Abstract
In a previous study we provided analytical and experimental evidence that some materials are able to store entropy-flow, of which the heat-conduction behaves as standing waves in a bounded region small enough in practice. In this paper we continue to develop distributed control of heat conduction in these thermal-inductive materials. The control objective is to achieve subtle temperature distribution in space and simultaneously to suppress its transient overshoots in time. This technology concerns safe and accurate heating/cooling treatments in medical operations, polymer processing, and other prevailing modern day practices. Serving for distributed feedback, spatiotemporal H ∞ /μ control is developed by expansion of the conventional 1D-H ∞ /μ control to a 2D version. Therein 2D geometrical isomorphism is constructed with the Laplace-Galerkin transform, which extends the small-gain theorem into the mode-frequency domain, wherein 2D transfer-function controllers are synthesized with graphical methods. Finally, 2D digital-signal processing is programmed to implement 2D transfer-function controllers, possibly of spatial fraction-orders, into DSP-engine embedded microcontrollers.
Highlights
In the works [1,2] and the references cited therein, we learned that many materials possess measurable entropy-flow storage, and that heat conduction in these thermally inductive materials behaves as a standing wave whenever the heat conduction region is small enough to some extent of engineering practice
Tracking of spatially subtle distributions of temperature under feedback control with pointed actuation will always be accompanied by temporally abrupt transients, resulting in unsafe or inaccurate heating/cooling treatments in medical operations, polymer processes, and other prevailing modern day practices
Distributed control is capable of tracking precise local-temperatures with slow heating over the entire region of heat conduction, which is otherwise beyond the nature of pointed control
Summary
In the works [1,2] and the references cited therein, we learned that many materials possess measurable entropy-flow storage, and that heat conduction in these thermally inductive materials behaves as a standing wave whenever the heat conduction region is small enough to some extent of engineering practice. An infinite-dimensional transfer function from pointed input to pointed output can further be identified in the frequency domain with fraction order [24,25,26,27,28], serving for 1DH∞/μ feedback loopshaping These dimension-infinitely elegant tools were not originally developed for distributed sensing and actuation. The Laplace-Galerkin transform [1,38,39,40] or Fourier-Galerkin transform [41,42] is justified to model the non-Fourier heat conduction and its controllers for bounded space regions that are of real concern in heat conduction practice Such an integral transform is obtained through the composite of Laplace transform in time and modal decomposition in space. For convenient computation in 2D or 3D space regions, the
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