Abstract: - A nonlinear state feedback input-output linearizing control with a tracking of the rotor magnetizing current and the rotor speed for an induction motor is ...
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
Input Output feedback linearization Control of Induction Motor with Varying Parameters F. Mehazzem* , A. Reama*, Y. Hamam* and H. Benalla** *Groupe ESIEE, Paris Est University, France, **Department of Electrical Engineering, Constantine University, Algeria,
Abstract: - A nonlinear state feedback input-output linearizing control with a tracking of the rotor magnetizing current and the rotor speed for an induction motor is presented. An accurate knowledge of the rotor magnetizing current, the load torque and the rotor resistance is the key factor in obtaining a high-performance and high-efficiency inductionmotor drive. For that, a stable model reference adaptive system (MRAS) rotor-resistance estimator and a load torque observer have been designed and added to the main control structure. Thus, the continuous adaptive update of the machine parameters ensures accurate decoupled control and high-performance operation. Simulation results are presented to validate the efficiency of the induction-motor drive in various operating modes. Key-Words: - Induction motor, Input Output feedback linearization design, parameters estimation. machine torque, thus enabling the torque and the flux to be controlled independently. A disadvantage of the Blaschke field-oriented controller is that the method assumes that the magnitude of the rotor flux is regulated to a constant value. Therefore, the rotor speed is only asymptotically decoupled from the rotor flux. Krzeminski [3], Marino et al. [4], [5] have developed an input-output decoupling controller which decouples the control of the magnitude of the rotor flux from the regulation of the rotor speed. In [6], an adaptive, exact decoupling, and linearizing control scheme has been developed in which the rotor resistance, the stator resistance and the load torque are estimated.
NOMENCLATURE
Rr , Rs
rotor , stator resis tan ce
ir ,is
rotor , stator current
imR
rotor magnetizing current
λ r ,λ s
rotor , stator flux linkage
u r ,u s
rotor , stator voltage input
wmR
magnetizing current vector angular speed
w
rotor angular speed
ρ
angle between the stationary
α −β
and the rotating d − q reference frame p number of pole pairs Lr , Ls rotor , stator induc tan ce Lm
mutual induc tan ce
J
inercia
TL
load torque
In this paper, a nonlinear input-output linearization and decoupling control scheme for the induction-motor drive has been developed. This later is completed with a stable model reference adaptive system (MRAS) rotorresistance estimator and a load torque observer to increase the tracking performance of the induction motor drive.
(.)d ,(.)q (.) in (d ,q ) frame (.)a ,(.)b (.) in (a ,b ) frame (ˆ.) estimate of (.) σR =
Lr − Lm L − Lm ,σ S = S Lm Lm
2 Model Description Assuming linear magnetic circuits and balanced three phase windings, the fifth-order nonlinear model of IM, expressed in the rotor flux reference frame is [7]:
1 Introduction Early control schemes implementing decoupled control of machine flux and torque are exemplified by fieldoriented control (FOC) as proposed by Blaschke [1] and Leonhard [2]. The basic goal of the FOC is to resolve the stator current vector into two components: one used to control the machine flux and the other to control the
ISSN: 1790-5117
139
ISBN: 978-960-474-026-0
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
disd 1 = u sd − (Rs + RR )isd + wmR Lσ isq + RR imR dt Lσ
(
disq dt
=
1 u sq − (Rs + RR )isq − wmR Lσ isd − wΜimR Lσ
(
dimR RR (isd − imR ) = dt Μ dρ R isq = w+ R dt Μ imR
Since the estimated rotor magnetizing current is sensitive to rotor-resistance variation, a stable rotor-resistance MRAS estimator can be developed.
) )
5 MRAS Rotor Resistance Estimation We can rewrite the dynamic model of an induction motor in stator frame as a compact form given in (W.Leonhard, 1984[7], A.V.Pavlov, and A.T.Zaremba 2001[9]) by:
(1)
dw p 2 = p Μ imR isq − t L dt J 3
dw 3 pLm T T i s Nλr − L = dt 2 JLr J dλr Rr R I + pwN λr + r Jis = − dt Lr Lr dis J Rr − I + pwN λr =− σLs Lr Lr dt
Where Lm Μ= 1+σ R
Lσ =
σ 1−σ
Μ
1 σ = 1− (1 + σ S )(1 + σ R ) RR =
Rr
wmR
(1 + σ R )2
dρ = dt
−
(2)
The load torque is needed for the input-output linearizing control. It is calculated by the load torque observer [8]: (3)
2 Rs + Lm Rr − Rr , β1 = p, 2 Lr Lr RR pR α2 = − r s , β2 = s , σLr Ls σLs
α1 = −
4 Adaptive Rotor Magnetizing Current Estimator
1 σLs
R
p
1
r , β3 = − , α4 = . σLs Lr σLs σLs Adding cdis/dt, c > 0, to both sides of (9) and formally dividing by (d/dt + c) transforms (9) to
α3 =
The estimator of the rotor magnetizing current vector requires only the measurement of the stator currents and the rotor speed. It is given by [7]:
dis = a + wb + ε dt
(4)
(10)
Where ε → 0 exponentially and the functions a and b are linear combinations of the filtered stator current and stator voltage command:
(5)
ˆi mR is the estimated magnitude of the vector imR and
a = (c + α1 )i11 + α 2i10 + α 3u10 + α 4u11
ρ is the estimated angle of the same vector.
b = N (β1i11 + β 2i10 + β 3u10 )
Where
ISSN: 1790-5117
(9)
Where
constant and z is the observer state.
)
(8)
0 − 1 N = 1 0
d 2 is di = (α1 I + wβ1 N ) s + (α 2 I + wβ 2 N )is + 2 dt dt (α 3 I + wβ3 N )us + α 4 dus dt
ˆt L is the observed load torque, τ 0 is the observer time
(
is + 1 u s σLs
In order to design a rotor resistance identifier, equations (7), (8) are transformed to eliminate the unobservable rotor flux. At this point an assumption is made that the rotor speed changes significantly slower relative to the rotor flux. Thus it may be treated as a constant parameter. First differentiating (8) and eliminating λr gives
3 Load Torque Observer
ˆ dˆimR R = R isd − ˆimR dt Μ ˆ dρ Rˆ isq = w+ R . dt Μ ˆimR
2 Rs + J Rr L2r
1 0 , I = 0 1
shaft :
ˆt L = 1 z − J w p τ 0 dz 2 = p Μ ˆimRiSq − ˆt L dt 3
1 σLs
(7)
Where
t L is the load torque and te is the motor torque on the
2 te = p Μ imR isq 3
(6)
140
ISBN: 978-960-474-026-0
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
s 1 is is , i11 = s+c s+c s 1 u10 = us u s , u11 = s+c s+c i10 =
The control design procedure is based on the aforementioned model (1). The angle ρ is computed by the equations (4) and (5). The rotor magnetizing current imR and the rotor speed w were chosen as the outputs, defining the output vector:
(11)
Here s denotes d/dt. To obtain a reference model for the rotor resistance identification the part of the right-hand side of (10) containing Rr is separated: dis = f1 + Rr f 2 + wNf 3 + ε dt
y = [ y1 y 2 ]T = [imR w]T
(17) The components of the stator voltage u sd and u sq were chosen as the inputs, defining the input vector:
(12)
[
u = u sd u sq
Where
x = [x1 x2 x3 x4 ]T = isd isq imR w T
[
x& = f (x ) + G .u
1 Lr
2 1 + Lm σL L s r
Where f (x ) = [ f1 f1 =
, γ 2 = ρ1 , γ 3 = ρ 2 . Lr Lr
(
)
)
1 Lσ
1 0 0 Lσ
T
h(x ) = [h1 h2 ]T , and the terms h1 and h2 are: h1 = imR , h2 = w
The directional (or Lie) derivate of the scalar function h( x ) in the direction of the vector function f ( x ) in the state x [11] will be needed in the following sections. It is defined as: L f h = ∇h. f ( x )
(15)
Higher Lie derivatives are defined recursively:
(
L(fi )h = L f L(fi −1 )h
Let the new state vector state transformation:
(16)
6 Control Design
)
z = [z1 z 2 z3 z 4 ]T
be defined by the
z1 = h1 = imR
Dynamic performances of the IM drive can be improved using the input-output linearizing control. If the motor and the model parameters are closely matched, this control structure provides exact decoupling and linearization of the model even during the rotor flux transients. In this case, the rotor magnetizing current and the rotor speed control are decoupled, while their dynamics are linear.
ISSN: 1790-5117
2 − (R + R )i + wL i + L R R isq + R i S R sd R mR σ sq σ Μ imR
T
)
(
f 4 ]T
f3
1 g1 = 0 0 0 , g 2 = 0 Lσ
diˆs = − L iˆs − is + f1 + wNf 3 + Rˆ r f 2 (14) dt Where L > 0 is a constant and Rˆ r is the estimate of Rr. The dynamics of the error e = iˆs − is is the following
de = − Le + Rˆ r − Rr f 2 − ε dt The adjustment equation for Rˆ r is T dRˆ r = −γ iˆs − is f 2 , γ > 0. dt
1 Lσ
f2
− (R + R )i − wL i − L RR isd isq − wΜi S R sq σ sq σ mR Μ imR R p 2 f 3 = R (isd − imR ), f 4 = p Μ imR isq − t L Μ J 3 G = [g1 g 2 ] , and the vectors g1 and g 2 are: f2 =
Such ω is available for measurement the design of an Rridentifier is straightforward. It is based on the MRAS identification approach [10] with (12) being a reference model. Within this approach an identifier consists of a tuning model depending on an estimate of the unknown parameter and a mechanism to adjust the estimate. This adjustment is performed to make the output of the tuning model asymptotically match the output of the reference model. In our case the tuning model is given by
(
(20)
y = h( x )
Rs 1 , ρ2 = , σLs σLs
pRs p β1 = p, β 2 = , β3 = − , σLs σLs
γ1 = −
]
(19) The IM model can now be written in the following matrix form:
(13)
f 3 = β1i11 + β 2i10 + β 3u10 Coefficients in (13) are calculated according to the following formulae:
ρ1 = −
(18)
The state vector is :
f1 = (c + ρ1 )i11 + ρ 2u11 f 2 = γ 1i11 + γ 2i10 + γ 3u10
]T
z 2 = L f (h1 ) =
dimR dt
(21)
z3 = h2 = w z 4 = L f (h2 ) =
dw dt
The state variables z1 , z 2 , z3 and z 4 must be linearly independent. This condition is fulfilled if they are chosen as defined by (21). Tracking the time derivative of state
141
ISBN: 978-960-474-026-0
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
variables zi yields the state space model in new coordinates:
reference trajectories
controllers defined by equations (27) (28), were chosen:
z&1 = z 2 z&2 =
L2f h1
(
+ Lg1L f h1.u sd + Lg 2 L f h1.u sq
t
+ K DI
(22)
t
+ K QI
(27)
)
− yˆ1 dτ
)
∫ (y 0
2 ref
(
) (28)
)
− yˆ 2 dτ
realization of the tracking controller (28) also requires the shaft acceleration y& 2 to be known. Since this variable is not always measured, it is calculated by the observer, described in previous section. If the actual and model parameters are closely matched the input-output linearizing control with the tracking controllers provides decoupled control of the rotor magnetizing current and rotor speed while their dynamics are linear. Tracking controllers also assures perfect tracking of the reference trajectories.
(23)
2 L isq 1 RR (−( RS + RR )isd + we Lσ isq + RR σ . ΜLσ 1−σ M imR
1 RR imR ) 1−σ
2 p 2Μ 1 imR (−( RS + RR )isq − we Lσ isd − wΜimR ) 3JLσ 1−σ
Lg1L f h1 =
)
Equation (27) and (28) reveal that the reference trajectories y1ref and y 2 ref must be C 2 functions. The
Where
+
1ref
0
(
Taking into account the definition of the input vector (17) the model can be rewritten as: 2 &y&1 L f h1 Lg1L f h1 Lg 2 L f h1 u sd + = && 2 y 2 L f h2 Lg1L f h2 Lg 2 L f h2 u sq = d (x ) + E (x ) . u
∫ (y
(
vq = &y&2ref + K Q3 y& 2 ref − yˆ& 2 + K Q 2 y2 ref − yˆ 2
z&4 = L2f h2 + Lg1L f h2 .u sd + Lg 2 L f h2 .u sq
L2f h2 =
)
vd = &y&1ref + K D 2 y&1ref − yˆ&1 + K D1 y1ref − yˆ1
z&3 = z 4
L2f h1 =
and wref , the tracking
imRref
RR ΜLσ
v1 = z&2
∫
v2 = z&4
∫
z 2 = z&1
∫
z1 = y1 = imR
Lg 2 L f h1 = 0 Lg1L f h2 = 0 2 p 2Μ Lg 2 L f h2 = imR 3JLσ
z 4 = z&3
∫
z3 = y2 = w
Fig. 1: Decoupled and linearized model of the IM
The obtained model is still nonlinear and coupled. The matrix E is nonsingular everywhere except for imR = 0 . Let the control inputs u = [u sd u sq ]T be chosen as: u = − E −1. d + E −1. v
Where v is the vector of new inputs v = [vd
(24) vq
7 Simulations Results
]T .
The overall configuration of the control system for IM is shown in Fig.2. The effectiveness of the proposed controller combined with the load torque observer and the rotor resistance estimation has been verified by simulations in Matlab/Simulink. The parameters of the induction motor used are given in Appendix. First, the simulation results have been obtained under a constant load of 10 Nm. The controller gains are : K D 2 = K Q 3 = 1200 ,
Equation (24) can now be rewritten by the components: 2 L isq 1 1 u sd = RS + RR isd − we Lσ isq − RR σ . − RR iˆmR ˆ 1−σ Μ imR 1 − σ Μ Lσ + vd RR
3JLσ 1 u sq = RS + RR isq + we Lσ isd + wΜ iˆmR + vq 1−σ 2 p 2 Μ iˆmR
K D1 = KQ 2 = 64.104 and K DI = K QI = 128.106 .
By applying the control inputs (24) to the motor model (23), the overall dynamics of the obtained closed loop system are given by the following model: &y&1 = vd (25) &y&2 = vq (26)
The load torque observer time constant is τ 0 = 0.03 . Parameters of the MRAS identifier are: γ = 100 , L=21000 and c=5000. Results obtained are shown in Fig.3. The reference speed is set to 80 rad/s until t=4s, when it is reversed to -80 rad/s to allow drive to operate in the generating mode. The reference magnetizing rotor current is set to 4.7A. The load torque is changed from 0 to 10 Nm at
The block diagram of the closed loop system (25) (26) is shown in Fig. 1. It is decoupled and linear but unstable. In order to stabilize it and provide tracking of the
ISSN: 1790-5117
142
ISBN: 978-960-474-026-0
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
t=0.6s. The rotor resistance estimator and the load torque observer are activated. It can be seen that the observed load torque converges very quickly to the actual value. In addition, the estimated values of Rr follow its actual value very closely. In order to show the convergence capability of the MRAS rotor resistance estimator, at t=1.2 s, the interne value of Rr in the MAS model has been disturbed and varied intentionally 100% of its initial value and held constant until t=2.5s, at the same moment, the rotor resistance estimator is disconnected from the control structure. It can be noted that the error in the estimated value of Rr produces a steady state error in the speed and rotor magnetizing current control and also generates an error in the observed value of the load torque. At t=2.5s, the initial value of Rr in MAS model has restored and the rotor resistance estimator has been reconnected and the rotor resistance is seen to converge to the actual value and also the other system variables. At t=4s, the reference speed is reversed to -80 rad/s to allow the drive to operate in the generating mode. The results show a stable operation of the drive in the various operating modes. Furthermore, simulation results have been performed to show the capability of the load torque observer to track the rapid load torque changes and also to show the decouple control of rotor magnetizing current and speed performance. The results are shown in Fig. 4. These results have been obtained using the same parameters used for the results in Fig. 3. The rotor resistance and load torque estimators are activated at t=0.2s. The load torque has been reversed from 10 to -10 Nm at t=2s. It can be seen from Fig. 4 that the observed load torque converges rapidly to its actual value and the rotor resistance estimator is stable. In addition, the results in Fig. 4 show an excellent decouple control of rotor magnetizing current and speed. Thus, the simulation results confirm the robustness of the proposed scheme with respect to the variation of the rotor resistance and load torque.
torque observer have been used. In simulation results, we have shown that the proposed nonlinear adaptive control algorithm achieved very good tracking performance within a wide range of the operation of the IM. The proposed method also presented a very interesting robustness properties with respect to the extremely variation of the rotor resistance and reversal of the load torque. The other interesting feature of the proposed method is that it is simple and easy to implement in real time.
APPENDIX INDUCTION MOTOR DATA Stator resistance Rotor resistance Mutual inductance Rotor inductance Stator inductance Number of pole pairs Motor load inertia
References: [1] F. Blaschke, “The principle of field orientation applied to the transvector closed-loop control system for rotating field machines,” Siemens Rev.,vol.34, no.5,pp. 217-220, May 1972. [2] W. Leonhard, “Microcomputer control of high dynamic performance ac-drives—A survey,” Automatica, vol. 22, no. 1, pp. 1–19, 1986. [3] Z. Krzeminski, “Nonlinear control of the induction motor,” in Proc. 10th IFAC World Congress, Munich, Germany, 1987, pp.349-354. [4] R. Marino, S. Peresada, and P. Valigi, “Adaptive partial feedback linearizing of induction motors,” in Proc. 29th Conf Decision and Control, Honolulu, HI, Dec. 1990. [5] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearizing control of induction motors,” IEEE Transactions on Automatic Control, Vol. 38, No. 2, pp. 208- 221, 1993. [6] Mohamed Rashed, Peter F. A. MacConnell, and A. Fraser Stronach, “Nonlinear Adaptive State-Feedback Speed Control of a Voltage-Fed Induction Motor With Varying Parameters”, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 3, MAY/JUNE 2006. [7] Leonhard, W. (1984). Control of electric drives. Springer Verlag. [8] S. Endo, H. Kobayashi, Y. Yoshida, S. Kobayashi, “Robust Digital Tracking Controller Design for High speed Positioning System –New Experimental Results, 3rd International Workshop on Advanced Motion Control, University of California, Berkeley, March 1994, pp. 643-647.
8 Conclusion In this paper, a novel scheme for speed and rotor magnetizing current control of induction motor using online estimations of the rotor resistance and load torque has been described. The nonlinear controller presented provides voltage inputs on the basis of rotor speed and stator currents measurements and guarantees rapid tracking of smooth speed and rotor magnetizing current references for unknown parameters (rotor resistance and load torque) and non-measurable state variables (rotor magnetizing current). MRAS rotor resistance and load
ISSN: 1790-5117
1.34 ohms; 1.24 ohms; 0.17 H; 0.18 H; 0.18 H; 2 0.0153 kgm2;
143
ISBN: 978-960-474-026-0
Proceedings of the 8th WSEAS International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES (POWER '08)
[9] Pavlov, A. V., & Zaremba, A. T. (2001). Real-time rotor and stator resistances estimation of an induction motor", Proceedings of NOLCOS-01, St-Petersbourg. [10] Sastry, S. and M. Bodson (1989). Adaptive control: stability, convergence ,and robustness. Prentice Hall. New Jersey. [11] Slotine, J. J., & Li, W. (1991). Applied nonlinear control. Prentice Hall. New York.
Fig. 2 Overall bloc diagram of the control scheme for IM
160
100
ω ref
120
60
Speed (rad/s)
Speed (rad/s)
ω ref
80
ω
80
40
0
-40
ω
40 20 0 -20 -40 -60
-80 -80
1
2
3
12
5
6
-100
7
imRref 10
imR 8
6
4
2
0 0
1
2
3
4
5
6
7
(b)
20
Load torque(N.m)
(a)
4
Rotor magnetizing current (A)
0
TL Tˆ
15
L
10
5
0
-5
1
2
3
(a)
4
5
6
7
imRref 10
imR 8
6
4
2
0 0
1
2
3
(b)
4
5
6
7
TL Tˆ
10
L
5
0
-5
-10
-15 1
2
3
(c)
4
5
6
7
0
Rr Rˆ
2.4 2.2
r
2 1.8 1.6 1.4 1.2 1 0.8
0
1
2
3
4
5
6
2
3
(c)
4
5
6
7
1.3
1.2
1.1
1
Rr Rˆ
0.9
0.8
r
0.7 0
7
1
2
3
4
5
6
7
(d) Time (s)
(d) Time (s)
Fig. 3 Tracking Performance and parameters estimates: a) reference and actual rotor speed, b) reference and actual rotor magnetizing current, c) TL and TˆL (N.m), d) Rr and Rˆr ( Ω )
ISSN: 1790-5117
1
1.4
Rotor Resistance(ohms)
0 2.6
Rotor Resistance(ohms)
0
12
15
Load torque(N.m)
Rotor magnetizing current (A)
-120
Fig. 4 Tracking Performance and parameters estimates: a) reference and actual rotor speed, b) reference and actual rotor magnetizing current, c) TL and TˆL (N.m), d) Rr and Rˆr ( Ω )
144
ISBN: 978-960-474-026-0
