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Use Delayed Feedback to Reduce Oscillations

Time delays can be intentionally added to controllers to improve system performance. The active vibration absorber below can be controlled using delayed state feedback.

Wolfram Language code: {eq1, eq2} = {u[t] + (Subscript[k, 1] + Subscript[k, 2]) Subscript[x, 1][t] - Subscript[k, 2] Subscript[x, 2][t] + (Subscript[b, 1] + Subscript[b, 2]) Derivative[1][Subscript[x, 1]][t] - Subscript[b, 2] Derivative[1][Subscript[x, 2]][t] + Subscript[m, 1]Derivative[2][Subscript[x, 1]][t] == 0, -u[t] + Subscript[k, 2] (-Subscript[x, 1][t] + Subscript[x, 2][t]) + Subscript[b, 2] (-Derivative[1][Subscript[x, 1]][t] + Derivative[1][Subscript[x, 2]][t]) + Subscript[m, 2] Derivative[2][Subscript[x, 2]][t] == 0};
Wolfram Language code: absorber = StateSpaceModel[{eq1, eq2}, {Subscript[x, 1][t], Subscript[x, 1]'[t], Subscript[x, 2][t], Subscript[x, 2]'[t]}, {u[t]}, {Subscript[x, 1][t]}, t];
Wolfram Language code: csys = SystemsModelStateFeedbackConnect[absorber, {{g SystemsModelDelay[τ]}}, {1}, {1}]
The closed-loop vibration absorber has smaller oscillations near the resonant frequency with a delayed controller than with a delay-free controller:
Wolfram Language code: values = {Subscript[k, 2] -> .1, Subscript[b, 2] -> .02, Subscript[m, 2] -> .1, Subscript[m, 1] -> 1, Subscript[b, 1] -> .05, Subscript[k, 1] -> 1, g -> 0.043};
Wolfram Language code: ynodelay = OutputResponse[csys /. values /. τ -> 0, Sin[ 1.1 t], {t, 0, 80}]; ydelay = OutputResponse[csys /. values /. τ -> 1.82, Sin[1.1t], {t, 0, 80}];
Wolfram Language code: Plot[{ynodelay, ydelay}, {t, 0, 80}, PlotRange -> All, PlotStyle -> {Purple, Thick}, PlotLegends -> {"Delay–Free", "Delayed"}, ImageSize -> Medium]