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SystemModelLinearize

SystemModelLinearize[model]

给出均衡 model 的线性化的 StateSpaceModel.

SystemModelLinearize[model,op]

在运算点 op 处进行线性化.

范例

打开所有单元关闭所有单元

基本范例 (3)

在平衡点线性化一个直流电机模型：

Wolfram Language code: SystemModelLinearize[[image]]

在给定状态和输入限制条件下，在平衡点线性化一个搅拌池模型：

Wolfram Language code: SystemModelLinearize[[image], {{{"H", 0.75}}, {{"u1", 5.23}}}]

线性化包含的 introductory hierarchical 示例中的一个模型：

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

Wolfram Language code: SystemModelLinearize[model]

范围 (19)

模型类型 (5)

线性化文本形式的 RLC 电路模型：

Wolfram Language code: SystemModelLinearize[[image]]

线性化 RLC 电路框图模型：

Wolfram Language code: SystemModelLinearize[[image]]

线性化因果 RLC 电路：

Wolfram Language code: SystemModelLinearize[[image]]

线性化一个 DAE 模型：

Wolfram Language code: SystemModelLinearize[\!\(\*GraphicsBox[«8»]\)]

符号式线性化一个 DAE 模型：

Wolfram Language code:

SystemModelLinearize[\!\(\*GraphicsBox[«8»]\), Method -> {"SymbolicDerivative", "SymbolicParameters" -> {"R1", "R2", "L"}}]

极限情况 (3)

线性化一个没有输入的模型：

Wolfram Language code: SystemModelLinearize[[image]]

线性化一个没有状态的模型：

Wolfram Language code: SystemModelLinearize[[image]]

线性化一个没有输出的模型：

Wolfram Language code: SystemModelLinearize[[image]]

线性化值 (5)

在平衡点处线性化：

Wolfram Language code: SystemModelLinearize[[image], "EquilibriumValues"]

在初值处线性化：

Wolfram Language code: SystemModelLinearize[[image], "InitialValues"]

在给定状态和输入限制条件下，在平衡点处进行线性化：

Wolfram Language code:

SystemModelLinearize[[image], {{{"H1", 2.03}, {"T1", 318.15}, {"T2", 318.15}}, {{"u1", 5}, {"u2", 5}}}, "EquilibriumValues"]

在给定状态限制条件下，在平衡点处进行线性化：

Wolfram Language code: SystemModelLinearize[[image], {{{"H1", 5}, {"T1", 318.15}, {"T2", 318.15}}, {}}, "EquilibriumValues"]

在给定部分状态和输入的情况下进行线性化，剩下的值则使用初值：

Wolfram Language code: SystemModelLinearize[[image], {{{"H1", 3.03}, {"T1", 318.15}, {"T2", 318.15}}, {{"u1", 5}, {"u2", 5}}}, "InitialValues"]

运算点 (6)

均衡附近的线性化：

Wolfram Language code: SystemModelLinearize[[image], "EquilibriumValues"]

初始值附近的线性化：

Wolfram Language code: SystemModelLinearize[[image], "InitialValues"]

带有给定状态和输入限制的均衡附近的线性化：

Wolfram Language code:

SystemModelLinearize[[image], {{{"H1", 2.03}, {"T1", 318.15}, {"T2", 318.15}}, {{"u1", 5}, {"u2", 5}}}, "EquilibriumValues"]

来自于仿真的初始值附近的线性化：

Wolfram Language code: sim = SystemModelSimulate[[image], All, {0, 0}]

Wolfram Language code: SystemModelLinearize[[image], {sim, "StartTime"}]

来自于仿真的最终值附近的线性化：

Wolfram Language code: sim = SystemModelSimulate[[image], All]

Wolfram Language code: SystemModelLinearize[[image], {sim, "StopTime"}]

来自于平稳状态仿真的最终值附近的线性化：

Wolfram Language code: sim = SystemModelSimulate[[image], All, Method -> {"StopAtSteadyState" -> True}]

Wolfram Language code: SystemModelLinearize[[image], {sim, "StopTime"}]

推广和延伸 (1)

隐藏所得 StateSpaceModel 中的标签：

Wolfram Language code: SystemModelLinearize[[image], SystemsModelLabels -> None]

选项 (5)

Method (4)

"SymbolicDerivative" 方法使用来自于仿真的完全指定的运算点：

Wolfram Language code: sim = SystemModelSimulate[[image], All, {0, 0}]

Wolfram Language code: SystemModelLinearize[[image], sim, Method -> "SymbolicDerivative"]

"NumericDerivative" 方法使用由状态和输入点指定的运算点：

Wolfram Language code: SystemModelLinearize[[image], {{{"capacitor.v", 0.}, {"inductor.i", 0.}}, {{"v", 0.}}}, Method -> "NumericDerivative"]

符号线性化运行保持某些符号参数：

Wolfram Language code: SystemModelLinearize[[image], Method -> {"SymbolicDerivative", "SymbolicParameters" -> {"R", "L", "C"}}]

用 "ReduceIndex" 在符号线性化时关闭指标约简：

Wolfram Language code: sim = SystemModelSimulate[[image], All, {0, 0}];

关闭指标约简得到一个描述 StateSpaceModel：

Wolfram Language code: SystemModelLinearize[[image], sim, Method -> {"SymbolicDerivative", "ReduceIndex" -> False}]

ProgressReporting (1)

用 ProgressReporting 控制进度汇报：

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

应用 (10)

分析线性系统 (5)

比较一个模型在平衡点处的响应和线性化：

Wolfram Language code: model = [image];

Wolfram Language code: {Subscript[x, 0], Subscript[u, 0], Subscript[y, 0]} = FindSystemModelEquilibrium[model];

在平衡点处进行线性化：

Wolfram Language code: ss = SystemModelLinearize[model, "EquilibriumValues"];

将平稳输出响应与非线性模型进行比较：

Wolfram Language code: {linY1, linY2} = OutputResponse[ss, {UnitStep[t], UnitStep[t]}, {t, 0, 300}];

Wolfram Language code:

nonLin = SystemModelSimulate[model, {0, 300}, <|"InitialValues" -> Rule@@@Subscript[x, 0], "Inputs" -> {"u1" -> Function[t, Subscript[u, 0][[1, 2]] + UnitStep[t]], "u2" -> Function[t, Subscript[u, 0][[2, 2]] + UnitStep[t]]}|>];

比较 y1 输出：

Wolfram Language code: Show[SystemModelPlot[nonLin, {"y1"}], Plot[Tooltip[Subscript[y, 0][[1, 2]] + linY1, "linear y1"], {t, 0, 300}]]

根据系统矩阵的特征值测试线性化系统的稳定性：

Wolfram Language code: ss = SystemModelLinearize[[image], "InitialValues"]

由于存在具有正实部的特征值，所以该系统是不稳定的：

Wolfram Language code: {a, b, c, d} = Normal[ss];

Wolfram Language code: Re@Eigenvalues[a]

输出响应曲线也表明系统不稳定：

Wolfram Language code: Plot[Evaluate[OutputResponse[ss, UnitStep[t], {t, 1}]], {t, 0, 1}]

根据传递函数的极点测试线性化系统的稳定性：

Wolfram Language code: tf = TransferFunctionModel@SystemModelLinearize[[image]];

由于存在具有正实部的极点，因此系统不稳定：

Wolfram Language code: TransferFunctionPoles[tf]

使用线性模型进行频率分析：

Wolfram Language code: model = \!\(\*GraphicsBox[«8»]\);

通过绘制线性化传递函数的进行频率分析：

Wolfram Language code: ss = SystemModelLinearize[model];

Wolfram Language code: BodePlot[ss, {0.5, 3}, PlotLayout -> "Magnitude"]

对仿真数据使用 Fourier 以验证结果：

Wolfram Language code: nonLin = SystemModelSimulate[model, {0, 200}, <|"Inputs" -> {"u" -> Function[t, UnitStep[t] - UnitStep[t - 2]]}|>];

根据计算：

Wolfram Language code:

odata = Abs@Fourier@Table[nonLin[{"inertia3.w"}, t][[1]], {t, 0, 200, 0.25}];
idata = Abs@Fourier@Table[UnitStep[t] - UnitStep[t - 2], {t, 0, 200, 0.25}];

Wolfram Language code:

ListLogLinearPlot[odata[[1 ;; 200]] / idata[[1 ;; 200]], Joined -> True, DataRange -> {0, 2Pi}, PlotRange -> {{0.5, 3}, {0, 50}}]

另外，特征值的虚部给出共振峰值：

Wolfram Language code: {a, b, c, d} = Normal[ss];

Wolfram Language code: Im[Eigenvalues[a]]

线性化发生在时间点 0 处：

Wolfram Language code: model = [image];

开关在时间点 0 时连上，进行线性化：

Wolfram Language code: model2 = SystemModel[model, {"t" -> 0}];

Wolfram Language code: rlc = SystemModelLinearize[model2]

如果开关没有在时间点 0 时连上，结果会不同：

Wolfram Language code: model3 = SystemModel[model2, {"t" -> 1}];

Wolfram Language code: rl = SystemModelLinearize[model3]

线性系统的控制器设计 (5)

用线性化模型设计一个 PID 控制器：

Wolfram Language code: tf = TransferFunctionModel@SystemModelLinearize[[image]];

定义一个 PID 控制器和闭环传递函数：

Wolfram Language code: pidTf = TransferFunctionModel[kp (1 + 1 / (τi ) + τd ), ];

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

选择 PID 参数以获得合适的阶跃响应：

Wolfram Language code:

Manipulate[
	Plot[OutputResponse[cltf /. {kp -> gain, τi -> iTime, τd -> dTime}, UnitStep[t], {t, 20}]//Evaluate, {t, 0, 20}, PlotStyle -> Thick, GridLines -> Automatic, Frame -> True, PlotRange -> {0, 1.2}], 
	{{gain, 1, "SubscriptBox[k, p]"}, 0.01, 10}, {{dTime, 0.01, "SubscriptBox[τ, d]"}, 0.01, 10}, {{iTime, 1.9, "SubscriptBox[τ, i]"}, 0.01, 10}, SaveDefinitions -> True]

基于其线性化设计一个直流电机的 lead-based 控制器：

Wolfram Language code: tf = TransferFunctionModel@SystemModelLinearize[[image]];

定义 PI-lead 控制器的传递函数：

Wolfram Language code: piLeadTf = TransferFunctionModel[kp (1 + 1 / τi (τd  + 1) / (α τd  + 1)), ];

开环传递函数：

Wolfram Language code: optf = SystemsModelSeriesConnect[piLeadTf, tf];

选择控制器参数：

Wolfram Language code:

Manipulate[BodePlot[newOptf = optf /. {kp -> Kp, τi -> Τi, τd -> Τd, α -> Α}, StabilityMargins -> True, PlotLayout -> "List"], 
	{{Kp, 4.5, "SubscriptBox[k, p]"}, 0.01, 10}, {{Τi, 2, "τi"}, 0.01, 10}, {{Τd, 0.01, "τd"}, 0.01, 10}, {{Α, 0.07, "α"}, 0.01, 0.9}, SaveDefinitions -> True]

运用所选参数，并用 PI-lead 控制器完成闭环部分：

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

Wolfram Language code:

Plot[OutputResponse[cltf, UnitStep[t], {t, 20}]//Evaluate, {t, 0, 20}, PlotStyle -> Thick, GridLines -> Automatic, Frame -> True, PlotRange -> All]

通过极点配置设计控制器：

Wolfram Language code: ss = SystemModelLinearize[[image]];

放置闭环极点：

Wolfram Language code: k = StateFeedbackGains[ss, {-0.251, -0.222, -0.222}]

计算闭环状态-空间模型：

Wolfram Language code: clss = SystemsModelStateFeedbackConnect[ss, k];

显示阶跃响应：

Wolfram Language code: Plot[OutputResponse[clss, {UnitStep[t], UnitStep[t]}, {t, 100}]//Evaluate, {t, 0, 100}, PlotRange -> All]

设计一个 LQ 控制器：

Wolfram Language code: ss = SystemModelLinearize[[image]];

定义状态和输入加权矩阵：

Wolfram Language code: {q1, q2} = {DiagonalMatrix[{1, 6 10^-4, 6 10^-4}], DiagonalMatrix[{0.0002, 0.0002}]};

定义 LQ 控制器的增益：

Wolfram Language code: k = LQRegulatorGains[ss, {q1, q2}]

闭环状态-空间模型：

Wolfram Language code: clss = SystemsModelStateFeedbackConnect[ss, k];

闭环阶跃响应：

Wolfram Language code: Plot[OutputResponse[clss, {UnitStep[t], UnitStep[t]}, {t, 100}]//Evaluate, {t, 0, 100}, PlotRange -> All]

设计一个状态估计器：

Wolfram Language code: model = [image];

Wolfram Language code: ss = SystemModelLinearize[model];

计算估计器增益和估计器状态-空间模型：

Wolfram Language code: l = EstimatorGains[ss, {-0.6020, -0.5856, -0.5697}]

Wolfram Language code: {a, b, c, d} = Normal[ss];

Wolfram Language code: ssobsv = StateSpaceModel[{a - l.c, ArrayFlatten[{{b - l.d, l}}], IdentityMatrix[3]}];

单位阶跃输入的状态和输出响应：

Wolfram Language code: sresp = StateResponse[ss, {UnitStep[t], UnitStep[t]}, {t, 200}];

Wolfram Language code: y = OutputResponse[ss, {UnitStep[t], UnitStep[t]}, {t, 200}];

观察状态响应：

Wolfram Language code: oresp = OutputResponse[ssobsv, {UnitStep[t], UnitStep[t], y[[1]], y[[2]]}, {t, 200}];

比较每个状态及其估计值：

Wolfram Language code: states = model["StateVariables"];

Wolfram Language code: Table[Plot[{sresp[[i]], oresp[[i]]}, {t, 0, 200}, PlotLabel -> states[[i]]], {i, 3}]

属性和关系 (7)

使用 SystemModel 的属性在初值点进行线性化：

Wolfram Language code: model = [image];

Wolfram Language code: ival = model["GroupedInitialValues"]

比较结果：

Wolfram Language code: SystemModelLinearize[model, ival, "InitialValues"] == SystemModelLinearize[model, "InitialValues"]

用 FindSystemModelEquilibrium 在平衡点进行线性化：

Wolfram Language code: model = [image];

Wolfram Language code: eqval = FindSystemModelEquilibrium[model];

比较结果：

Wolfram Language code: SystemModelLinearize[model, eqval] == SystemModelLinearize[model, "EquilibriumValues"]

在平衡点比较模型的响应和线性化：

Wolfram Language code: model = [image];

Wolfram Language code:

{Subscript[x, 0], Subscript[u, 0], Subscript[y, 0]} = FindSystemModelEquilibrium[model, {"H" == 0.85, "Tu" == 38, "Tx" == 38}]

在平衡点处线性化：

Wolfram Language code: ss = SystemModelLinearize[model, {{{"H", 0.85}, {"Tu", 38}, {"Tx", 38}}, {}}, "EquilibriumValues"]

将平稳输出响应与非线性模型进行比较：

Wolfram Language code: {linY1, linY2} = OutputResponse[ss, {UnitStep[t], UnitStep[t]}, {t, 0, 300}];

Wolfram Language code:

nonLin = SystemModelSimulate[model, {0, 300}, <|"InitialValues" -> Rule@@@Subscript[x, 0], "Inputs" -> {"u1" -> Function[t, Subscript[u, 0][[1, 2]] + UnitStep[t]], "u2" -> Function[t, Subscript[u, 0][[2, 2]] + UnitStep[t]]}|>];

比较第一个输出：

Wolfram Language code: Show[SystemModelPlot[nonLin, {"y1"}], Plot[Evaluate@Tooltip[Subscript[y, 0][[1, 2]] + linY1, "linear y1"], {t, 0, 300}]]

用 TransferFunctionModel 转换为传递函数表示：

Wolfram Language code: SystemModelLinearize[[image]]

Wolfram Language code: TransferFunctionModel[%]

用 ToDiscreteTimeModel 离散化线性化模型：

Wolfram Language code: SystemModelLinearize[[image]]

用采样间隔进行离散化：

Wolfram Language code: ToDiscreteTimeModel[%, 0.10, z]

线性化的状态空间模型不是唯一的：

Wolfram Language code: linearize1 = [image];

Wolfram Language code: linearize1["ModelicaDisplay"]

Wolfram Language code: ss1 = SystemModelLinearize[linearize1]

改变变量 x1 和 x2 定义的顺序：

Wolfram Language code: linearize2 = [image];

Wolfram Language code: linearize2["ModelicaDisplay"]

Wolfram Language code: ss2 = SystemModelLinearize[linearize2]

这些模型是等效的，并具有相同的传递函数：

Wolfram Language code: TransferFunctionModel[ss1] == TransferFunctionModel[ss2]

StateSpaceModel 可以线性化常微分方程组：

Wolfram Language code: ClearAll[t, x, θ, i, vin, g, r, lin, mc, mp, l, k, dc, dp, a];

Wolfram Language code:

eqs = {(mc + mp)x''[t] - mp l Cos[θ[t]] θ''[t] + mp l Sin[θ[t]] θ'[t]^2 == k i[t] - dc x'[t], θ''[t]mp (4l^2  / 3 + a^2 / 4) - mp l x''[t] Cos[θ[t]] == mp g l Sin[θ[t]] - dp θ'[t], 0 == vin[t] - r i[t] - lin i'[t] - k x'[t]};

Wolfram Language code: opssm = Sequence[{{i[t], 0}, {θ[t], 0}, {θ'[t], 0}, {x[t], 0}, {x'[t], 0}}, {{vin[t], 0}}];

Wolfram Language code: outputs = {θ[t], x[t]};

Wolfram Language code: ssm = StateSpaceModel[eqs, opssm, outputs, t]//Simplify

使用近似的数字参数值：

Wolfram Language code:

parVals = {g -> 9.80665, r -> 13, lin -> 1, mc -> 445 * 0.1^3, mp -> 7700 * π * 0.005^2 * 0.61, l -> 0.61 / 2, a -> 0.005, k -> 3.7 * 157.48 * 1.6, dc -> 2, dp -> 0.01};

Wolfram Language code: ssm /. parVals

用 SystemModelLinearize 线性化同一系统的 SystemModel：

Wolfram Language code: model = SystemModel[[image], <|"ParameterValues" -> {"electricalMotor.J" -> 0}|>];

Wolfram Language code: op = FindSystemModelEquilibrium[model]

Wolfram Language code: SystemModelLinearize[model, op]

根据方程创建一个 NonlinearStateSpaceModel，并与 SystemModel 相比较：

Wolfram Language code: nssm = NonlinearStateSpaceModel[eqs, opssm, outputs, t]//Simplify

Wolfram Language code:

plotComparison[nssmVar_, smVar_] := Show[SystemModelPlot[nssm, {nssmVar}, 20, <|"ParameterValues" -> parVals, "Inputs" -> {vin -> UnitBox}|>, PlotLabel -> nssmVar, ...], SystemModelPlot[model, {smVar}, 20, <|"Inputs" -> {"V" -> UnitBox}|>, TargetUnits -> "Unit", ...]];

Wolfram Language code: Row@MapThread[plotComparison, {{θ, x}, {"pendulum.pendulumDamper.phi_rel", "pendulum.sliderConstraint.s"}}]

Wolfram Language code: plotComparison[x, "pendulum.sliderConstraint.s"]

可能存在的问题 (1)

有些模型不能被符号式线性化：

Wolfram Language code: SystemModelLinearize["DocumentationExamples.Modeling.ElectricCircuit.CircuitWithEvent", Method -> "SymbolicDerivative"]

Wolfram Language code: SystemModelLinearize["DocumentationExamples.Basic.StateOut", Method -> "SymbolicDerivative"]

Wolfram Language code: SystemModelLinearize["DocumentationExamples.Basic.InOutNoState", Method -> "SymbolicDerivative"]

用 "NumericDerivative" 进行数字式线性化：

Wolfram Language code: SystemModelLinearize["DocumentationExamples.Modeling.ElectricCircuit.CircuitWithEvent", Method -> "NumericDerivative"]

Top

	"InitialValues"	来自 model 的初始值
	"EquilibriumValues"	FindSystemModelEquilibrium[model]
	sim or {sim,"StopTime"}	来自 SystemModelSimulationData sim 的最终值
	{sim,"StartTime"}	sim 的初始值
	{sim,time}	来自 sim 在 time 的值
	{{{x₁,x₁₀},…},{{u₁,u₁₀},…}}	状态值 x_i0 和输入值 u_i0

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SystemModelLinearize

更多信息和选项

范例

基本范例 (3)

范围 (19)

模型类型 (5)

极限情况 (3)

线性化值 (5)

运算点 (6)

推广和延伸 (1)

选项 (5)

Method (4)

ProgressReporting (1)

应用 (10)

分析线性系统 (5)

线性系统的控制器设计 (5)

属性和关系 (7)

可能存在的问题 (1)

文本

CMS

APA

BibTeX

BibLaTeX

	Method	Automatic	线性化算法的方法
	ProgressReporting	$ProgressReporting	控制进程的显示

SystemModelLinearize

更多信息和选项

范例

基本范例 (3)

范围 (19)

模型类型 (5)

极限情况 (3)

线性化值 (5)

运算点 (6)

推广和延伸 (1)

选项 (5)

Method (4)

ProgressReporting (1)

应用 (10)

分析线性系统 (5)

线性系统的控制器设计 (5)

属性和关系 (7)

可能存在的问题 (1)

参见

相关指南

相关链接

历史

文本

CMS

APA

BibTeX

BibLaTeX