COMSOL 6.4 - Extracting Reduced-Order State-Space Matrices

Extracting Reduced-Order State-Space Matrices

The tutorial in this section illustrates how to use the mphreduction function to extract the state-space matrices for a model. The studied heat transfer model contains a heat source, which is set as input in the state-space system. The temperature at a specific point is used as output. Follow the step-by-step instructions below to obtain the state-space matrices and to solve the system in Simulink.

Model Definition in the COMSOL Desktop

1

Start the COMSOL Desktop.

2

In the menu, select Application Libraries.

3

In the Application Libraries, browse to > , and select model_tutorial_llmatlab. Click .

Model Parameters

The loaded model is used as base model for documentation tutorial. The geometry, mesh, and physics is set, but for this specific problem you need to edit the model to use parameters for the initial and external temperature as they will be used later in the reduced model.

1

Under , select node.

2

In the window for , update the table according to the settings below:


Name	Expression	Description
power	30[W]	Total power
Temp	300[K]	Base temperature
Text	300[K]	External temperature
T0	293.15[K]	Initial temperature

Physics Settings — Heat Transfer in Solids

In this section you will set up the Heat Transfer in Solids physics interface. If you do not have the Heat Transfer Module or MEMS Module skip the instructions below and go directly to Physics Settings — General Form PDE.

1

In the , select > node. In the window set the field to T0.

2

In the , select > node. In the window enter Text in field.

Even though you can generate ROM including constraint models in COMSOL, the resulting reduced matrices are not suited for Simulink state-space system. In this model the bottom temperature is imposed as a constraint, you will change it to heat flux condition instead.

3

Right-click > node, and select .

4

Right-click node, and select .

5

Select boundary 3.

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Under , select .

7

In the field enter 1e6.

8

In the field enter Temp.

You have now set up the physics, and can go directly to Study Settings to continue.

Physics Settings — General Form PDE

In this section you will find the instruction to set up the physics using the General Form PDE interface. If you have the Heat Transfer Module or MEMS Module skip the instructions in this section and go directly to Study Settings.

1

In the , right-click node and select .

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In the toolbar select .

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In the window, expand > . Select and click .

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In the window for , under section click .

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In the window, expand , and select . Click .

6

In the window for , under section click .

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In the window, expand , and select . Click .

8

In the , expand and select node.

9

In the window for , under section in the Γ table enter:


-material.k11*ux
-material.k22*uy
-material.k33*uz

10

In the window for , under section in the f text field enter 0.

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In the window for , under section in the da text field enter material.rho*material.Cp.

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In the , expand and select node.

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In the window for , under section in the field enter T0.

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In the , right-click and select .

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In the window for , select domain 2.

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In the window for , in the field enter power/(1[cm]*1[cm]*1[mm]).

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In the , right-click and select .

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In the window for , select boundaries 7 8 10 11 12.

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In the window for , in the field enter 10[W/(m^2*K)]*(Text-u).

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In the , right-click and select .

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In the window for , select boundaries 3.

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In the window for , in the field enter 1e6[W/(m^2*K)]*(Temp-u).

To create a reduced-order model, first add an unreduced model study and a training study to your model. In this section you will add a Time Dependent study and an eigenfrequency study that will be later be set as unreduced model and training studies, respectively.

1

In the toolbar, select .

2

In the window, select and click .

3

In the window for , enter range(0,1,50) in the field.

Add a domain point probe plot, which will define the output of the reduced-order model.

4

Under node, right-click and select > .

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In the window for , locate the field and enter: 1e-2 1e-2 1e-2.

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In the toolbar, select .

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In the window, expand and select . Then click .

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In the window for , select and enter 60.

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In the window for , select and leave it with the default value.

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In the toolbar, select .

Reduced-Order Model

Before adding a reduced-order study to your model you need to set the input, define the power variable power, the bottom temperature Temp, and the exterior temperature Text.

1

In the , click .

2

In the window, under , select .

3

Right-click and select > .

4

In the window for , edit the table as defined below:


Control name	Expression
power	30[W]
Temp	300[K]
Text	300[K]

In the step below you will now add a model reduction study.

5

In the window, select , and click .

6

Right-click , and select .

7

In the window for , under section enter the following settings:


name	Value in list
Training study for eigenmodes	Study 2
Unreduced model study	Study 1
Defined by study step	Time Dependent

8

Under the section, enter the table as defined below:


Variable	Expression	Description
Tout	comp1.ppb1	Temperature

9

Now click to generate the reduced-order model.

10

The first part the modeling is done. You can now save your Model MPH file and continue to follow the instruction from the MATLAB prompt.

Extracting the State Space Matrices in MATLAB

1

At the MATLAB prompt, to load the model you have just created, enter the command

model = mphload(<modelname>);

where <modelname> is the name of the Model MPH file. If the model file path is not set in MATLAB you can enter the full path of the model name.

2

A call to mphreduction creates the state-space matrices needed to simulate the reduced-order system. Enter:

M = mphreduction(model, 'rom1')

3

For later use, you can enter in MATLAB the value for the input of the dynamic system:

Run the Simulation in Simulink

1

At the MATLAB prompt, enter simulink to start Simulink.

2

In the window, select .

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From the drag the block from the COMSOL 6.4 library into the empty Simulink diagram.

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Double-click the block and make sure that the System variable name is

5

Continue editing the Simulink diagram by adding block, a block and a so it looks like in the figure below:

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In the Simulation toolbar, locate and enter .

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You can now click Run to run the simulation.