Spatial Aggregation: Modeling and Controlling Physical Fields

Christopher Bailey-Kellogg and Feng Zhao

Many important physical phenomena, such as temperature distribution, air flow, and acoustic waves, are described as continuous, distributed parameter fields. Controlling and optimizing these physical processes and systems are common design tasks in many scientific and engineering domains. However, the challenges are multifold: distributed fields are conceptually harder to reason about than lumped parameter models; computational methods are prohibitively expensive for complex spatial domains; the underlying physics imposes severe constraints on observability and controllability. This paper develops an ontological abstraction and an aggregation-disaggregation mechani.qm, in a f~amework collectively known as spatial aggregation (SA), for reasoning about and synthesizing distributed control schemes for physical fields. The ontological abstraction models physical fields as networks of spatial objects. The aggregation-disaggregation mechanism employs a set of data types and generic operators to find a feasible control structure, specifying control placement and associated actions that satisfy given constraints. SA abstracts common computational patterns of control design and optimization in a small number of operators to support modular programming; it builds concise and articulable structural descriptions for physical fields. We illustrate the use of the SA ontological abstraction and operators in an example of regulating a thermal field in industrial heat treatment.

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