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Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter
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Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter Fateh Mehazzem a,*, Ahmed Lokmane Nemmour a, Abdellatif Reama b Laboratoire d'electrotechnique de Constantine, Departement d'electrotechnique, Universite Constantine 1, 25000 Constantine, Algeria b ESIEE-Paris, Universite Paris-Est 2, Bd Blaise Pascal, Cite Descartes, 93162 Noisy le Grand Cedex, France a
article info
abstract
Article history:
A novel version of nonlinear induction motors (IM) control algorithm based on the back-
Received 14 November 2016
stepping approach has been proposed in this work. For that, the conventional mathe-
Received in revised form
matical model of IM is replaced by the multiscalar one, which is more suitable for
13 April 2017
application of the backstepping design. By using such combination, the system has fast
Accepted 5 May 2017
dynamic response, better load disturbance rejection capability, less parameters sensitivity
Available online xxx
and better tracking performance of rotor speed and rotor flux magnitude. The effectiveness of the proposed control structure is validated by simulation as well by experiment under
Keywords: Induction motor
critical disturbance conditions. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Backstepping control Multi-scalar induction machine model SVPWM voltage source inverter
Introduction Electric motors are nowadays the most popular electrical systems in the industry. Their function, to convert electrical energy into mechanical energy, gives them a big economic importance. So no installation designer, maker or operator can ignore them. Among all the types of existing motors, induction motors [1], especially phase squirrel cage motors are the most used in industrial applications due to their easy installation, robustness and reliability. Furthermore, their control is widely needed for many applications.
From an automatic viewpoint, induction motor is a dynamic system, which presents certain number of control constraints because of its characteristics: nonlinear system, multivariable and strongly coupled, the resistive and inductive parameters vary as well the load. We can notice also that some variables cannot be measured like flux, which involves the use of observers to fulfill the control structure. The development of the semiconductor components technology [2] has contributed to the development of static converters. Moreover, this allowed the real-time implementation of the most complex algorithms dedicated to control induction motor, whatever their execution time.
* Corresponding author. E-mail addresses: [email protected] (F. Mehazzem), [email protected] (A.L. Nemmour), [email protected] (A. Reama). http://dx.doi.org/10.1016/j.ijhydene.2017.05.035 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Nomenclature ðus ; ur Þ ðis ; ir Þ ðfs ; fr Þ p Lm ðLs ; Lr Þ ðRs ; Rr Þ J um ua us Tl
stator and rotor voltage vectors stator and rotor current vectors stator and rotor flux vectors number of pole pairs mutual inductance stator and rotor inductances respectively stator and rotor resistances respectively moment of inertia electrical rotor angular speed angular speed of reference frame electrical synchronous angular speed machine load torque
Vector control based flux oriented introduced by Blaschke in 1972 [3], was an important step for controlling induction motor with high dynamic performance in industrial applications. The principle of vector control is to reduce the behavior of the induction motor to that of a DC motor. This method is based on the transformation of electrical variables to a frame that rotates with the rotor flux vector. In this new frame, the dynamics of the rotor flux are asymptotically linear and decoupled. In general, we can say that two types of vector control are possible: firstly, direct vector control, where it is estimated the magnitude and the position of the rotor flux and secondly, the indirect vector control, where only the position of rotor flux is estimated. The principle drawback of vector control is its sensitivity to parameters variation. Since the end of 1980's years, adaptive control of nonlinear systems has boosted with the first version of the inputeoutput adaptive linearization [4]. Later, several adaptive state feedback control forms have been proposed in order to overcome drawbacks of vector control and improve IM control performance. Among them: sliding mode control [5,6], passivitybased control [7] and direct torque control [8]. In parallel, another class of algorithms has known great developments. The concept of these algorithms is based on emulating certain behaviors in nature and the use of artificial intelligence. Therefore, these approaches, qualified intelligent have demonstrated their efficiency in various applications fields (image processing, computer vision and speech recognition). In recent decades they have also attracted attention in the field of motor control. Among them, there are fuzzy logic, genetic algorithms and neural networks [9,10]. In last decades, much progress has been made in the area of control of nonlinear systems. The backstepping is one of these new popular techniques in this area. It was developed by , Ioannis Kanellakopoulos and Miroslav Krstic Peter Kokotovic in 1991 [11]. It provides a systematic method to perform the design of a controller for a class of non-triangular systems. The main advantage of this method is to ensure the stability of the association (controller-process). The objective of the application of the Backstepping technique for induction motor control is to establish a control law of the IM via a chosen Lyapunov function, in order to guarantee stability for the overall system. It has the advantage of being robust according to parametric variations and ensures good tracking references. The combination of Backstepping
technique with vector control gives the induction machine control interesting qualities of robustness [12]. In recent years, the concept of the backstepping approach has been widely exploited in literature either in control or in observation of the IM [13]. Combined with other techniques (such as: fuzzy logic or sliding mode), the backstepping control overcomes constraint related to parameter variations [14], and can be exploited in a sensorless control [15]. As mentioned before, classical mathematical induction machine model is non-linear, multi-variables and strongly coupled. An internal coupling appears between mechanical and electromagnetic variables. This contributes mainly to make induction motor control very complex. For that, multiscalar induction machine model gives a good alternative to reduce this complexity. Its structure as two separate linear decoupled subsystems makes it more suitable for applying backstepping control approach. The non linear control based on the multiscalar model or MM control was given for the first time in 1987 [16]. The multiscalar IM model is obtained by doing a variable change based on the choice of specific state variables in the conventional one. By applying this type of control for induction motor we can easily obtain in the same structure a non-linear control and decoupling between electromagnetic torque and the square of linear combination of a stator current vector and the vector of rotor linkage flux [17]. This was completely impossible with vector control using classical induction machine model. When using a current source inverter to supply IM makes application of control technique based on the proposed new multi-scalar variables difficult to implement in real time [18]. However, when induction motor is fed by voltage source inverter with a constant rotor flux modulus, non linear control becomes equivalent to vector control, but the application of the non linear backstepping control gives more system structure simplicity and good overall drive response [19]. In this paper, a novel structure of nonlinear backstepping control is used. Control laws are obtained using the multiscalar induction motor model. The use of variables transformation to obtain the multiscalar model makes the control approach more suitable to perform, because only four state variables have been obtained with a relatively simple nonlinearity form. The proposed speed and rotor flux magnitude control structure shows a perfect decoupled control during transient and steady state. So, an ideal tracking of reference trajectories can be achieved. This contribution is organized as follows. Section Material and methods gives the methodology. First, a description of the classical model of induction motor is given, followed by the multiscalar one, after that backstepping control design, applied to multiscalar induction motor model is presented. Finally, stability analysis for global system through Barbalat's lemma and Hurwitz criterion is studied. In Section Results and discussion, the application of the developed algorithm is illustrated for two tests: the first concerns speed inversion and step changing of the load torque, second test is for parameters variations. Simulations and experimental results are given with backstepping-multiscalar control for the both tests. Finally, the most important contributions of this paper are summarized in Section Conclusion.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Material and methods Classical induction machine mathematical model The induction machine as an electromechanical system is governed by the laws of electricity and mechanics. For that we can obtain two electrical equations relative to the stator and the rotor, and one mechanical equation. Consequently, the classical induction machine model can be written in a rotating frame with an angular speed ua by the following equations [20,21]:
d4s þ jua 4 s dt d4r þ jðua um Þ 4 r u r ¼ Rr i r þ dt us
¼ Rs i s þ
J dum ¼ Imag ð 4 s i r Þ Tl p dt
(1)
(3)
d4rx Rr Lm ¼ 4rx þ Rr isx þ ðus um Þ4ry dt Lr Lr
(4)
d4ry Rr Lm ¼ 4ry þ Rr isy ðus um Þ4rx dt Lr Lr
(5)
Lm Lr þ um 4ry þ usx wd wd disy 1 Rs Lm ¼ isy þ 4 us isx Td dt Ls wd sy Lm Lr um 4rx þ usy wd wd p dum Lm p 4ry isx þ 4rx isy Tl ¼ JLr J dt
constant Td in the rotor current set value channel by the following equations: disx 1 ¼ isx I*s (9) Td dt d4rx Rr Lm ¼ 4rx þ Rr isx þ ðus um Þ4ry (10) dt Lr Lr d4ry Rr ¼ 4ry ðus um Þ4rx dt Lr
(11)
p dum Lm p 4ry isx Tl ¼ JLr J dt
(12)
In order to obtain the state multi-scalar induction machine model, a variables transformation is used from the classical one. Four novel state variables are used, as follow [22]:
(2)
In order to give better and detailed representation of the IM, the model above can be rearranged to express the dynamical equations of current and flux vector components. For that, the mathematical model of IM as differential equations of state variables in the rotating frame xey with angular speed us is given by:
disx 1 Rr Lm ¼ isx þ 4 þ us isy Td dt Lr wd rx
3
(6)
(7)
(8)
x11 ¼ um
(13)
x12 ¼ 4rx isy 4ry isx
(14)
x21 ¼ 42rx þ 42ry
(15)
x22 ¼ 4rx isx þ 4ry isy
(16)
x11 : is interpreted as rotor angular speed, x12 , x22 : are interpreted as scalar and vector products of the stator current and rotor flux vectors. The first is proportional to the motor torque, second is proportional to the energy. x21 : is interpreted as the square of the rotor flux. The multi-scalar model of induction machine will be given then by calculating derivatives of new scalar variables using the differential equations of the classical model presented above. The new model will be expressed as fellow: dx11 pLm p ¼ x12 Tl J dt JLr
(17)
dx12 1 ¼ x12 þ v1 T dt
(18)
dx21 Rr Rr Lm ¼ 2 x21 þ 2 x22 dt Lr Lr
(19)
dx22 1 Rr Lm 2 ¼ x22 þ i þ v'2 T dt Lr sx
(20)
where:
where:
1 L2 Rs þ L2m Rr ¼ r ; wd ¼ Ls Lr L2m : Td Lr wd
1 Rr 1 ¼ þ T Lr Td
We can notice that the model above is nonlinear, highly coupled and complex. For that we need to simplify this model in order to obtain a new structure more suitable for the application of the control process.
Multiscalar induction machine model The classical mathematical model of the wound-rotor induction machine can be expressed in rotor current rotating frame (isy ¼ 0), taking into account first-order delay time with a time
(21)
v1 ¼ ðus um Þx22
1 4 I* Td ry s
(22)
v'2 ¼ ðus um Þx12 þ
1 4 I* Td rx s
(23)
The above model has significantly less non linearities than the classical model presented before. We can notice that v1 and v20 are inputs for the new model and contribute to determine the variables us and I*s.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Eq. (20) contains non-linearity which can be compensated by a non-linear feedback. This will give equation: Rr Lm 2 i (24) v2 ¼ v'2 Lr sx The resolution of the system Eq. (22) and (23) gives the expression of the stator current amplitude and slip frequency: 4rx v2 4ry v1 42r
(25)
* 4ry v2 þ 4rx v1 us um ¼ isx 42r
(26)
I*s ¼ Td
The following expression also can be obtained: I*s 2 ¼
x212 þ x222 x21
(27)
One can notice that this expression is a function only of multiscalar variables, and don't need any rotor flux estimation, and this will be useful for calculation of the inverter output voltages. From the new IM model presented by (17)e(20), and using non-linear feedback in Eq. (24) we can obtain two linear fully decoupled subsystems e mechanical and electromagnetic: Mechanical subsystem: dx11 pLm p ¼ x12 Tl J dt JLr
(28)
dx12 1 ¼ x12 þ v1 T dt
(29)
dx22 1 ¼ x22 þ v2 T dt
Backstepping approach The basic idea of backstepping technique is to select recursively appropriate functions as virtual control inputs for first order subsystems. In order to apply backstepping for the whole system, design is divided into several stages. Extended Lyapunov functions are associated step by step to ensure global system stability. The obtained multi-scalar IM model above is more appropriate then the classical one for the application of the backstepping control design. Tracking objectives are defined as the rotor angular speed x11 ¼ um and the rotor flux square x21 ¼ 42r . The backstepping design is divided in two steps: Step 1 We start by defining the tracking errors as: e1 ¼ x*11 x11 ; e3 ¼ x*21 x21
Electromagnetic subsystem: dx21 Rr Rr Lm ¼ 2 x21 þ 2 x22 dt Lr Lr
The mechanical subsystem is represented by a delayed first order system, followed in serial by an integral bloc. The load torque for this subsystem represents the disturbance. For that, and in order to limit it, we should use a cascade control structure. The electromagnetic subsystem is represented by two delayed first order blocs, linked in a series. Same case, in order to limit the square of rotor flux, we should again using a cascade control structure. That implies, when using multiscalar model, the control structure should be composed of PI cascade controllers with constant parameters. Tuning of these parameters is done according to the control theory of linear systems.
(30)
(31)
The above model is linear and fully decoupled, which makes it possible to use a linear cascaded controller, as shown in Fig. 1. We can notice that motor torque is only proportional to the state variable x12, which belongs to the mechanical side of control design when using multiscalar model form. The rotor flux should be controlled by the electromagnetic subsystem, and its command value can be calculated by taking it as criterion either in the minimization of energy losses or in the minimization of response time in the mechanical subsystem [16].
(32)
Then, their derivatives are:
*
pLm p x12 þ Tl; J JLr
*
Rr Rr Lm x21 2 x22 Lr Lr
e1 ¼ x11 2 e3 ¼ x21 þ 2
(33)
Since the objective requirements are that the two errors converge to zero, and the current must be regulated and limited. These requirements are satisfied by viewing x12 and x22 as “virtual controls variables” in the above equations and using them to control the errors e1, e3. When using the Lyapunov function: 1 1 V ¼ e21 þ e23 2 2
(34)
Fig. 1 e Cascaded structure for the multiscalar model control design. Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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The derivative of V is given by:
V ¼ e1 e1 þ e3 e3 * * pLm p Rr Rr Lm ¼ e1 x11 x12 þ Tl þ e3 x21 þ 2 x21 2 x22 J JLr Lr Lr * pLm p þ Tl ¼ k1 e21 k3 e23 þ e1 k1 e1 þ x11 J JLr Rr Rr Lm þe3 k3 e3 þ 2 x21 2 x22 ð35Þ Lr Lr
x*22
JLr p ¼ k1 e1 þ x*11 þ Tl ; J pLm Lr Rr ¼ k3 e3 þ 2 x21 2Rr Lr
¼ k1 e21 k2 e22 k3 e23 k4 e24 þ e2 Rr þe4 2 e3 þ k4 e4 þ j2 v2 Lr
pLm e1 þ k2 e2 þ j1 v1 JLr
(45) Finally, the control laws are obtained by: v1 ¼
(36)
þe4 ðj2 v2 Þ
where k1, k3 are positive design constants. Their correct tuning will improve closed loop dynamic. When the stabilizing virtual controls are chosen as x*12
Ve ¼ e1 e1 þ e2 e2 þ e3 e3 þ e4 e4
pLm Rr ¼ e1 k1 e1 þ e2 þ e2 ðj1 v1 Þ þ e3 k3 e3 þ 2 e4 JLr Lr
pLm e1 þ k2 e2 þ j1 ; JLr
v2 ¼ 2
Rr e3 þ k4 e4 þ j2 Lr
This gives:
We obtain:
Ve ¼ k1 e21 k2 e22 k3 e23 k4 e24 0
V ¼ k1 e21 k3 e23 0
(37)
Then, the virtual controls in Eq. (36) are selected to fulfill requirement of the control objectives and also give references for the next step of backstepping design. Step 2 Now, the new tracking objectives became x12 and x22. So we define again errors related to the desired variables in Eq. (36): e2 ¼ x*12 x12 ¼
JLr JLr Tl x12 k1 e1 þ x*11 þ pLm pLm
(38)
e4 ¼ x*22 x22 ¼
Lr Rr k3 e3 þ 2 x21 x22 2Rr Lr
(39)
pLm e2 ; JLr
(40)
2Rr e3 ¼ k3 e3 þ e4 Lr
Also, derivatives for errors e2 ; e4 could be expressed by
e 2 ¼ j1 v 1 ; e 4 ¼ j2 v 2
(41)
where ji 'sði ¼ 1; 2Þ are known variables. Their expressions are given by j1 ¼
JLr k1 pLm
Lr j2 ¼ k3 pRr
k1 e1 þ
pLm e2 JLr
Rr k3 e3 þ 2 e4 Lr
þ
JLr JLr Rr x 11 þ Tl þ x12 pLm pLm Lr
(42)
(43)
When extending the Lyapunov function of Eq. (42) to include the state variables e2, e4 we get: Ve ¼
1 2 e þ e22 þ e23 þ e24 2 1
In order to fulfill requirements of the precedent multiscalar model, we must give measurement of rotor fluxes 4rx, 4ry, which make control more complex. The stator fluxes components 4sx, 4sy can be obtained from currents and voltages measurements using: Z
usa Rs isa dt; Z
usb Rs isb dt 4sb ¼ 4sa ¼
(48)
Lr ð4 sLs isa Þ; Lm sa Lr 4 sLs isb 4rb ¼ Lm sb 4ra ¼
(44)
To derive the control algorithm, we again compute the derivative of Ve along with the error Eqs. 40e43:
(49)
For the proposed control structure, we will use a simple rotor fluxes components 4ra, and 4rb estimators using only measurements of the stator current components and the rotor speed um. Expression of estimators in stationary frame is given by d~ 4ra Rr Lm ~ um 4 ~ rb þ Rr isa ; ¼ 4 dt Lr ra Lr d~ 4rb Rr Lm ~ þ um 4 ~ ra þ Rr isb ¼ 4 dt Lr rb Lr
(50)
Relationship between the rotor flux components in the stationary and rotating frame is given by
Rr Rr 2 x21 þ ð2Lm þ 1Þx22 Lr Lr
(47)
Then, the rotor fluxes 4ra, 4rb can be computed directly as:
Then the error Eq. (40) is given by: e1 ¼ k1 e1 þ
(46)
~ rx 4 ~ ry 4
¼
cos q*s sin q*s
where : q*s ¼
Z
4 ~ ra sin q*s * ~ rb 4 cos qs
u*s dt
(51)
(52)
From Eq. (26), it is possible to calculate u*s : u*s ¼ um þ
4ry v2 þ 4rx v1 isx 42r
(53)
The inverter output voltages could be calculated as [23]:
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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d isy ref 1 ðisx Þref isx rLs s isy ref þ Ls s ; ¼ Kp þ Ki s dt
1 Lm ~ isy ref isy þ rLs sðisx Þref þ r 4 u*sy ¼ Kp þ Ki s Lr rx
environment. The results of the two tests are shown in Figs. 3 and 4.
u*sx
Experimental results (54)
Stability analysis In order to prove stability of the proposed control scheme, we should prove that all errors e1 ; e2 ; e3 and e4 are bounded and converge asymptotically to zero according to Barbalat's Lemma [24]. To show boundedness of all errors, we can rearrange the dynamical equation from (40) and (41) as: :
E ¼ A:E
(55)
where: 3 2 pLm 0 0 k1 7 6 JL r 7 6 7 6 2R 6 r 7 ek2 0 7 6 0 6 Lr 7 7; A¼6 7 6 pL m 7 6 0 k3 0 7 6 7 6 JLr 7 6 5 4 2Rr 0 0 k4 Lr 2 3 e1 6 e2 7 7 E¼6 4 e3 5; 4rd : constant ðtransient regime is neglectedÞ e4 The stability of the global system is obtained if and only if the matrix A is Hurwitz, (roots with a negative real parts), which is verified by a good selection of gains k1 ; k2 ; k3 ; k4 . This proves the boundedness of all the errors. Further, since the motor speed u and d-axis current id tracking objectives are equivalent to that the error variables converge to zero, it can be concluded that the design objectives have been satisfied.
Results and discussion The bloc structure of the overall control system for IM is given on Fig. 2. In order to improve effectiveness, robustness and limits of the proposed approach, two tests are done: Test 1: speed inversion and step changing of the load torque The speed reference is a step smoothed function equal to 400 rpm, followed by inversion at t ¼ 5 s. The reference flux is set to 0.5 Wb. A constant load torque of 14 N m is applied between t ¼ 3 s and t ¼ 7 s. Test 2: parameters variations We have introduced intentionally a critical variation for some IM parameters: 50% raise of (Rs, Rr) and 25% raise of (Ls, Lr and Lm) relative to the identified model parameters.
Simulation results The effectiveness for speed and rotor flux magnitude regulations has been verified by simulation using Matlab/Simulink
The bloc structure of the experimental setup is given in Fig. 5. Experimental setup consists of many blocs: - Wound rotor induction motor with the following nominal characteristics: Star/Delta 380 V/220 V AC, 8.9 A/15.5 A, 4 kW, p.f. 0.82, 1440 r.p.m, 50 Hz. - Tripolar variable rheostat connected to the stator or to the rotor in order to allow us a voluntary change in the value of the stator or the rotor resistances during operation of the induction motor. - Powder brake. - Optical position encoder with 1024 lines per rotation for measuring angular position and speed. - SVPWM inverter with the nominal characteristics: 1000 V, 30 A. - The dSPACE interface generates through its external panel the SVPWM pulses for the inverter and receives the signals of the measured currents of phase “a” and “b” through ADCs and the angular position signals through encoder. - Speed and voltage (for the SVPWM inverter) commands are given from the dSPACE “controldesk” software. For signals sampling, two sample times were used: 750 ms for angular positions and 200 ms for current signals. The control algorithm execution is done within a time step of 200 ms. The inverter switching frequency is kept at 10 kHz using the slave DSP. The control and estimation algorithms need as inputs (1) the stator currents and (2) the encoder position. For that filters are used: Digital low pass filter is used for filtering encoder position signal. Digital synchronous resonating filter [25] is used for reducing high-frequency noise in the stator current signals arising out of SVPWM and electrical grid. In experimental mode, load torque is applied between t ¼ 8 s and t ¼ 18 s for the two tests. The experimental results are shown in Figs. 6 and 7.
Discussion Based on the simulation and the experimental results shown in the below figures, we can note that the proposed backstepping-multiscalar control overcomes both tests successfully. In the first test, two types of disturbances have been applied: speed inversion and a large load torque. The responses of the new multi-scalar variables are satisfactory. References x11 and x21 are perfectly tracked, and perturbation is rejected instantly and completely. x12 and x22 give us an idea on product responses between stator currents and rotor fluxes. From these figures, we can deduce that current limitation is effective, and no overtaking is recorded in both cases.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Fig. 2 e Bloc structure diagram of IM control using backstepping controllers.
(b) scalar product of stator current and rotor flux vectors 100
X11* X11
50
X 12
X11 (rad/s)
(a) rotor speed 100 80 60 40 20 0 -20 -40 -60
0
-50
-80 -100 0
2
4
6
Time (s)
8
10
(c) square of the rotor flux
-100 0
2
4
Time (s)
6
8
10
(d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X22
X 21
2 1.5
0
1 -10 0.5 0 0
2
4
Time (s)
6
8
10
-20 0
2
4
Time (s)
6
8
10
Fig. 3 e Test 1: tracking performance (simulation results).
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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(a) rotor speed
(b) scalar product of stator current and rotor flux vectors
100
100
X11* X11
50
X 12
X 1 1 (rad/s )
50
0
0
-50
-100 0
-50
2
4
6
Time (s) (c) square of the rotor flux
8
10
-100 0
2
4
6
8
10
2
4
6
8
10
Time (s) (d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X22
X21
2 1.5
0
1 -10 0.5 0 0
2
4
Time (s)
6
8
10
-20 0
Time (s)
Fig. 4 e Test 2: tracking performance (simulation results).
Fig. 5 e The experimental setup.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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(b) scalar product of stator current and rotor flux vectors
(a) rotor speed 100 80 60 40
X 11 (rad/s )
100
X11 X11*
50
X 12
20 0 -20
0
-40
-50
-60 -80 -100 0
5
10
15
20
Time (s)
-100 0
25
(c) square of the rotor flux
5
10
15
Time (s)
20
25
(d) vector product of stator current and rotor flux vectors 20
3 2.5
10
X22
X 21
2 1.5
0
1 -10
0.5 0 0
5
10
15
Time (s)
20
-20 0
25
5
10
15
Time (s)
20
25
Fig. 6 e Test 1: tracking performance (experimental results).
(a) rotor speed
(b) scalar product of stator current and rotor flux vectors
100
100
X11 X11*
80 60
50
X11 (rad/s)
40
X12
20 0 -20
0
-40
-50
-60 -80 -100 0
5
10
15
Time (s)
20
25
(c) square of the rotor flux
-100 0
5
10
15
Time (s)
20
25
(d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X 22
X 21
2 1.5
0
1 -10 0.5 0 0
5
10
15
Time (s)
20
25
-20 0
5
10
15
Time (s)
20
25
Fig. 7 e Test 2: tracking performance (experimental results). Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Second test deals with robustness against parameters variation. We can see clearly that these changes have not affected multi-scalar variables responses. That implies robustness of the proposed control despite disturbances and parameters variations.
Conclusion In this paper, using multi-scalar induction motor model instead of the classical one gives an interesting feature for induction motor control fed by SVPWM inverter. Based on the obtained results, we can observe the improvement and effectiveness of the proposed control scheme. In such systems, the main advantage is the possibility to move the operating point with the rotor flux magnitude without changing the dynamic of the system. The new multiscalar variables allow to obtaining novel backstepping control structure that achieves a very good tracking performance with almost total rejection of the disturbances (load torque and model uncertainties) and presents very interesting global stability properties. This kind of control allow an economical operation of induction machine drive in which the flux is decreased if the load is reduced. The other interesting feature of the proposed algorithm is that it is simple and easy to implement in real time.
Acknowledgements This research was supported by Engineering school ESIEE Paris, France. The authors thank the engineering school for material resources that were given to them and that allowed the testing and validation of experimental results presented in this paper through an experimental test bench.
Appendix Induction motor data Stator resistance 1.34 U; Rotor resistance 1.18 U; Mutual inductance 0.17 H; Rotor inductance 0.18 H; Stator inductance 0.18 H; Number of pole pairs 2; Motor load inertia 0.0153 kgm2.
references
[1] Jacob Fantidis. The temperature measurement of the windings in a three-phase electrical motor under different conditions. Gazi Univ J Sci Part A Eng Innov July 2015;3(2):39e44. [2] Karmakar S, Chandy JA, Gogna M, Jain FC. Fabrication and circuit modeling of NMOS inverter based on quantum dot gate field-effect transistors. J Electron Mater August 2012;41(8):2184e92.
[3] Blaschke F. The principle of field oriented as applied to the new Tran vector closed-loop control system for rotating machine. Siemens Rev 1972;39(4):217e20. [4] Marino R, Peresada S, Valigi P. Adaptive inputeoutput linearizing control of induction motors. IEEE Trans Automatic Control 1993;38(2):208e21. [5] Evangelista CA, Valenciaga F, Puleston P. Multivariable 2sliding mode control for a wind energy system based on a double fed induction generator. Int J Hydrogen Energy July 2012;37(13):10070e5. [6] Chen Chyi-Tsong, Peng Shih-Tien. A sliding mode control scheme for non-minimum phase non-linear uncertain input-delay chemical processes. J Process Control January 2006;16(1):37e51. rez G. On speed control [7] Ortega R, Nicklasson PJ, Espinosa-Pe of induction motor. Automatica 1996;32(3):455e60. [8] Tazerart Farid, Mokrani Zahra, Rekioua Djamila, Rekioua Toufik. Direct torque control implementation with losses minimization of induction motor for electric vehicle applications with high operating life of the battery. Int J Hydrogen Energy 19 October 2015;40(39):13827e38. € kbulut Muammer, Dandil Bes‚ir. Neuro-fuzzy control of a [9] Go dynamometer for the emulation of nonlinear mechanical loads. J Polytech 2006;9(2):79e86. € khan AK, Cansever Galip, Delibas‚i Akın. Trajectory [10] Ayc¸a Go tracking control of an industrial robot manipulator using fuzzy SMC with RBFNN. Gazi Univ J Sci February 2015;28(1):141e8. [11] Kanellakopoulos I, Kokotovic PV, Morse AS. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Autom Contr 1991;36:1241e53. [12] Happyanto Dedid Cahya, Fauzi Rizana, Hair Jumaddil. Backstepping development as controller in fast response three phase induction motor based on indirect field oriented control. Bali, Indonesia: 2016 International Electronics Symposium (IES); September 29e30th, 2016. [13] Morawiec Marcin. Z-type observer backstepping for induction machines. IEEE Trans Ind Electron April 2015;62(4):2090e102. [14] Jinpeng Yu, Yumei Ma, Haisheng Yu, Chong Lin. Adaptive fuzzy dynamic surface control for induction motors with iron losses in electric vehicle drive systems via backstepping. Inf Sciences (Elsevier) January 2017;376(10):172e89. [15] Benlaloui, Drid S, Chrifi-Alaoui L, Benoudjit D. Sensorless speed backstepping control of induction motor based on sliding mode observer: experimental results. In: 15th international conference on sciences and techniques of automatic control and computer engineering (STA); December 21e23, 2014. Hammamet, Tunisia. [16] Abu_Rub Haitham, Iqbal Atif, Guzinski Jaroslaw. High performance control of AC drives with matlab_simulink models. John Wiley & Sons Ltd; 2012. [17] Krzeminski Zbigniew. Advanced control of electrical drives and power electronic converters. Volume 75 of the series studies in systems, decision and control. 01 October 2016. p. 3e26. [18] Morawiec Marcin. Sensorless control of induction machine supplied by current source inverter. Asian J Control November 2015;17(6):2403e8. [19] Mehazzem F, Nemmour AL, Reama A. Application of backstepping-multiscalar control to induction motor fed by voltage source inverter. In: The fourth European conference on renewable energy systems (ECRES 2016); August 2016. p. 28e31. Istanbul, Turkey. [20] Leonard W. Control of electrical drives. Springer-Verlag; 1985. [21] Vas P. Vector control of AC machines. Oxford Science Publishing; 1990.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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[22] Abu-Rub H, Guzinski J, Krzeminski Z, Toliyat H. Speed observer system for advanced sensor less control of induction motor. IEEE Trans energy Convers June 2003;18(2). [23] Krauss PC. Analysis of electric machinery. IEEE Press; 1995. [24] Slotine JJ, Li W. Applied nonlinear control. New Jersey: Prentice-Hall; 1991.
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[25] Syam P, Nandy PK, Chattopadhyay AK. Improvement in power quality and a simple method of subharmonic suppression for a cycloconverter-fed synchronous motor drive. IEE Proc B, EPA July 2002;(4):292e303.
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Available online at www.sciencedirect.com
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Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter Fateh Mehazzem a,*, Ahmed Lokmane Nemmour a, Abdellatif Reama b Laboratoire d'electrotechnique de Constantine, Departement d'electrotechnique, Universite Constantine 1, 25000 Constantine, Algeria b ESIEE-Paris, Universite Paris-Est 2, Bd Blaise Pascal, Cite Descartes, 93162 Noisy le Grand Cedex, France a
article info
abstract
Article history:
A novel version of nonlinear induction motors (IM) control algorithm based on the back-
Received 14 November 2016
stepping approach has been proposed in this work. For that, the conventional mathe-
Received in revised form
matical model of IM is replaced by the multiscalar one, which is more suitable for
13 April 2017
application of the backstepping design. By using such combination, the system has fast
Accepted 5 May 2017
dynamic response, better load disturbance rejection capability, less parameters sensitivity
Available online xxx
and better tracking performance of rotor speed and rotor flux magnitude. The effectiveness of the proposed control structure is validated by simulation as well by experiment under
Keywords: Induction motor
critical disturbance conditions. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Backstepping control Multi-scalar induction machine model SVPWM voltage source inverter
Introduction Electric motors are nowadays the most popular electrical systems in the industry. Their function, to convert electrical energy into mechanical energy, gives them a big economic importance. So no installation designer, maker or operator can ignore them. Among all the types of existing motors, induction motors [1], especially phase squirrel cage motors are the most used in industrial applications due to their easy installation, robustness and reliability. Furthermore, their control is widely needed for many applications.
From an automatic viewpoint, induction motor is a dynamic system, which presents certain number of control constraints because of its characteristics: nonlinear system, multivariable and strongly coupled, the resistive and inductive parameters vary as well the load. We can notice also that some variables cannot be measured like flux, which involves the use of observers to fulfill the control structure. The development of the semiconductor components technology [2] has contributed to the development of static converters. Moreover, this allowed the real-time implementation of the most complex algorithms dedicated to control induction motor, whatever their execution time.
* Corresponding author. E-mail addresses: [email protected] (F. Mehazzem), [email protected] (A.L. Nemmour), [email protected] (A. Reama). http://dx.doi.org/10.1016/j.ijhydene.2017.05.035 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Nomenclature ðus ; ur Þ ðis ; ir Þ ðfs ; fr Þ p Lm ðLs ; Lr Þ ðRs ; Rr Þ J um ua us Tl
stator and rotor voltage vectors stator and rotor current vectors stator and rotor flux vectors number of pole pairs mutual inductance stator and rotor inductances respectively stator and rotor resistances respectively moment of inertia electrical rotor angular speed angular speed of reference frame electrical synchronous angular speed machine load torque
Vector control based flux oriented introduced by Blaschke in 1972 [3], was an important step for controlling induction motor with high dynamic performance in industrial applications. The principle of vector control is to reduce the behavior of the induction motor to that of a DC motor. This method is based on the transformation of electrical variables to a frame that rotates with the rotor flux vector. In this new frame, the dynamics of the rotor flux are asymptotically linear and decoupled. In general, we can say that two types of vector control are possible: firstly, direct vector control, where it is estimated the magnitude and the position of the rotor flux and secondly, the indirect vector control, where only the position of rotor flux is estimated. The principle drawback of vector control is its sensitivity to parameters variation. Since the end of 1980's years, adaptive control of nonlinear systems has boosted with the first version of the inputeoutput adaptive linearization [4]. Later, several adaptive state feedback control forms have been proposed in order to overcome drawbacks of vector control and improve IM control performance. Among them: sliding mode control [5,6], passivitybased control [7] and direct torque control [8]. In parallel, another class of algorithms has known great developments. The concept of these algorithms is based on emulating certain behaviors in nature and the use of artificial intelligence. Therefore, these approaches, qualified intelligent have demonstrated their efficiency in various applications fields (image processing, computer vision and speech recognition). In recent decades they have also attracted attention in the field of motor control. Among them, there are fuzzy logic, genetic algorithms and neural networks [9,10]. In last decades, much progress has been made in the area of control of nonlinear systems. The backstepping is one of these new popular techniques in this area. It was developed by , Ioannis Kanellakopoulos and Miroslav Krstic Peter Kokotovic in 1991 [11]. It provides a systematic method to perform the design of a controller for a class of non-triangular systems. The main advantage of this method is to ensure the stability of the association (controller-process). The objective of the application of the Backstepping technique for induction motor control is to establish a control law of the IM via a chosen Lyapunov function, in order to guarantee stability for the overall system. It has the advantage of being robust according to parametric variations and ensures good tracking references. The combination of Backstepping
technique with vector control gives the induction machine control interesting qualities of robustness [12]. In recent years, the concept of the backstepping approach has been widely exploited in literature either in control or in observation of the IM [13]. Combined with other techniques (such as: fuzzy logic or sliding mode), the backstepping control overcomes constraint related to parameter variations [14], and can be exploited in a sensorless control [15]. As mentioned before, classical mathematical induction machine model is non-linear, multi-variables and strongly coupled. An internal coupling appears between mechanical and electromagnetic variables. This contributes mainly to make induction motor control very complex. For that, multiscalar induction machine model gives a good alternative to reduce this complexity. Its structure as two separate linear decoupled subsystems makes it more suitable for applying backstepping control approach. The non linear control based on the multiscalar model or MM control was given for the first time in 1987 [16]. The multiscalar IM model is obtained by doing a variable change based on the choice of specific state variables in the conventional one. By applying this type of control for induction motor we can easily obtain in the same structure a non-linear control and decoupling between electromagnetic torque and the square of linear combination of a stator current vector and the vector of rotor linkage flux [17]. This was completely impossible with vector control using classical induction machine model. When using a current source inverter to supply IM makes application of control technique based on the proposed new multi-scalar variables difficult to implement in real time [18]. However, when induction motor is fed by voltage source inverter with a constant rotor flux modulus, non linear control becomes equivalent to vector control, but the application of the non linear backstepping control gives more system structure simplicity and good overall drive response [19]. In this paper, a novel structure of nonlinear backstepping control is used. Control laws are obtained using the multiscalar induction motor model. The use of variables transformation to obtain the multiscalar model makes the control approach more suitable to perform, because only four state variables have been obtained with a relatively simple nonlinearity form. The proposed speed and rotor flux magnitude control structure shows a perfect decoupled control during transient and steady state. So, an ideal tracking of reference trajectories can be achieved. This contribution is organized as follows. Section Material and methods gives the methodology. First, a description of the classical model of induction motor is given, followed by the multiscalar one, after that backstepping control design, applied to multiscalar induction motor model is presented. Finally, stability analysis for global system through Barbalat's lemma and Hurwitz criterion is studied. In Section Results and discussion, the application of the developed algorithm is illustrated for two tests: the first concerns speed inversion and step changing of the load torque, second test is for parameters variations. Simulations and experimental results are given with backstepping-multiscalar control for the both tests. Finally, the most important contributions of this paper are summarized in Section Conclusion.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Material and methods Classical induction machine mathematical model The induction machine as an electromechanical system is governed by the laws of electricity and mechanics. For that we can obtain two electrical equations relative to the stator and the rotor, and one mechanical equation. Consequently, the classical induction machine model can be written in a rotating frame with an angular speed ua by the following equations [20,21]:
d4s þ jua 4 s dt d4r þ jðua um Þ 4 r u r ¼ Rr i r þ dt us
¼ Rs i s þ
J dum ¼ Imag ð 4 s i r Þ Tl p dt
(1)
(3)
d4rx Rr Lm ¼ 4rx þ Rr isx þ ðus um Þ4ry dt Lr Lr
(4)
d4ry Rr Lm ¼ 4ry þ Rr isy ðus um Þ4rx dt Lr Lr
(5)
Lm Lr þ um 4ry þ usx wd wd disy 1 Rs Lm ¼ isy þ 4 us isx Td dt Ls wd sy Lm Lr um 4rx þ usy wd wd p dum Lm p 4ry isx þ 4rx isy Tl ¼ JLr J dt
constant Td in the rotor current set value channel by the following equations: disx 1 ¼ isx I*s (9) Td dt d4rx Rr Lm ¼ 4rx þ Rr isx þ ðus um Þ4ry (10) dt Lr Lr d4ry Rr ¼ 4ry ðus um Þ4rx dt Lr
(11)
p dum Lm p 4ry isx Tl ¼ JLr J dt
(12)
In order to obtain the state multi-scalar induction machine model, a variables transformation is used from the classical one. Four novel state variables are used, as follow [22]:
(2)
In order to give better and detailed representation of the IM, the model above can be rearranged to express the dynamical equations of current and flux vector components. For that, the mathematical model of IM as differential equations of state variables in the rotating frame xey with angular speed us is given by:
disx 1 Rr Lm ¼ isx þ 4 þ us isy Td dt Lr wd rx
3
(6)
(7)
(8)
x11 ¼ um
(13)
x12 ¼ 4rx isy 4ry isx
(14)
x21 ¼ 42rx þ 42ry
(15)
x22 ¼ 4rx isx þ 4ry isy
(16)
x11 : is interpreted as rotor angular speed, x12 , x22 : are interpreted as scalar and vector products of the stator current and rotor flux vectors. The first is proportional to the motor torque, second is proportional to the energy. x21 : is interpreted as the square of the rotor flux. The multi-scalar model of induction machine will be given then by calculating derivatives of new scalar variables using the differential equations of the classical model presented above. The new model will be expressed as fellow: dx11 pLm p ¼ x12 Tl J dt JLr
(17)
dx12 1 ¼ x12 þ v1 T dt
(18)
dx21 Rr Rr Lm ¼ 2 x21 þ 2 x22 dt Lr Lr
(19)
dx22 1 Rr Lm 2 ¼ x22 þ i þ v'2 T dt Lr sx
(20)
where:
where:
1 L2 Rs þ L2m Rr ¼ r ; wd ¼ Ls Lr L2m : Td Lr wd
1 Rr 1 ¼ þ T Lr Td
We can notice that the model above is nonlinear, highly coupled and complex. For that we need to simplify this model in order to obtain a new structure more suitable for the application of the control process.
Multiscalar induction machine model The classical mathematical model of the wound-rotor induction machine can be expressed in rotor current rotating frame (isy ¼ 0), taking into account first-order delay time with a time
(21)
v1 ¼ ðus um Þx22
1 4 I* Td ry s
(22)
v'2 ¼ ðus um Þx12 þ
1 4 I* Td rx s
(23)
The above model has significantly less non linearities than the classical model presented before. We can notice that v1 and v20 are inputs for the new model and contribute to determine the variables us and I*s.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Eq. (20) contains non-linearity which can be compensated by a non-linear feedback. This will give equation: Rr Lm 2 i (24) v2 ¼ v'2 Lr sx The resolution of the system Eq. (22) and (23) gives the expression of the stator current amplitude and slip frequency: 4rx v2 4ry v1 42r
(25)
* 4ry v2 þ 4rx v1 us um ¼ isx 42r
(26)
I*s ¼ Td
The following expression also can be obtained: I*s 2 ¼
x212 þ x222 x21
(27)
One can notice that this expression is a function only of multiscalar variables, and don't need any rotor flux estimation, and this will be useful for calculation of the inverter output voltages. From the new IM model presented by (17)e(20), and using non-linear feedback in Eq. (24) we can obtain two linear fully decoupled subsystems e mechanical and electromagnetic: Mechanical subsystem: dx11 pLm p ¼ x12 Tl J dt JLr
(28)
dx12 1 ¼ x12 þ v1 T dt
(29)
dx22 1 ¼ x22 þ v2 T dt
Backstepping approach The basic idea of backstepping technique is to select recursively appropriate functions as virtual control inputs for first order subsystems. In order to apply backstepping for the whole system, design is divided into several stages. Extended Lyapunov functions are associated step by step to ensure global system stability. The obtained multi-scalar IM model above is more appropriate then the classical one for the application of the backstepping control design. Tracking objectives are defined as the rotor angular speed x11 ¼ um and the rotor flux square x21 ¼ 42r . The backstepping design is divided in two steps: Step 1 We start by defining the tracking errors as: e1 ¼ x*11 x11 ; e3 ¼ x*21 x21
Electromagnetic subsystem: dx21 Rr Rr Lm ¼ 2 x21 þ 2 x22 dt Lr Lr
The mechanical subsystem is represented by a delayed first order system, followed in serial by an integral bloc. The load torque for this subsystem represents the disturbance. For that, and in order to limit it, we should use a cascade control structure. The electromagnetic subsystem is represented by two delayed first order blocs, linked in a series. Same case, in order to limit the square of rotor flux, we should again using a cascade control structure. That implies, when using multiscalar model, the control structure should be composed of PI cascade controllers with constant parameters. Tuning of these parameters is done according to the control theory of linear systems.
(30)
(31)
The above model is linear and fully decoupled, which makes it possible to use a linear cascaded controller, as shown in Fig. 1. We can notice that motor torque is only proportional to the state variable x12, which belongs to the mechanical side of control design when using multiscalar model form. The rotor flux should be controlled by the electromagnetic subsystem, and its command value can be calculated by taking it as criterion either in the minimization of energy losses or in the minimization of response time in the mechanical subsystem [16].
(32)
Then, their derivatives are:
*
pLm p x12 þ Tl; J JLr
*
Rr Rr Lm x21 2 x22 Lr Lr
e1 ¼ x11 2 e3 ¼ x21 þ 2
(33)
Since the objective requirements are that the two errors converge to zero, and the current must be regulated and limited. These requirements are satisfied by viewing x12 and x22 as “virtual controls variables” in the above equations and using them to control the errors e1, e3. When using the Lyapunov function: 1 1 V ¼ e21 þ e23 2 2
(34)
Fig. 1 e Cascaded structure for the multiscalar model control design. Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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The derivative of V is given by:
V ¼ e1 e1 þ e3 e3 * * pLm p Rr Rr Lm ¼ e1 x11 x12 þ Tl þ e3 x21 þ 2 x21 2 x22 J JLr Lr Lr * pLm p þ Tl ¼ k1 e21 k3 e23 þ e1 k1 e1 þ x11 J JLr Rr Rr Lm þe3 k3 e3 þ 2 x21 2 x22 ð35Þ Lr Lr
x*22
JLr p ¼ k1 e1 þ x*11 þ Tl ; J pLm Lr Rr ¼ k3 e3 þ 2 x21 2Rr Lr
¼ k1 e21 k2 e22 k3 e23 k4 e24 þ e2 Rr þe4 2 e3 þ k4 e4 þ j2 v2 Lr
pLm e1 þ k2 e2 þ j1 v1 JLr
(45) Finally, the control laws are obtained by: v1 ¼
(36)
þe4 ðj2 v2 Þ
where k1, k3 are positive design constants. Their correct tuning will improve closed loop dynamic. When the stabilizing virtual controls are chosen as x*12
Ve ¼ e1 e1 þ e2 e2 þ e3 e3 þ e4 e4
pLm Rr ¼ e1 k1 e1 þ e2 þ e2 ðj1 v1 Þ þ e3 k3 e3 þ 2 e4 JLr Lr
pLm e1 þ k2 e2 þ j1 ; JLr
v2 ¼ 2
Rr e3 þ k4 e4 þ j2 Lr
This gives:
We obtain:
Ve ¼ k1 e21 k2 e22 k3 e23 k4 e24 0
V ¼ k1 e21 k3 e23 0
(37)
Then, the virtual controls in Eq. (36) are selected to fulfill requirement of the control objectives and also give references for the next step of backstepping design. Step 2 Now, the new tracking objectives became x12 and x22. So we define again errors related to the desired variables in Eq. (36): e2 ¼ x*12 x12 ¼
JLr JLr Tl x12 k1 e1 þ x*11 þ pLm pLm
(38)
e4 ¼ x*22 x22 ¼
Lr Rr k3 e3 þ 2 x21 x22 2Rr Lr
(39)
pLm e2 ; JLr
(40)
2Rr e3 ¼ k3 e3 þ e4 Lr
Also, derivatives for errors e2 ; e4 could be expressed by
e 2 ¼ j1 v 1 ; e 4 ¼ j2 v 2
(41)
where ji 'sði ¼ 1; 2Þ are known variables. Their expressions are given by j1 ¼
JLr k1 pLm
Lr j2 ¼ k3 pRr
k1 e1 þ
pLm e2 JLr
Rr k3 e3 þ 2 e4 Lr
þ
JLr JLr Rr x 11 þ Tl þ x12 pLm pLm Lr
(42)
(43)
When extending the Lyapunov function of Eq. (42) to include the state variables e2, e4 we get: Ve ¼
1 2 e þ e22 þ e23 þ e24 2 1
In order to fulfill requirements of the precedent multiscalar model, we must give measurement of rotor fluxes 4rx, 4ry, which make control more complex. The stator fluxes components 4sx, 4sy can be obtained from currents and voltages measurements using: Z
usa Rs isa dt; Z
usb Rs isb dt 4sb ¼ 4sa ¼
(48)
Lr ð4 sLs isa Þ; Lm sa Lr 4 sLs isb 4rb ¼ Lm sb 4ra ¼
(44)
To derive the control algorithm, we again compute the derivative of Ve along with the error Eqs. 40e43:
(49)
For the proposed control structure, we will use a simple rotor fluxes components 4ra, and 4rb estimators using only measurements of the stator current components and the rotor speed um. Expression of estimators in stationary frame is given by d~ 4ra Rr Lm ~ um 4 ~ rb þ Rr isa ; ¼ 4 dt Lr ra Lr d~ 4rb Rr Lm ~ þ um 4 ~ ra þ Rr isb ¼ 4 dt Lr rb Lr
(50)
Relationship between the rotor flux components in the stationary and rotating frame is given by
Rr Rr 2 x21 þ ð2Lm þ 1Þx22 Lr Lr
(47)
Then, the rotor fluxes 4ra, 4rb can be computed directly as:
Then the error Eq. (40) is given by: e1 ¼ k1 e1 þ
(46)
~ rx 4 ~ ry 4
¼
cos q*s sin q*s
where : q*s ¼
Z
4 ~ ra sin q*s * ~ rb 4 cos qs
u*s dt
(51)
(52)
From Eq. (26), it is possible to calculate u*s : u*s ¼ um þ
4ry v2 þ 4rx v1 isx 42r
(53)
The inverter output voltages could be calculated as [23]:
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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d isy ref 1 ðisx Þref isx rLs s isy ref þ Ls s ; ¼ Kp þ Ki s dt
1 Lm ~ isy ref isy þ rLs sðisx Þref þ r 4 u*sy ¼ Kp þ Ki s Lr rx
environment. The results of the two tests are shown in Figs. 3 and 4.
u*sx
Experimental results (54)
Stability analysis In order to prove stability of the proposed control scheme, we should prove that all errors e1 ; e2 ; e3 and e4 are bounded and converge asymptotically to zero according to Barbalat's Lemma [24]. To show boundedness of all errors, we can rearrange the dynamical equation from (40) and (41) as: :
E ¼ A:E
(55)
where: 3 2 pLm 0 0 k1 7 6 JL r 7 6 7 6 2R 6 r 7 ek2 0 7 6 0 6 Lr 7 7; A¼6 7 6 pL m 7 6 0 k3 0 7 6 7 6 JLr 7 6 5 4 2Rr 0 0 k4 Lr 2 3 e1 6 e2 7 7 E¼6 4 e3 5; 4rd : constant ðtransient regime is neglectedÞ e4 The stability of the global system is obtained if and only if the matrix A is Hurwitz, (roots with a negative real parts), which is verified by a good selection of gains k1 ; k2 ; k3 ; k4 . This proves the boundedness of all the errors. Further, since the motor speed u and d-axis current id tracking objectives are equivalent to that the error variables converge to zero, it can be concluded that the design objectives have been satisfied.
Results and discussion The bloc structure of the overall control system for IM is given on Fig. 2. In order to improve effectiveness, robustness and limits of the proposed approach, two tests are done: Test 1: speed inversion and step changing of the load torque The speed reference is a step smoothed function equal to 400 rpm, followed by inversion at t ¼ 5 s. The reference flux is set to 0.5 Wb. A constant load torque of 14 N m is applied between t ¼ 3 s and t ¼ 7 s. Test 2: parameters variations We have introduced intentionally a critical variation for some IM parameters: 50% raise of (Rs, Rr) and 25% raise of (Ls, Lr and Lm) relative to the identified model parameters.
Simulation results The effectiveness for speed and rotor flux magnitude regulations has been verified by simulation using Matlab/Simulink
The bloc structure of the experimental setup is given in Fig. 5. Experimental setup consists of many blocs: - Wound rotor induction motor with the following nominal characteristics: Star/Delta 380 V/220 V AC, 8.9 A/15.5 A, 4 kW, p.f. 0.82, 1440 r.p.m, 50 Hz. - Tripolar variable rheostat connected to the stator or to the rotor in order to allow us a voluntary change in the value of the stator or the rotor resistances during operation of the induction motor. - Powder brake. - Optical position encoder with 1024 lines per rotation for measuring angular position and speed. - SVPWM inverter with the nominal characteristics: 1000 V, 30 A. - The dSPACE interface generates through its external panel the SVPWM pulses for the inverter and receives the signals of the measured currents of phase “a” and “b” through ADCs and the angular position signals through encoder. - Speed and voltage (for the SVPWM inverter) commands are given from the dSPACE “controldesk” software. For signals sampling, two sample times were used: 750 ms for angular positions and 200 ms for current signals. The control algorithm execution is done within a time step of 200 ms. The inverter switching frequency is kept at 10 kHz using the slave DSP. The control and estimation algorithms need as inputs (1) the stator currents and (2) the encoder position. For that filters are used: Digital low pass filter is used for filtering encoder position signal. Digital synchronous resonating filter [25] is used for reducing high-frequency noise in the stator current signals arising out of SVPWM and electrical grid. In experimental mode, load torque is applied between t ¼ 8 s and t ¼ 18 s for the two tests. The experimental results are shown in Figs. 6 and 7.
Discussion Based on the simulation and the experimental results shown in the below figures, we can note that the proposed backstepping-multiscalar control overcomes both tests successfully. In the first test, two types of disturbances have been applied: speed inversion and a large load torque. The responses of the new multi-scalar variables are satisfactory. References x11 and x21 are perfectly tracked, and perturbation is rejected instantly and completely. x12 and x22 give us an idea on product responses between stator currents and rotor fluxes. From these figures, we can deduce that current limitation is effective, and no overtaking is recorded in both cases.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Fig. 2 e Bloc structure diagram of IM control using backstepping controllers.
(b) scalar product of stator current and rotor flux vectors 100
X11* X11
50
X 12
X11 (rad/s)
(a) rotor speed 100 80 60 40 20 0 -20 -40 -60
0
-50
-80 -100 0
2
4
6
Time (s)
8
10
(c) square of the rotor flux
-100 0
2
4
Time (s)
6
8
10
(d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X22
X 21
2 1.5
0
1 -10 0.5 0 0
2
4
Time (s)
6
8
10
-20 0
2
4
Time (s)
6
8
10
Fig. 3 e Test 1: tracking performance (simulation results).
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 1
(a) rotor speed
(b) scalar product of stator current and rotor flux vectors
100
100
X11* X11
50
X 12
X 1 1 (rad/s )
50
0
0
-50
-100 0
-50
2
4
6
Time (s) (c) square of the rotor flux
8
10
-100 0
2
4
6
8
10
2
4
6
8
10
Time (s) (d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X22
X21
2 1.5
0
1 -10 0.5 0 0
2
4
Time (s)
6
8
10
-20 0
Time (s)
Fig. 4 e Test 2: tracking performance (simulation results).
Fig. 5 e The experimental setup.
Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 1
(b) scalar product of stator current and rotor flux vectors
(a) rotor speed 100 80 60 40
X 11 (rad/s )
100
X11 X11*
50
X 12
20 0 -20
0
-40
-50
-60 -80 -100 0
5
10
15
20
Time (s)
-100 0
25
(c) square of the rotor flux
5
10
15
Time (s)
20
25
(d) vector product of stator current and rotor flux vectors 20
3 2.5
10
X22
X 21
2 1.5
0
1 -10
0.5 0 0
5
10
15
Time (s)
20
-20 0
25
5
10
15
Time (s)
20
25
Fig. 6 e Test 1: tracking performance (experimental results).
(a) rotor speed
(b) scalar product of stator current and rotor flux vectors
100
100
X11 X11*
80 60
50
X11 (rad/s)
40
X12
20 0 -20
0
-40
-50
-60 -80 -100 0
5
10
15
Time (s)
20
25
(c) square of the rotor flux
-100 0
5
10
15
Time (s)
20
25
(d) vector product of stator current and rotor flux vectors
3
20
2.5 10
X 22
X 21
2 1.5
0
1 -10 0.5 0 0
5
10
15
Time (s)
20
25
-20 0
5
10
15
Time (s)
20
25
Fig. 7 e Test 2: tracking performance (experimental results). Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035
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Second test deals with robustness against parameters variation. We can see clearly that these changes have not affected multi-scalar variables responses. That implies robustness of the proposed control despite disturbances and parameters variations.
Conclusion In this paper, using multi-scalar induction motor model instead of the classical one gives an interesting feature for induction motor control fed by SVPWM inverter. Based on the obtained results, we can observe the improvement and effectiveness of the proposed control scheme. In such systems, the main advantage is the possibility to move the operating point with the rotor flux magnitude without changing the dynamic of the system. The new multiscalar variables allow to obtaining novel backstepping control structure that achieves a very good tracking performance with almost total rejection of the disturbances (load torque and model uncertainties) and presents very interesting global stability properties. This kind of control allow an economical operation of induction machine drive in which the flux is decreased if the load is reduced. The other interesting feature of the proposed algorithm is that it is simple and easy to implement in real time.
Acknowledgements This research was supported by Engineering school ESIEE Paris, France. The authors thank the engineering school for material resources that were given to them and that allowed the testing and validation of experimental results presented in this paper through an experimental test bench.
Appendix Induction motor data Stator resistance 1.34 U; Rotor resistance 1.18 U; Mutual inductance 0.17 H; Rotor inductance 0.18 H; Stator inductance 0.18 H; Number of pole pairs 2; Motor load inertia 0.0153 kgm2.
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Please cite this article in press as: Mehazzem F, et al., Real time implementation of backstepping-multiscalar control to induction motor fed by voltage source inverter, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.05.035