Robust backstepping vector control for the doubly fed induction motor S. Drid, M. Tadjine and M.-S. Naı¨t-Saı¨d Abstract: A robust vector control intended for a doubly fed induction motor (DFIM) mode is considered. The state-all-flux induction machine model with a flux orientation constraint is replaced by a simpler control model. The double-flux orientation leads to orthogonality between the stator and rotor fluxes, resulting in a linear and decoupled machine control and an optimal developed torque. The inner flux controllers are designed using the Lyapunov linearisation approach. This flux control is exponentially stabilised independently of the speed. Associated with sliding-mode control, this solution shows good robustness with respect to parameter variations, measurement errors and noisse. Finally, a speed controller is designed using two methods: the first with a PI controller and the second with the Lyapunov method associated with a backstepping procedure, especially employed for the unknown load torques. This second solution shows good robustness with respect to inertia variation and guarantees torque and speed tracking. The global asymptotic stability of the overall system is proven theoretically. The simulation and experimental results largely confirm the effectiveness of the proposed DFIM system control.
Nomenclature Cel , Cn , Cmax d d J Kupl Ls , Lr M P q Rs , Rr Ts , Tr d u rs , rr s V v vc vr vs
Electromagnetic torque, rate torque and maximal torque unknown load torque subscript indicating direct orthogonal component Inertia induction machine upper-load coefficient stator and rotor inductances mutual inductance number of pairs of poles subscript indicating quadratic orthogonal component stator and rotor resistances Stator and rotor time-constants (Ts,r ¼ Ls,r/Rs,r) torque angle absolute rotor position stator and rotor flux absolute positions leakage flux total coefficient (s ¼ 1 2 M 2/LrLs) rotor speed, rad/s; mechanical rotor frequency, rad/s; injected rotor current frequency, rad/s induced rotor current frequency, rad/s stator current frequency (rd/s) symbol indicating a complex number
# The Institution of Engineering and Technology 2007 doi:10.1049/iet-cta:20060053 Paper first received 4th February and in revised form 23rd July 2006 S. Drid, M.-S. Naı¨t-Saı¨d are with the LSPIE Research Laboratory, Electrical Engineering Department, University of Batna, Rue M.E.H., Boukhlof, Algeria M. Tadjine is with the Process Control Laboratory, Electrical Engineering Department, ENP Alger, 10 Avenue Hassen Badi, Alger B182, Algeria E-mail:
[email protected] IET Control Theory Appl., 2007, 1, (4), pp. 861 –868
ˆ
1
symbol indicating measured value symbol indicating estimated value symbol indicating command value, or complex conjugate of a complex number
Introduction
The doubly fed induction machine (DFIM) is a very attractive solution for variable-speed applications such as electric vehicles and electrical energy production [1 – 6]. Obviously, the required variable-speed domain and the desired performance depends on the application [1 –9]. The use of a DFIM offers the opportunity to modulate power flow into and out the rotor winding. In general, when the rotor is fed through a cycloconverter, the power range can reach the order of megawatts – a level usually confined to synchronous machines [1 – 12]. The DFIM has some distinct advantages compared with the conventional squirrel-cage machine. It can be controlled from the stator or rotor by various possible combinations. The disadvantage of using two converters for stator and rotor supply can be compensated by the better control performances of the powered system [3]. Indeed, the input commands are by means of four precise degrees of control freedom, compared with to the simple control of the squirrel cage induction machine. A flux orientation strategy can transform the nonlinear coupled DFIM mathematical model to a linear model, leading to as attractive solution in generating or motoring operations [1 – 17]. In the search for more powerful control, we rely on the backstepping technique, applied for the first time to induction motor control, making it possible to obtain a higher output. The application of these new configurations to the DFIM machine leads to novel control problems, because the torques created by the machine are dynamically related by nonlinear equations. Additional dynamics is thus necessary to study the stabilisation of the differential 861
equations resulting from the Lagrangian mechanical equations and the electrical equations. An additional difficulty arises owing to the fact that the load torque imposed by the load is unknown. The main idea behind all flux-orientation control strategies is that the machine flux position and vector flux components are computed from direct physical measurements. In a DFIM, both stator and rotor currents are easily measured, and the flux vectors can be calculated using nondifferential equations, with the flux current constituting a simpler flux estimator. It depends on inductive parameters (inductances) that are functions of the machine saturation state. Its robustness against saturation can be guaranteed by associating a sliding-mode controller with nonlinear feedback control, with the latter allowing flux orientation via a simply obtained linear control model. The result is that the stator flux oriented in the q axis becomes the active power input command from which the developed torque will be controlled, while the rotor flux assumes the role of a reactive power input command acting on the magnetising machine system. The main motivation of the present paper is the design of a vector control for DFIM using a nonlinear vector control associated with a sliding-mode controller with a Lyapunov function guaranteeing global system stability with improved control robustness. In addition, a speed controller is designed using backstepping techniques allowing determination of the unknown load torque. The results demonstrate that the proposed systems control presents an interesting alternative in the DFIM framework. 2
The DFIM model
The DFIM model, expressed in the synchronous reference frame, is as follows: the voltage equations are df¯ s þ jvs f¯ s dt df¯ r þ jvr f¯ r u r ¼ Rrir þ dt
u s ¼ Rs is þ
f¯ r ¼ Lr¯ir þ M ¯is
1 ¯ M ¯ df¯ s þ jvs f¯ s fs fr þ sTs sTs Lr dt
M ¯ 1 ¯ df¯ r u r ¼ þ jvr f¯ r fs þ fr þ sTr Ls sTr dt
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(5)
f3 ¼ g3 fsd þ g4 frd vr frq f4 ¼ g3 fsq þ g4 frq þ vr frd with
g1 ¼
1 , sTs
g2 ¼
M , sTs Lr
g3 ¼
M , sTr Ls
g4 ¼
1 sTr
The electromagnetic torque is given by Cel ¼
PM (f f fsd frq ) sLs Lr sq rd
(6)
and its associated equation of motion is J
dv ¼ Cel d dt
(7)
In DFIM operations, the stator and rotor magnetomotive force (MMF) rotations are imposed directly by the two external voltage source frequencies. Hence the rotor speed depends on a linear combination of these frequencies, and will be constant if they too are constants for any load torque (given, of course, in the machine stability domain). In DFIM modes, synchronisation between the two MMFs is mainly required in order to guarantee natural machine stability [5 – 7]. This is a similar situation to the synchronous machine stability problem, where, without recourse to the strict control of the DFIM MMFs’ relative position, there in a rests of machine instability or breakdown mode.
3.1
(2)
Vector control strategy Vector control by double flux orientation
This consists in orienting, at the same time, the stator flux and rotor flux as indicated in Fig. 1. Thus, it results the constraints given below in (8). The rotor flux is oriented on the d axis, and the stator flux is oriented on the q axis. Conventionally, the d axis is the magnetising axis and the q axis is the torque axis. Consequently, d will be 908 and the two fluxes become orthogonal, so we can write the following expressions [18, 19]
fsq ¼ fs (3)
Separating the real and imaginary parts of (3), we can write dfsd dt dfsq dt dfrd dt dfrq dt
f2 ¼ g1 fsq g2 frq þ vs fsd
3
From (1) and (2), the state all-flux model is rewritten as u s ¼
f1 ¼ g1 fsd g2 frd vs fsq
(1)
and the flux equations are
f¯ s ¼ Ls¯is þ M ¯ir
where f1 , f2 , f3 and f4 are given by
frd ¼ fr fsd ¼ frq ¼ 0
(8)
Using (8), the developed torque given by (6) can be rewritten as Cel ¼ kc fs fr
¼ f1 þ usd ¼ f2 þ usq (4) ¼ f3 þ urd ¼ f4 þ urq
Fig. 1 DFIM vector diagram after flux orientation IET Control Theory Appl., Vol. 1, No. 4, July 2007
3.3 Robust feedback Lyapunov linearisation control
where kc ¼
PM sLs Lr
(9)
fs appears as the input command of the active power or simply of the developed torque, and fr appears as the input command of the reactive power or simply of the main machine magnetisation. 3.2 Vector control based on the Lyapunov feedback linearisation approach The following result can be stated Proposition 1: Consider the all-flux state model (4). Then, the double-flux orientation constraints (8) are satisfied provided that the following control laws are used usd ¼ f1 K1 fsd urq ¼ f4 K2 frq usq ¼ f2 þ f˙ s K3 (fsq fs )
(10)
urd ¼ f3 þ f˙ r K4 (frd fr ) where the Ki are positive gains. Proof: Let the Lyapunov function related to the flux dynamics be defined by 1 1 1 1 V ¼ f2sd þ f2rq þ (fsq fs )2 þ (frd fr )2 . 0 (11) 2 2 2 2 This function is globally positive-definite over the whole state space. Its derivative is given by V_ ¼ fsd f˙ sd þ frq f˙ rq þ (fsq fs )(f˙ sq f˙ s ) þ (frd fr )(f˙ rd f˙ r )
(12)
Substituting (4) into (11) gives V_ ¼ fsd ( f1 þ usd ) þ frq ( f4 þ urq ) þ (fsq fs )( f2 þ usq f˙ s ) þ (frd fr )( f3 þ urd f˙ r )
(13)
In practice, the nonlinear functions fi involved in the statespace model (4) are strongly affected by the conventional effects of induction motors, such as temperature, saturation and skin effects, in addition of the different nonlinearities related to harmonic pollution due to supply converters and noise measurements. Since the control law developed in the preceding section is based on exact knowledge of these functions fi , one can expect that in a real situation the control law (10) may fail to ensure flux orientation. In this section, our objective is to determine a new vector control law making it possible to maintain double-flux orientation in the presence of physical parameter variations and measurement noises. Globally, we can write fi ¼ f^i þ Dfi
(16)
where ˆfi is the true nonlinear feedback function (NLFF), fi is the effective NLFF and Dfi is the NLFF variation around fi . The Dfi can be generated from all of the parameters and variables as indicated above. We assume that all of the D fi are bounded as follows: jDfij , bi , where the bi are known bounds. Knowledge of the bi is not difficulties obtain, since one can use a sufficiently large number to satisfy the constraint jDfij , bi . The Dfi can be generated from the all of the parameters and variations as indicated above. Inserting (16) in (4), we obtain dfsd dt dfsq dt dfrd dt dfrq dt
¼ f^1 þ Df1 þ usd ¼ f^2 þ Df2 þ usq (17) ¼ f^3 þ Df3 þ urd ¼ f^4 þ Df4 þ urq
The following result can be stated:
Inserting the control law (10) in (13), we obtain V_ ¼ K1 f2sd K2 f2rq K3 (fsq fs )2 K4 (frd fr )2 , 0
(14)
The function given in (14) is globally negative-definite. Hence, using Lyapunov’s theorem [20], we conclude that t!þ1
usq ¼ f^2 þ f˙ s K3 (fsq fs ) K33 sgn(fsq fs )
lim frq ¼ 0
lim (frd fr ) ¼ 0
usd ¼ f^1 K1 fsd K11 sgn(fsd ) urq ¼ f^4 K2 frq K22 sgn(frq )
lim fsd ¼ 0
t!þ1
Proposition 2: Consider the realistic all-flux state model (17). Then, the double-flux orientation constraints (8) are satisfied provided that the following control laws are used
(15)
urd ¼ f^3 þ f˙ r K4 (frd fr ) K44 sgn(frd fr ) (18)
t!þ1
lim (fsq
t!þ1
fs )
¼0
In (15), the first and second equations concern the doubleflux orientation constraints applied to the DFIM model, which are defined above by (8), and the third and fourth equations define the errors after feedback flux control. This latter offers the possibility to control the main machine magnetisation on the d axis by frd and the developed torque on the q axis by fsq . A IET Control Theory Appl., Vol. 1, No. 4, July 2007
where Kii bi and Kii . 0 for i ¼ 1, . . . , 4. Proof: Let the Lyapunov function related to the flux dynamics (17) be defined by 1 1 1 1 V1 ¼ f2sd þ f2rq þ (fsq fs )2 þ (frd fr )2 . 0 (19) 2 2 2 2 863
we have V_ 1 ¼ fsd (Df1 K11 sgn(fsd )) þ frq (Df2 K22 sgn(frq )) þ (fsq fs )(Df3 K33 sgn(fsq )) þ (frd fr )(Df4 K44 sgn(frd )) þ V_ , 0
(20)
where V˙ is given by (14). Hence the Dfi variations can be absorbed if we take K11 . jDf1 j K22 . jDf2 j
(21)
K33 . jDf3 j K44 . jDf4 j These inequalities are jDfij , bi , Kii . Finally, we can write
satisfied
since
Ki . 0
and
Fig. 3 General block diagram of DFIM control scheme
4.2
V_ 1 , V_ , 0
(22)
Hence, using Lyapunov’s theorem [20], we conclude that lim fsd ¼ 0
t!þ1
Constraints
Two constraints must be satisfied: 1. The load torque d is unknown but limited; that is, max (d) Kupl Cmax (Fig.4). 2. The magnetising flux frd must be non-zero (remanence flux) for the following control development.
lim frq ¼ 0
t!þ1
lim (frd fr ) ¼ 0
(23)
t!þ1
lim (fsq fs ) ¼ 0
t!þ1
The design of these robust controllers, resulting from (18), is given in Fig. 2. Fig. 3 is a general block diagram of the suggested DFIM control scheme. As shown in this figure, we can see that only one PI speed controller is used and the fluxes are nonlinear feedback-controlled in association with a slidingmode controller. We can also note the placement of the estimator block, which first evaluates the modulus and position flux fs , fr , rs and rr from the measured currents using (2) and they the feedback functions f1, f2, f3 and f4 given by (5). To avoid the presence of the PI speed controller in the scheme shown in Fig. 3, we can design another controller as explained in the following section.
4.3
Control law
The following result can be stated: Proposition 3: Consider the all-flux state model (4) and the equation of motion. Then the double-flux orientation constraints (8) are satisfied provided that the following control laws are used usd ¼ f1 K1 fsd 1 usq ¼ ( e kc urd fsq þ kc urq fsd þ kc usd frq kc frd þ C_ ref ) fc K6 (Cel Cref ) urq ¼ f4 K2 frq urd ¼ f3 þ f˙ r K4 (frd fr ) (24)
4 4.1
Robust backstepping speed controller design Control objective
and Cref ¼ J v_ ref K5 (v vref ) K6 sign(v vref )
(25)
In order to remove the PI speed controller with respect to the double-flux orientation strategy, we will reformulate the all-control system using backstepping techniques. This designed control system is based on the use of a Lyapunov function that allows control of all transient electromagnetic phenomena in the machine.
where
Fig. 2 Design of a robust controller
Fig. 4 Mechanical curve of the induction machine (speed – torque)
(w ¼ sd, sq, rd, or rq; i ¼ 1, . . . , 4) 864
fc ¼ kc ðfsd f2 þ fsq f3 fsd f4 frq f1 Þ and the Ki are positive gains.
IET Control Theory Appl., Vol. 1, No. 4, July 2007
Proof: The control law is constructed by using a two-step backstepping approach as follows Step 1: The control law is designed using this approach such that the adequate torque Cel which satisfies Cel ! Cref . Let the Lyapunov function related to the speed dynamics (7), considered as the lower dynamics in the DFIM system, be defined by 1 V2 ¼ J (v vref )2 . 0 2
Fig. 6 represents the effect of inertia variation on the system response and stability for the two controllers. Step 2: Now, we find the control voltages usd , usq , urd and urq to ensure that Cel ! Cref and to obain the double flux. Let the Lyapunov function related to the speed dynamic and the flux dynamics (4– 7) be defined by 1 1 1 V3 ¼ V2 þ f2sd þ f2rq þ (frd fr )2 2 2 2 1 þ (Cel Cref )2 . 0 2
(26)
This function is globally positive-definite over the whole state space. Its derivative is given by
This function is globally positive-definite over the whole state space. Its derivative is given by
V_ 2 ¼ J (v vref )(v˙ v˙ ref ) ¼ (v vref )(Cel d J v˙ ref )
(27)
If it is supposed that Cel ! Cref then the control law is given by (25), with K6 . max (d )
(32)
(28)
V_ 3 ¼ J (v vref )(v˙ v˙ ref ) þ fsd f˙ sd þ frq f˙ rq þ (frd fr )(f˙ rd f˙ r ) þ (Cel Cref )(C˙ el C˙ ref ) (33) with
Let us replace the control laws (25) in (27), we obtain V_ 2 ¼ K5 (v vref )2 þ (v vref )[d K6 sign(v vref )] (29) The term (v 2 vref)[2d 2 K6 sign(v 2 vref)] , 0 8(v 2 vref) and 8d; then V_ 2 , K5 (v vref ) , 0 2
(30)
The function given in (30) is globally negative-definite. Hence, using Lyapunov is theorem [20], we conclude that Cel ! Cref lim (v vref ) ¼ 0
J (v vref )(v˙ v˙ ref ) ¼ (v vref )(Cel Cref ) þ (v vref )(Cref d J v˙ ref ) (34) If we insert the control law (25) in (34), we can write J (v vref )(v˙ v˙ ref ) (v vref )(Cel Cref ) k5 (v vref )2
(35)
The derivative of the Lyapunov function (33) becomes (31)
t!1
Fig. 5 represents the two speed regulator configurations, with Ck ¼ 1/kcfrd
V_ 3 (v vref )(Cel Cref ) k5 (v vref )2 þ fsd f˙ sd þ frq f˙ rq þ (frd fr )(f˙ rd f˙ r ) þ (Cel Cref )(C_ el C_ ref )
(36)
The torque is giving by (6), and its derivative is C_ el ¼ fc þ kc (usq frd þ urd fsq urq fsd usd frq )
(37)
Inserting the control law (24) in (36), we obtain Fig. 5 Designs of the speed controller a PI controller b Robust backstepping controller
V_ 3 K5 e2 K1 f2sd K2 f2rq K4 (frd fr )2 K6 (Cel Cref )2 , 0
(38)
Fig. 6 Inertia variation effect for the two controllers a PI controller b Robust backstepping controller c Zoom IET Control Theory Appl., Vol. 1, No. 4, July 2007
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Hence, using Lyapunov’s theorem [20], we conclude that lim fsd ¼ 0
t!þ1
lim frq ¼ 0
t!þ1
lim (frd fr ) ¼ 0
t!þ1
(39)
lim (v vref ) ¼ 0
t!þ1
lim (Cel Cref ) ¼ 0
t!þ1
5 5.1
A
Simulation and experimental results Reference profiles and machine parameters
To test the speed tracking, we applied a sinusoidal speed reference of magnitude 160 rad/s with frequency 5 Hz. The machine was loaded at 10 mN. The inertia system could reach five times the rated inertia value (see Figs. 7 and 8). To test the torque tracking, we applied a sinusoidal speed reference having a magnitude 10 mN with a frequency 5 Hz. The machine speed was 160 rad/s. The inertia system could reached five times the rated inertia value (see Figs. 9 and 10). The machine parameters were as follows: Rs ¼ 1.2 V, Ls ¼ 0.158 H, Lr ¼ 0.156 H, Rr ¼ 1.8 V, M ¼ 0.15 H, P ¼ 2 J ¼ 0.07 Kg m2, Pn ¼ 4 kW, 220/380 V, 50 Hz, 1440 tr/min, 15/8.6 A, cos w ¼ 0.85 5.2
Fig. 8 Speed-tracking response with backstepping controller
Simulation results
Figs. 7 and 8 show speed response versus time; we observe that with the backstepping speed controller, a good tracking speed was achieved without any effect of variation of parameters, is contrast on the PI controller. The same holds for torque tracking shown in Figs. 9 and 10. 5.3
Experimental results
Fig. 9 Torque-tracking response with PI controller
In order to validate our approach, experimental and simulation tests were carried out using the proposed control DFIM scheme. The testing conditions were as follows. The unloaded DFIM started with a constant acceleration a0 ¼ 10 000 rpm/s, after 0.1 s, the speed was maintained to 1000 rpm. At this speed, at 0.35 s, a sudden load torque
was applied (step 10 Nm) to the controlled DFIM. After that, a reversal speed test was applied to the loaded machine (10 Nm) at 0.7 s, where the speed changed from 1000 rpm to 21000 rpm, exhibiting a linear speed time evolution with slope a0 ¼ 20 000 rpm/s. Fig. 11 presents the speed response versus time according to the profile described above. Figs. 12a and 13a show,
Fig. 7 Speed-tracking response with PI controller
Fig. 10 Torque-tracking response with backstepping controller
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IET Control Theory Appl., Vol. 1, No. 4, July 2007
Fig. 11 Speed response versus time a Experimental b Simulation
Fig. 12 Stator current versus time a Experimental b Simulation
Fig. 13 Rotor current versus time a Experimental b Simulation IET Control Theory Appl., Vol. 1, No. 4, July 2007
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respectively, the stator and rotor currents versus time during the test. To compare the results of the simulation with the experimental data, the machine operated under the same test conditions. It can be seen that the simulation results are in agreement with the experimental results (Figs. 11b, 12b, and 13b). 6
Conclusion
In spite of the presence of rubbing contacts (rings–brushes), a DFIM can be fed and controlled from the stator and/or the rotor. This offers more convenience for identification of rotor parameters and for direct measurement of rotor current. Consequently, it leads to a simpler flux estimator. It depends on inductive parameters (inductances), which are functions of the machine’s saturation state. In this paper, we have presented a robust vector control intended for a DFIM mode. The use of a state all-flux induction machine model with a flux orientation constraint gives way to a simpler control model. The stability of the robust nonlinear feedback control is proved using a Lyapunov function. Control robustness is achieved by a sliding-mode controller, and is guaranteed in this work in order to reduce the effects of parametric variations, uncertainties and measurement noise. The simulation results for the DFIM control proposed here demonstrate clearly a satisfactory performance according to the references profiles defined above. A Lyapunov-like method and backstepping procedure are used to design the speed controller and especially to determine the unknown load torque. This solution shows good robustness with respect to inertia variation and guarantees torque and speed tracking. Therefore the high control performances can be achieved. The experimental and simulation results generally confirm the effectiveness of the proposed DFIM control system. 7
References
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