2b-272
Copyright © 1996 IFAC 13th Triennial World Congress, San Fmncist.:o. USA
FLATNESS AND SAMPLING CONTROL OF INDUCTION MOTORS Philippe Martin" Pierre Rouchon t
• Cenlre AUlomalique el Syslemes, teole des Mines de Paris 35, rue Saint-Honore, 77305 Fonlainebleau Cedex, France
[email protected].•mpfr t Cenlre Automalique el Systemes, t eole des Mines de Paris 60, Bd. Saint-Michel, 75272 Paris, France
[email protected]
Abstract. A sampling control strategy for flat systems, based on computing open-loop references, is proposed. It is used to control the induction motor, shown to be flat. Keywords. Nonlinear control, sampling control, flaUless, induction mOlOrs.
1. INTRODUCTION
trol community, see e.g. [8,15,12,1l,3,4] and the references therein. We show the induction molor is flat -this means its dynamics is nOl as complicated as it may look-and apply our sampling control. Simulations iUustrate that a good performanoe is achieved even with a fairly large sampling period. It is worth mentioning that our computations are straightforward mainly because they are carried out using the complex variable notation (time-varying phasor) model of the induction motor, contrary lO what is usually done.
In this paper we propose a sampling control strategy for Hat systems. It essentially relies on the fundamental property that the motion planning for a flat system is trivial, due to the fact that its dynamical behavior can be completely described by differentially independent functions [7.13,5]. The control scheme simply consists in computing an open loop-+,) to steer the system from the state at time It to the state i k +1 at time ll ot I, "Sensible" means on the one hand that the error to the reference trajeclOry must
x,
decrease. i.e.,
IIx,+] - i'+l1I ::: KTllx, - i,lI.
with U(I) (and possibly some of its derivatives) nollOO large. • The control U(I) is hold to a constant U, E u((l" 1,+ ,)) and applied to the true system (starting from x,). We then have the following simple robustness result (we recall the notation 'P(a) = 0(0") means there exists a constant '" such that 11'1'(0')11 :!: "110"11 fora ->- 0):
Propo.iliiition 1 The error to the reference trajeclory decreases up 10 measurement, control M{d and model errors:
IIx,+, - x>+]1I ::: KT IIx~ - x,ll
IIx'+,- >h,1I :::KTllx, -x,1I + IIx, - i, 110(1)
for some KT < I, and on the other hand thal the control u(1) (and possibly some of its derivatives) is not too large. There is a considerable freedom of choice at this stage, but it should be emphasized that this boils down to fixing the initial and
+ lI~ull,O(T) + .O(T)
end poinlS of the flat outpUL and some of their derivatives and
connecting these "boundary constrainLS" by an otherwise un-
KT < t,
where
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(measuremelll error)
(colllrol hold error)
(model error),
lIoull,:= sup«!•. ".,] IIU, - u(I)II.
Wc now show that holding the conlrolto a suitable value can reduce the corresponding elTor:
The control hold and model errors can lhus be made arbitrarily small by shrinking the sampling period T.
Proprn;ition 3 The error due 10 lhe control hold can be made 118ull.0(T 2) by choosing U, := ~ u,«)d/111(1) =
I,(x(s), U,)ds
lo(i(s), u(r»ds,
IIW)II = IIh, +
,
!O lIox,lI +
(4)>/1 - ob)H(s)
- lo(i(s), u(s))) dsll
f (LII~(s)1I
+ PIIH(s)1I + NE)ds
(f,(x(s), If,) - lo(i(s), u(s)))dsll
!O 115xlII +
!O 115x, 11 +
whereP :=
sup 1I>/I-obllandIlHII.:= 4.9 ~ r.+l
(LI15x(s) 11 +
f
L 118x(s) lids,
"
1I/, (x(s), U,) - lo(i(s), u(s» lids
sup
IIH(t)ll .
ftJr,
LT
L,
d>Jr, R, +K(dl M
1
R,L,
where - := - J( RrM
M .
+-
M'
+ (R, + R,-, )i. = LT .M
(13)
u, . •
+ jnplJ-)>Jr,=u " '*' L,
..
u" (14)
.
IS a dlmenslon1ess coeffiCient. We LT are thus left with only one (complex) differential equation. It is possible, though far more tedious and seldom done, to recover the same equations by a singular perturbalion analysis on the state equations for >Jr, and i , obtained from (1)-(4) by
eliminating 1/16 and i r . The common interpretation for the two time scales is that the transfer from VOltage to current is fast compared to the transfer from cUlTent to flux .
The model of the "perfect" motor thus boils down to the electro-magnetic equation (12) and the mechanical equation (6), which remains unchanged. It involves only the state variables >Jr" IJ, (} and the input u, and is obviously flat, the other variables being given by the static equa tions (11)-(13). Notice that (13) can be seen as an observer since it provides the usually non-measured ftux 1frr from the measured curreOl t6 and voltage U,. We will use thi s "perfect" model to design the control law.
lime,.)
Fig. I. open-loop control (no parameter error).
l ·.;. ~'}S:;- 1 :1 ~.1
-1.8.,
The nominal angular velocity is 150 radls and the load torque is IONm.
Figure I corresponds to a purely open-loop control on the whole simulation time: the sample value of u, is simply equal to the mean value of u;(1) over thc sampling interval. The perfOlTRance of this open-loop control is quite good when the
model parameters arc well ~known. because the motor is naturally not unstable. Nevertheless, when the parameters are not I Lcroy Some' A3L instaUcd 1l11lU: Lltboratoirc d' AUlOmatique de Nanles, France.
OA
0.5
0.6
':~~'
L, = 0.425 H, M = (U72 H, np = 2, L, = 0.076 H, J = 0.0293 Nm/rad/s2 .
The simulations illustrate a change of the angular velocity set-point: wc want to go from 150 rad/s to 75 rad/s in 0.5 s. The slip velocity is kept and equal to R, /L,. The samplin period T is equal 10 10 ms, which is very large compared to sampling pcriods usually considered [9,101.
SIAf6~current I ~~ (A)
:~~
following parameters:
R, = 0.93 Ohm,
0.1
" . 8 - -
For simulation, we consider a 1.5 kW motor 1 having the
R, = 4.90 Ohm,
0
0
0.'
0.2
0.3
004
0.5
0.6
time (I'
Fig. 2. closed-loop control (50% error on R,).
precisely known (for instanCe R" the rotor resistance, varies a lot in operation), it is much beuer to use the control strategy proposed in this paper. On figure 2, the control has been computed with a rotor resistance overestimated by 50%. On each sampling interval, the control steering back to the reference trajectory, corresponds to a stable second order linear error dynamics with time constants 0.10 s and 0.08 s. In spite of this large parameter error, the pcrformance is very good.
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Acknowledgments
C13] Ph. Martin.
Contribution d /'etude des Systil1Jl!s Plats. PhD thesis, Ecole des Mines de Paris, France, 1992. [14] S. Monaco and D. Normaod-Cyrol. A unified representation fornonlinear discrele-time and sampled dYnamics. 1. Moth. Systems Estim. Control, 5(1):103-106,1995. (15] R. Onega and G. Espinosa. Torque regulation of induction motors. Automatica, 29:621-633, 1993. DiJJ~rentiellemenl
We are thanlc:ful to J. Chiasson for inleresting remarks about
a prior version of this paper and also 10 A.K. Chelouah et E. DeIaleau for fruitful discussions on electrical mOlOrs.
7. REFERENCES (1) l.-P. Barbot,S.Mooaco,D.Normand-Cyrot,and N. Pan-
talos. Some comments about Iinearization under sampling. In Proc. oJ tile 31nd IEEE Con! on Decision and Control,pages 2392-2397, Tucsoo, 1992. (2] A.K. Chelouah. Sur la commande en temps diserel de systimes non lineaires continus: applications a I' eIectrotechnique et ala robotique mobile. PhD thesis, Universilt Paris XI, Orsay, France, 1994. (31 J. Chiasson. Dynamic feedback linearization of the induction motor. IEEE Trans. Automat. Control, 38:15881594, 1993. (4] J. Chiasson. A new approach to dynamic feedback linearization control of an induction motor. In Proc. of tile 34ndIEEE Con[. on Decision and Control, pages 21732178, New Orleans, 1995. [5] M. F1iess,J. L~vinc, Ph. Martin, andP. Rouchon. Nonlinear control and Lie·B acklund transformations: IOwards a new differential geometric standpoint. In Proc. of the 33nd IEEE Con! on Decision and Control, pages 329344, Lake Buena Visla. 1994. [6] M. Fliess, J. Levine, Ph. Manin, and P. Rouchon. Design of trajectory slabilizing feedback for driftless Hat systems. In Proc. of the 3nd European Control Con[., 1995. To appear. (1) M. F1iess, J. Uvine, Ph. Manin, and P. Rouchon. flatness and defect ofnonlinearsyslems: introductory theory and applications. Int.1. ofConlrol, 61(6):1321-1361 , 1995. [8] W. Leonhard. Control of Electrical Drives. SpringerVetlag, 1985. 19] W. Lconhard. 30 years space vectors, 20 years field orientation, 10 years digital signal processing with control ac-drivcs, a review (pan I). ErE Journal, 1(1 ): 13-20, 1991. (10] W. Leonhard. 30 years space vectors, 20 years field oricnlation, 10 years digital signal processing wi1l1 control ac-drives, a review (part 2). ErE Journal, 1(2):89-100, 1991. [11] R. Marino, S. Percsada, and P. Tomci . Output feedback control of current-fed induction mOlors with unknown rotor resistance. IEEE Trans. Automat. Control, 1996. To appear. [12J R. Marino, S. Peresada. and P. Valigi. Adaptive inputoutpUl lineari zi ng control of induction motors. IEEE Trans. Automat. Control, 38:208- 221 , 1993.
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