| Simulink® Control Design™ | |
| On this page… |
|---|
What are Linearization Points? |
Before linearizing the model, you must first configure the model diagram. This involves selecting input and output linearization points (also called analysis I/Os), and then, if necessary, inserting open-loop points for performing open-loop analysis. You can inspect the selected linearization points using the Analysis I/Os pane within the Linearization Task node of the Control and Estimation Tools Manager.
A linearization input point defines an input to the linearized model while a linearization output point defines an output of the linearized model. Additionally, when the linearized models are computed using numerical perturbation, an input point is the point on the diagram where the small perturbation to the input signal is introduced and an output point is the point on the diagram where the small perturbation to the output signal is measured. Linearization input and output points are not the same as operating points which define the state of the model at the point of linearization.
The region between the input and output points defines the portion of the model that you want to linearize, unless a feedback loop feeds the output signal back into another section of the model. In some cases you might want to remove the effect of a feedback signal. For example, you might want to linearize only the plant model within a feedback control loop. When such a feedback loop is present, you can remove the effect of the loop without manually breaking signal lines by inserting an open-loop point. Instructions for inserting open-loop points are in Performing Open-Loop Analysis.
You can use the Simulink® Control Design™ software to linearize the whole model, or any blocks or subsystems within the model. To define the system you are linearizing, place linearization points before and after it in the model diagram.
This section continues the magball example from Creating Operating Points from Simulation. At this stage in the example, a linearization task has been created and operating points have been specified.
For a definition of linearization points, also known as analysis I/Os, see What are Linearization Points?.
In the magnetic ball model, the nonlinearities are all contained within the Magnetic Ball Plant. To linearize this subsystem:
Select the Linearization Task node within the Control and Estimation Tools Manager.
On the magball model diagram, position the mouse on the signal line between the Magnetic Ball Plant and the Controller. Right-click and select Linearization Points > Input Point from the menu.
The model diagram now contains a small arrow pointing toward a circle just above the signal line between the Controller and the Magnetic Ball Plant, as in the following figure. This symbol indicates an input point for linearization has been placed there.
Right-click the signal line after the Magnetic Ball Plant and select Linearization Points > Output Point from the menu.
A small arrow pointing away from a circle on the signal line appears after the Magnetic Ball Plant indicating an output point for linearization has been placed there. The diagram should now look like that in the following figure.
To inspect the linearization points in the Control and Estimation Tools Manager, see Inspecting Analysis I/Os.
To remove a linearization point from a signal line in your model, repeat the same process as for inserting a linearization point. For example, to remove an input point, right-click the signal line containing the input point and select Linearization Points > Input Point from the menu. The input point disappears from the diagram.
Due to the presence of feedback loops in a model, the input and output points might not completely define the portion of the model you want to linearize. In these cases, to remove the effect of signals feeding back into the portion of the model you are linearizing, you might choose to insert an open-loop point. What Is Open-Loop Analysis? discusses the concepts behind open-loop analysis. Inserting Loop Openings continues the magball example by inserting open-loop linearization points in the magball model. Open-Loop Analysis Using Functions in the online documentation gives methods for assigning open-loop points in Simulink models using functions.
Many control systems contain feedback loops. An example of such a system is shown in this figure.
The model represented in this figure is at equilibrium. Consider linearizing the plant, P, about this equilibrium operating point by changing the input signal, U, by a small amount, u, and measuring the change in the output signal, y. The portion of the system that you want to linearize is shown in the following figure.
However, due to the presence of the feedback loop, the change
in the output signal will feed back into the controller, C, and then
into the plant. This affects the behavior of the system you are linearizing.
In fact, if C and P were linear, the linearized model between U and Y would be
rather than P(s).
You could manually remove the feedback signal from the model in an attempt to resolve this issue. However, as shown in the following figure, this changes the operating point of the system since the error signal, E, is now equal to the reference signal, R. Linearizing about this new operating point would change the linearization results. Of course, this only makes a difference for nonlinear models. When the model is already linear, it has the same form regardless of the operating point.
When linearizing Simulink models, label an input or output
point as open loop. Doing so ensures that the output signal is not
fed back into the model but keeps the operating point the same. In
other words, in the linear case, you would compute P(s) rather than
.
This section continues the example from Inserting Linearization Points. At this stage in the example, a linearization task has already been created for the model, operating points have been specified, and linearization input and output points have been inserted.
To linearize only the Magnetic Ball Plant, right-click the signal line containing the output point and select Linearization Points > Open Loop. This inserts a small x next to the output point in the diagram, representing a loop opening.
Note You do not need to place a loop opening in the same place as an input or output point. For example, in the following figure the highlighted blocks are included in the linearization. The loop opening is placed after the gain on the outer feedback loop, which removes the effect of this loop from the linearization. Placing a loop opening at the same place as the output point would have removed the effect of the inner loop from the linearization as well. |
This section continues the example from Inserting Loop Openings. At this stage in the example, a linearization task has already been created for the model, operating points have been specified, linearization input and output points have been inserted, and loop openings have been created.
To view the linearization points, click the Analysis I/Os tab in the Linearization Task node in the Control and Estimation Tools Manager, as shown in the following figure. Use this pane to inspect and make changes to your linearization points.
Note You cannot make changes to the Analysis I/Os or perform open-loop analysis when you select Numerical Perturbation as the Linearization Algorithm in the Linearization Task Options window (accessed by selecting Tools > Options from the Control and Estimation Tools Manager window). See Changing Linearization Settings and Algorithms in the online Simulink Control Design User's Guide for information. |
| What Is Linearization? | Linearizing the Model | |
| © 1984-2008- The MathWorks, Inc. - Site Help - Patents - Trademarks - Privacy Policy - Preventing Piracy - RSS |