Control System Analysis, Matemática aplicada - Appd Math

Example 1 – Linearization of non-linear systems.

Linearize a function

Suppose we have a system represented by the following function:
nullOur task is to linearize f (x) around xo = π / 2. As:
nullWe find the following values and substitute them in the previous equation:
nullThen we can represent our nonlinear system by means of the following negative line equation:
null

The result of the linearization of f (x) around xo = π / 2 can be seen in Figure 2.48:

null

null

Linearize a differential equation

Suppose now that our system is represented by the following differential equation:
nullThe 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:
nullSo:nullWe proceed then to the substitution in the differential equation:nullWe now apply the derivation rules:nullAnd 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:

NEXT: Example 2 – Linearization of a Magnetic Levitation (MAGLEV) system – sphere. 

 

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