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:
NEXT: Example 2 – Linearization of a Magnetic Levitation (MAGLEV) system – sphere.
Literature review by:
Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer
Se hacen trabajos, se resuelven ejercicios!!
WhatsApp: +34633129287 Atención Inmediata!!
Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
Contacto: España. +34633129287
Caracas, Quito, Guayaquil, Cuenca.
WhatsApp: +34633129287 +593998524011
3 comentarios sobre “Example 1 – Linearization of non-linear systems.”