Proportional-Derivative Controller 

Model the position of a moving body with 1 kg mass and use a sampled proportional-derivative (PD) controller to keep the position at a constant reference value.

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X system = {x''[t] == u[t], x'[0] == x[0] == 0, u[0] == 1}; control = WhenEvent[Mod[t, \[Tau]], u[t] -> kp (xref - x[t] - td x'[t])]; params = {kp -> 1, td -> 1, \[Tau] -> 1, xref -> 1};

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X sol = NDSolve[{system, control} /. params, {x, u}, {t, 0, 12}, DiscreteVariables -> u];

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X Plot[Evaluate[{xref /. params, x[t], u[t]} /. sol], {t, 0, 12}, PlotLegends -> {"xref", "x(t)", "u(t)"}, ImageSize -> Medium]

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