Documentation

Specifying Portion of Model to Linearize

Specifying Subsystem, Loop, or Block to Linearize

Simulink® Control Design™ lets you specify the subsystem, loop, or block to linearize using linearization input and output points (linearization I/O points).

A linearization input point defines the additive input signal to the linear model. A linearization output point defines the output signal of the linear model.

You can linearize:

• Closed- or open-loop responses using a linearization input point on the input signal to the portion of the model you want to linearize, and a linearization output point at the output signal of that portion of the model.

• Specific subsystem or block.

In this case, linearization I/O points are the input and output signals corresponding to the subsystem or block.

You can define other linear models using additional types of linear analysis points:

• Loop Transfer — Specifies an output point before a loop opening followed by an input. Use this input/output type to compute the open-loop transfer function around the loop.

• Loop Break — Specifies a loop opening. Use to compute open-loop transfer function around a loop. Typically, you use this input/output type when you have nested loops or to ignore the effect of some loops.

• Sensitivity — Specifies an additive input followed by an output measurement. Use to compute sensitivity transfer function for an additive disturbance at the signal.

• Complementary Sensitivity — Specifies an output followed by an additive input. Use to compute closed-loop transfer function around the loop.

Linearization I/O points are pure annotations and do not impact model simulation.

Opening Feedback Loops

If your model contains one or more feedback loops, you can choose to linearize an open-loop or a closed-loop system.

Simulink Control Design lets you remove the effects of the feedback loop by inserting an open loop point without having to manually break signal lines. In fact, for nonlinear models, do not open the loop by manually removing the feedback signal from the model; this action changes the model operating point and produces a different linear model.

 Note:   If a model is already linear, it has the same form regardless of the operating point.

Correct placement of the loop opening is critical to obtaining the right linear model. For example, you might want to linearize only the plant model in a feedback control loop.

To understand the difference between open-loop and closed-loop analysis, consider this single-loop control system.

Suppose you want to linearize the plant, P, about an equilibrium operating point of the model.

To linearize only the plant P, you must open the loop at the output of the P block. If you do not open the loop, and if C and P are linear, the linearized model between U and Y is $\frac{P\left(s\right)}{1+P\left(s\right)C\left(s\right)}$.

The loop opening does not need to be in the same location as the linearization input or output point. For example, the next figure shows a loop opening after the gain on the outer feedback loop, which removes the effect of this loop from the linearization. In the figure, the blocks colored blue are included in the linearization. The block colored white is not included.

In this example, if you place a loop opening at the same location as the linearization output point, the effect of the inner loop from the linearization is also removed.

Ways to Specify Portion of Model to Linearize

There are several ways to specify linearization inputs, outputs, and loop-opening locations (linear analysis points, linearization I/O sets, or, simply, I/O sets) that define the portion of the model you want to linearize. Each method has its own advantages. You can:

• Specify Portion of Model to Linearize in Simulink Model — An advantage of this method is that the locations of linearization I/O points and loop openings are shown graphically in the model. When you specify linearization I/O sets this way and save the model, the I/O set persists in the model.

• Specify Portion of Model to Linearize in Linear Analysis Tool — The Create linearization I/O set dialog box in the Linear Analysis Tool allows you to define multiple open-loop or closed-loop transfer functions for your model interactively. This approach does not make changes to the model.

• Define linearization I/O sets at the command line using `linio`. This method allows you to define multiple open-loop or closed-loop transfer functions without changing the model.

• Define analysis points and openings for an `slLinearizer` interface at the command line. This method allows you to obtain multiple open-loop or closed-loop transfer functions without changing the model.