Evaluate a Linear Controller Design Using Nonlinear Simulation 

Starting with a nonlinear model, design a linear controller and evaluate the performance of the controller on the nonlinear system.

The model of an inverted pendulum is nonlinear, but linear in the input force .

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X eqns = {(M + m) x''[t] + m l Sin[\[Theta][t]] \[Theta]'[t]^2 - m l Cos[\[Theta][t]] \[Theta]''[t] == F[t], l \[Theta]''[t] - x''[t] Cos[\[Theta][t]] - g Sin[\[Theta][t]] == 0};

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X pars = {l -> 1, m -> 0.2, M -> 0.3, g -> 9.8};

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X pend = AffineStateSpaceModel[ eqns, {\[Theta][t], x[t]}, {F[t]}, {\[Theta][t], x[t]}, t]

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Design a controller based on the linear model.

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X k = StateFeedbackGains[ StateSpaceModel[pend /. pars], {-2, -3 + I, -3 - I, -4}]

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Connect the linear controller to the nonlinear model.

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X csys = SystemsModelStateFeedbackConnect[pend, k]

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Simulate the nonlinear closed-loop system.

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X res = OutputResponse[{csys /. pars, {40 \[Degree], 0, 0, 0}}, 0, {t, 0, 4}];

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X Plot[res, {t, 0, 4}, PlotRange -> All, PlotLegends -> {\[Theta][t], x[t]}]

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