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FindSystemModelEquilibrium

FindSystemModelEquilibrium[model]

搜索模型 model 的平衡点.

FindSystemModelEquilibrium[model,{{{x₁,x₁₀},…},{{u₁,u₁₀},…},{{y₁,y₁₀},…}}]

从点 x_i=x_i0、u_i=u_i0 和 y_i=y_i0 开始搜索平衡点.

FindSystemModelEquilibrium[model,{x₁v₁,…},…]

搜索平衡点，其中变量 x_i 被限制为值 v_i 等.

范例

打开所有单元关闭所有单元

基本范例 (3)

从初始值开始搜索，找出平衡点：

Wolfram Language code: FindSystemModelEquilibrium[[image]]

用给定的状态初始值：

Wolfram Language code: FindSystemModelEquilibrium[[image], {{{"H", 55}, {"Tu", 60}, {"Tx", 60}}, {}, {}}]

找出已有 introductory hierarchical 中一个实例的平衡点：

Wolfram Language code: model = Last@SystemModelExamples["Models", "IntroductoryExamples.Hierarchical.*"]

Wolfram Language code: FindSystemModelEquilibrium[model]

范围 (3)

给出状态、输入和输出初始值：

Wolfram Language code:

FindSystemModelEquilibrium[[image], {{{"H", 0.75}, {"Tu", 38}, {"Tx", 38}}, {{"u1", 5.23}, {"u2", 3.08}}, {{"y1", 5.23}, {"y2", 3.08}}}]

限制输入和输出，给出状态的初始值：

Wolfram Language code: FindSystemModelEquilibrium[[image], {"u1" == 5.23, "y1" == 3.75}, {{{"H", 0.75}, {"Tu", 38}, {"Tx", 38}}, {}, {}}]

找出限制条件下的平衡点：

Wolfram Language code: FindSystemModelEquilibrium[[image], {"u1" == 5.23, "u2" == 3.08}]

选项 (1)

ProgressReporting (1)

使用 ProgressReporting 控制进度报告：

Wolfram Language code: FindSystemModelEquilibrium[[image], ProgressReporting -> False]

应用 (5)

找出一个同时有水流入和流出的水池的平衡点：

Wolfram Language code: FindSystemModelEquilibrium[[image], {"h" == 1}]

在平衡点附近线性化模型：

Wolfram Language code: model = [image];

Wolfram Language code: {state, input, output} = FindSystemModelEquilibrium[model, {"H" == 0.75, "Tu" == 38, "Tx" == 38}];

Wolfram Language code: SystemModelLinearize[model, {state, input}]

在平衡点附近线性化模型并分析稳定性：

Wolfram Language code: model = [image];

Wolfram Language code: {state, input, output} = FindSystemModelEquilibrium[model, {"H" == 0.75, "Tu" == 38, "Tx" == 38}];

Wolfram Language code: ss = SystemModelLinearize[model, {state, input}];

Wolfram Language code: NyquistPlot[ss, PlotRange -> Full]

Wolfram Language code: Eigenvalues[ss[[1, 1]]]

设计一个 PI 控制器，以保持同时有流入和流出的池中的液位不变：

Wolfram Language code: model = [image];

将液位 "h" 限制为 2，找出平衡点：

Wolfram Language code: h0 = 2;

Wolfram Language code: {state, input, output} = FindSystemModelEquilibrium[model, {"h" == h0}]

线性化并完成 PI 控制器的闭环部分：

Wolfram Language code: tf = TransferFunctionModel@SystemModelLinearize[model, {state, input}];

Wolfram Language code: piTf = TransferFunctionModel[kp (1 + 1 / (Subscript[τ, i] )), ];

Wolfram Language code: cltf = SystemsModelFeedbackConnect[SystemsModelSeriesConnect[piTf, tf]];

显示一类 PI 控制器的闭环步进响应：

Wolfram Language code:

Manipulate[
	Plot[Evaluate[h0 + OutputResponse[cltf /. {kp -> gain, Subscript[τ, i] -> iTime}, UnitStep[t], {t, 20}]], {t, 0, 20}, PlotStyle -> Thick, GridLines -> Automatic, Frame -> True, PlotRange -> {h0, 4}], 
	{{gain, 1.4, "SubscriptBox[k, p]"}, 0.01, 10}, {{iTime, 1.6, "SubscriptBox[τ, i]"}, 0.01, 10}, SaveDefinitions -> True]

摆动到任意角度的单摆：

单摆垂直垂下时的平衡点：

Wolfram Language code: model = [image];

Wolfram Language code: FindSystemModelEquilibrium[model]

竖直向上、位于轴线之上的单摆：

Wolfram Language code: FindSystemModelEquilibrium[model, {{{"x", -3.1}, {"v", 0}}, {}, {}}]//Chop

Wolfram Language code: FindSystemModelEquilibrium[model, {{{"x", +3.1}, {"v", 0}}, {}, {}}]//Chop

首次积分的 level 曲线给出系统的势能：

Wolfram Language code: model["SystemEquations"]

Wolfram Language code: fi = (v ^ 2/2) - 9.81Cos[x];

单摆有一个稳定平衡点，有两个不稳定平衡点：

Wolfram Language code:

Show[
	Plot3D[fi, {x, -2π, 2π}, {v, -2.5, 2.5}, MeshFunctions -> {#3&}, AxesLabel -> Automatic, Mesh -> 5, PlotPoints -> 50], Graphics3D[{Blue, PointSize[Large], Point[{{0, 0, -9.2}, {-Pi, 0, 10}, {Pi, 0, 10}}]}]]