Nonlinear Multivariable Strategy for Induction Motor Drives Control M. K. Maaziz, E. Mendes*, P. Boucher, D. Dumur Service Automatique Supélec, F 91192 Gif-sur-Yvette cedex / France *
Lgep,Cnrs-UMR8507-Supélec-Université Paris XI-Paris VI, France E-mail:
[email protected]; phone: + 33 (0)1 69 85 13 75; fax: + 33 (0)1 69 85 13 89
ABSTRACT. The increasing interest of the induction motor is due to its large industrial applications field as well as its properties of reliability. Low maintenance, simplicity and relatively low cost are the main reasons to explain the development of modern electrical drives based on induction motors. From the control point of view, they represent a complex multivariable nonlinear problem: the dynamic model of the system is multivariable, coupled and highly nonlinear; the rotor flux is not usually measurable and the electrical parameters variations of the resistance and inductance, due to temperature, significantly change the system dynamics. Thus, the induction motors constitute an important application area for nonlinear control theory. In this paper, an original algorithm is proposed, based on a reference control in open loop combined with PI controllers. Since the considered outputs, the rotor speed and the rotor flux magnitude are planned, it yields that the reference control can be completely numerically calculated and applied to the machine. For the closed-loop strategy, the P.I controllers are added for the stabilisation around the reference trajectories. The proposed control law allows to solve some previous problems of the induction machines control and to optimize on an easier way trajectories tracking problems. It will be shown that advantages of this innovative structure include robustness with respect to electrical parameters variations or load torque variations. Experimental results are given, including estimation of the rotor flux norm of the induction motor, and considering robustness against resistance and inductance variations. KEYWORDS: Non linear control, electrical drive, a.c. machines.
Introduction The increasing interest of the induction motor is due to its large industrial applications field as well as its properties of reliability. Coupled to new fields of applications with severe specifications (e.g. high speed machining), low maintenance, simplicity and relatively low cost are the main reasons to explain the development of modern electrical drives based on induction motors. From the control point of view, they represent a complex multivariable nonlinear problem: the dynamic model of the system is multivariable, coupled and highly nonlinear; the rotor flux is not usually measurable and the electrical parameters variations of the resistance and inductance, due to temperature, significantly change the system dynamics. Thus, the induction motors constitute an important application area for nonlinear control theory. In the last years, many strategies have been studied to control induction motors. However, even if traditional control techniques of induction machine are adopted: scalar control, field oriented control, direct torque control, … . improvements can be achieved in order to extend its operating range, to reduce the influence of electrical and mechanical parameter variations, and improve transient performance. During the last years, many nonlinear control design approaches have been applied to this problem for better performances. Among latest works, like singular perturbation methods [DJE 93], backstepping control [TAN 99], many approaches concern inputoutput linearisation [BOD 94], [CHI 93], based on recent studies on nonlinear systems [FOS 93], [ISI 89], Von Raumer [VON 94] has proposed a controller designed to track torque and rotor flux references, Marino et al. [MAR 90] have developed an input-output decoupling controller which decouples the regulation of the rotor speed and the rotor flux norm. The robustness to parameter variations and load disturbances has been achieved with cascaded predictive control (GPC) combined with input-output torque and flux linearisation [BEN 95], [BOU 97]. In this paper, an original algorithm is proposed, based on a reference control in open loop combined with PI controllers. Since the considered outputs, the rotor speed and the rotor flux magnitude are planned, it yields that the reference control can be completely numerically calculated and applied to the machine. For the closed-loop strategy, the P.I controllers are added for the stabilisation around the reference trajectories. The proposed control law allows solving some previous problems of the induction machines control and better optimisation of trajectories tracking problems. It will be shown that advantages of this control scheme include robustness with respect to electrical parameters variations. The organisation of this paper proceeds as follows. First, a mathematical model of the induction motor is presented. Then, a brief overview of the open loop control strategy is shown. The closed loop control scheme is developed to allow tracking of the desired reference trajectories. Finally, experimental results are given, including an estimation of the rotor flux norm of the induction motor.
2. Model of the induction motor The reader is referred to [BOS 90], [LEO 85] for the general theory of electrical drives and induction motors and for related control problems. The three phases-two phases Park's transformation is used to determine the model of the motor in the stator fixed α - β reference frame. Under the assumption of linearity of the magnetic circuits, a two phases equivalent machine representation is given by the following fifth-order model, which includes both the electrical and mechanical dynamics: x& = f (x) + g u With:
[ u =[ u sα
(1)
x = i sα i sβ φrα φrβ ω
]
]T
u sβ T i sα , i sβ : stator currents, φrα , φrβ : rotor fluxes, ω : speed, u sα , u sβ : stator voltages.
Vector function f(x) and constant matrix g are defined as follows: k φrα + p ω k φrβ − γi sα + Tr k φrβ − p ω k φrα − γi sβ + Tr L 1 φrα − p ω φrβ f (x) = m i sα − Tr Tr L m i sβ − 1 φrβ + p ω φrα Tr Tr L T + f ω ( ) L p m φ i − φ i − rα sβ rβ sα JT J r
(
g=[ g1
)
1 σ L s g 2 ]= 0
T
0 1 σ Ls
0 0 0 0 0 0
All parameters above have the following meanings: s = 1−
L2m Ls Lr
; k=
2 Lm 1 R + R Lm ; ?= s r 2 s L s Lr s Ls Lr
Where:
L s , Lr are stator and rotor inductances, Lm is the mutual inductance, Rs , Rr are stator and rotor resistances, Tr = L r Rr is the rotor time constant, p is the pole pair number, J is the inertia of the machine, f is the friction coefficient, T L is the load torque considered as an unknown disturbance.
3. Open-loop control strategy The aim of the proposed approach is to exploit the general equations of the two phases induction motor representation to design an open loop control, derived from a finite number of time derivatives and first integrals of the chosen outputs: y = ( y1 , y 2 ) = (ω , ρ )
(2)
Where ω is the velocity of the machine, ρ the rotor flux magnitude. From this two phases mathematical model of the induction machine [LEO 85] (for more details with respect to the notations and equations, see for example [MAR 96]), and using the mechanical equation, the rotor and stator electrical and flux equations, it becomes possible, after some theoretical mathematical developments, to derive the stator voltage complex input u s under the following form: . . .. = u sα us = us y , y , y , y , y + jusβrefBO = f NL1BO + j f NL 2 BO ∫ ∫ refBO
(3)
See [MAA 99], [MAA 00] for more details about these mathematical developments. This global expression proved that all states and the stator voltage input u s of the motor are completely expressed in the desired formulation, with a finite number of time derivatives and first integrals of the chosen outputs. That is to say that, with a particular selection of flux magnitude and velocity reference trajectories (which can be therefore called ‘planned’trajectories), an openloop reference control signal can be derived, which, applied on the motor under ‘nominal’ behaviour assumptions, will provide the required outputs. Of course, this first aspect of the control signal will be completed in the next section by a closedloop term, taking into account the effects of non adaptation between the system and the model, influence of disturbances, …
Let us try now to examine more in details what is the knowledge required to compute this open-loop reference control signal. In the previous framework of the two phases mathematical model of the induction machine, the rotor flux φr is now written under its complex form φr = ρ exp( jα ) where ρ is again the rotor flux magnitude and α the argument of the complex rotor flux. With this notation, the mechanical equation takes the following form: J
dω p 2 . = ρ α − T L − fω dt Rr
(4)
Thus leading to: α& =
Rr 1 [Jω& + fω + TL ] p ρ2
(5)
As a matter of fact, with the assumption that outputs y = ( y1 , y 2 ) are planned, the complex rotor flux φr could be entirely determined if a nominal profile of the load torque T L is considered. For the open loop strategy, it is assumed to correspond to the dry friction torque ( fω / p ). Furthermore, the objective of the induction machine flux control is to maintain the flux constant, even with torque variations. Considering all these remarks, Eq. 5 may be rewritten [MAA 00] to provide the following planned argument of the complex rotor flux:
[
R 1 α&BO = r 2 J y&1 ref + f y1ref + TL p y2ref
]
(6)
In open-loop the reference control u s is thus schematically obtained following the structure below, showing that the only knowledge of α&BO is required, which is automatically performed as soon as the outputs trajectories are planned. Notice that α&BO is simply the slip velocity. y1 ref = ω ref y 2 ref = ρ ref
α&BO
f NL1BO = u sα refBO f NL 2 BO = u sβ refBO
Figure 1. Elaboration of the open-loop reference control signal u s To conclude this open-loop strategy, it must be noticed that this new approach in the field of induction machine control is really interesting. The fact that the user can plan flux and velocity trajectories enables to take into account torque or current saturation or other particular behaviours, by simply designing an ‘admissible’ reference control signal. This explicit consideration of saturation for example is not easy to implement with structures mentioned in the introduction.
4. Stabilisation around the desired trajectories The purpose of this section is to define a complementary term to the previous openloop reference control signal, which may definitely stabilise the system around the desired trajectories. In other words, a closed-loop architecture must be superimposed to the open-loop structure, to overcome problems due to non adaptation between system and model, neglected dynamics, parameters variations with temperature, load torque variations, … The most important advantage of this two steps strategy is that this scheme is very general, since every kind of closed-loop structures can be implemented, either ‘classical’ ones (such as P.I. controllers, as illustrated in our further developments), or ‘advanced’ones (such predictive control see [MAA 99], [MAA 00]. Generally speaking, if two monovariable controllers are designed, the global open/closed-loop strategy becomes the following: u s BF = u s α BF + ju sβBF = f NL1BF + j f NL 2 BF
(7)
With the two components of the complex control signal defined by the relation:
usα BF , usβBF = f (α&BF = α&BO +
Rr 1 v y1 , v y2BF ) . p y22ref BF 14 243 KTV
Where v y1BF is the output of the velocity controller, homogenous to a torque, and v y 2 BF the output of the flux controller, homogenous to the flux derivative. This structure is represented on figure 2. y1 ref = ω ref
α&BO
y 2 ref = ρ ref ν y1BF
f NL1BF = u sα refBF
α&BF ν y2 BF
f NL 2 BF = u sβrefBF
Figure 2. Elaboration of the closed-loop reference control signal u s Considering now P.I. controllers, the closed-loop strategy requires the definition of two monovariable P.I.’s, one for the velocity loop, another for the flux loop. These controllers are designed according to the following classical equations: t
v y1BF (t ) = K pω (ω ref (t ) − ω meas (t )) + Kiω ∫(ω ref (τ ) − ω meas (τ )) dτ
(8)
0 t
v y 2 BF (t ) = K pf ( ρref (t ) − ρˆ(t )) + Kif ∫( ρref (τ ) − ρˆ(τ )) dτ 0
(9)
Where ω ref is the planned velocity signal and ω meas the measured velocity, ρref is the planned flux signal and ρˆ the measured flux magnitude. These two control laws require the measurement of the speed and the flux magnitude. In fact, the velocity information is indirectly obtained through position measurement, and the rotor flux norm is derived through a flux observer. 5. Flux observer As a matter of fact, only the two components i sα , i sβ of the stator current and the rotor position θ are measured. These two components of the stator current can be obtained in the d – q reference frame of rotor flux through the following equation:
(
)
i sd + j i sq = i sα + j i sβ exp(− jδ)
(10)
δ= pθ + α
With:
Thus, the following relations give the flux observer: dρ = Misd dt dα M i sq = dt Tr ρ
ρ + Tr
(11) (12)
For real time implementation purposes, a discrete version of this flux observer is obtained through Eq. 10, Eq. 11 and Eq. 12, with a sampling period Ts = 153.2 µs . The structure of the global control scheme including the open-loop structure, the closed-loop P.I. controllers and the flux observer is given on figure 3. Reference model ∆y1
ω ref ρref
+
+ -
α&BO
+ P.I. K TV ν y1BF
α&BF
Voltage source inverter
PWM
u sβref BF
ν y2 BF
P.I.
Speed and flux controllers
FNLBF
+
∆y 2
-
u sα ref BF
ρˆ
Rotor flux observer
I. M.
ω
θ is β is α
d dt
θ
Position encoder
Figure 3. Block diagram of the complete control scheme including speed and flux control with rotor flux estimation
6. Experimental results The proposed control law has been applied on a small 1.1 kW induction motor, which corresponds to the benchmark a.c. machine of the LGEP (Laboratoire de Génie Électrique de Paris), with main features: R r = 3.6 Ω
J = 0.015 Kg.m 2
R s = 8.0 Ω
f = 0.01 N.m.s -1
L r = 0.47 H L s = 0.47 H
p=2 ω nom = 73.3038 rad/s
M = 0.452 Η
Tnom = 5 Nm
φr αβ = 1.14 wb
The choice of the tuning parameters of the two P.I. controllers is given in table 1 below, together with the corresponding response time of each loop. t r (s)
Kp
Ki
Flux
0.1
100
2500
Speed
0.08
125
3906
Table 1. Choice of the tuning parameters Figures 4 to 9 correspond to the experimental results of the open-loop reference control without closed loop, figures 10 to 20 to the complete control law, reference control, rotor flux observer and P.I controllers. Figures 10 to 15 correspond to experimental results without load, on figures 16 to 20 the machine is loaded with a load torque of 5 Nm, which sign corresponds to the speed one. All experimental results, in closed-loop, include 30% variations of Rr in the control part and in the observer part. [rad/s] 100
Filtered speed 50
0
No filtered speed -50
-100
0
1
2
Figure 4. Motor speed setpoints
3
4
5
6
7 Time [s] 8
[Wbs]
1.5
1
0.5
0
Filtered flux setpoint
No filtered flux setpoint
0
1
2
3
4
5
6
7 Time [s] 8
Figure 5. Rotor flux norm setpoints
[rad/s]
120 100 80
Filtered setpoint
60 40 20 0 -20
Motor.Speed
-40 -60 -80
0
1
2
3
4
5
6
7 Time [s] 8
Figure 6. Motor speed
[Wbs]
1.5
1
0.5
Rotor flux norm Filtered flux setpoint
0
0
Figure 7. Rotor flux norm
1
2
3
4
5
6
7 Time [s] 8
[A]
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
0
1
2
3
4
5
6
7 Time [s] 8
2
3
4
5
6
7 Time [s] 8
3
4
5
6
7 Time [s] 8
Figure 8. Stator current norm
[V]
250 200 150 100 50 0 -50 -100 -150 -200 -250
0
1
Figure 9. Open-loop stator voltage
[rad/s]
1.5 1 0.5 0
-0.5 -1 -1.5 -2 -2.5
0
1
Figure 10. Motor speed error
2
[Wbs] 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
0
1
2
3
4
5
6
7 Time [s] 8
2
3
4
5
6
7 Time [s] 8
2
3
4
5
6
7 Time [s] 8
Figure 11. Rotor flux norm error
[A]
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
0
1
Figure 12. Stator current norm
[Nm]
5 4 3 2 1 0 -1 -2 -3 -4 -5
0
1
Figure 13. Output of the speed controller ( v y1BF )
[V]
10
5
0
-5
0
1
2
3
4
5
6
7 Time [s] 8
5
6
7 Time [s] 8
5
6
7 Time [s] 8
Figure 14. Output of the flux controller ( v y2BF ) [V]
300 200 100 0 -100 -200 -300
0
1
2
3
4
Figure 15. Closed-loop stator voltage u sα ref BF [rad/s]
1.5 1 0.5 0
-0.5 -1 -1.5 -2
0
1
Figure 16. Motor speed error
2
3
4
[Wbs]
0.03 0.02 0.01 0
-0.01 -0.02 -0.03 0
1
2
3
4
5
6
7 Time [s] 8
2
3
4
5
6
7 Time [s] 8
2
3
4
5
6
7 Time [s] 8
Figure 17. Rotor flux norm error
[A]
6 5 4 3 2 1 0
0
1
Figure 18. Stator current norm
[Nm]
6 4 2 0 -2 -4 -6 0
1
Figure 19. Output of the speed controller ( v y1BF )
[V]
10 8 6 4 2 0 -2 0
1
2
3
4
5
6
7 Time [s] 8
Figure 20. Output of the flux controller ( v y2BF ) 7. Conclusion A new real-time design approach for speed and flux tracking, called PID / VSRC (Voltage single reference control) is proposed, which also allows a reduction of the influence of electrical parameters and load torque variations, which are the main drawbacks of traditional controllers. The methodology is based on an original combination between a reference control and PI controllers. The reference control is on one hand elaborated through a reference model using the planned trajectories of the rotor speed and the rotor flux norm, and on the other hand expressed only from the chosen outputs. It results an open-loop reference control entirely numerically calculable. For the closed-loop strategy, the stabilisation around the speed and the rotor flux norm trajectories is obtained with two independent P.I. controllers. This particular closed-loop strategy may be easily extended to more ‘advanced’control laws. The global control law is applied on an industrial a.c. machine benchmark. The good results show that this technique may overcome some of the difficulties related to the control of induction motors, and is very well adapted to trajectory tracking problem. This method appears to be really interesting for this kind of motor drives, and opens up new fields of investigation to systems requiring severe specifications, such as high speed machining for example. The control of induction machines may become more simple, providing better performance than more ‘classical’d.c. motors. Furthermore, the same approach can be extended to position tracking through non linear open/closed loop control in the same way as performed for the velocity. Constraints on the position may also be taken into account in a very easy way [MAA 00].
8. References [BOD 94] BODSON, M., CHIASSON, J., NOVOTNAK, R., "High-performance induction motor control via input-output linearisation", IEEE Control System Magazine, pp. 25-33, 1994. [BOS 90] BOSE, B.K, Power electronics and ac drives, Prentice-Hall, New Jersey, 1990. [BEN 95] BENYAHIA, A., BOUCHER, P., DUMUR, D., "Nonlinear multivariable predictive control of induction machine", Proceedings of the 2nd International Symposium MMAR, vol.1, pp. 267-272, Miedzyzdroje, 1995. [BOU 97] BOUCHER, P., BENYAHIA, A., DUMUR, D., "Feedback linearisation and multivariable cascaded predictive structure control of induction machine", Proceedings of the 7th European Conference on Power Electronics and Applications, pp. 3825-3830, Trondheim, 1997. [CHI 93] CHIASSON, J., "Dynamic feedback linearisation of the induction motor", IEEE Transactions Automatic Control, vol.38, no10, pp. 1588-1593, 1993. [DJE 93] DJEMAI, M., HERNANDEZ, J., BARBOT, J.P., "Nonlinear control with flux observer for a singularity perturbed induction motor", Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, 1993. [FOS 93] FOSSARD, A.J., NORMAND-CYROT, D., Collective Book, Systèmes non linéaires n 3. Commande, Masson, 1993. [ISI 89]
ISIDORI, A., Nonlinear control Systems, 2nd edition, Springer-Verlag, 1989.
[LEO 85] LEONHARD, W., Control of Electrical drives, Springer-Verlag, 1985. [MAA 99] MAAZIZ, M.K., BOUCHER, P., DUMUR, D., "Flux and speed tracking of an induction motor based on nonlinear predictive control and feedback linearisation", Proceedings of the 18th American Control Conference, pp. 2148-2152, San Diego, June 1999. [MAA 00] MAAZIZ, M.K., MENDES, E., BOUCHER, P., DUMUR, D., "Validation expérimentale des schémas de commande GPC/SRC et PID/SRC : Application au benchmark ‘transitique rapide’ de commande de la machine asynchrone", Proceedings of the 1st International Conference on Control CIFA’2000, Lille, July 2000. [MAR 90] MARINO, R., PERESADA, S., VAGILI, P., "Adaptive partial feedback linearisation of induction motors", Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, 1990. [MAR 96] MARTIN, Ph., ROUCHON, P., "Two remarks on induction motors", Proceedings of the CESA’96, pp. 76-79, Lille, 1996. [VON 94] VON RAUMER, T., Commande adaptative non linéaire de machine asynchrone, Thèse de l’Institut National Polytechnique de Grenoble, July, 1994. [TAN 99] TAN, H., CHANG, J., "Adaptive backstepping control of induction motor with uncertainty", Proceedings of the 18th American Control Conference, pp. 1-5, San Diego, June 1999.
