Sometimes it might not be possible to design a controller with PIDTune (unstable model, etc).
In the following model we instead design a controller with
TransferFunctionModel and dynamically adjust it to a suitable output response with Manipulate.
The plant I want to control is an inverted pendulum driven by a simple motor device. I read the angle values and feed them back to a PID Controller.
linearizedModel = WSMLinearize["SimplePendulum.Components.Plant"];
tf = TransferFunctionModel@linearizedModel;
pidTf = TransferFunctionModel[
kp (1 + 1/(\[Tau]i \[FormalS]) + \[Tau]d \[FormalS]), \[FormalS]];
cltf = SystemsModelFeedbackConnect[
SystemsModelSeriesConnect[pidTf, tf]];
Manipulate[
Plot[{OutputResponse[
cltf /. {kp -> gain, \[Tau]i -> iTime, \[Tau]d -> dTime}, \[Pi]/
2 UnitStep[t], {t, 20}], \[Pi]/2} // Evaluate, {t, 0, 2},
PlotStyle -> Thick, GridLines -> Automatic, PlotRange -> {-1, 2}],
{{gain, 108, "\!\(\*SubscriptBox[\(k\), \(p\)]\)"}, 10,
400}, {{dTime, 0.06, "\!\(\*SubscriptBox[\(\[Tau]\), \(d\)]\)"},
0.01, 10}, {{iTime, .85, "\!\(\*SubscriptBox[\(\[Tau]\), \(i\)]\)"},
0.01, 10}, SaveDefinitions -> True]
Which creates this manipulate:
Once I've found appropriate values in the Manipulate I simulate the model with
WSMSimulate and can visualize the results with a plot:
{pars, parVals} = {{"PID.k", "PID.Ti", "PID.Td"}, {108, 0.85, 0.06}};
sim = WSMSimulate["SimplePendulum.SimpleExample", {0, 10},
WSMParameterValues ->
Join[Thread[pars -> parVals], {"PID.yMax" -> 100, "PID.wd" -> 1}]];
Plot[{Evaluate@sim[{"plant1.y"}, t][[1]], \[Pi]/2}, {t, 0, 2},
GridLines -> Automatic, PlotRange -> {-1, 2}]
And just to confirm we can see the 3D-animation of the system as well:
The Modelica models used:
package SimplePendulum
extends Modelica.Icons.Library;
package Components
extends Modelica.Icons.Library;
model Plant
annotation(Icon(coordinateSystem(extent={{-100.0,-100.0},{100.0,100.0}}, preserveAspectRatio=true, initialScale=0.1, grid={10,10}), graphics={Rectangle(visible=true, fillColor={255,255,255}, extent={{-100.0,-100.0},{100.0,100.0}})}), Diagram(coordinateSystem(extent={{-148.5,-45.0},{148.5,105.0}}, preserveAspectRatio=true, initialScale=0.1, grid={5,5}), graphics));
Modelica.Blocks.Interfaces.RealInput u annotation(Placement(visible=true, transformation(origin={-120.0,80.0}, extent={{-20.0,-20.0},{20.0,20.0}}, rotation=0), iconTransformation(origin={-120.0,-0.0}, extent={{-20.0,-20.0},{20.0,20.0}}, rotation=0)));
Modelica.Blocks.Interfaces.RealOutput y annotation(Placement(visible=true, transformation(origin={140.0,-0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0), iconTransformation(origin={110.0,-0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.Rotational.Sources.Torque torque1 annotation(Placement(visible=true, transformation(origin={-65.0,80.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.Rotational.Components.Inertia inertia1(J=0.01) annotation(Placement(visible=true, transformation(origin={-40.0,80.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
inner Modelica.Mechanics.MultiBody.World world annotation(Placement(visible=true, transformation(origin={-120.0,0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.MultiBody.Parts.BodyBox bodyBox1(r={0.5,0,0}) annotation(Placement(visible=true, transformation(origin={30.0,-0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.MultiBody.Joints.Revolute revolute1(useAxisFlange=true) annotation(Placement(visible=true, transformation(origin={-20.0,0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.Rotational.Sensors.AngleSensor angleSensor1 annotation(Placement(visible=true, transformation(origin={40.0,80.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Mechanics.Rotational.Components.Damper damper1(d=0.1) annotation(Placement(visible=true, transformation(origin={-40.0,40.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
equation
connect(angleSensor1.phi,y) annotation(Line(visible=true, origin={111.1375,40.0}, points={{-60.1375,40.0},{15.6375,40.0},{15.6375,-40.0},{28.8625,-40.0}}, color={0,0,127}));
connect(u,torque1.tau) annotation(Line(visible=true, origin={-98.5,80.0}, points={{-21.5,0.0},{21.5,0.0}}, color={0,0,127}));
connect(revolute1.axis,angleSensor1.flange) annotation(Line(visible=true, origin={-3.3333,63.3333}, points={{-16.6667,-53.3333},{-16.6667,16.6667},{33.3333,16.6667}}));
connect(torque1.flange,inertia1.flange_a) annotation(Line(visible=true, origin={-52.5,80.0}, points={{-2.5,-0.0},{2.5,0.0}}));
connect(revolute1.axis,inertia1.flange_b) annotation(Line(visible=true, origin={-23.3333,56.6667}, points={{3.3333,-46.6667},{3.3333,23.3333},{-6.6667,23.3333}}));
connect(damper1.flange_a,revolute1.support) annotation(Line(visible=true, origin={-41.6,23.29}, points={{-8.4,16.71},{-11.4,16.71},{-11.4,-10.065},{15.6,-10.065},{15.6,-13.29}}));
connect(damper1.flange_b,revolute1.axis) annotation(Line(visible=true, origin={-23.3333,30.0}, points={{-6.6667,10.0},{3.3333,10.0},{3.3333,-20.0}}));
connect(world.frame_b,revolute1.frame_a) annotation(Line(visible=true, origin={-70.0,0.0}, points={{-40.0,0.0},{40.0,0.0}}));
connect(revolute1.frame_b,bodyBox1.frame_a) annotation(Line(visible=true, origin={5.0,-0.0}, points={{-15.0,0.0},{15.0,-0.0}}));
end Plant;
end Components;
model SimpleExample
extends Modelica.Icons.Example;
annotation(Diagram(coordinateSystem(extent={{-148.5,-105.0},{148.5,105.0}}, preserveAspectRatio=true, initialScale=0.1, grid={5,5})));
Modelica.Blocks.Sources.Constant const(k=3.1415/2) annotation(Placement(visible=true, transformation(origin={-80.0,0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
Modelica.Blocks.Continuous.LimPID PID(k=108, Ti=0.85, Td=0.06, yMax=100) annotation(Placement(visible=true, transformation(origin={-40.0,0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
SimplePendulum.Components.Plant plant1 annotation(Placement(visible=true, transformation(origin={0.0,0.0}, extent={{-10.0,-10.0},{10.0,10.0}}, rotation=0)));
equation
connect(PID.y,plant1.u) annotation(Line(visible=true, origin={-20.5,0.0}, points={{-8.5,-0.0},{8.5,0.0}}, color={0,0,127}));
connect(plant1.y,PID.u_m) annotation(Line(visible=true, origin={-8.2,-8.4}, points={{19.2,8.4},{22.2,8.4},{22.2,-6.6},{-31.8,-6.6},{-31.8,-3.6}}, color={0,0,127}));
connect(const.y,PID.u_s) annotation(Line(visible=true, origin={-60.5,0.0}, points={{-8.5,-0.0},{8.5,0.0}}, color={0,0,127}));
end SimpleExample;
end SimplePendulum;
