A Design Method of Pi Controller For An Induction Motor With Par PDF
A Design Method of Pi Controller For An Induction Motor With Par PDF
A Design Method of Pi Controller For An Induction Motor With Par PDF
-
(3)
Fig. I. The system characteristic for inertia variation. (a) Unit-step response
according to inerlia variation. (b) The movement afpole according lo inertia
"?.ri81iO".
For pole-zero cancellation, substitute ( 5 ) to (4) and then get ( 6 ) .
is*T
Fig. 2. Block diagram of induction motor using PI speed controller
H K,+ & is
How to select PI current controller gain in (6) is that the
cut-off frequency of closed-loop transfer function passes
through the -3db from its zero-frequency. It can yield gains of
PI current controller with (7).
409
polynomial. If this Kharitonov polynomial is Hunvitz stable, BiK BtK K
a 2 = l , a l = [ , ' , e ] , a o = [ z Jm," '1-J K,,
every polynomial in the P(s) set of interval polynomials is
Hunvitz stable. However, it is general that the individual F2=(~-~2,6=(~-6),Fo=l
coefficients change interactively in the ordinary closed-loop P2 = a 2 , 4 = a , , 6 = a,,
system, is suggested the generalized kharitonov theory to solve
Applying generalized Kharitonov's robust stability theory to
such a problem.[6]
( I 2 ) , obtain four Kharitonov polynomials. And applying
K'(s)=xo+x,s+y2s2+y,S3+XqS4+xSS5+yhS6+...
Hunvitz stability criterion to (13), find feedback gain region for
K~(s)=Xo+yls+y2S*+X,S3+x4S4+ySS5+y6s6+... satisfying stability condition. That is, if select the controller
(9) gains in this area, clos8:d-loop system keeps stability
K 3 ( s ) = y o + ~ I ~ + ~ 2 ~ 2 + y 3 ~ 3 + y 4 ~ 4 + ~ S ~ 5 + ~ ~ ~ 6 + ~ ~ ~
margin 6 about specific variation of system parameter
K + ( S ) = yo + ylS +X2S2+ 3s) + y 4 2 + yss5 + X6S +...
6
F0aomi, + h a h i " +F2a2max
The previously stated Kharitonov robust control method is to
place the system poles in the left half of the s-plane. This F~aomin+Flalmm +F2~12max
(13)
method is not considering the stability gain margin and the Foaomsx + 6ah" + F2L12min
system damping. By extending this method when the parameters Foaom,u + 4a1max +F202,i"
vary within the specific range, can ensure the minimum speed For discrimination of stability about (12), can be expressed as
and current control characteristics for the closed-loop system A(S - 6 )= a2sz + (al - 26a2)s+ a2B2- a , 6 + a.
that has additional gain margin. Assuming the gain margin is 6 , (14)
this range in the s-plane is which is called 6 Hunvitz range and = C*S2 +CIS+CO
is shown in fig.5. If we take have an interest in the 2-mass where c 2 = a 2 , cI = a , - 2 6 a 2 , c , , ' = a 2 6 2 - a i 6 + a o
system with torsional vibration, consider phase margin 6 Applying Routh-Hurwitz stability criterion to (14), equation
(15) indicates the absolute stability condition with variation of
inertia.
c2 = a 2 = I > O
B .tK
c, = a i -26a2 = - 2 6 + - 2 > 0 (15)
J
c o = a 2 62 - a , 6 + a o = 6 - PB +SK+ - > KO.
J J
The stability condition obtained from (15) according to speed
Fig 5 . S Hwwitz stability area &in s-planes controller gain.
K. >26J-B
IV. DESIGN OF THE ROBUST PI CONTROLLER
From fig.2, the transfer functions of motor input torque and Replacing (16) into (13) yisslds the K , ,Kf conditions as
angular velocityG(s) and of PI controller CJs) are (IO), (11)
K , > 265,- -B
respectively.
K i >-62Jm,,+6(B+.K,)
Therefore, if we choose gains of current controller using (l7),
I B can secure stability margin about variation of inertia.
where N(s) = no =-,O(s) = dls + do = s +-
J - J The system goveming equation consists of outer mechanical
and subsystem parameters. Because of the physical properties, the
outer mechanical subsystem has much slower dynamic response
speed in comparison with inner electrical subsystem. The
mechanical parameters affect the limit of stability margin 6 .
The characteristic equation of closed-loop is (12). This
We obtain the limit of 6 through position of dominant pole.
equation includes the stability margin 6,and the parametric
Multiplying dominant pole by IO in initial condition, get the
variation range.
limit of stability margin. Simulating from 6 Hunvitz stability
A(s-6) = g ( S I ) point of view, also obtain 6 . Considering the characteristics of
= (s-d)D(s-6)+ K,(s -6+wP,)N(s-6) experimental set, stability margin 6 is selected as 5 .
2 (12) Fig.6 illustrates speed controller gain area satisfying stability
= a2 (sI - 6 ) + a , (s, - 6 )+ a o
condition. We assumed the controller gains W p , , Kp exist from
=P2F2+qF;+pOFo 0 to IO and the total inertia of system is varying between -70%
where
410
and +70%. Considering the hardware specification, we select TABLE1. PI CURRENT CONRRGLLERGAM USE FOR
the value of Wpiand Kp is selected as 9. SIMULATION AND EXPERlMENTATlGN
For obtain robust current controller, applied kbaritonov Simulation Simulation
robust control theory to current control loop. The closed-loop (Classical method) (Proposed method)
transfer function derived from (6). The characteristic equation Kp 4.5813 K p 5.57
of current control loop represented in (1 8). The closed-loop K; 790.6769 K, 10545
feedback system characteristic equation considering stability
margin is (19)
TABLE 2. SYSTEM PARAMETERS
K,s +K j
G,’(s)= Jw Induction motor Inertia 0.0088 kg-m’
uL,s2 + ( R + K J s + K,
JB dc motor Inertia 0.009 kg.m’
J n Round disk Inertia 0.02 kern'
A ( S ) = U L ~ S+~( R + K , ) ~ +K , R, Stator resistance 0.385 n
R, Rotor resistance 0.342 I2
=a2s2+als+ao
L, Stator inductance 0.03257 H
a2 =[(G)),i.,(oLJml3 L, Rotor inductance 0.03245 H
where L., Muhlal inductance 0.03132 H
ut =[Rmi, +K,, R, +K,l,ao =K,
How to achieve robust current controller gain is the same
procedure as speed controller. Then stable condition according
to gain of current controller is as following.
K . > 2 6 ~ L-.R
-
original system locate between 1300 and 1400. If parameter
variation exists, the modified pole must be designed to get into 07 lm.
this range for stability margin. If choose the current controller (
I.
1
gain as exact stability margin, 1400, the problem occurs to
practical system. The exact stability margin causes excessive
46
111.
-,ia ......., .
+I
411
(4 (b)
Fig. IO. Characteristics of current control with parameter vanation. (a)
Conventional design scheme (b) Proposed design scheme
- -50 L
0
UIO
i.n. I ".
. ..
('4 Frequency (rad!%)
Fig. 9. Bade diagram of current control system with cur-off frequency 2000
radis. (a) Conventional design scheme (b) Proposed design scheme
0
V. SIMULATION
E 5 : r 7
In this section, the speed and current controller gain best-tit
__..___
method and its control performance are verified with computer
?
simulations. Tahlel,2 show the mechanical, the electrical c" -30
parameters and PI controller gains used in this simulation. ' 0 5 IO
time [roe]
These simulations have been implemented using Matlab.
The RLS method is generally accepted for estimating' v i l o o z
100 -
electrical and mechanical parameters and load characteristics.
0 .~~~ .____
The effectiveness of RLS is proved in several papers. Although
-100
1
we omitted this procedure, these results are transferred to design
PI controller.
Fig.lO(a), (b) show phase current trajectory of current
controllers, which are designed classical PI controller technique
and proposed method respectively. The condition of resistance
variation is 50% from its rated value. In the case of conventional
2P -30 0 3cE3F!3 5
time [.eel
10
412
z o
B
R e o
A . . -
(Io
Ek o h . . p-?---
C ao
fI o +
? e o
REFERENCES
B.K. Bose, "Technology Trends in Microcompuler Control of Electrical
Machines."IEEE Tmm.Ind Elecf.,vol35,No.l,pp 160-177, 1988
K. Nandam. Pradeep, "Analog and Digital Speed Control of DC Drives
Using Propotional-htegral and Integral-ProportionalContol techniques', ,
IEEE. Trans. Ind Elecf., vol. IE-34, No.2, pp 227-233, 1987
,
/
P. De Wit, R. Ortega, and 1. Mareek, indirect field oriented control of
induction motors is robustly globally stable, Aufomolicu, vol. 32, no.10,
pp. 1393.1402, 1996.
Masayasu Shimakage, Antonio Mora", Masao Nagai, and Minoru
Hayase, " Design OfVibration Suppression H-infconlral System for
High-Rise Building", IPEC-Yokohama '95, p.464, 1995
Katsujiko Ogata, Modern Control Engirierring, Znd, Prentice-Hall, 1990,
595-605.
S.P. Bhattachalyya, H. Chapellat, and L.H. Kcel, Robust Confml The
Porumebic Approach, Prentice- all, 1995.
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