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Create a Controller for a Lathe by Approximating the Time Delays

    
The force on the cutting tool in a lathe depends on the cutting depth from the previous rotation of the work piece. This can be modeled as a internal delay in the system:
Wolfram Language code: lathe = StateSpaceModel[10.Derivative[2][x][t] == -100 x[t] - Derivative[1][x][t] - 200 (f[t] + x[t] - x[t - 1]), {x[t], x'[t]}, {f[t]}, x[t], t]
The uncontrolled system is unstable:
Wolfram Language code: OutputResponse[lathe, DiracDelta[t - 1], {t, 0, 5}]; Plot[%, {t, 0, 5}]
Creating a delay-free approximation for a time-delay system allows for all of the standard controller design techniques:
Wolfram Language code: latheApprox = SystemsModelDelayApproximate[lathe, 3]
Design an estimator-regulator by moving the unstable open-loop poles:
Wolfram Language code: openlooppoles = Eigenvalues[Normal[latheApprox][[1]]]; l = EstimatorGains[latheApprox, Join[Drop[Sort[openlooppoles], -2], {-4 + I, -4 - I}]]; k = StateFeedbackGains[latheApprox, Join[Drop[Sort[openlooppoles], -2], {-4 + I, -4 - I}]]; controller = EstimatorRegulator[latheApprox, {l, k}];
The closed-loop system response is stable:
Wolfram Language code: closedloopLathe = SystemsModelFeedbackConnect[lathe, controller];
Wolfram Language code: OutputResponse[closedloopLathe, DiracDelta[t - 1], {t, 0, 20}]; Plot[%, {t, 0, 20}]