lathe=StateSpaceModel[10.x''[t]==-100 x[t]- 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}]

latheApprox=SystemsModelDelayApproximate[lathe,3]

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}];

closedloopLathe=SystemsModelFeedbackConnect[lathe,controller];

OutputResponse[closedloopLathe,DiracDelta[t-1],{t,0,20}];
Plot[%,{t,0,20}]