构建非线性估计器 

使用捕食与被捕食模型构建一个非线性估计器，根据猞猁的种群数量来估计白靴兔的数量. »

模型的输入为  且输出值为 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;

初始种群数值和定期变化的食物供给值.

In[3]:=

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

种群数量也会定期变化.

In[4]:=

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

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该极限环行为可以在参数绘图中观察.

In[5]:=

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

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根据线性模型计算估计器增益集合.

In[6]:=

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

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使用计算的增益构建非线性估计器.

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

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观测值输入为  和 ℓ.

In[8]:=

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

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在初始瞬变后，估计值向实际值收敛.

In[9]:=

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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