![ ISSN: 2088-8708
IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23
13
useful nonlinearities in pursuing the objectives of stabilization and tracking. A nonlinear backstepping
control design scheme is developed for the speed tracking control of PMSM that has exact model knowledge.
The asymptotic stability of the resulting closed loop system is guaranteed according to Lyapunov stability
theorem.
The speed variation of the PMSM is widely used in high-performance applications. The PMSM has
very large power density, high power factor and high efficiency. In a high-performance control of PMSM,
the information of rotor position and speed is very important. In the speed control loop, for the field oriented
control, the coordinate transformation has needs precise rotor position. Rotor position and speed can be
measured by a shaft encoder or other type of sensors, in other case the speed is measured with an Encoder
resolver connected to the PMSM machine drive. However, the presence of such sensors is not acceptable for
cost, maintenance and reliability reasons. The concept of sensorless control was proposed in the 1970s and
has been continually developed for PMSM rotor position and speed estimation. The basic principle of
sensorless control is to deduce the rotor speed and position using various information and means, including
direct calculation, parameter identification, condition estimation, indirect measuring and so on. The stator
currents and voltages are generally used to calculate the information of speed and rotor position.
The FPGA technology is now used by an increasing number of designers in various fields of
application such as signal processing, telecommunication, video, embedded control systems, and electrical
control systems. This last domain, i.e. the studies of control of electrical machines, will be presented in this
paper [1]. Indeed, these components have already been used with success in many different applications such
as Pulse Width Modulation (PWM), control of induction machine drives and multimachine system control.
This is because the FPGA-based implementation of controllers can efficiently answer current and future
challenges of this field.
Considering the complexity of the diversity of the electric control devices of the machines, it is
difficult to define with universal manner a general structure for such systems. However, by having a reflexion
compared to the elements most commonly encountered in these systems, it is possible to define a general
structure of an electric control device of machines which is show in Figure 1:
Figure 1. Architecture of the Control
This paper presents the realization of a platform for not adaptative and adaptative Backstepping
control of PMSM using FPGA based controller. This realization is especially aimed for future high
performance applications. In this approach, not only the architecture corresponding to the control algorithm is
studied, but also architecture and the ADC interface, Encoder interface and RS232 UART architecture [2].
2. PMSM MODEL SYSTEM
In this paper, we apply the different algorithms control on a machine type PMSM (Permanent Magnet
Synchronous Motor) [3], which consists of three stator windings and a rotor magnet. This motor is described
by the following equation.](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-2-320.jpg)
![IJPEDS ISSN: 2088-8694
FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI)
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.
..
.
.
..
..
p
Cf
dt
d
JC
iL
iL
dt
d
irV
dt
d
irV
re
sqsqsq
fsdsdsd
sd
sq
sqssq
sq
sd
sdssd
(1)
Where Ω is the rotation's speed, p the Number of pairs of poles, J the moment of inertia, f the
Coefficient of viscous friction, Cr the resistive torque, Φf the flux produced by the permanent magnet, Lsd and
Lsq the d-q axis stator inductance, Vsd and Vsq the d-q axis stator voltage, rs the stator winding resistance and
Ce the electromagnetic torque.
3. NONLINEAR BACKSTEPPING APPROACH
The Backstepping approach algorithm is control techniques that can linearize a nonlinear system
such as the PMSM machine drive in the presence of uncertainties. Unlike other feedback linearization
techniques, adaptive Backstepping has the flexibility of keeping useful non linearity’s intact during
stabilization. The essence of Backstepping is the stabilization of a virtual control state. Hence, it generates a
corresponding error variable which can be stabilized by carefully selecting proper control inputs. These
inputs can be determined from Lyapunov stability analysis [4].
It is obvious that the dynamic model of PMSM is highly nonlinear because of the coupling between
the speed and the stator currents (equation (1)). According to the vector control principle, the direct axis
current id is always forced to be zero in order to orient all the linkage flux in the d axis and achieve
maximum torque per ampere.
J
C
J
f
iiLLi
J
p
dt
d
L
V
p
L
ip
L
L
i
L
r
dt
di
L
V
ip
L
L
i
L
r
dt
di
r
sqsdsqsdsqf
sq
sq
sq
f
sd
sq
sd
sq
sq
ssq
sd
sd
sq
sd
sq
sd
sd
ssd
))((
2
3
.
.
(2)
The vector T
sqsd iix choice as state vector is justified by the fact that currents and speed
are measurable and that the control of the instantaneous torque can be done comfortable via the currents isd
and/or isq. And stator voltages as control variables T
sqsd VVu .
The principal objective of the backstepping controller is to regulate the speed of the PMSM drive to its
reference value Ωref whatever external disturbances. We assume that the engine parameters are known and
invariant.
3.1. Backstepping Speed Controller
The first step is defined the tracking errors:
refe (3)
The derivative of (3) is:
rsqsdsqsdsqfrefref CfiiLLi
p
Jdt
de
e ))((
2
31 (4)
We define the following quadratic function:](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-3-320.jpg)
![ ISSN: 2088-8708
IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23
15
2
1
2
1
eV (5)
Its derivative along the solution of (5), is given by:
rsqsdsqsdsqfref CfiiLLi
p
J
eeeV ))((
2
31
1
(6)
Using the Backstepping design method, we consider the d-q axes currents components isd and isq as
our virtual control elements and specify its desired behavior, which are called stabilizing function in the
backstepping design terminology as follows:
)..(
3
2
0
ekJCf
p
i
i
r
f
sqref
sdref
(7)
With kΩ is a positive constant
Substituting (7) in (6) the derivative of V1:
02
1 ekV (8)
3.2. Backstepping Current Controller
We have the asymptotic stability of the origin of the system (1). We defined current following
errors:
sqsqrefq
sdrefsdsdrefd
iie
iwithiie 0
(9)
Their dynamics can be written:
sd
sd
sq
f
sd
sq
sd
sq
sq
s
r
f
sqsqrefq
sd
sd
sq
sd
sq
sd
sd
s
sdsdrefd
L
V
L
p
ip
L
L
i
L
r
ekJCf
p
iie
L
V
ip
L
L
i
L
r
iie
.)..(
3
2
.
(10)
To analyze the stability of this system we propose the following Lyapunov function:
)(
2
1 222
2 qd eeeV (11)
Its derivative along the trajectories (8), (9) and (10) is:
]
2
3
))(
2
3
2
3
.(
3
)(2
[
])(
2
3
[222
2
sq
f
sd
sq
sd
sq
sq
s
sq
sqf
sqdsqsdq
f
f
qqq
sqsqsdsq
sd
sq
sd
s
sd
sd
dddqqddqqdd
L
i
L
L
i
L
r
L
V
e
J
p
ekieLL
J
p
e
J
p
p
fJk
eke
ieLL
J
p
i
L
L
L
r
L
V
ekeekekekeeeeeeV
(12)
The expression (12) found above requires the following control laws:
qsqqfsdsdsqs
sqf
sqdsqsdq
f
f
sq
sq
sqsqsd
sd
sqsqsdsdsddsd
eLkiLire
J
Lp
ekieLL
J
p
e
J
p
p
fJkL
V
ieLL
J
pL
iLireLkV
2
3
)(
2
3
2
3
3
)(2
)(
2
3
(13)](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-4-320.jpg)
![IJPEDS ISSN: 2088-8694
FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI)
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With this choice the derivatives of (13) become:
02 qqdd ekekekV (14)
4. NONLINEAR ADAPTATIVE BACKSTEPPING APPROACH CONTROL
4.1. Principle
In the previous section, the control laws are developed under the assumption that the machine
parameters are known and invariants. This assumption is not always true. In fact, the flow created by the
magnet varies with increasing temperature and with the fields created by the stator currents. Stator resistance
also varies with temperature. Also, the change in operating conditions is implicitly load torque and hence the
coefficient of friction and inertia. Adaptive Backstepping version takes into account the variations of these
parameters.
In equation (7), we do not know exactly the value of the load torque Cr, it will be replaced by its
estimate rCˆ .
)..ˆ.(
3
2ˆ
ekJCf
p
i r
f
sqref
(15)
From (13) and (15), we deduce the dynamics of the speed error as follows:
ekJieLL
p
e
p
C
Jdt
de
sqdsqsd
q
f
r
..)(
2
3
2
3~
1 (16)
With rrr CCC ˆ~ is the error of the estimated load torque.
The Dynamic errors and direct currents quadratic write:
sq
sd
sq
sd
sd
s
sd
sdsd
i
L
L
i
L
R
L
V
dt
di
dt
de
(17)
r
fsq
f
sd
sq
sd
sd
sq
s
sq
sd
sqdsqsdq
f
f
sqsqrefq
Ck
J
f
pL
i
L
L
i
L
R
L
V
ekieLL
J
p
e
J
p
p
fJk
dt
di
dt
di
dt
de
~
)(
3
2
.)(
2
3
2
3
3
)(2
(18)
To analyze the stability of this system we propose the following Lyapunov function:
3
2
2
2
1
2
222
2
~~~
2
1
fsr
qd
RC
eeeV (19)
Its derivative along the trajectories (16), (17) and (18) is:
q
sq
q
f
qffsqq
sq
sdd
sd
ss
f
q
f
q
rr
sq
f
sd
sq
sd
sq
sq
s
sq
sqf
sqdsqsdq
f
f
qqq
sqsqsdsq
sd
sq
sd
s
sd
sd
dddqqdd
ffssrrqqdd
e
L
e
J
fJk
ee
J
p
ie
L
ie
L
RR
J
e
pJ
fe
p
ek
CC
L
i
L
L
i
L
R
L
V
e
J
p
ekieLL
J
p
e
J
p
p
fJk
eke
ieLL
J
p
i
L
L
L
R
L
V
ekeekekek
RRCCeeeeeeV
1
ˆ2
3~1~11~1~
ˆ3
2
ˆ3
2~1~
]
ˆ
2
ˆ3
))(
2
3
2
ˆ3
.(
ˆ3
)(2
[
)(
2
3
~~1~~1~~1
2
321
222
321
2
(20)](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-5-320.jpg)
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FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI)
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5. FPGA-BASED IMPLEMENTATION OF AN ROBUST BACKSTEPPING CONTROL SYSTEM
5.1. Development of the Implementation
There are several manufacturers of FPGA components such: Actel, Xilinx and Altera…etc. These
manufacturers use different technologies for the implementation of FPGAs. These technologies are attractive
because they provide reconfigurable structure that is the most interesting because they allow great flexibility
in design. Nowadays, FPGAs offer the possibility to use dedicated blocks such as RAMs, multipliers wired
interfaces PCI and CPU cores. The architecture designing was done using with CAD tools. The description is
made graphically or via a hardware description language high level, also called HDL (Hardware Description
Language). Is commonly used language VHDL and Verilog. These two languages are standardized and
provide the description with different levels, and especially the advantage of being portable and compatible
with all FPGA technologies previously introduced [7].
The simulation procedure begins by verifying the functionality of the control algorithm by trailding
a functional model using Simulink’s System Generator for Xilinx blocks. For this application, the functional
model consists in a Simulink timeis discretired model of the No adaptative Backstepping algorithm
associated with a voltage inverter and PMSM model.
The Figure 8 summarizes the different steps of programming an FPGA. The synthesizer generated
with CAD tools first one Netlist which describes the connectivity of the architecture. Then the placement-
routing optimally place components and performs all the routing between different logic. These two steps are
used to generate a configuration file to be downloaded into the memory of the FPGA. This file is called
bitstream. It can be directly loaded into FPGA from a host computer.
Figure 8. Programming FPGA devisees
In this work an FPGA XC3S500E Spartan3E from Xilinx is used. This FPGA contains 400,000 logic
gates and includes an internal oscillator which issuer a 50MHz frequency clock. The map is composed from a
matrix of 5376 slices linked together by programmable connections.
5.2. Simulation Procedure
Figure 9. Functional Model for No adaptative Backstepping Controller from SYSTEM GENERATOR](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-9-320.jpg)
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FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI)
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(a) (b)
Figure 12. (a) Speed performance rotor of PMSM, (b) Error Speed performance
The Experimental results show the performance of PMSM machine, using two approaches control
nonlinear. This two control algorithms show the robustness and effecacité of the system.
The work presented in this paper shows a good robustness of the backstepping control vis a vis the
disturbances. It is noted that it is very dificult implementer of a non-linear control of a FPGA seen that there
are two current loops and speed.
With this new aproach was able to implement this order through the logiciele generator system
which facilitates this task. The Results obtained show the validation of this work.
Nonlinear backstepping control is very effective as orders that exists in literature (Sliding Mode,
Direct Torque Control ...), it has improved the performance of the PMSM machine at the current and speed,
response time, system speed (excuster to 40.5s for the program), and system stability regardless of the
disturbance and the parametric variations of the machine.
7. CONCLUSION
In this paper a robust continuous approachs Nonlinear not Adaptative Backstepping Control and
Adaptative Backstepping Control strategy for permanent-magnet synchronous motor (PMSM) drive systems
is presented. The FPGA based implementation is detailed, a bench test was realized by a prototyping platform,
the experimental results obtained show the effectiveness and the benefit of our contribution and the different
steps of implementation for the control FPGA.
REFERENCES
[1] Melicio R, Mendes VMF, Catalao JPS. “Modeling, Control and Simulation of Full- Power Converter Wind
Turbines Equipped with Permanent Magnet SynchronousGenerator”. 2010; 5: 397-408.
[2] JJ Chen and KP Chin. "Automatic flux-weakening control of permanent magnet synchronous motors using a
reduced-order controller". IEEE Trans. Power Electron. 2000; 15: 881-890.
[3] M Rodic, K Jezernik. “Speed Sensoless Sliding Mode Torque Control of Induction Motor “. IEEE Trans on.
Industrial Electronics. February 2002.
[4] Hisn-Jang Shieh and Kuo-Kai Shyu. “Non Linear Sliding Mode Torque Control With Adaptive Backstepping
Approach For Induction Motor Drive”. IEEE Transactions on Industrial Electronics. 1999; 46(2): 380-388.
[5] D Zhang and H Li. “A stochastic-based FPGA controller for an induction motor drive with integrated neural
network algorithms”. IEEE Trans. Ind. Electron. 2008; 55(2): 551–561.
[6] CF Jung and JS Chen. “Water bath temperature control by a recurrent fuzzy controller and its FPGA
implementation”. IEEE Trans. Ind. Electron. 2006; 53(3): 941–949.
[7] E Monmasson and M Cirstea. “FPGA Design Methodology for Industrial Control Systems – A Review”. IEEE
Trans Ind. Electron. 2007; 54(4): 1824-1842.
[8] B Bossoufi, M Karim, S Ionita, A Lagrioui. “Low-Speed Sensorless Control of PMSM Motor drive Using a
NONLINEAR Approach BACKSTEPPING Control: FPGA-Based Implementation”. Journal of Theoretical and
Applied Information Technology JATIT. 2012; 36(1): 154-166.
[9] B BOSSOUFI, M KARIM, S IONITA, A LAGRIOUI. “Nonlinear Non Adaptive Backstepping with Sliding-Mode
Torque Control Approach for PMSM Motor”. Journal of Journal of Electrical Systems JES. 2012; 8(2): 236-248.
[10] A Lagrioui, H Mahmoudi, B Bossoufi. “DISCRETE LINEAR PREDICTIVE CONTROL OF PERMANENT
MAGNET SYNCHRONOUS MOTOR (PMSM)”. Journal of Theoretical and Applied Information Technology
JATIT. 2011; 31(1): 21-28.
[11] B Bossoufi, M Karim, S Ionita, A Lagrioui. “The Optimal Direct Torque Control of a PMSM drive: FPGA-Based
Implementation with Matlab & Simulink Simulation”. Journal of Theoretical and Applied Information Technology
JATIT. 2011; 28(2): 63-72.](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-11-320.jpg)
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[12] B Bossoufi, M Karim, S Ionita, A Lagrioui, G Iana. “Matlab & Simulink Simulation with FPGA-Based
Implementation Sliding Mode Control of a Permanent Magnet Synchronous Machine Drive”. WSEAS
TRANSACTIONS on SYSTEMS and CONTROL. 2011; 3(6): 92-103.
[13] TC Lee, KT Song, CH Lee, and CC Teng. “Tracking control of unicycle-modeled mobile robots using a saturation
feedback controller”. IEEE Trans. Control Syst. Technol. 2001; 9(2): 305–318.
BIOGRAPHIES OF AUTHORS
Badre BOSSOUFI was born in Fez city, Morocco, on May 21, 1985. He received the Ph.D.
degree in Electrical Engineering from University Sidi Mohammed Ben Abdellah, Faculty of
Sciences, Morocco and PhD. degree from University of Pitesti, Faculty of Electronics and
Computer, Romanie and Montefiore Institute of electrical engineering, Luik, Belgium, in 2013.
He was an Assistant Professor of Electrical Engineering, at the fHigher School of technologie,
Oujda Morocco.His research interests include static converters, electrical motor drives, power
electronics, Smart Grid, Renewable Energy and Artificial Intelligence
Ahmed LAGRIOUI was born in taounate city –Morocco, on 1971. He received the aggregation
degree in electrical engineering from the ENSET School, in 2003. He received the DESA degree
in industrial electronics from the Mohammadia school’s of engineers. He received the Ph.D.
degree in Electrical Engineering from the Mohammadia school’s of engineers, Rabat, Morocco.
His research interests include static converters, electrical motor drives and power electronics.](https://image.slidesharecdn.com/0224nov13paperijpedsbadre-171212150859/85/FPGA-Based-Implementation-Nonlinear-Backstepping-Control-of-a-PMSM-Drive-12-320.jpg)
FPGA-Based Implementation Nonlinear Backstepping Control of a PMSM Drive
- 1. International Journal of Power Electronics and Drive System (IJPEDS) Vol.4, No.1, March 2014, pp. 12~23 ISSN: 2088-8694 12 Journal homepage: http://iaesjournal.com/online/index.php/IJPEDS/ FPGA-Based Implementation Nonlinear Backstepping Control of a PMSM Drive Badre Bossoufi*, Mohammed Karim**, Ahmed Lagrioui**, Mohammed Taoussi** * Laboratory of Electrical Engineering and Maintenance, Higher School of Technology, EST-Oujda, University of Mohammed I, Morocco ** STIC Team, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco Article Info ABSTRACT Article history: Received Oct 7, 2013 Revised Dec 18, 2013 Accepted Jan 9, 2014 In this paper, we present a new contribution of FPGAs (Field-Programmable Gate Array) for control of electrical machines. The adaptative Backstepping control approach for a permanent magnet synchronous motor drive is discussed and analyzed. We present a Matlab&Simulink simulation and experimental results from a benchmark based on FPGA. The Backstepping technique provides a systematic method to address this type of problem. It combines the notion of Lyapunov function and a controller procedure recursively. First, the adaptative and no adaptative Backstepping control approach is utilized to obtain the robustness for mismatched parameter uncertainties. The overall stability of the system is shown using Lyapunov technique. The simulation results clearly show that the proposed scheme can track the speed reference. Secondly, some experimental results are demonstrated to validate the proposed controllers. The experimental results carried from a prototyping platform are given to illustrate the efficiency and the benefits of the proposed approach and the various stages of implementation of this structure in FPGA. Keyword: Adaptive backstepping control Backstepping design technique FPGA Lyapunov stability Permanent magnet synchronous machine (PMSM) Copyright © 2014 Institute of Advanced Engineering and Science. All rights reserved. Corresponding Author: Badre BOSSOUFI, Departement of Electrical and Computer Engineering, Mohammed I University, 451, Adarissa, Fès, Morocco Email: badre_isai@hotmail.com 1. INTRODUCTION Three-phase Permanent Magnet Synchronous Motor (PMSMs) is strongly used in industry and consumes more than 70% of industrial electricity. This is why considerable efforts and different searches are being done to improve their performances and their efficiency. The efficiency of electrical machine drives is greatly reduced at light loads, where the flux magnitude reference is held on its initial value. The loss minimization is realized using high-quality materials and excellent design procedures in the manufacturing process. Moreover, expert control algorithms are employed in order to improve machine performance. In this paper we are interested in two mode controls for PMSM drive, the not adaptative and adaptative backstepping. The not adaptive backstepping approach offers a choice of design tools for accommodation of uncertainties nonlinearities. And can avoid wasteful cancellations. However, the not adaptive backstepping approach is capable of keeping almost all the robustness properties of the mismatched uncertainties. The not adaptive backstepping is a rigorous and procedure design methodology for nonlinear feedback control. The principal idea of this approach is to recursively design controllers for machine torque constant uncertainty subsystems in the structure and ‘‘step back’’ the feedback signals towards the control input. This approach is different from the approach of the conventional feedback linearization in that it can avoid cancellation of
- 2. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 13 useful nonlinearities in pursuing the objectives of stabilization and tracking. A nonlinear backstepping control design scheme is developed for the speed tracking control of PMSM that has exact model knowledge. The asymptotic stability of the resulting closed loop system is guaranteed according to Lyapunov stability theorem. The speed variation of the PMSM is widely used in high-performance applications. The PMSM has very large power density, high power factor and high efficiency. In a high-performance control of PMSM, the information of rotor position and speed is very important. In the speed control loop, for the field oriented control, the coordinate transformation has needs precise rotor position. Rotor position and speed can be measured by a shaft encoder or other type of sensors, in other case the speed is measured with an Encoder resolver connected to the PMSM machine drive. However, the presence of such sensors is not acceptable for cost, maintenance and reliability reasons. The concept of sensorless control was proposed in the 1970s and has been continually developed for PMSM rotor position and speed estimation. The basic principle of sensorless control is to deduce the rotor speed and position using various information and means, including direct calculation, parameter identification, condition estimation, indirect measuring and so on. The stator currents and voltages are generally used to calculate the information of speed and rotor position. The FPGA technology is now used by an increasing number of designers in various fields of application such as signal processing, telecommunication, video, embedded control systems, and electrical control systems. This last domain, i.e. the studies of control of electrical machines, will be presented in this paper [1]. Indeed, these components have already been used with success in many different applications such as Pulse Width Modulation (PWM), control of induction machine drives and multimachine system control. This is because the FPGA-based implementation of controllers can efficiently answer current and future challenges of this field. Considering the complexity of the diversity of the electric control devices of the machines, it is difficult to define with universal manner a general structure for such systems. However, by having a reflexion compared to the elements most commonly encountered in these systems, it is possible to define a general structure of an electric control device of machines which is show in Figure 1: Figure 1. Architecture of the Control This paper presents the realization of a platform for not adaptative and adaptative Backstepping control of PMSM using FPGA based controller. This realization is especially aimed for future high performance applications. In this approach, not only the architecture corresponding to the control algorithm is studied, but also architecture and the ADC interface, Encoder interface and RS232 UART architecture [2]. 2. PMSM MODEL SYSTEM In this paper, we apply the different algorithms control on a machine type PMSM (Permanent Magnet Synchronous Motor) [3], which consists of three stator windings and a rotor magnet. This motor is described by the following equation.
- 3. IJPEDS ISSN: 2088-8694 FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI) 14 . .. . . .. .. p Cf dt d JC iL iL dt d irV dt d irV re sqsqsq fsdsdsd sd sq sqssq sq sd sdssd (1) Where Ω is the rotation's speed, p the Number of pairs of poles, J the moment of inertia, f the Coefficient of viscous friction, Cr the resistive torque, Φf the flux produced by the permanent magnet, Lsd and Lsq the d-q axis stator inductance, Vsd and Vsq the d-q axis stator voltage, rs the stator winding resistance and Ce the electromagnetic torque. 3. NONLINEAR BACKSTEPPING APPROACH The Backstepping approach algorithm is control techniques that can linearize a nonlinear system such as the PMSM machine drive in the presence of uncertainties. Unlike other feedback linearization techniques, adaptive Backstepping has the flexibility of keeping useful non linearity’s intact during stabilization. The essence of Backstepping is the stabilization of a virtual control state. Hence, it generates a corresponding error variable which can be stabilized by carefully selecting proper control inputs. These inputs can be determined from Lyapunov stability analysis [4]. It is obvious that the dynamic model of PMSM is highly nonlinear because of the coupling between the speed and the stator currents (equation (1)). According to the vector control principle, the direct axis current id is always forced to be zero in order to orient all the linkage flux in the d axis and achieve maximum torque per ampere. J C J f iiLLi J p dt d L V p L ip L L i L r dt di L V ip L L i L r dt di r sqsdsqsdsqf sq sq sq f sd sq sd sq sq ssq sd sd sq sd sq sd sd ssd ))(( 2 3 . . (2) The vector T sqsd iix choice as state vector is justified by the fact that currents and speed are measurable and that the control of the instantaneous torque can be done comfortable via the currents isd and/or isq. And stator voltages as control variables T sqsd VVu . The principal objective of the backstepping controller is to regulate the speed of the PMSM drive to its reference value Ωref whatever external disturbances. We assume that the engine parameters are known and invariant. 3.1. Backstepping Speed Controller The first step is defined the tracking errors: refe (3) The derivative of (3) is: rsqsdsqsdsqfrefref CfiiLLi p Jdt de e ))(( 2 31 (4) We define the following quadratic function:
- 4. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 15 2 1 2 1 eV (5) Its derivative along the solution of (5), is given by: rsqsdsqsdsqfref CfiiLLi p J eeeV ))(( 2 31 1 (6) Using the Backstepping design method, we consider the d-q axes currents components isd and isq as our virtual control elements and specify its desired behavior, which are called stabilizing function in the backstepping design terminology as follows: )..( 3 2 0 ekJCf p i i r f sqref sdref (7) With kΩ is a positive constant Substituting (7) in (6) the derivative of V1: 02 1 ekV (8) 3.2. Backstepping Current Controller We have the asymptotic stability of the origin of the system (1). We defined current following errors: sqsqrefq sdrefsdsdrefd iie iwithiie 0 (9) Their dynamics can be written: sd sd sq f sd sq sd sq sq s r f sqsqrefq sd sd sq sd sq sd sd s sdsdrefd L V L p ip L L i L r ekJCf p iie L V ip L L i L r iie .)..( 3 2 . (10) To analyze the stability of this system we propose the following Lyapunov function: )( 2 1 222 2 qd eeeV (11) Its derivative along the trajectories (8), (9) and (10) is: ] 2 3 ))( 2 3 2 3 .( 3 )(2 [ ])( 2 3 [222 2 sq f sd sq sd sq sq s sq sqf sqdsqsdq f f qqq sqsqsdsq sd sq sd s sd sd dddqqddqqdd L i L L i L r L V e J p ekieLL J p e J p p fJk eke ieLL J p i L L L r L V ekeekekekeeeeeeV (12) The expression (12) found above requires the following control laws: qsqqfsdsdsqs sqf sqdsqsdq f f sq sq sqsqsd sd sqsqsdsdsddsd eLkiLire J Lp ekieLL J p e J p p fJkL V ieLL J pL iLireLkV 2 3 )( 2 3 2 3 3 )(2 )( 2 3 (13)
- 5. IJPEDS ISSN: 2088-8694 FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI) 16 With this choice the derivatives of (13) become: 02 qqdd ekekekV (14) 4. NONLINEAR ADAPTATIVE BACKSTEPPING APPROACH CONTROL 4.1. Principle In the previous section, the control laws are developed under the assumption that the machine parameters are known and invariants. This assumption is not always true. In fact, the flow created by the magnet varies with increasing temperature and with the fields created by the stator currents. Stator resistance also varies with temperature. Also, the change in operating conditions is implicitly load torque and hence the coefficient of friction and inertia. Adaptive Backstepping version takes into account the variations of these parameters. In equation (7), we do not know exactly the value of the load torque Cr, it will be replaced by its estimate rCˆ . )..ˆ.( 3 2ˆ ekJCf p i r f sqref (15) From (13) and (15), we deduce the dynamics of the speed error as follows: ekJieLL p e p C Jdt de sqdsqsd q f r ..)( 2 3 2 3~ 1 (16) With rrr CCC ˆ~ is the error of the estimated load torque. The Dynamic errors and direct currents quadratic write: sq sd sq sd sd s sd sdsd i L L i L R L V dt di dt de (17) r fsq f sd sq sd sd sq s sq sd sqdsqsdq f f sqsqrefq Ck J f pL i L L i L R L V ekieLL J p e J p p fJk dt di dt di dt de ~ )( 3 2 .)( 2 3 2 3 3 )(2 (18) To analyze the stability of this system we propose the following Lyapunov function: 3 2 2 2 1 2 222 2 ~~~ 2 1 fsr qd RC eeeV (19) Its derivative along the trajectories (16), (17) and (18) is: q sq q f qffsqq sq sdd sd ss f q f q rr sq f sd sq sd sq sq s sq sqf sqdsqsdq f f qqq sqsqsdsq sd sq sd s sd sd dddqqdd ffssrrqqdd e L e J fJk ee J p ie L ie L RR J e pJ fe p ek CC L i L L i L R L V e J p ekieLL J p e J p p fJk eke ieLL J p i L L L R L V ekeekekek RRCCeeeeeeV 1 ˆ2 3~1~11~1~ ˆ3 2 ˆ3 2~1~ ] ˆ 2 ˆ3 ))( 2 3 2 ˆ3 .( ˆ3 )(2 [ )( 2 3 ~~1~~1~~1 2 321 222 321 2 (20)
- 6. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 17 The expression (16) found above requires the following control laws: qsqqfsdsdsqs sqf sqdsqsdq f f sq sq sqsqsd sd sqsqsdsdsddsd eLkiLiRe J Lp ekieLL J p e J p p fJkL V ieLL J pL iLiReLkV ˆˆ 2 ˆ3 )( 2 3 2 ˆ3 3 )(2 )( 2 3ˆ (21) Therefore the dynamics of the Lyapunov function can be simplified as follows: q sq q f qffsqq sq sdd sd ss f q f q rrqqdd e L e J fJk ee J p ie L ie L RR J e pJ fe p ek CCekekekV 1 ˆ2 3~1~11~1~ ˆ3 2 ˆ3 2~1~ 2 32 1 222 2 (22) Hence the adaptation laws as follows: J e pJ fe p ek C f q f q r ˆ3 2 ˆ3 2~ 1 (23) sqq sq sdd sd s ie L ie L R 11~ 2 (24) q sq q f qf e L e J fJk ee J p 1 ˆ2 3~ 2 3 (25) With this choice, the expression (19) becomes: 0222 2 qqdd ekekekV (26) So the system is globally asymptotically stable in the presence of parametric uncertainties. 4.2. Simulation and Test Performance Figure 2. System configuration of adaptive Backstepping Control
- 7. IJPEDS ISSN: 2088-8694 FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI) 18 The following results are obtained by choosing the following values: Gains of the control law: 15.0k , 01.0dk , 01.0qk . Adaptation gains: 15.01 , 01.02 , 015.03 . Follow of the trajectory (a) (b) (c) (d) Figure 3. Test performance of the adaptive controller for trajectory tracking, (a) Speed response trajectory (b) Error Speed response (c) d-q axis current without uncertainties (d) abc axis current Disturbance rejection (a) (b) (c) Figure 4. Test performance of the adaptive controller for rejecting disturbance torque load applied at t = 0.3s. (a) Speed response trajectory (b) d-q axis current without uncertainties (c) Electromagnetic Torque
- 8. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 19 Parametric uncertainties (a) (b) (c) (d) Figure 5. Test performance of the adaptive controller following a change in Rs (a) Speed response trajectory (b) d-q axis current without uncertainties (c) Electromagnetic Torque (d) current isa (a) (b) Figure 6. Test performance of the adaptive controller following a change in Φf (a) Speed response trajectory (b) d-q axis current without uncertainties (a) (b) Figure 7. Test performance of the adaptive controller following a change in Lsd and Lsq, (a) Speed response trajectory (b) d-q axis current without uncertainties
- 9. IJPEDS ISSN: 2088-8694 FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI) 20 5. FPGA-BASED IMPLEMENTATION OF AN ROBUST BACKSTEPPING CONTROL SYSTEM 5.1. Development of the Implementation There are several manufacturers of FPGA components such: Actel, Xilinx and Altera…etc. These manufacturers use different technologies for the implementation of FPGAs. These technologies are attractive because they provide reconfigurable structure that is the most interesting because they allow great flexibility in design. Nowadays, FPGAs offer the possibility to use dedicated blocks such as RAMs, multipliers wired interfaces PCI and CPU cores. The architecture designing was done using with CAD tools. The description is made graphically or via a hardware description language high level, also called HDL (Hardware Description Language). Is commonly used language VHDL and Verilog. These two languages are standardized and provide the description with different levels, and especially the advantage of being portable and compatible with all FPGA technologies previously introduced [7]. The simulation procedure begins by verifying the functionality of the control algorithm by trailding a functional model using Simulink’s System Generator for Xilinx blocks. For this application, the functional model consists in a Simulink timeis discretired model of the No adaptative Backstepping algorithm associated with a voltage inverter and PMSM model. The Figure 8 summarizes the different steps of programming an FPGA. The synthesizer generated with CAD tools first one Netlist which describes the connectivity of the architecture. Then the placement- routing optimally place components and performs all the routing between different logic. These two steps are used to generate a configuration file to be downloaded into the memory of the FPGA. This file is called bitstream. It can be directly loaded into FPGA from a host computer. Figure 8. Programming FPGA devisees In this work an FPGA XC3S500E Spartan3E from Xilinx is used. This FPGA contains 400,000 logic gates and includes an internal oscillator which issuer a 50MHz frequency clock. The map is composed from a matrix of 5376 slices linked together by programmable connections. 5.2. Simulation Procedure Figure 9. Functional Model for No adaptative Backstepping Controller from SYSTEM GENERATOR
- 10. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 21 The simulation procedure begins by verifying the functionality of the control algorithm by trailding a functional model using Simulink’s System Generator for Xilinx blocks. For this application, the functional model consists in a Simulink time discretired model of the No adaptative Backstepping algorithm associated with a voltage inverter and PMSM model. The Figure 8 shows in detail the programming of the control shown in Figure 9 in the SYSTEM GENERATOR environment from Xilinx, we will implement it later in the memory of the FPGA for the simulation of PMSM. The second step of the simulation is the determination of the suitable sampling period and fixed point format. 5.3. Prototyping platform To test the FPGA based controller, a prototyping platform for the control of a Permanent magnet Synchronous Machine was assembled (Figure 10). Figure 10. Prototyping platform control 6. EXPERIMENTAL RESULTS The implementation of the indirect control by sliding mode on FPGA devices is characterized by a reduced operation time. The Figure 11 shown the experimental results of Indirect Sliding Mode PMSM with the FPGA platform are shown. Update frequency for this implementation is 20 kHz. All results were extracted from the FPGA by the ChipScope tool of Xilinx. (a) (b) (c) Figure 11. (a) Stator current locus for ISMC, (b) abc-axis current in the PMSM, (c) d-axis and q-axis current in the PMSM In Figure 11.a the experimental results No Adaptative Backstepping Control of PMSM with the FPGA platform are shows the evolution of the stator current isd which shows that the output follows the reference isdref and isq. The Figure 11.b shows the stator current isa and isb. Update frequency for this implementation is 20 kHz. LOAD PMSMFPGA Inverter (IGBT)
- 11. IJPEDS ISSN: 2088-8694 FPGA-Based Implementation Nonlinear Backstepping control of a PMSM Drive (Badre BOSSOUFI) 22 (a) (b) Figure 12. (a) Speed performance rotor of PMSM, (b) Error Speed performance The Experimental results show the performance of PMSM machine, using two approaches control nonlinear. This two control algorithms show the robustness and effecacité of the system. The work presented in this paper shows a good robustness of the backstepping control vis a vis the disturbances. It is noted that it is very dificult implementer of a non-linear control of a FPGA seen that there are two current loops and speed. With this new aproach was able to implement this order through the logiciele generator system which facilitates this task. The Results obtained show the validation of this work. Nonlinear backstepping control is very effective as orders that exists in literature (Sliding Mode, Direct Torque Control ...), it has improved the performance of the PMSM machine at the current and speed, response time, system speed (excuster to 40.5s for the program), and system stability regardless of the disturbance and the parametric variations of the machine. 7. CONCLUSION In this paper a robust continuous approachs Nonlinear not Adaptative Backstepping Control and Adaptative Backstepping Control strategy for permanent-magnet synchronous motor (PMSM) drive systems is presented. The FPGA based implementation is detailed, a bench test was realized by a prototyping platform, the experimental results obtained show the effectiveness and the benefit of our contribution and the different steps of implementation for the control FPGA. REFERENCES [1] Melicio R, Mendes VMF, Catalao JPS. “Modeling, Control and Simulation of Full- Power Converter Wind Turbines Equipped with Permanent Magnet SynchronousGenerator”. 2010; 5: 397-408. [2] JJ Chen and KP Chin. "Automatic flux-weakening control of permanent magnet synchronous motors using a reduced-order controller". IEEE Trans. Power Electron. 2000; 15: 881-890. [3] M Rodic, K Jezernik. “Speed Sensoless Sliding Mode Torque Control of Induction Motor “. IEEE Trans on. Industrial Electronics. February 2002. [4] Hisn-Jang Shieh and Kuo-Kai Shyu. “Non Linear Sliding Mode Torque Control With Adaptive Backstepping Approach For Induction Motor Drive”. IEEE Transactions on Industrial Electronics. 1999; 46(2): 380-388. [5] D Zhang and H Li. “A stochastic-based FPGA controller for an induction motor drive with integrated neural network algorithms”. IEEE Trans. Ind. Electron. 2008; 55(2): 551–561. [6] CF Jung and JS Chen. “Water bath temperature control by a recurrent fuzzy controller and its FPGA implementation”. IEEE Trans. Ind. Electron. 2006; 53(3): 941–949. [7] E Monmasson and M Cirstea. “FPGA Design Methodology for Industrial Control Systems – A Review”. IEEE Trans Ind. Electron. 2007; 54(4): 1824-1842. [8] B Bossoufi, M Karim, S Ionita, A Lagrioui. “Low-Speed Sensorless Control of PMSM Motor drive Using a NONLINEAR Approach BACKSTEPPING Control: FPGA-Based Implementation”. Journal of Theoretical and Applied Information Technology JATIT. 2012; 36(1): 154-166. [9] B BOSSOUFI, M KARIM, S IONITA, A LAGRIOUI. “Nonlinear Non Adaptive Backstepping with Sliding-Mode Torque Control Approach for PMSM Motor”. Journal of Journal of Electrical Systems JES. 2012; 8(2): 236-248. [10] A Lagrioui, H Mahmoudi, B Bossoufi. “DISCRETE LINEAR PREDICTIVE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR (PMSM)”. Journal of Theoretical and Applied Information Technology JATIT. 2011; 31(1): 21-28. [11] B Bossoufi, M Karim, S Ionita, A Lagrioui. “The Optimal Direct Torque Control of a PMSM drive: FPGA-Based Implementation with Matlab & Simulink Simulation”. Journal of Theoretical and Applied Information Technology JATIT. 2011; 28(2): 63-72.
- 12. ISSN: 2088-8708 IJPEDS Vol. 4, No. 1, March 2014 : 12 – 23 23 [12] B Bossoufi, M Karim, S Ionita, A Lagrioui, G Iana. “Matlab & Simulink Simulation with FPGA-Based Implementation Sliding Mode Control of a Permanent Magnet Synchronous Machine Drive”. WSEAS TRANSACTIONS on SYSTEMS and CONTROL. 2011; 3(6): 92-103. [13] TC Lee, KT Song, CH Lee, and CC Teng. “Tracking control of unicycle-modeled mobile robots using a saturation feedback controller”. IEEE Trans. Control Syst. Technol. 2001; 9(2): 305–318. BIOGRAPHIES OF AUTHORS Badre BOSSOUFI was born in Fez city, Morocco, on May 21, 1985. He received the Ph.D. degree in Electrical Engineering from University Sidi Mohammed Ben Abdellah, Faculty of Sciences, Morocco and PhD. degree from University of Pitesti, Faculty of Electronics and Computer, Romanie and Montefiore Institute of electrical engineering, Luik, Belgium, in 2013. He was an Assistant Professor of Electrical Engineering, at the fHigher School of technologie, Oujda Morocco.His research interests include static converters, electrical motor drives, power electronics, Smart Grid, Renewable Energy and Artificial Intelligence Ahmed LAGRIOUI was born in taounate city –Morocco, on 1971. He received the aggregation degree in electrical engineering from the ENSET School, in 2003. He received the DESA degree in industrial electronics from the Mohammadia school’s of engineers. He received the Ph.D. degree in Electrical Engineering from the Mohammadia school’s of engineers, Rabat, Morocco. His research interests include static converters, electrical motor drives and power electronics.