IJMET 04 03 021.pdf


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME

The poles of the transfer function exemplifies an unstable system, since one of the
poles has a positive value. Therefore, the active magnetic bearing necessitates a control
system. Stable fuctioning can be realized with decentralized PID controller, with the transfer
function [3]:
GPID ( s ) = K p + K I s + sK D

(12)

The block diagram of the control system with PID controller for y-axis is illustrated
in Figure 12, whereyr(t) connotes the reference value of the rotor position (generally equal to
zero),icy(t) is the reference control current and y(t) is position of the rotor.

Fig. 12. Block diagram of the control system
Laplace transfer function of the closed loop is described by the transmittance:

K D kiy 2 Kpkiy
Kk
S +
S + I iy
m
m
m
GCL ( s) =
K
k
Kpk
Kk
iy
S 3 + D iy S 2 +
S + I iy
m
m
m

(13)

The closed-loop system with the PID controller has three polesλ1,λ2,λ3. To deduce the
magnitude of KP, KI, KD, the coefficients of the denominator of GCL(s) in Eq. 13 should be
evaluated against coefficients of the polynomial form:
s 3 + (λ1 + λ 2 + λ3 ) s 2 + (λ1λ 2 + λ 2 λ3 + λ1λ3 ) s + λ1 λ 2 λ3

(14)

As a result, the specifications of the PID controller are equal to:

Kp =

(λ1 λ 2 + λ 2 λ3 + λ3 λ1 )
k iv
λλ λ m
KI = 1 2 3
kiv
K D = {( −λ1 − λ2 − λ3 ) m}kiy

(15)

Position of the polesλ1,λ2,λ3 in the s-plane influences the characteristics of the transients. According to
the pole placement method [4] two poles can be determined from:

λ1 = −ωnζ + iωn 1 − ζ 2
λ2 = −ωnζ + iωn 1 − ζ 2

(16)
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