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A linearized model is an approximation that is valid in a region around the operating point of the system where the linearization took place. Near the operating point the approximation is good, while far away it may be poor. A linearized model of a car being operated at 3000 ft. is very accurate at elevations close to 3000 ft. but less accurate as the car travels higher or lower.
The following figure shows a nonlinear function,
, and a linear function,
. The linear function is an approximation
to the nonlinear function about the operating point x=1, y=1.
Near this operating point, the approximation is good. Away from this
operating point, the approximation is poor. The precise boundaries
of this region are often somewhat arbitrary. The following figure
shows a possible region of good approximation for the linearization
of
.
Your choice of operating point is important when you:
Linearize a Simulink model. The choice of operating point determines the accuracy of the linear approximation.
Designing compensators with Simulink Control Design software. A Compensator Design Task uses linearization when analyzing a Simulink model.
Tip Choose an operating point that is very close to the expected operating values of the system. One option is to use an equilibrium operating point, described in Equilibrium Operating Points. |
A model can have two entirely different linearizations when the linearization is performed about different operating points. The following model can be linearized using the Simulink Control Design software.
The linearization result for this model is shown in the following figure.
When this linearization is performed about two different operating points, two different linearization results occur as shown in the following table.
| Operating Point | Linearization Result |
|---|---|
| Initial Condition = 5, State x1 = 5 | 30/s |
| Initial Condition = 0, State x1 = 0 | 0 |
Note The operating point consists of values for all the states in the model, although only those states between the linearization points are linearized. The whole model contributes to the operating point values of the states, inputs, and outputs of the portion of the model you are linearizing. |
| What Are Operating Points? | Creating Operating Points | |
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