherms are moresparse along the narrow channel of the dumbbell, indi-cating a very small temperature decay. Intuitively, theheat flux is constrained by the narrow geometry of thechannel. More formally, physics tells us this directioncorresponds to the weakest coupling direction of thetemperature field s. In order to control the field usinga collection of decentralized sources, the field must bedecoupled into constituent regions. We need to developabstractions and operators that formalize the physicalintuition so as to explicitly model field granularity andpermit principled and programmed trade-offs amongdifferent design choices.

SA provides the field ontology that represents aphysical field as a network of spatial objects (Yip Zhao 1996; Bailey-Kellogg, Zhao, & Yip 1996). Eachspatial object comprises a geometric description for itsspatial extent and a feature description for its values.For instance, a spatial object in a temperature fielddescribes a location and the temperature at that loca-tion. The spatial objects are governed by interactionrules relating objects to each other, along with con-straints on feature values. The major elements of theSA ontology are summarized in Table 1. The field on-tology is specialized for the control and optimizationtasks discussed in this paper. We deploy two partic-ularly useful field discretizations that represent a fieldeither as a grid of points (Figure 3(a)) or as a angular mesh of elements (Figure 3(b)). Each tial object abstracts geometry and physics over a finiteregion. A neighborhood graph encodes spatial adja-cencies among the objects. Local constraints, derivedfrom a finite difference approximation to the Laplaceoperator at grid points or conservation properties overelements in a mesh, govern the evolution of the objects.The field values are determined by a local relaxationmethod that iteratively updates spatial objects usingthe local constraints. With these two field partitions,the SA abstraction mechanism subsumes the finite dif-ference and finite element models commonly used inengineering and science for numerically approximatinga field (Vichnevetsky 1981). We note that SA sup-ports many important reasoning and controlling tasks

2To be precise, this gives the smallest positive eigenvalueof the Laplacian graph of the field (Spielman & Teng 1996).

¯ Spatial Objects- Geometric description

- Feature descriptionExamples: temperature; pressure; velocity

¯ Constitutive LawsExamples:- Fourier’s law: heat.flux = -k~T-,T

- Ohm’s law: charge-flux = 7dv- -~__ du- Hooke’s law: stress - E~

¯ Spatial Neighborhood StructuresExamples:- Minimal spanning tree- Regular grid neighborhood graph- Delaunay mesh neighborhood graph

¯ Interaction RulesExamples:- Causal interaction- Consistency rules- Update rules

Table 1: Spatial Aggregation field ontology.

for physical fields, in addition to approximating thefield. SA can also take advantage of existing numericalanalysis softwares.

Building upon the spatial object abstraction, SAprovides an aggregation-disaggregation mechanism forreasoning about structures in control and optimiza-tion of physical fields. Figure 4 illustrates the bi-directional mapping between lower- and higher-levelobjects within each spatial aggregation layer. SA em-ploys a sequence of such bi-directional mappings tomediate the input fields and higher-level compact de-scriptions. In the aggregation process, SA operatorsaggregate, classify, and redescribe build a hierar-clay of increasingly more abstract spatial object neigh-borhood graphs (N-graphs) and equivalence classes.During disaggregation, the operators disaggregateand flatten open up higher-level aggregate objectsto permit control over finer component objects of thefield. The interpolate operator transfers informationbetween coarser and finer objects4. Additional opera-tors act on the objects, fields, and N-graphs, to search,map, filter, update, form correspondences, and main-tain consistency.

For control and optimization tasks, the input to SA

aWe have already incorporated the widely used numer-ical linear algebra software package LAPACK in SAL.

4The new SA operators disaggregate, interpolate,and update are introduced here for the purpose of control.

3

(a)

(b)

Figure 3: A field modeled by a network of local spatialobjects: (a) Finite difference grid; (b) Finite elementmesh. The grid or mesh defines spatial adjacency be-tween a spatial object (solid circle) and its neighbors(empty circles). The shaded region in the mesh repre-sents spatial extent of the object. Higher-dimensionalfields can be likewise modeled.

Higher-Leve! Oblects

I Oi~gg~g~

Search) ConsL~ncy-check

Search

Low~-Level Oblocts

Figure 4: Spatial aggregation and disaggregation sup-port bidirectional mapping between lower- and higher-level spatial objects. SA aggregates lower-level ob-jects to form a neighborhood graph, classifies theminto equivalence classes, and redescribes the classesas higher-level objects. Reciprocally, SA disaggregateshigher-level objects into component objects and formsa neighborhood graph of the objects.

is a physical field modeled as a collection of spatial ob-jects, a control objective, and a set of optimality con-straints. SA determines a spatially distributed controlthat effectively steers the physical process to meet thedesired criteria s. To determine the structure and pa-rameters of the distributed control, i.e., the number,location, and values of control sources, one needs tosearch the large design space subject to performanceconstraints. Although many numerical methods existfor parametric design problems that optimize controlvalues, structural design that determines the numberand locations of distributed control has remained anad hoc practice.

Illustration: Control of a Thermal FieldAs an example of how spatial aggregation operatorssupport modular programming for control applica-tions, consider the temperature regulation problem fora rectangular piece of material, similar to that of Fig-ure 1. The temperature field is represented by thermalspatial objects (therm_obj) indicating locations andtemperature values. The control objective is to estab-lish a uniform temperature distribution over the entirefield, using a small set of discrete heat sources, sub-ject to constraints on the number of control sources,maximum source output, and acceptable temperaturefluctuations. This global control objective can be for-mulated locally by constraining each thermal objectto have a temperature within some error toleranceof its desired temperature. The available control au-thority consists of point sources, each of which regu-lates temperature in a local neighborhood. The de-sign task is to determine the number, locations, andvalues of heat sources required to satisfy the objec-tive. A separate field represents control sources withspatial objects (src_obj) indicating locations, controlvalues, and (possibly overlapping) regions of tempera-tures that each is attempting to control. Neighborhoodgraphs explicate spatial adjacencies among thermal ob-jects and among source objects.

The control design is accomplished by an iterativeaggregation-disaggregation process. Given a sourceconfiguration, aggregation groups sources according tolocality and similarity in control authority, and re-places a group with a more global source if the re-placement adequately controls the combined controlregions. Disaggregation splits a single source that doesnot adequately control its region, replacing it with mul-tiple more localized sources that better control sub-regions. An aggregation-based approach repeatedlyaggregates sources, starting from a dense set, whilea disaggregation-based approach repeatedly disaggre-gates sources, starting from a sparse set. The trade-offs

5A dual problem concerns placement of sensors for max-imum observability. We will only consider the controllabil-ity problem in this paper and note that the observabilityproblem can be likewise solved.

4

while control error is too largefor each unsatisfactory control region r

disaggregate:generate candidate control partitions:

find weak coupling directionspartition roptimize control locations/values

replace control with best configurationupdate field values

Figure 5: Algorithm for disaggregation-based controlconfiguration design.

between aggregation and disaggregation include thetype of knowledge required (initial configurations; howto group vs. how to split) and the complexity (poten-tial amount of work at each level; fan-in vs. fan-out).An even more advanced strategy interleaves aggrega-tion and disaggregation so as to exploit the advantagesof both.

The spatial aggregation language providesaggregate, classify, and redescribe opera-tors for hierarchically building source aggregates(Bailey-Kellogg, Zhao, & Yip 1996). For brevity,we omit the code for aggregation-based design,concentrating instead on the disaggregation-basedapproach.

Figure 5 outlines the algorithm for disaggregation-based design. Disaggregation-based design is an it-erative process, repeatedly checking the control errorand disaggregating unsatisfactory sources. To disag-gregate a source, multiple possible disaggregations areconsidered, and the one with the best error profile ischosen. To form disaggregations, we make use of thephysical knowledge discussed in the preceding sectionand illustrated in Figure 2: small gradients indicateweak coupling. By placing new sources along directionsof weak coupling, a design can optimally "divide-and-conquer" the problem. The remainder of this sectionillustrates how the spatial aggregation language sup-ports the implementation of this algorithm, providingoperators at the right level of abstraction and encap-sulating the domain-specific physical knowledge.

The spatial aggregation operator update specifies it-erative processes by repeatedly applying user-specifiedupdate rules until convergence is reached. The updaterule for source disaggregation checks whether a givensource adequately controls its region and disaggregatesit if the error is too large (see Figure 6(a)). Note the algorithm presented here is greedy, but could beextended to maintain multiple possibilities in a searchtree, or to allow backtracking through a dependencynetwork.

The disaggregal;e operator replaces a spatial objectwith a set of constituent objects, exploring the struc-ture of the domain and taking advantage of locality

updaSe_eource_by_disaggregation(src_obj)if error(region(src_obj)) > max_error

new_src_objs = disaggregate(src_obj)replace(source_field, src_obj,

new_src_objs)

(a)partition_ctrl_region(src_obj)

ctrl_reg s region(src_obj)if area(ctrl_reg) < min_area

return empty_setelse

subregs = decouple_along_gradient(ctrl.reg)new_src_obj_sets =

map(subregs,create_src_objs_at_sample_points)

returncreate_sets_with_one_member_from_each

(new_src_obj_sets)

(b)update_source_by_gradient_descent(src_obj)

grad = gradient(error_in_region, src_obj)if abe(grad) > min_change

new_ctrl_val -ctrl_value(src_obj) + stop.size*grad

set_c~rl_value(src_obj, new_ctrl_val)

(c)update_value_by_fd(therm_obj)

old_temp - temp(therm_obj)new_temp ~

avg(neighbors(therm_ngraph, therm_obj))+ alpha*source.at(position(therm_obj)))

if abs(old_temp - new_temp) > min_changeset_temp(therm.obj, new_temp)

(d)refine(therm_obj)

for_each W, S, SW nbr inneighbors(therm_ngraph, ~herm.obj)

new_pos - midpoint(posi~ion(therm_obj),position(nbr))

new_temp - interpolate(new_poe, therm_ngraph,temperature))

add(therm_field,create_therm_obj(new_pos, nev_temp))

(e)

Figure 6: User-level SA pseudocode for disaggregation-based temperature regulation design. (a) Update rulefor disaggregating an unsatisfactory source. (b) Gen-eration of partitions for disaggregation, using physi-cal knowledge. (c) Update rule for optimizing sourcevalue. (d) Update rule for evaluating thermal field us-ing finite difference approximation. (e) Update rule formultigrid-style refinement of temperature field.

(a)

(b)

(c)

Figure 7: Control a thermal field by successive sourcedisaggregations, exploiting gradient information. Darkcircles indicate source positions, vertical lines delineateregions of control, and curves describe isotherms.

and scales determined by geometry and physics. Differ-ent implementations use different user-specified knowl-edge to help disaggregate an object. In Figure 6(b), user-defined partition function divides a source’s con-trol region into subregions, decoupling along the gra-dient as discussed above. It then places new sources atvarious points in the new regions and forms the crossproduct of the sets of possible new sources, returningsets of possible disaggregations with one new sourcefor each subregion. Disaggregate tests the resultingerror in each partition to choose an optimal disaggre-gation for a source. Figure 7 shows source locationsand isotherms after the first few disaggregations forthe temperature regulation problem. The decouplingalgorithm used here splits in half along the weaker cou-pling axis when a source does not adequately controlits region. More advanced disaggregation would parti-tion the field using more refined knowledge of gradientsand other scales; the next section illustrates how spa-tial aggregation operators can support such knowledge.

The initial values of the new sources in Figure 6(b)can be determined by proportioning the parent source’svalue according to the areas. Aa alternative is to ap-proximate the required source density by applying fi-nite differences to the control region, and then scalingby the area the source covers. In either case, the ini-tial guess can likely be improved. A local adjustmentmethod, such as gradient descent, can be specified interms of an update rule. Figure 6(c) provides code foran update rule to adjust the value of a source in or-der to minimize the error in its control region. Theposition of a source can be likewise adjusted.

To measure the quality of control, the temperaturefield is computed for the specified source configuration.A local relaxation method is specified by the update

! ! : : ! !: i i .~. ,.~... : : ii i i : ! i i

Figure 8: Spatial object refinement using a finite differ-ence grid approximation. New objects (empty circles)are inserted between old objects (solid circles); dottedlines show the finer grid.

operator. The rule in Figure 6(d) updates a thermalobject based on values of neighbors and any sourcelocated gt that point. Different implementations ofupdate update objects sequentially or in parallel; inthis case, that yields the Ganss-Seidel or Jacobi nu-merical relaxation method.

The computational efficiency of the tempera-ture solver could be improved by a multi-levelmethod (Briggs 1987) that uses a solution for coarsespatial objects as an initial guess for a solution for finerspatial objects, as shown in Figure 8. The code in Fig-ure 6(e) refines a coarse thermal object, adding pointsbetween the object and its west, south, and southwestneighbors in a finite difference grid. The interpolateoperator fills in temperatures for the new thermal ob-jects based on values of neighboring objects in the grid.Various implementations of inl;erpolate support lin-ear interpolation, quadratic interpolation, and so forth.

In summary, the spatial aggregation operators pro-vide powerful building blocks for developing programsfor distributed control problems of thermal fields. Theoperators modularize user-specified knowledge and ab-stract away many low-level implementation details.Table 2 summarizes the SA operators and their imple-mentations currently available in the C+÷-based com-piler for SAL. The disaggregation-based design pro-gram runs reasonably efficiently in SAL: using a 500-object field discretization, 8 candidate source locationsper disaggregation, and fairly strict convergence crite-ria, it takes 1 to 10 minutes on a 100MHz Pentiumto design the source configuration plotted in Figure 7,depending on the constraints placed on source values.

/k More Complex ExampleAs we have illustrated for a simple rectangular domain,the spatial aggregation-disaggregation approach to de-centralized control exploits the locality of a field to finda rough solution by divide-and-conquer and then lo-cally refine the control placement and actions. Control-ling complex spatial domains requires more detailedknowledge about the field. In particular, SA operatorsare employed to extract information about the thermalgradient.

For example, consider the problem in Figure 9(a)"an irregularly-shaped metal sheet has a hole near itsright end and a single heat source near the center ofmass. Figure 9(b) shows temperature gradient field di-

¯ Disaggregate: object -~ objectsRefines object into a set of component objects.-Generate a single refinement, using user-

provided generator- Enumerate multiple refinements and choose the

best one, using user-provided generator and se-lector

- Generate multiple refinements as needed andchoose the first satisfactory one, using user-provided stream generator and tester

¯ Interpolate: geometry ̄ objects ¯ value_function--+ valueDetermines value for geometry based on values ofobjects given by value~unction.- Average of members of a set- Locally weighted average in a field- Average of neighbors in an ngraph

¯ Update: objects * update_rule --+ objectsModifies objects by repeatedly applyingupdate_rule to each.- Object collection: a set or field- Temporal order: sequential or parallel- Convergence criteria: fixed number of times or

until predicate satisfied¯ Existing SA data types and operators:

- Spatial object, field, ngraph-Aggregate, classify, redescribe, search,

map, filter

Table 2: SA data types and operators.

/

(b)I I l i

(c)

Figure 9: A complex thermal domain: (a) Geometryand heat source location; (b) Directions of the resultingtemperature gradient field; (c) Gradient trajectoriesaggregated by SA operators.

rections (a gradient vector is normal to the isothermcurve) computed from a pointwise description of thetemperature. The gradient vectors are then groupedinto equivalence classes and redescribed as trajectoriesusing the SA operators classify and redescribe, asshown in Figure 9(c). Average gradient magnitudesalong trajectories through the source are compared tofind directions of weaker decay; the disaggregate op-erator splits source objects along such directions as pre-viously discussed.

DiscussionThe SA aggregntion-disaggregation mechanism modu-larizes common computational ideas in data interpre-tation and control tasks and permits reuse of genericoperators. Using these operators, a problem can be de-scribed at a level closer to one’s intuitive understand-ing of the physics. The resulting programs are eas-ier to understand and modify. The multi-layer spatialaggregates can be used for automatically generating

7

explanations for why higher-level control decisions aremade. For instance, the field geometry might induce aweak gradient direction; hence, SA refines spatial ob-jects along that direction in order to decouple the field.In contrast, traditional numerical simulations requirehumans to interpret and explain the results.

Traditionally, qualitative physics has focused onlumped parameter models of physical systems (DeK-leer & Brown 1984; Forbus 1984; Kuipers 1986). Forinstance, the device ontology describes an electricalcircuit as a network of components, while abstract-ing away the spatial dimension. However, many im-portant physical phenomena are modeled as spatiallydistributed parameter systems. The SA field ontologydescribes such physical phenomena by encapsulatingthe important spatial information in the spatial objectsand the neighborhood structures, permitting efficientreasoning about these phenomena.

Unlike control design for lumped parameter linearsystems, few analytic design techniques have beendeveloped for distributed control of large physicalfields (Sandell Jr. et al. 1978). In practice, the designis often accomplished by brute-force numerical simu-lations. SA offers a powerful modeling framework andan alternative mechanism for synthesizing decentral-ized control. The multi-resolutional SA model is par-ticularly useful for formulating structural design prob-lems that are not well addressed by numerical methods.In addition, the SA mechanism can integrate existingnumerical techniques to solve parametric design prob-lems.

We believe SA is applicable to a wide variety of prob-lems in data interpretation and control. For instance,the parti-game algorithm for learning control strate-gies in high-dimensional state-spaces can be formulatedin terms of SA operators (Moore & Atkeson 1995).Similarly, Bradley and Zhao presented several meth-ods for synthesizing nonlinear control laws in phasespaces (Bradley & Zhao 1993); their methods partitionphase spaces into manageable subspaces. SA can alsodescribe adaptive aggregation in dynamic program-ming where states and their dependencies are groupedinto meta-states according to residual errors (Bertsekas& Castanon 1989).

We have described steady-state control problems fora class of physical fields such as heat conduction, grav-ity, electrostatics, and incompressible fluid flow thatare governed by elliptic partial differential equations.To apply SA to transient problems, we need to de-velop techniques for tracking and correlating temporalobjects. Another important class of problems involveswave phenomena (such as sound) governed by hyper-bolic partial differential equations. Extending SA toaddress wave control problems remains as a future re-search topic.

Other researchers have developed related frame-works and systems for reasoning about spatial, ana-logue representations of the physical world. Williams

and Nayak discussed model-based methods for model-ing, configuring and programming distributed physicalsystems (Williams & Nayak 1996). Lundell presenteda qualitative model for distributed parameter physi-cal fields (Lundell 1996). Forbus et al. developed theMetric Diagram/Place Vocabulary (MD/PV) frame-work for qualitative spatial reasoning (Forbus, Nielsen,& Faltings 1991). In computer scene analysis, Hum-mel and Zucker formalized a class of problems calledrelaxation labeling that computes a consistent label-ing for a network of objects (Hummel & Zucker 1983).In comparison to the above work, the SA frameworkfocuses on structure recovery and control of spatialfields. It differs from relaxation labeling in that SAbuilds multi-layer spatial aggregates to exploit a vari-ety of spatial and temporal constraints. It also differsfrom many neural net based learning and optimizationmethods in that SA explicitly models and exploits fieldtopological structures. This paper develops SA mech-anisms for controlling physical fields that significantlyextend the framework developed in (Yip & Zhao 1996;Bailey-Kellogg, Zhao, & Yip 1996).

Conclusions

This paper advances the state-of-the-art in qualita-tive physics and spatial reasoning in several ways. Ithas developed the SA ontological abstraction for dis-tributed parameter physical fields and an aggregation-disaggregation mechanism for synthesizing structuresand parameters for distributed control of the fields. Ithas illustrated how a modular program can be writtenusing the SA generic operators for the control and op-timization of a thermal field regulation problem. SAextracts meaningful structures from continuous fieldsso that design decisions can be articulated using dis-crete spatial objects. Furthermore, because spatial ag-gregates comprise mixed numeric, geometric, and sym-bolic descriptions, we expect SA to take advantageof and complement the arsenal of existing numericalmethods.

This research addresses several scientific and engi-neering concerns central to artificial intelligence. Weexpect SA to shed light on bi-directional mappingsbetween macroscopic decisions and local actions inbiological and engineered systems. In addition, SAaims to systematize design principles and program-ming methodologies for interpreting and controllingdistributed parameter physical fields, and to provideuseful tools for practicing engineers.

Acknowledgment

We thank the following people for helpful discussions:Andy Berlin, Geoff Chase, Tad Hogg, Xingang Huang,Shiou Loh, Jeff May, Pandu Nayak, Ivan Ordonez, E1-isha Sacks, Gerry Sussman, Brian Williams, and KenYip.

This research is supported in part by an NSF Na-tional Young Investigator Award CCR-9457802, an Al-

fred P. Sloan Foundation Research Fellowship, a grantfrom Xerox Palo Alto Research Center, and an AT&Tgrant.

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