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FindSystemModelEquilibrium

FindSystemModelEquilibrium[model]

searches for an equilibrium to the model model.

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

searches for an equilibrium, starting from the points x_i=x_i0, u_i=u_i0 and y_i=y_i0.

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

searches for an equilibrium, with variable x_i constrained to have the value v_i etc.

FindSystemModelEquilibrium

FindSystemModelEquilibrium[model]

searches for an equilibrium to the model model.

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

searches for an equilibrium, starting from the points x_i=x_i0, u_i=u_i0 and y_i=y_i0.

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

searches for an equilibrium, with variable x_i constrained to have the value v_i etc.

Details and Options

FindSystemModelEquilibrium is typically used to find values of states, inputs and outputs for which the dynamics of a system model stop.
The model can be a SystemModel object, a full model name string or a shortened model name accepted by SystemModel.
FindSystemModelEquilibrium returns a list {{{x₁,},…},{{u₁,},…},{{y₁,},…}}, where , and are the computed equilibrium values for states, inputs and outputs.
With no explicit starting point given, SystemModel[model]["GroupedInitialValues"] is used.
An equilibrium for a differential algebraic system is a value and such that .
FindSystemModelEquilibrium will attempt to find a local equilibrium point. In general, many equilibrium points may exist for a system.
The following option can be given:
ProgressReporting $ProgressReporting control display of progress

Examples

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Basic Examples (3)

Find an equilibrium, starting the search at initial values:

Wolfram Language code: FindSystemModelEquilibrium[[image]]

Use given start values for states:

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

Find an equilibrium for one of the included introductory hierarchical examples:

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

Wolfram Language code: FindSystemModelEquilibrium[model]

Scope (3)

Give start values for states, inputs and outputs:

Wolfram Language code:

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

Use constraints on inputs and outputs, and start values for states:

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

Find an equilibrium point with given constraints:

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

Options (1)

ProgressReporting (1)

Control progress reporting with ProgressReporting:

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

Applications (5)

Find an equilibrium point for a single water tank with inflow and outflow:

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

Linearize a model around an equilibrium point:

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}]

Linearize around an equilibrium point and analyze the stability:

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]]]

Design a PI controller for keeping the level in a tank with inflow and outflow constant:

Wolfram Language code: model = [image];

Find the equilibrium where the level "h" is constrained to be 2:

Wolfram Language code: h0 = 2;

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

Linearize and close the loop around a PI controller:

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]];

Show the closed-loop step response for a family of PI controllers:

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]

Simple pendulum swinging through any angle:

Equilibrium with the pendulum hanging straight down:

Wolfram Language code: model = [image];

Wolfram Language code: FindSystemModelEquilibrium[model]

Pendulum standing straight up above its axis:

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

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

Level curves of the first integral give the potential energy of the system:

Wolfram Language code: model["SystemEquations"]

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

The pendulum has one stable and two unstable equilibrium points:

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}}]}]]