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A Neutral Time-Delay System

    
Connecting a first-order-plus-time-delay model with a PID controller gives a neutral time-delay system. Neutral time-delay systems have delays on the highest-order term in the denominator.
Wolfram Language code: {foptd, pid} = {TransferFunctionModel[Unevaluated[{{(1. E^-s/1 + 0.125s)}}], s, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[1 + (1/10 s) + (s/8), s]};
Wolfram Language code: nsys = SystemsModelFeedbackConnect[foptd, pid]
When the input has a discontinuity, it is repeated at each multiple of the delay:
Wolfram Language code: OutputResponse[nsys, UnitStep[t - 1], {t, 0, 10}]; Plot[{%, UnitStep[t - 1]}, {t, 0, 10}, PlotStyle -> {Automatic, Dashed}]
Wolfram Language code: OutputResponse[nsys, Cos[t], {t, 0, 10}]; Plot[{%, Cos[t]}, {t, 0, 10}, PlotStyle -> {Automatic, Dashed}]
Wolfram Language code: OutputResponse[nsys, SquareWave[t / 8.5], {t, 0, 10}]; Plot[{%, SquareWave[t / 4]}, {t, 0, 10}, PlotStyle -> {Automatic, Dashed}]