Construct a Nonlinear Estimator 

Construct a nonlinear estimator using a predator-prey model to estimate the snowshoe hare population based on the lynx population. »

The input to the model is  and the output is ℓ.

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X pars = {\[ScriptR] -> 2.1, \[ScriptA] -> 3.2, \[ScriptB] -> 0.4, \[ScriptC] -> 50, \[ScriptD] -> 0.55, \[Kappa] -> 150};

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X sys = AffineStateSpaceModel[ {{-((\[ScriptA] \[ScriptH] \[ScriptL])/(\ \[ScriptC] + \[ScriptH])) + \[ScriptH] \[ScriptR] (1 - \[ScriptH]/\ \[Kappa]), -\[ScriptD] \[ScriptL] + (\[ScriptA] \[ScriptB] \[ScriptH] \ \[ScriptL])/(\[ScriptC] + \[ScriptH])}, {{ \[ScriptH] (1 - \ \[ScriptH]/\[Kappa]) }, {0 }}, {\[ScriptL]}, {{ 0 }}}, {{\[ScriptH], \ -((\[ScriptC] \[ScriptD])/(-\[ScriptA] \[ScriptB] + \[ScriptD]))}, {\ \[ScriptL], (\[ScriptB] \[ScriptC] \[ScriptR] (-\[ScriptC] \[ScriptD] \ + \[ScriptA] \[ScriptB] \[Kappa] - \[ScriptD] \ \[Kappa]))/((-\[ScriptA] \[ScriptB] + \[ScriptD])^2 \[Kappa])}}, \ {Subscript[\[FormalU], 1]}, {Automatic}, Automatic , SamplingPeriod -> None] /. pars;

The values of the initial population and a periodically varying food supply.

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X ic = {80, 25}; \[ScriptF] = 1.1 + Cos[2 \[Pi] t];

The populations also vary periodically.

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X sr = StateResponse[{sys, ic}, \[ScriptF], {t, 0, 40}]; Plot[sr, {t, 0, 40}, PlotLegends -> {"hare", "lynx"}]

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This limit-cycle behavior can also be seen on a parametric plot.

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X ParametricPlot[sr, {t, 0, 40}, Frame -> True, FrameLabel -> {"hare", "lynx"}]

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Compute a set of estimator gains based on the linearized model.

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X \[ScriptG] = EstimatorGains[StateSpaceModel[sys], {-1, -2}]

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Construct the nonlinear estimator using the computed gains.

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X \[ScriptCapitalL] = StateOutputEstimator[sys, \[ScriptG]]

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The observer inputs are  and ℓ.

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X \[ScriptCapitalU] = Join[{\[ScriptF]}, OutputResponse[{sys, ic}, \[ScriptF], {t, 0, 30}]]

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After initial transients, the estimated value converges to the actual one.

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X Subscript[\[ScriptH], estim] = OutputResponse[\[ScriptCapitalL], \[ScriptCapitalU], {t, 0, 30}][[ 1]]; Plot[{sr[[1]], Subscript[\[ScriptH], estim]}, {t, 0, 5}, PlotLegends -> {"Actual", "Estimated"}]

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