Linearize a function
Suppose we have a system represented by the following function:
Our task is to linearize f (x) around xo = π / 2. As:
We find the following values and substitute them in the previous equation:
Then we can represent our nonlinear system by means of the following negative line equation:
The result of the linearization of f (x) around xo = π / 2 can be seen in Figure 2.48:
Linearize a differential equation
Suppose now that our system is represented by the following differential equation:
The presence of the term cosx makes the previous one a non-linear equation. It is requested to linearize said equation for small excursions around x = π / 4.
To replace the independent variable x with the excursion δx, we take advantage of the fact that:
So:We proceed then to the substitution in the differential equation:We now apply the derivation rules:And for the term that involves the cosx function we apply the same methodology that we have just seen in the previous example for a given function, that is, linearize f (x) around xo = π / 4:
Note that in the previous equation the excursion is zero when the function is evaluated exactly at the point xo. The same happens when the slope is evaluated in xo:So:
Therefore, we can rewrite the differential equation in a linear fashion around the point xo =π /4 as follows:
That is to say:
Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.
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