Remotely Control a Cooling Fan 

The model for controlling a ventilation fan can include several delays.

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X {fan, sensing, actuator, com, controller} = {StateSpaceModel[{{{-(1/10)}}, {{1}}, {{1/ 10}}, {{0}}}, SamplingPeriod ->None, SystemsModelLabels -> None], TransferFunctionModel[Unevaluated[{{E^(-s \[Delta])}}], s, SamplingPeriod ->None, SystemsModelLabels -> None], TransferFunctionModel[{{{E^(Rational[-1, 10] s)}}, 1}, s, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[{{{E^(Rational[-1, 2] s)}}, 1}, s, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[Unevaluated[{{(25 + 22 s)/s}}], s, SamplingPeriod ->None, SystemsModelLabels -> None]};

A controller that works on the fan directly may fail when delays are present.

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X y0 = OutputResponse[ SystemsModelFeedbackConnect[ SystemsModelSeriesConnect[controller, fan]], UnitStep[t - 2], {t, 0, 20}]; Plot[y0, {t, 0, 20}, PlotRange -> All]

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X remotesys = SystemsModelSeriesConnect[actuator, SystemsModelSeriesConnect[fan, sensing]]; comsys = SystemsModelSeriesConnect[ SystemsModelSeriesConnect[com, remotesys], com]

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X remotesys = SystemsModelSeriesConnect[actuator, SystemsModelSeriesConnect[fan, sensing]]; comsys = SystemsModelSeriesConnect[ SystemsModelSeriesConnect[com, remotesys], com]; y1 = OutputResponse[ SystemsModelFeedbackConnect[ SystemsModelSeriesConnect[controller, comsys /. \[Delta] -> 1/5]], UnitStep[t - 2], {t, 0, 10}]; Plot[%, {t, 0, 10}, PlotRange -> All]

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The closed-loop fan with a Smith predictor has the same characteristics as the delay-free fan.

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X smith = SmithDelayCompensator[comsys, controller]; cl2 = SystemsModelFeedbackConnect[ SystemsModelSeriesConnect[smith, comsys]];

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X responses = Prepend[Table[ OutputResponse[cl2 /. {\[Delta] -> del}, UnitStep[t - 2], {t, 0, 20}], {del, {1/5, 5, 10, 15}}], y0];

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X Plot[responses, {t, 0, 20}, PlotRange -> All, PlotStyle -> Prepend[Table[Dashed, {4}], Thick], ImageSize -> Medium]

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