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.

In[1]:=

X lathe = StateSpaceModel[ 10. (x^\[Prime]\[Prime])[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]

Out[1]=

The uncontrolled system is unstable.

In[2]:=

X lathe = StateSpaceModel[ 10. (x^\[Prime]\[Prime])[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]; OutputResponse[lathe, DiracDelta[t - 1], {t, 0, 5}]; Plot[%, {t, 0, 5}]

Out[2]=

Creating a delay-free approximation for a time-delay system allows for all of the standard controller design techniques.

In[3]:=

X latheApprox = SystemsModelDelayApproximate[lathe, 3]

Out[3]=

Design an estimator-regulator by moving the unstable open-loop poles.

In[4]:=

X 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.

In[5]:=

X closedloopLathe = SystemsModelFeedbackConnect[lathe, controller];

In[6]:=

X closedloopLathe = SystemsModelFeedbackConnect[lathe, controller];; OutputResponse[closedloopLathe, DiracDelta[t - 1], {t, 0, 20}]; Plot[%, {t, 0, 20}]

Out[6]=