Estimator Design Using Feedback Linearization 

Feedback linearization also allows for the design of nonlinear estimators using linear techniques. Estimate the yaw angle and the yaw rate for a ship with hull length and speed , based on Norrbin's model and using feedback linearization. »

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X pars = {Subscript[n, 3] -> 1, Subscript[n, 1] -> 1, Subscript[n, 2] -> 0, Subscript[n, 0] -> 0};

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X asys = AffineStateSpaceModel[(2 l/v) \[Psi]''[t] + Subscript[n, 3] \[Psi]'[t]^3 + Subscript[n, 2] \[Psi]'[t]^2 + Subscript[n, 1] \[Psi]'[t] + Subscript[n, 0] == (v/l) \[Delta][t], \[Psi][t], \[Delta][t], \[Psi][t], t] /. pars /. {l -> 45}

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The model is completely feedback linearizable.

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X \[ScriptCapitalF] = FeedbackLinearize[asys]

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Compute a set of estimator gains using pole placement.

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X EstimatorGains[\[ScriptCapitalF]["LinearSystem"], {-2 + I, -2 - I}]

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Assemble the nonlinear estimator.

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X EstimatorGains[\[ScriptCapitalF]["LinearSystem"], {-2 + I, -2 - I}]; \[ScriptL] = \[ScriptCapitalF][{"OriginalSystemEstimator", %}]

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Plot the estimated state trajectories for a specific ship speed.

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X inp = -UnitStep[t] + UnitStep[t - 1]; ics = {-2, 1};

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X esr = OutputResponse[ SystemsModelDelete[\[ScriptL] /. v -> 5, None, -1], Join[{inp}, OutputResponse[{asys /. v -> 5, ics}, inp, {t, 0, 10}]], {t, 0, 10}];

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X Plot[esr, {t, 0, 10}, PlotStyle -> Dashed, PlotRange -> All, PlotLegends -> {"Yaw (\[Psi])", "Yaw rate (\[Psi]')"}]

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Compute the actual state trajectories.

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X sr = StateResponse[{asys /. v -> 5, ics}, inp, {t, 0, 10}]

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Compare the actual and estimated yaw angles and yaw rates.

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X Table[Plot[{sr[[i]], esr[[i]]}, {t, 0, 5}, PlotStyle -> {Automatic, Dashed}, PlotLegends -> {"Actual", "Estimated"}, PlotRange -> All, PlotLabel -> Switch[i, 1, "Yaw Angle", _, "Yaw rate"]], {i, {1, 2}}]

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