Design a Smith Predictor for a Tank Reactor 

An isothermal continuous stirred-tank reactor (CSTR) can be modeled by a first-order-plus-time-delay system.

In[1]:=

X reactor = TransferFunctionModel[Unevaluated[{{E^(-\[Delta] s)/(1 + s)}}], s, SamplingPeriod ->None, SystemsModelLabels -> None];

Designing a PI controller with the delay set to zero results in an unstable closed-loop system.

In[2]:=

X pi = PIDTune[reactor /. \[Delta] -> 0, "PI"]; closedloop = SystemsModelFeedbackConnect[SystemsModelSeriesConnect[pi, reactor]]

Out[2]=

In[3]:=

X pi = PIDTune[reactor /. \[Delta] -> 0, "PI"]; closedloop = SystemsModelFeedbackConnect[SystemsModelSeriesConnect[pi, reactor]]; OutputResponse[closedloop /. \[Delta] -> 2, UnitStep[t - 1], {t, 0, 12}]; Plot[%, {t, 0, 12}, PlotRange -> All, ImageSize -> Medium]

Out[3]=

Use SmithDelayCompensator to create a stabilizing controller for the time-delay system.

In[4]:=

X smith = SmithDelayCompensator[reactor, pi]; closedloopSmith = SystemsModelFeedbackConnect[ SystemsModelSeriesConnect[smith, reactor]]

Out[4]=

In[5]:=

X smith = SmithDelayCompensator[reactor, pi]; closedloopSmith = SystemsModelFeedbackConnect[ SystemsModelSeriesConnect[smith, reactor]]; OutputResponse[closedloopSmith /. \[Delta] -> 2, UnitStep[t - 1], {t, 0, 12}]; Plot[%, {t, 0, 12}, PlotRange -> All, ImageSize -> Medium]

Out[5]=