Page 757 - The Mechatronics Handbook

0066_Frame_C24 Page 5 Thursday, January 10, 2002 3:43 PM

Linearization
Since we will concentrate on time invariant systems, (24.1) and (24.2) can be rewritten as

dx
(
------- = Fx t(), u t()) (24.17)
dt
(
y t() = Gx t(), u t()) (24.18)
We assume that the model (24.17) and (24.18) has at least one equilibrium point given by {x Q , u Q , y Q }.
This is a triad conformed by three constant vectors satisfying

0 = F x Q , u Q ) (24.19)
(
(
y Q = Gx Q , u Q ) (24.20)

Note that the equilibrium point is defined by the state derivatives equal to zero.
If we now consider a neighborhood around the equilibrium point, then we can approximate the model
(24.17) and (24.18) by a truncated Taylor’s series having the form

∂F
∂F
(
x ˙ t() ≈ Fx Q , u Q ) + ------- x=x ( x t() x Q ) + ------- x=x ( u t() u Q ) (24.21)
–
–
∂ x Q ∂ u Q
u=u Q u=u Q
y t() ≈ Gx Q , u Q ) + ∂G ( x t() x Q ) + ∂G ( u t() u Q ) (24.22)
(
-------
-------
–
–
∂ x x=x Q ∂ u x=x Q
u=u Q u=u Q
Equations (24.21) and (24.22) can then be written as
d∆x t() B∆u t() (24.23)
----------------- =
dt A∆x t() +
∆y t() = C∆x t() + D∆u t() (24.24)
where
∆x t() = x t() x Q , ∆u t() = u t() u Q , ∆y t() = y t() y Q (24.25)
–
–
–
and

∂F
∂G
A = ∂F , B = ------- x=x , C = ∂G x=x , D = -------- x=x (24.26)
-------
-------
∂ x x=x Q ∂ u Q ∂ x Q ∂ u Q
u=u u=u u=u Q u=u
Q Q Q
The linearization ideas presented above are illustrated in the following example.
Example 24.2
Consider the magnetic levitation system shown in Fig. 24.2.
The metallic sphere is subject to two forces: its own weight, mg, and the attraction force generated
by the electromagnet, f(t). The electromagnet is commanded through a voltage source, e(t) > 0, ∀t.

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