- Email: [email protected]
Magnetic saturation effect in the achievement of wide speed range control for induction machine
Magnetic saturation effect in the achievement of wide speed range control for induction machine H. Ouadi+, F. Giri++, A. Elfadili++ L. Dugard+++, * Hassan II University, Sciences faculty of Ain Chock, Casablanca, Morocco (e-mail: hamidouadi3@ yahoo.fr). ** GREYC Lab, University of Caen, Caen, France (email: fouad.giri@ unicaen.fr) *** GIPSA Lab, ENSIEG, Grenoble, France (e-mail: [email protected]} Abstract: The problem of induction machine speed control is considered. Most previous works considered the machine magnetic characteristic to be linear and the flux reference to be constant. Generally, the constant flux reference is given the flux nominal value located in elbow zone of the magnetic characteristic. Doing so, the machine operation mode is not optimal in presence of small load torques. Similarly, if the flux reference is given a small value, the machine operation is again not optimal in presence of large load torques. As a matter of fact, an optimal operation mode can not be achieved with a constant flux reference. But, varying flux control strategies can not be designed without accounting for the nonlinearity of the machine magnetic characteristic. In the present work, we precisely develop a speed control strategy based on a machine model that accounts for the magnetic characteristic saturation. A speed controller is designed using a nonlinear design technique. The performances of the developed control strategy are formally analysed and their supremacy with respect to standard control solutions (assuming linear magnetic characteristic) is illustrated through simulation. Keywords: ac machine, speed control, flux control, magnetic saturation, non linear control design, Lyapunov function. 1. INTRODUCTION It is widely recognized that the induction motor is going to become the main actuator for industrial purposes. Indeed, as compared to the DC machine, it provides a better power/mass ratio, simpler maintenance (as it includes no mechanical commutators) and a relatively lower cost. However, the problem of controlling the induction motor is more complex, due to its multivariable and highly nonlinear features. Furthermore, the model parameters are time-varying, even during normal operation conditions, and not all its state variables are accessible to measurement. Most previous control strategies for induction machine speed regulation were based on the standard model and involved constant flux references. The standard model is obtained supposing the machine magnetic characteristic to be static and linear. As a matter of fact, such characteristic is nonlinear in physical machines, it exhibits several nonlinear features e.g. flux saturation (Fig 1). On the other hand, the reference flux involved in previously proposed controller is given the nominal value, generally located in the elbow of the machine magnetic characteristic (Hu and al 1996), (Robert and al 1999). In such situation, machine efficiency is maximal when the machine load torque is close to its nominal value. But, in practical applications (Leonard 1985), the load torque is usually not a priori fixed and may be subject to large variations. Then, if the load torque is small (with respect to the nominal load), there is a useless energy stored in stator inductances reducing the machine efficiency. On the other hand, if the flux reference is given a small value, the achievable machine motor torque will be insufficient to
balance large load torques. Therefore, it is impossible with constant flux reference controllers to achieve a speed control guaranteeing optimal machine performances. Optimality is intended in the sense of efficiency, power factor, maximal torque, etc. To overcome the above shortcomings, it is necessary to develop new speed control strategies where the flux reference is variable (dependent on both the speed reference and load torque). As these are allowed to be varying in a large interval (ranging from 0 to their nominal values), it is expected that the optimal flux reference is in turn subject to large excursions, covering the linear and nonlinear parts of the magnetic characteristic. Therefore, the development of speed control strategies, guaranteeing optimal machine performances, must rely on a machine model that takes into account the nonlinear feature of the magnetic characteristic. Fortunately, an example of such models exists; it was developed and experimentally validated in Ouadi and al (2004). In the present paper, we develop a new controller (NC) involving variable flux reference generator and speed regulators obtained from the model developed in Ouadi and al (2004) (that accounts for magnetic saturation). A speed regulator is designed using the backstepping technique. The speed controller thus obtained turns out to be quite different from those obtained through the standard controller (SC) that assumes linear static magnetic characteristic. It is shown that the NC ensures global asymptotic stability of the closed-loop system and enforces the machine speed to perfectly track its varying reference trajectory, despite the changing load torque. These performances are also illustrated through simulations
This paper is organized as follows: Section 2 describes the induction machine model; machine speed controller is designed and analyzed in Section 3; the designed controller performances are illustrated by simulation in Section 4; a conclusion and reference list end the paper. 2. INDUCTION MOTOR MODELLING In Ouadi and al (2004), a model that accounts for the saturation feature of the machine magnetic characteristic (fig 1) has been developed and experimentally validated for the considered induction motor. Such magnetically saturated model is defined by the following state-space representation:
def
(1)
y = h( x) = [h1( x) h2 ( x) h3 ( x)] with:
T
x = [ x1, x2 , x3 , x4 , x5 ]T = [isα , isβ , φrα , φrβ , Ω]T
u = [u sα , u sβ ]T , y = [ x1, x2 , x5 ]T
f1( x, u ) − a2isα + δ (t )φrα + a3 pΩφrβ + a3usα f ( x, u ) − a2isβ − a3 pΩφrα + δ (t )φrβ + a3usβ 2 a1isα − Lseqδ (t )φrα − pΩφrβ f 3 ( x, u ) = (3a) δ ( ) φ φ a i − L t + p Ω 1 β β α s seq r r f 4 ( x, u ) p T f 5 ( x, u ) (φrα isβ − φrβ isα ) − L J J h1( x) x1 h ( x ) = x (3b) 2 2 h3 ( x) x5 where: . δ (t ) is a varying parameter that depends on the machine magnetic state (see Fig 2). In Ouadi and al (2004), this dependence was given a polynomial approximation, i.e.: δ (t ) = W (Φ r ) (4) where:
(5) W (Φ r ) = b0 + b1Φ r + ... + bm Φ m r The involved coefficients have been experimentally identified in Ouadi and al (2004) using Fig 1. . Φ r denotes the amplitude of the (instantaneous) rotor flux, denoted φr . Consequently, one has: Φ r = φr2α + φr2β
1
0.5
0
(6)
stator current and stator voltage, respectively . Ω represents the motor speed, . Rs , Rr denote the stator and rotor resistances; . TL represents the load torque; . p is the number of pole pairs; . L seq is the equivalent inductance (of both stator and rotor
0
20
40 60 Magnetic c urrent (A)
80
100
Fig 1. Magnetic characteristic experimentally obtained in Ouadi and al (2004). for a 7.5KW induction motor. The rotor flux norm (Wb) versus the magnetic current (A), 4000 3500 3000 2500 2000 1500 1000 500
where φrα ,φrβ denote the rotor flux αβ -components. . ( isα , isβ ) and ( usα , u sβ ) are the αβ-components of the
leakage) as this is seen from the stator,
1.5
(2) Rotor flux norm (Wb)
f5 ( x, u )]T
1 Rs + Rr , a3 = Lseq Lseq
The numerical values of the model parameters are those of Ouadi and al (2004) where the model is experimentally validated using an induction motor of 7.5 KW power (Table 1). Finally, notice that the standard model, which has been widely used in previous works (Leonard and al (1985)-Ortega and al (1996)), is readily obtained from the above satureted model letting δ (t ) = δ L = cte be a constant parameter in (3a). The value of the latter depends on the operation point on the machine magnetic characteristic. In the latter, the Standard-model will be considered with respect to the nominal flux value of 1.1Wb . In Section 4, the numerical value of δ L is given for the induction machine with the characteristics in Table I, operating around this nominal point.
delta
def
x = f ( x, u ) = [ f1 ( x, u ) ...
. a1 = Rr , a2 =
0
0.5 1 R otor flux norm (W b)
1.5
Fig.2. Characteristic ( δ , Φ r ) directly computed points(++) and polynomial interpolation (solid). Unities: δ(ΩH −2 ) , Φ r (Wb) 3. SPEED REGULATOR DESIGN AND ANALYSIS We are interested in the problem of controlling the machine speed and the rotor flux norm for the saturated induction machine described by (1). The speed reference Ω ref is any bounded and derivable function of time and its two first derivatives are available and bounded. These properties can always be achieved by filtering the reference. The controller design will now be performed in two steps using the backstepping technique Krstic and al (1995). First, introduce the tracking errors:
e1 = Ωref − Ω
(7)
z1 = Φ 2ref − (φr2α + φr2β )
(8)
Step 1. The dynamics of the errors e1 and z1 are obtained deriving these with respect to time. Using (1)-(3a), it follows that: p TL f (9) e1 = Ω − Ω) ref − ( (φrα isβ − φrβ isα ) − J J J z = 2Φ Φ − 2φ φ − 2φ φ 1
ref
ref
rα rα
rβ rβ
= 2Φ ref Φ ref − 2φrα ( − Lseqδ (t )φrα + a1isα − ωφrβ ) − 2φrβ (− Lseqδ (t )φrβ + a1isβ + ωφrα ) 2 2 = 2Φ ref Φ ref − 2a1 (φrα isα + φrβ isβ ) + 2 Lseqδ (t )(φrα + φrβ ) = 2Φ Φ − 2a (φ i + φ i )
ref
ref
1
rα sα
rβ sβ
+ 2 Lseqδ (t )(Φ 2ref − z1 )
(10)
In equations (9) and (10), the quantities p (φrα isβ − φrβ isα ) and 2a1 (φrα isα + φrβ isβ ) stands up as J virtual control signals. If these were actual controls, the error system (9)-(10) could be globally asymptotically stabilized p letting (φrα isβ − φrβ isα ) = μ1 and 2a1 (φrα isα + φrβ isβ ) = ν1 J with: def TL f (11) μ1 = c1e1 + Ω + Ω ref + J J def
ν 1 = d1 z1 + 2Φ ref Φ ref + 2 Lseqδ (t )( Φ 2ref − z1 )
(12)
where c1 and d1 are any positive real design parameters. Indeed, considering the function Lyapunov candidate: 1 V1 = (e12 + z12 ) (13) 2 p it follows from (9)-(12) that, letting (φrα isβ − φrβ isα ) = μ1 J and 2a1 (φrα isα + φrβ isβ ) = ν1 would give:
V1 = −c1e12 − d1z12 (14) which proves the global asymptotic stability of the system p (9)-(12). As the quantities (φrα isβ − φrβ isα ) and J 2a1 (φrα isα + φrβ isβ ) are not actual control signals, they cannot be let equal to μ1 and ν 1 . Nevertheless, we retain μ1 and ν 1 as first stabilizing functions. Step 2. Introduce the new errors: p (15) e2 = μ1 − (φrα isβ − φrβ isα ) J (16) z2 = ν1 − 2a1 (φrα isα + φrβ isβ ) Then, the dynamics of the errors e1 and z1 , described by (9)-(12), can be rewritten as follows: e1 = −c1e1 + e2 (17a) z1 = −d1z1 + z2 (17b)
Also, the time-derivative of the function V1 can be expressed in function of the new errors as follows: V = −c e2 − d z 2 + e e + z z (18) 1
11
1 1
1 2
1 2
The second design step consists in choosing the actual control signals, u sα and u sβ , so that all errors ( e1, z1, e 2 , z 2 ) converge to zero. p (19) (φrα isβ + φrα isβ − φrβ isα − φrβ isα ) J Using (1)-(3a) and (11) one successively gets from (19): + TL + f Ω e2 = c1e1 + Ω ref J J p p − (φrα isβ − φrβ isα ) − (φrα isβ − φrβ isα ) J J + TL = c1(−c1e1 + e2 ) + Ω ref J f p T f + ( (φrα isβ − φrβ isα ) − L − Ω) J J J J p − ((− Lseqδ (t )φrα + a1isα − ωφrβ )isβ J −(− Lseqδ (t )φrβ + a1isβ + ωφrα )isα ) e2 = μ1 −
p φrα (δ (t )φrβ − a3ωφrα − a2isβ + a3u sβ ) J p + φrβ (δ (t )φrα + a3ωφrβ − a2isα + a3usα ) J
−
(20)
This can be condensed as follows: e2 = μ2 +
p a3 (φrβ usα − φrα usβ ) J
(21)
with
+ μ2 = c1(−c1e1 + e2 ) + Ω ref + − + − +
TL J
f p T f ( (φrα isβ − φrβ isα ) − L − Ω) J J J J p ( − Lseqδ (t )φrα + a1isα − ωφrβ )isβ J p (− Lseqδ (t )φrβ + a1isβ + ωφ rα )isα J p φrα (δ (t )φrβ − a3ωφrα − a2isβ ) J p φ rβ (δ (t )φ rα + a 3ωφ rβ − a 2 i sα ) J
Similarly, It follows from (16) and (1)-(3a) z 2 undergoes the following differential equation: z2 = ν1 − 2a1(φrα isα + φrα isα + φrβ isβ + φrβ isβ ) Using (1)-(3a) and (12) one successively gets from (23):
(22) that, (23)
V2 = e1e1 + e2e2 + z1z1 + z2 z2
+ 2Φ 2 + 2 L δ (t )(2Φ Φ z2 = d1z1 + 2Φ ref Φ ref ref seq ref ref − z1 ) + 2 Lseqδ(t )(Φ 2ref − z1 )
( (
Using (18), (21) and (26a), equation (31) implies:
)
− 2a1[ − Lseqδ (t )φrα + a1isα − ωφ rβ isα + φrα δ (t )φrα + a3ωφ rβ − a2isα + a3usα
(
)
)
(24)
)
φrβ φrα φrα + φrβ Φr Φr
2
(
)
(26a)
2 + 2 Lseqδ (t )(2Φ ref Φ ref − z1 ) + 2 Lseqδ (t )(Φ ref − z1 )
)
− 2a1( − Lseqδ (t )φrα + a1isα − ωφrβ isα
(
)
(26b)
The derivatives Φ ref and Φ ref are obtained using the relation: Φ ref = ξ ( I s ) = h0 + h1I s + h2 I s2 + ... + hn I sn
(27)
Specifically, one has: dξ ( I s ) dξ ( I s ) dI s Φ = ref = dt dI s dt dξ ( I s ) isα isα + isβ isβ = dI s Is
(28a)
2 d 2ξ ( I s ) isα isα + isβ isβ Φ = ref Is dI s 2 2 2 dξ ( I s ) (isα ) + isα isα + (isβ ) + isβ isβ + dI s Is
−
2 dξ ( I s ) (isα isα + isβ isβ ) dI s I s3
(28b)
To analyze the error system, composed of equations (17a-b), (21) and (26a), let us consider the following augmented Lyapunov function candidate: 1 V2 = V1 + (e22 + z22 ) 2
1 1
2 2
[
]
(32b)
where c 2 and d 2 are new positive real design parameters. Equation (32b) suggests that the control signals u sα , u sβ should be chosen so that the two quantities between brackets on the right side of (32) equal zero. Letting these quantities equal to zero and solving the resulting equation with respect to ( u sα , u sβ ), gives the following control law:
+ 2Φ 2 ν 2 = d1 (− d1 z1 + z 2 ) + 2Φ ref Φ ref ref
+ − Lseqδ (t )φrβ + a1isβ + ωφrα isβ ) + φ rβ (δ (t )φ rβ − a 2 isβ ) + φrα (δ (t )φrα − a2isα ))
2 2
+ z2 z1 + d 2 z2 + ν 2 − 2a1a3 (φrα usα + φrβ usβ )
with
(
11
to the right side of
p + e2 e1 + c2e2 + μ2 + a3 (φrβ usα − φrα usβ ) J
(25)
Equation (24) is given the following compact form: z2 = ν 2 − 2a1a3 φrβ u sβ + φrα u sα
)
Adding c2e22 − c2e22 + d 2 z22 − d 2 z22 (32a) and rearranging terms, yields: V = −c e2 − c e2 − d z 2 − d z 2
where the derivative of δ (t ) is obtained from (4): dδ dΦ r dδ = δ(t ) = dΦ r dt dΦ r
p V2 = −c1e12 − d1z12 + e2 e1 + μ2 + a3 (φrβ u sα − φrα usβ ) J (32a) + z2 z1 + ν 2 − 2a1a3 (φrα u sα + φrβ u sβ )
(
+ − Lseqδ (t )φrβ + a1isβ + ωφ rα )isα isβ + φrβ δ (t )φrβ − a3ωφ rα − a2isβ + a3usβ ]
(
(31)
(30)
Its time-derivative along the trajectory of the state vector (e1, z1, e2, z2) is:
usα λ0 u = sβ λ2
λ1 λ3
−1
− e − c e − μ2 1 2 2 − z1 − d 2 z2 − ν 2
(33)
with: p p a3φrβ , λ1 = − a3φrα , λ2 = −2a1a3φrα and J J λ3 = −2a1a3φrβ
λ0 =
(34)
λ λ It is worth noting that the matrix 0 1 is nonsingular λ2 λ3 p because its determinant D = λ0λ3 − λ1λ2 = −2 a1a3Φ 2r J never vanish because of the existence of the remnent flux. Substituting the control law (33) to u sα , u sβ in the right side
of (32) yields: V2 = −c1e12 − c2e22 − d1z12 − d 2 z22 (35) which is a negative function definite of the state vector ( e1, z1, e 2 , z 2 ). The results obtained until now are summarized in the following theorem: Theorem 1 (Main result). Consider the closed-loop system composed of the induction machine, described by model (1), and the nonlinear controller defined by the control law (33). Then, one has the following properties: 1) The closed-loop error system undergoes the following equations: e1 = −c1e1 + e2 ; z1 = −d1z1 + z2 (36a) e2 = −e1 − c2e2 ;
(36b) z2 = − z1 − d 2 z 2 2) The above system is globally asymptotically stable and consequently all errors vanish asymptotically, whatever the initial conditions.
V2 = (e12 + z12 + e22 + z22 ) / 2 is Lyapunov function of the error system (36a-d). Part 2 follows from the fact that V2 is negative definite. This completes the proof of Theorem 1. 4. SIMULATIONS RESULTS In this section, we will first illustrate the suppremacy of the new controller (NC) that accounts for magnetic saturation, over the standard controller (SC) that assumes linear static magnetic characteristic. The comparison is performed using a 7.5 kW induction machine whose characteristics are summarized in Table I. Then, we will show that perfect tracking of a varying flux reference is always achieved by the NC controller but generally not by the SC The experimental protocol is conceived in a way to make the machine operate in different conditions. These are fixed through the load variation law (Fig. 3) and speed reference (Fig. 4). The NC controller defined by the control law (33) is given the following design parameters values which proved to be convenient: c1 = 5
c 2 = 500
d 1 = 10
d 2 = 1000
The SC controller is obtained from the NC-controller simply letting the parameter δ (t ) in (33) be constant equal to 720. The other design parameters are given the following values that proved to be convenient for this controller: c1 = 25 c 2 = 420 d 1 = 100 d 2 = 200 The above value of δ = 720 characterizes the model (3) when the machine operates around the nominal operation point corresponding to a flux value of 1 .1Wb (on the magnetic characteristic). Obviously, the SC controller thus obtained is expected to perform well when the machine operates around the previous nominal working i.e. when the flux reference is sufficiently close to 1.1 Wb . Fig. 4 shows that the two controllers have an identical tracking quality as long as speed regulation is concerned. When it comes to flux tracking, the new controller is quite better (Figs 5 and 6). Fig 7 illustrate the benefit one gets from letting the flux reference be varying. For instance, in presence of a small load ( C r = 0 ), a flux setpoint of Φ ref = 0.4 Wb leads to lower stator current consumption than in the case of a flux setpoint of Φ ref = 1.1 Wb . Indeed, the absorbed current in the last case is 16 A while it is only of 6 A in the first case.
1
KW V Wb Ω Ω Kgm2 N m.s rd-1 mH
Equivalent inductance of stator and rotor leakage seen from the stator
5. CONCLUSIONS In this paper, we have considered the problem of speed and flux control of induction machine in presence of the magnetic characteristic saturation. A new controller described by (33) is designed using the backstepping technique, based on model (1). Besides accounting for the saturation feature of the magnetic characteristic, the proposed controller can involves a state-dependent optimized flux reference. It is formally shown (Theorem 1) that the proposed controller e1 = Ω ref − Ω and guarantees that the errors 2 z1 = Φ ref − (φ r2α + φ r2β ) globally converge to zero. That is, the
tracking objective is perfectly achieved both for the machine speed and rotor flux. 40 35 30 25 Load torque (Nm)
respectively. This proves Part 1. On the other hand, it is readily seen from (13), (30) and (35) that
TABLE I. MOTOR CHARACTERISTICS PN 7.5 Nominal power 380 Nominal voltage Usn 1.1 Nominal flux Φrn 0.63 stator resistance Rs 0.4 rotor resistance Rr 0.22 Inertia moment J 0.001 Friction coefficient f 2 Number of pole pairs p 7 Leakage equivalent inductance1 Lseq
20 15 10 5 0 -5 -10
0
5
10
15 time(s)
20
25
30
Fig.3. Applied load torque 160 150 140 130 120 Speed (rd/s)
Proof. Equations (36a) are immediately obtained from (17ab). Equations (36b) are obtained substituting the control law (33) to u sα , u sβ on the right side of (21) and (26a),
110 100 90 80 70 60
Remarks 1.a) It is worthy pointing out that the difference between the two control strategies, is more significant in presence of low load torques, because it is then that the SC reference is most different from the nominal flux value. b) Furthermore, there is no difference between the SC and NC controllers whenever a flux reference is considered and given the nominal value 1.1 Wb .
50
0
5
10
15 time(s)
20
25
30
Fig 4: Speed reference (dotted) and speed responses (solid): the NC and SC are indistinguishable. Unities: time (s); speed (rd/s)
1.1 1
flux norm (Wb)
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0
5
10
15 time(s)
20
25
30
Fig 5: Rotor flux norm reference (dotted) and flux response with standard controller (solid). Flux unity: Wb. 1.1 1
flux (Wb)
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0
5
10
15 time(s)
20
25
30
Fig 6: Flux reference (dotted) and flux response with new controller (33) (solid). 70
60
Stator current norm (A)
50
40
30
20
10
0
0
5
10
15 time(s)
20
25
30
Fig 7: Stator current
REFERENCES Ayman, M., EL-Refaie, Jahns Thomas, M., (2005) “Optimal Flux Weakening in Surface PM Machines Using Fractional-Slot Concentrated Windings” transactions on industry applications, vol. 41, no3 Edwin, C.K.P. and Stranislaw, H., (2001) “An Introduction to optimization” John Wilay & Sons. Hu, J., Dawson M. and Qu, Z., (1996) 'Robust tracking control of an induction motor', Int. J. Robust and Nonlinear Control, Vol. 6, pp. 201-219 Khalil, H.K., (1996) “Nonlinear systems” second edition, Prentice Hall, Upper Saddle River Krstic, M., Kanellakopoulos, I., Kokotovic, P., (1995) “Nonlinear and adaptive control design” , John Wilay & Sons, Inc.
Leonard, W. (1985) “Control of Electrical Drives” SpringerVerlag Berlin Heidelberg New York Tokyo Ouadi, H., Giri, F., and Dugard, L., (2004) “ Modelling saturated induction motors”. IEEE Conference on Control Applications (CCA’04), Taipei, Taiwan. Vol.1, pp. 75 – 80. Ortega, R., Nicklasson, P.J., and Espinosa-Perez, G. (1996) “On speed control of induction motors”, Automatica, Vol 32, N 3 Robert, T.N., Chiasson, J., Bodson, M., (1999)’High Performance Motion Control of an induction motor with magnetic saturation’ IEEE transactions on control systems technology, Vol 7, N0 3 Seguier, G. and Notelet, F., (2005) 'Electrotechique Industrielle', Editions Tec Doc, Lavoisier, Paris, ISBN 2-7430-0791-5. Xingyi, X., Novotny, D. W., (1992) “Selection of the flux reference for induction machine drives in the field weakening region” IEEE Transactions on Industry Applications, vol. 28, pp. 1353-1358.
balance large load torques. Therefore, it is impossible with constant flux reference controllers to achieve a speed control guaranteeing optimal machine performances. Optimality is intended in the sense of efficiency, power factor, maximal torque, etc. To overcome the above shortcomings, it is necessary to develop new speed control strategies where the flux reference is variable (dependent on both the speed reference and load torque). As these are allowed to be varying in a large interval (ranging from 0 to their nominal values), it is expected that the optimal flux reference is in turn subject to large excursions, covering the linear and nonlinear parts of the magnetic characteristic. Therefore, the development of speed control strategies, guaranteeing optimal machine performances, must rely on a machine model that takes into account the nonlinear feature of the magnetic characteristic. Fortunately, an example of such models exists; it was developed and experimentally validated in Ouadi and al (2004). In the present paper, we develop a new controller (NC) involving variable flux reference generator and speed regulators obtained from the model developed in Ouadi and al (2004) (that accounts for magnetic saturation). A speed regulator is designed using the backstepping technique. The speed controller thus obtained turns out to be quite different from those obtained through the standard controller (SC) that assumes linear static magnetic characteristic. It is shown that the NC ensures global asymptotic stability of the closed-loop system and enforces the machine speed to perfectly track its varying reference trajectory, despite the changing load torque. These performances are also illustrated through simulations
This paper is organized as follows: Section 2 describes the induction machine model; machine speed controller is designed and analyzed in Section 3; the designed controller performances are illustrated by simulation in Section 4; a conclusion and reference list end the paper. 2. INDUCTION MOTOR MODELLING In Ouadi and al (2004), a model that accounts for the saturation feature of the machine magnetic characteristic (fig 1) has been developed and experimentally validated for the considered induction motor. Such magnetically saturated model is defined by the following state-space representation:
def
(1)
y = h( x) = [h1( x) h2 ( x) h3 ( x)] with:
T
x = [ x1, x2 , x3 , x4 , x5 ]T = [isα , isβ , φrα , φrβ , Ω]T
u = [u sα , u sβ ]T , y = [ x1, x2 , x5 ]T
f1( x, u ) − a2isα + δ (t )φrα + a3 pΩφrβ + a3usα f ( x, u ) − a2isβ − a3 pΩφrα + δ (t )φrβ + a3usβ 2 a1isα − Lseqδ (t )φrα − pΩφrβ f 3 ( x, u ) = (3a) δ ( ) φ φ a i − L t + p Ω 1 β β α s seq r r f 4 ( x, u ) p T f 5 ( x, u ) (φrα isβ − φrβ isα ) − L J J h1( x) x1 h ( x ) = x (3b) 2 2 h3 ( x) x5 where: . δ (t ) is a varying parameter that depends on the machine magnetic state (see Fig 2). In Ouadi and al (2004), this dependence was given a polynomial approximation, i.e.: δ (t ) = W (Φ r ) (4) where:
(5) W (Φ r ) = b0 + b1Φ r + ... + bm Φ m r The involved coefficients have been experimentally identified in Ouadi and al (2004) using Fig 1. . Φ r denotes the amplitude of the (instantaneous) rotor flux, denoted φr . Consequently, one has: Φ r = φr2α + φr2β
1
0.5
0
(6)
stator current and stator voltage, respectively . Ω represents the motor speed, . Rs , Rr denote the stator and rotor resistances; . TL represents the load torque; . p is the number of pole pairs; . L seq is the equivalent inductance (of both stator and rotor
0
20
40 60 Magnetic c urrent (A)
80
100
Fig 1. Magnetic characteristic experimentally obtained in Ouadi and al (2004). for a 7.5KW induction motor. The rotor flux norm (Wb) versus the magnetic current (A), 4000 3500 3000 2500 2000 1500 1000 500
where φrα ,φrβ denote the rotor flux αβ -components. . ( isα , isβ ) and ( usα , u sβ ) are the αβ-components of the
leakage) as this is seen from the stator,
1.5
(2) Rotor flux norm (Wb)
f5 ( x, u )]T
1 Rs + Rr , a3 = Lseq Lseq
The numerical values of the model parameters are those of Ouadi and al (2004) where the model is experimentally validated using an induction motor of 7.5 KW power (Table 1). Finally, notice that the standard model, which has been widely used in previous works (Leonard and al (1985)-Ortega and al (1996)), is readily obtained from the above satureted model letting δ (t ) = δ L = cte be a constant parameter in (3a). The value of the latter depends on the operation point on the machine magnetic characteristic. In the latter, the Standard-model will be considered with respect to the nominal flux value of 1.1Wb . In Section 4, the numerical value of δ L is given for the induction machine with the characteristics in Table I, operating around this nominal point.
delta
def
x = f ( x, u ) = [ f1 ( x, u ) ...
. a1 = Rr , a2 =
0
0.5 1 R otor flux norm (W b)
1.5
Fig.2. Characteristic ( δ , Φ r ) directly computed points(++) and polynomial interpolation (solid). Unities: δ(ΩH −2 ) , Φ r (Wb) 3. SPEED REGULATOR DESIGN AND ANALYSIS We are interested in the problem of controlling the machine speed and the rotor flux norm for the saturated induction machine described by (1). The speed reference Ω ref is any bounded and derivable function of time and its two first derivatives are available and bounded. These properties can always be achieved by filtering the reference. The controller design will now be performed in two steps using the backstepping technique Krstic and al (1995). First, introduce the tracking errors:
e1 = Ωref − Ω
(7)
z1 = Φ 2ref − (φr2α + φr2β )
(8)
Step 1. The dynamics of the errors e1 and z1 are obtained deriving these with respect to time. Using (1)-(3a), it follows that: p TL f (9) e1 = Ω − Ω) ref − ( (φrα isβ − φrβ isα ) − J J J z = 2Φ Φ − 2φ φ − 2φ φ 1
ref
ref
rα rα
rβ rβ
= 2Φ ref Φ ref − 2φrα ( − Lseqδ (t )φrα + a1isα − ωφrβ ) − 2φrβ (− Lseqδ (t )φrβ + a1isβ + ωφrα ) 2 2 = 2Φ ref Φ ref − 2a1 (φrα isα + φrβ isβ ) + 2 Lseqδ (t )(φrα + φrβ ) = 2Φ Φ − 2a (φ i + φ i )
ref
ref
1
rα sα
rβ sβ
+ 2 Lseqδ (t )(Φ 2ref − z1 )
(10)
In equations (9) and (10), the quantities p (φrα isβ − φrβ isα ) and 2a1 (φrα isα + φrβ isβ ) stands up as J virtual control signals. If these were actual controls, the error system (9)-(10) could be globally asymptotically stabilized p letting (φrα isβ − φrβ isα ) = μ1 and 2a1 (φrα isα + φrβ isβ ) = ν1 J with: def TL f (11) μ1 = c1e1 + Ω + Ω ref + J J def
ν 1 = d1 z1 + 2Φ ref Φ ref + 2 Lseqδ (t )( Φ 2ref − z1 )
(12)
where c1 and d1 are any positive real design parameters. Indeed, considering the function Lyapunov candidate: 1 V1 = (e12 + z12 ) (13) 2 p it follows from (9)-(12) that, letting (φrα isβ − φrβ isα ) = μ1 J and 2a1 (φrα isα + φrβ isβ ) = ν1 would give:
V1 = −c1e12 − d1z12 (14) which proves the global asymptotic stability of the system p (9)-(12). As the quantities (φrα isβ − φrβ isα ) and J 2a1 (φrα isα + φrβ isβ ) are not actual control signals, they cannot be let equal to μ1 and ν 1 . Nevertheless, we retain μ1 and ν 1 as first stabilizing functions. Step 2. Introduce the new errors: p (15) e2 = μ1 − (φrα isβ − φrβ isα ) J (16) z2 = ν1 − 2a1 (φrα isα + φrβ isβ ) Then, the dynamics of the errors e1 and z1 , described by (9)-(12), can be rewritten as follows: e1 = −c1e1 + e2 (17a) z1 = −d1z1 + z2 (17b)
Also, the time-derivative of the function V1 can be expressed in function of the new errors as follows: V = −c e2 − d z 2 + e e + z z (18) 1
11
1 1
1 2
1 2
The second design step consists in choosing the actual control signals, u sα and u sβ , so that all errors ( e1, z1, e 2 , z 2 ) converge to zero. p (19) (φrα isβ + φrα isβ − φrβ isα − φrβ isα ) J Using (1)-(3a) and (11) one successively gets from (19): + TL + f Ω e2 = c1e1 + Ω ref J J p p − (φrα isβ − φrβ isα ) − (φrα isβ − φrβ isα ) J J + TL = c1(−c1e1 + e2 ) + Ω ref J f p T f + ( (φrα isβ − φrβ isα ) − L − Ω) J J J J p − ((− Lseqδ (t )φrα + a1isα − ωφrβ )isβ J −(− Lseqδ (t )φrβ + a1isβ + ωφrα )isα ) e2 = μ1 −
p φrα (δ (t )φrβ − a3ωφrα − a2isβ + a3u sβ ) J p + φrβ (δ (t )φrα + a3ωφrβ − a2isα + a3usα ) J
−
(20)
This can be condensed as follows: e2 = μ2 +
p a3 (φrβ usα − φrα usβ ) J
(21)
with
+ μ2 = c1(−c1e1 + e2 ) + Ω ref + − + − +
TL J
f p T f ( (φrα isβ − φrβ isα ) − L − Ω) J J J J p ( − Lseqδ (t )φrα + a1isα − ωφrβ )isβ J p (− Lseqδ (t )φrβ + a1isβ + ωφ rα )isα J p φrα (δ (t )φrβ − a3ωφrα − a2isβ ) J p φ rβ (δ (t )φ rα + a 3ωφ rβ − a 2 i sα ) J
Similarly, It follows from (16) and (1)-(3a) z 2 undergoes the following differential equation: z2 = ν1 − 2a1(φrα isα + φrα isα + φrβ isβ + φrβ isβ ) Using (1)-(3a) and (12) one successively gets from (23):
(22) that, (23)
V2 = e1e1 + e2e2 + z1z1 + z2 z2
+ 2Φ 2 + 2 L δ (t )(2Φ Φ z2 = d1z1 + 2Φ ref Φ ref ref seq ref ref − z1 ) + 2 Lseqδ(t )(Φ 2ref − z1 )
( (
Using (18), (21) and (26a), equation (31) implies:
)
− 2a1[ − Lseqδ (t )φrα + a1isα − ωφ rβ isα + φrα δ (t )φrα + a3ωφ rβ − a2isα + a3usα
(
)
)
(24)
)
φrβ φrα φrα + φrβ Φr Φr
2
(
)
(26a)
2 + 2 Lseqδ (t )(2Φ ref Φ ref − z1 ) + 2 Lseqδ (t )(Φ ref − z1 )
)
− 2a1( − Lseqδ (t )φrα + a1isα − ωφrβ isα
(
)
(26b)
The derivatives Φ ref and Φ ref are obtained using the relation: Φ ref = ξ ( I s ) = h0 + h1I s + h2 I s2 + ... + hn I sn
(27)
Specifically, one has: dξ ( I s ) dξ ( I s ) dI s Φ = ref = dt dI s dt dξ ( I s ) isα isα + isβ isβ = dI s Is
(28a)
2 d 2ξ ( I s ) isα isα + isβ isβ Φ = ref Is dI s 2 2 2 dξ ( I s ) (isα ) + isα isα + (isβ ) + isβ isβ + dI s Is
−
2 dξ ( I s ) (isα isα + isβ isβ ) dI s I s3
(28b)
To analyze the error system, composed of equations (17a-b), (21) and (26a), let us consider the following augmented Lyapunov function candidate: 1 V2 = V1 + (e22 + z22 ) 2
1 1
2 2
[
]
(32b)
where c 2 and d 2 are new positive real design parameters. Equation (32b) suggests that the control signals u sα , u sβ should be chosen so that the two quantities between brackets on the right side of (32) equal zero. Letting these quantities equal to zero and solving the resulting equation with respect to ( u sα , u sβ ), gives the following control law:
+ 2Φ 2 ν 2 = d1 (− d1 z1 + z 2 ) + 2Φ ref Φ ref ref
+ − Lseqδ (t )φrβ + a1isβ + ωφrα isβ ) + φ rβ (δ (t )φ rβ − a 2 isβ ) + φrα (δ (t )φrα − a2isα ))
2 2
+ z2 z1 + d 2 z2 + ν 2 − 2a1a3 (φrα usα + φrβ usβ )
with
(
11
to the right side of
p + e2 e1 + c2e2 + μ2 + a3 (φrβ usα − φrα usβ ) J
(25)
Equation (24) is given the following compact form: z2 = ν 2 − 2a1a3 φrβ u sβ + φrα u sα
)
Adding c2e22 − c2e22 + d 2 z22 − d 2 z22 (32a) and rearranging terms, yields: V = −c e2 − c e2 − d z 2 − d z 2
where the derivative of δ (t ) is obtained from (4): dδ dΦ r dδ = δ(t ) = dΦ r dt dΦ r
p V2 = −c1e12 − d1z12 + e2 e1 + μ2 + a3 (φrβ u sα − φrα usβ ) J (32a) + z2 z1 + ν 2 − 2a1a3 (φrα u sα + φrβ u sβ )
(
+ − Lseqδ (t )φrβ + a1isβ + ωφ rα )isα isβ + φrβ δ (t )φrβ − a3ωφ rα − a2isβ + a3usβ ]
(
(31)
(30)
Its time-derivative along the trajectory of the state vector (e1, z1, e2, z2) is:
usα λ0 u = sβ λ2
λ1 λ3
−1
− e − c e − μ2 1 2 2 − z1 − d 2 z2 − ν 2
(33)
with: p p a3φrβ , λ1 = − a3φrα , λ2 = −2a1a3φrα and J J λ3 = −2a1a3φrβ
λ0 =
(34)
λ λ It is worth noting that the matrix 0 1 is nonsingular λ2 λ3 p because its determinant D = λ0λ3 − λ1λ2 = −2 a1a3Φ 2r J never vanish because of the existence of the remnent flux. Substituting the control law (33) to u sα , u sβ in the right side
of (32) yields: V2 = −c1e12 − c2e22 − d1z12 − d 2 z22 (35) which is a negative function definite of the state vector ( e1, z1, e 2 , z 2 ). The results obtained until now are summarized in the following theorem: Theorem 1 (Main result). Consider the closed-loop system composed of the induction machine, described by model (1), and the nonlinear controller defined by the control law (33). Then, one has the following properties: 1) The closed-loop error system undergoes the following equations: e1 = −c1e1 + e2 ; z1 = −d1z1 + z2 (36a) e2 = −e1 − c2e2 ;
(36b) z2 = − z1 − d 2 z 2 2) The above system is globally asymptotically stable and consequently all errors vanish asymptotically, whatever the initial conditions.
V2 = (e12 + z12 + e22 + z22 ) / 2 is Lyapunov function of the error system (36a-d). Part 2 follows from the fact that V2 is negative definite. This completes the proof of Theorem 1. 4. SIMULATIONS RESULTS In this section, we will first illustrate the suppremacy of the new controller (NC) that accounts for magnetic saturation, over the standard controller (SC) that assumes linear static magnetic characteristic. The comparison is performed using a 7.5 kW induction machine whose characteristics are summarized in Table I. Then, we will show that perfect tracking of a varying flux reference is always achieved by the NC controller but generally not by the SC The experimental protocol is conceived in a way to make the machine operate in different conditions. These are fixed through the load variation law (Fig. 3) and speed reference (Fig. 4). The NC controller defined by the control law (33) is given the following design parameters values which proved to be convenient: c1 = 5
c 2 = 500
d 1 = 10
d 2 = 1000
The SC controller is obtained from the NC-controller simply letting the parameter δ (t ) in (33) be constant equal to 720. The other design parameters are given the following values that proved to be convenient for this controller: c1 = 25 c 2 = 420 d 1 = 100 d 2 = 200 The above value of δ = 720 characterizes the model (3) when the machine operates around the nominal operation point corresponding to a flux value of 1 .1Wb (on the magnetic characteristic). Obviously, the SC controller thus obtained is expected to perform well when the machine operates around the previous nominal working i.e. when the flux reference is sufficiently close to 1.1 Wb . Fig. 4 shows that the two controllers have an identical tracking quality as long as speed regulation is concerned. When it comes to flux tracking, the new controller is quite better (Figs 5 and 6). Fig 7 illustrate the benefit one gets from letting the flux reference be varying. For instance, in presence of a small load ( C r = 0 ), a flux setpoint of Φ ref = 0.4 Wb leads to lower stator current consumption than in the case of a flux setpoint of Φ ref = 1.1 Wb . Indeed, the absorbed current in the last case is 16 A while it is only of 6 A in the first case.
1
KW V Wb Ω Ω Kgm2 N m.s rd-1 mH
Equivalent inductance of stator and rotor leakage seen from the stator
5. CONCLUSIONS In this paper, we have considered the problem of speed and flux control of induction machine in presence of the magnetic characteristic saturation. A new controller described by (33) is designed using the backstepping technique, based on model (1). Besides accounting for the saturation feature of the magnetic characteristic, the proposed controller can involves a state-dependent optimized flux reference. It is formally shown (Theorem 1) that the proposed controller e1 = Ω ref − Ω and guarantees that the errors 2 z1 = Φ ref − (φ r2α + φ r2β ) globally converge to zero. That is, the
tracking objective is perfectly achieved both for the machine speed and rotor flux. 40 35 30 25 Load torque (Nm)
respectively. This proves Part 1. On the other hand, it is readily seen from (13), (30) and (35) that
TABLE I. MOTOR CHARACTERISTICS PN 7.5 Nominal power 380 Nominal voltage Usn 1.1 Nominal flux Φrn 0.63 stator resistance Rs 0.4 rotor resistance Rr 0.22 Inertia moment J 0.001 Friction coefficient f 2 Number of pole pairs p 7 Leakage equivalent inductance1 Lseq
20 15 10 5 0 -5 -10
0
5
10
15 time(s)
20
25
30
Fig.3. Applied load torque 160 150 140 130 120 Speed (rd/s)
Proof. Equations (36a) are immediately obtained from (17ab). Equations (36b) are obtained substituting the control law (33) to u sα , u sβ on the right side of (21) and (26a),
110 100 90 80 70 60
Remarks 1.a) It is worthy pointing out that the difference between the two control strategies, is more significant in presence of low load torques, because it is then that the SC reference is most different from the nominal flux value. b) Furthermore, there is no difference between the SC and NC controllers whenever a flux reference is considered and given the nominal value 1.1 Wb .
50
0
5
10
15 time(s)
20
25
30
Fig 4: Speed reference (dotted) and speed responses (solid): the NC and SC are indistinguishable. Unities: time (s); speed (rd/s)
1.1 1
flux norm (Wb)
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0
5
10
15 time(s)
20
25
30
Fig 5: Rotor flux norm reference (dotted) and flux response with standard controller (solid). Flux unity: Wb. 1.1 1
flux (Wb)
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0
5
10
15 time(s)
20
25
30
Fig 6: Flux reference (dotted) and flux response with new controller (33) (solid). 70
60
Stator current norm (A)
50
40
30
20
10
0
0
5
10
15 time(s)
20
25
30
Fig 7: Stator current
REFERENCES Ayman, M., EL-Refaie, Jahns Thomas, M., (2005) “Optimal Flux Weakening in Surface PM Machines Using Fractional-Slot Concentrated Windings” transactions on industry applications, vol. 41, no3 Edwin, C.K.P. and Stranislaw, H., (2001) “An Introduction to optimization” John Wilay & Sons. Hu, J., Dawson M. and Qu, Z., (1996) 'Robust tracking control of an induction motor', Int. J. Robust and Nonlinear Control, Vol. 6, pp. 201-219 Khalil, H.K., (1996) “Nonlinear systems” second edition, Prentice Hall, Upper Saddle River Krstic, M., Kanellakopoulos, I., Kokotovic, P., (1995) “Nonlinear and adaptive control design” , John Wilay & Sons, Inc.
Leonard, W. (1985) “Control of Electrical Drives” SpringerVerlag Berlin Heidelberg New York Tokyo Ouadi, H., Giri, F., and Dugard, L., (2004) “ Modelling saturated induction motors”. IEEE Conference on Control Applications (CCA’04), Taipei, Taiwan. Vol.1, pp. 75 – 80. Ortega, R., Nicklasson, P.J., and Espinosa-Perez, G. (1996) “On speed control of induction motors”, Automatica, Vol 32, N 3 Robert, T.N., Chiasson, J., Bodson, M., (1999)’High Performance Motion Control of an induction motor with magnetic saturation’ IEEE transactions on control systems technology, Vol 7, N0 3 Seguier, G. and Notelet, F., (2005) 'Electrotechique Industrielle', Editions Tec Doc, Lavoisier, Paris, ISBN 2-7430-0791-5. Xingyi, X., Novotny, D. W., (1992) “Selection of the flux reference for induction machine drives in the field weakening region” IEEE Transactions on Industry Applications, vol. 28, pp. 1353-1358.