Connecting Time-Delay Systems 

When two time-delay systems are connected in series, the delays can sum in the resulting system.

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X {tfm, ssm} = {TransferFunctionModel[ Unevaluated[{{E^(-5 s)/(1 + 2 s)}}], s, SamplingPeriod ->None, SystemsModelLabels -> None], StateSpaceModel[{{{0, 1}, {-3, -(2/3)}}, {{0}, {SystemsModelDelay[ 7/3]}}, {{1, 0}}, {{0}}}, SamplingPeriod ->None, SystemsModelLabels -> None]};

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X series = SystemsModelSeriesConnect[tfm, ssm] // Simplify

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X Plot[Evaluate[ Table[OutputResponse[sys, UnitStep[t], {t, 0, 20}], {sys, {tfm, ssm, series}}]], {t, 0, 20}, PlotLegends -> {"tfm", "ssm", "series"}]

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Time delays can sum in a feedback connection as well.

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X feedback = SystemsModelFeedbackConnect[ssm, tfm] // Simplify

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X Plot[Evaluate[ Table[OutputResponse[sys, UnitStep[t], {t, 0, 20}], {sys, {tfm, ssm, feedback}}]], {t, 0, 20}, PlotLegends -> {"tfm", "ssm", "feedback"}]

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