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  1. 1. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org E-ISSN: 1817-3195 NONLINEAR ADAPTIVE BACKSTEPPING CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR (PMSM) 1 A. LAGRIOUI, 2H. MAHMOUDI 1,2 Department of Electrical Engineering, Powers Electronics Laboratory Mohammadia School of Engineering- BP 767 Agdal- Rabat -Morocco E-mail: 1 lagrioui71@gmail.com , 2 mahmoudi@emi.ac.ma ABSTRACT In this paper, a nonlinear adaptive speed controller for a permanent magnet synchronous motor (PMSM) based on a newly developed adaptive backstepping approach is presented. The exact, input-output feedback linearization control law is first introduced without any uncertainties in the system. However, in real applications, the parameter uncertainties such as the stator resistance and the rotor flux linquage and load torque disturbance have to be considered. In this case, the exact input-output feedback linearization approach is not very effective, because it is based on the exact cancellation of the nonlinearity. To compensate the uncertainties and the load torque disturbance, the input-output feedback linearization approach is first used to compensate the nonlinearities in the nominal system. Then, nonlinear adaptive backstepping control law and parameter uncertainties, and load torque disturbance adaptation laws, are derived systematically by using adaptive backstepping technique. The simulation results clearly show that the proposed adaptive scheme can track the speed reference s the signal generated by a reference model, successfully, and that scheme is robust to the parameter uncertainties and load torque disturbance. Keywords: Feedback input-output linearization – Adaptive Backstepping control, Permanent Magnet Synchronous Motor (PMSM) cancellations. In addition, the adaptive backstepping approach [11][16] is capable of keeping almost all the robustness properties of the mismatched uncertainties. The adaptive backstepping is a systematic and recursive design methodology for nonlinear feedback control. The basic idea of backstepping design is to select recursively some appropriate functions of state variables as pseudo control inputs for lower dimension subsystems of overall system. Each backstepping stage results in a new pseudo control design, expressed in terms of the pseudo control designs from preceding design stages. When the procedure terminates, a feedback design for the true control input results in achieving of a final Lyapunov function by an efficient original design objective. The latter is formed by summing up the Lyapunov functions associates with each individual design stage [3][6][9]. In this paper, the backstepping approach is used to design state feedback non linear control, first under an assumption of known electrical parameters, and then with the possibility of 1. INTRODUCTION Permanent magnet synchronous motors (PMSM) are widely used in high performance servo applications due to their high efficiency, high power density, and large torque to inertia ratio [1], [2]. However, PMSMs are nonlinear multivariable dynamic systems and, without speed sensors and under load and parameter perturbations, it is difficult to control their speed with high precision, using conventional control strategies. Linearization and/or high-frequency switching based nonlinear speed control techniques, such as feedback linearization control and sliding mode control, have been implemented for the PMSM drives [13]. However, it is more efficient to use a nonlinear control method that is based on minimizing a cost function and allows one to tradeoff between the control accuracy and control effort. The adaptive backstepping design offers a choice of design tools for accommodation of uncertainties an nonlinearities. And can avoid wasteful 1
  2. 2. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org parametric uncertainly and bounded disturbances included. The steady state performances of the backstepping based controller and the problem of the disturbance rejection are enhanced via the introduction of an integral action in the controller. c3 = did = a1id + a2 iq Ω + a3 vd dt diq = b1iq + b2 id Ω + b3 Ω + b4 vq dt dΩ = c1Ω + c2 id iq + c3iq + c4TL dt The dynamic model of a typical surface-mounted PMSM can be described in the well known (d-q) frame through the park transformation as follows [5][7][11]: Lq did R v = − s id + pΩiq + d dt Ld Ld Ld dt =− REPRESENTATION [11],[12] Φf Rs L v id − d pΩid − pΩ+ d Lq Lq Lq Ld ⎡id ⎤ ⎢ ⎥ The vector [ X ] = ⎢iq ⎥ choice as state vector is ⎢Ω ⎥ ⎣ ⎦ (1) 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 i sd Where v d , v q Direct-and quadrature-axis stator voltages and/or i sq . Such representation of the nonlinear Direct-and quadrature-axis stator currents state corresponds to: Ld , Lq Direct -and quadrature-axis inductance P Rs Φf Ω ω f TL J d[X ] = F [ X ] + [ G ].[ U ] dt [ y] = H [X ] Number of poles Stator resistance (3) With Rrotor magnet flux linkage ⎡ x1 ⎤ ⎡id ⎤ ⎢ ⎥ [ X ] = ⎢ x 2 ⎥ = ⎢iq ⎥ Three order state vector ⎢ ⎥ ⎢ x 3 ⎥ ⎢Ω ⎥ ⎣ ⎦ ⎣ ⎦ Viscous friction coefficient Load torque Moment of Inertia a1 = − Lq Rs ; a2 = p ; Ld Ld b1 = − Φf L Rs ; b2 = − p d ; b3 = − p Lq Lq Lq a3 = 1 Ld Control vector (5) ⎡ y1 ⎤ ⎡i ⎤ [ y] = ⎢ ⎥ = ⎢ d ⎥ ⎣ y 2 ⎦ ⎣Ω ⎦ p.Ω ) (4) ⎡v d ⎤ [U ] = ⎢ ⎥ ⎣v q ⎦ Eelectrical rotor speed Mechanical rotor speed ( ω = We assume : b4 = (2) 3. PMSM NONLINEAR STATE T f dΩ 3p = (Φ f iq + (Ld −Lq )id iq ) − Ω+ L dt 2J J J id , iq 3p 1 .Φ f et c 4 = − 2J J The (1) equation become: 2. MODEL OF THE PMSM diq E-ISSN: 1817-3195 Output vector ⎤ ⎡ f1(x) ⎤ ⎡a1.x1 + a2.x2.x3 ⎥ ⎢ f (x)⎥ = ⎢b .x + b .x .x + b .x F[ X ] = ⎢ 2 ⎥ ⎢ 1 2 2 1 3 3 3 ⎥ ⎢ f3(3) ⎥ ⎢c1.x3 + c2.x1.x2+c3.x2 + c4.Cr ⎥ ⎣ ⎦ ⎣ ⎦ 1 f 3p ; c1 = − ; c2 = ( Ld − Lq ) Lq J 2J 2 (6)
  3. 3. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org & ⎡vd ⎤ ⎡v1 ⎤ ⎡ y1 ⎤ ⎢v ⎥ = ⎢ && ⎥ = [ξ ( x)] + [D( x)].⎢v ⎥ ⎣ 2 ⎦ ⎣ y2 ⎦ ⎣ q⎦ ⎤ ⎡ f1 ( x) ⎤ ⎡a1.id + a2 .iq .Ω ⎢ ⎥ ⎢ f ( x)⎥ = b .i + b .i .Ω + b .Ω ⎢1 q 2 d ⎥ (7) 2 3 ⎢ ⎥ ⎢ f 3 ( x)⎥ ⎢c1.Ω + c2 .id .iq + c3 .iq + c4 .Cr ⎥ ⎣ ⎦ ⎣ ⎦ (12) Where ⎡ f1 ( x) ⎤ [ξ ( x)] = .⎢ ⎥ ⎣c 2 x 2 f1 ( x) + (c 2 x1 + c3 ) f 2 ( x) + c1 f 3 ( x)⎦ 0 ⎡ a ⎤ [ D( x)] = .⎢ 3 a 3 c 2 x 2 b4 (c 2 x1 + c 3 )⎥ ⎣ ⎦ Now, the standard pole-assignment technique can be adopted to design the state feedback control output as follows: Nonlinear functions of our model: ⎡ g11 g12 ⎤ ⎡a3 0⎤ [G] = ⎢ g 21 g 22 ⎥ = ⎢0 b4 ⎥ Control matrix (8) ⎥ ⎥ ⎢ ⎢ ⎢ g31 g 32 ⎥ ⎢0 0 ⎥ ⎦ ⎣ ⎣ ⎦ ⎡h (x) ⎤ ⎡x1 ⎤ ⎡i ⎤ H[X] = ⎢ 1 ⎥ = ⎢ ⎥ = ⎢ d ⎥ Output vector ⎣h2(x)⎦ ⎣x3⎦ ⎣Ω⎦ E-ISSN: 1817-3195 d v1 = ki (idref −id ) + idref dt (13) 2 d d v2 = 2 Ωref + kΩ1. (Ωref −Ω) + kΩ2.(Ωref −Ω) dt dt Where the idref and Ωref are the commands for the (9) The model set up in (2) can be represented by the following functional diagram (Figure1) speed and the d-axis current, respectively. Figure 1: Nonlinear diagram block of PMSM in d-q frame. 4. INPUT-OUTPUT FEEDBACK LINEARIZATION Figure2 : simulation scheme of input-output feedback linearization control The input-output feedback linearization control for a PMSM is introduced first. The control objective is to make the mechanical speed follow the desired speed asymptotically. 5. NONLINEAR BACKSTEPPING CONTROLLER In order to make the actual speed follow The desired one and avoid the zero dynamic, w and isd are chosen as the control outputs. Then the new variables are defined as follows: 5.1 Non adaptive case It is assumed that the engine parameters are known and invariant. y1 = id equation (1) the mathematical model of the machine. The objective is to regulate the speed to its reference value . We begin by defining the tracking errors: By choosing (10) y2 = Ω Then the new state equations in the new state coordinates can be written as follows: . [i d ] T iq Ω as variable states and ew = Ω ref − Ω ed = idref − id with idref = 0 . y1 = id = f1 ( x) + a3 .vd & & (11) y 2 = Ω = f 3 ( x) && &&2 = Ω = c1 f 3 ( x) + c2 iq ( f1 ( x) + a3 vd ) + y (14) eq = eqref − iq And its dynamics derived from (1): (c 2 id + c3 )( f 2 ( x) + a 4 .v q ) dew dΩ ref dΩ dΩ = − =− dt dt dt dt To linearize the equations, define the new control variable as follows: We replace (1) in (15), we obtain: 3 (15)
  4. 4. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org dew f T 3p = − [(Ld − Lq )idiq + Φf iq ] + Ω+ L dt J J 2J Its derivative along the trajectories (21) (22) and (23) is: (16) We define the following quadratic function: 1 2 V1 = ew 2 & & & & V2 = e w e w + e d e d + e q e q (17) Lq ⎤ ⎡ v R 3p ed ⎢kd ed − d + s − Ω iq + (Ld − Lq )ewiq ⎥ Ld Ld Ld 2J ⎦ ⎣ ⎡ 2(k J − f ) ⎛ 3 pΦ f ⎜ eq + + eq ⎢kq eq + w 3 pΦ f ⎜ 2J ⎢ ⎝ ⎣ & & V1 = ewew T ⎤ (18) f ⎡ 3p = ew.⎢− [(Ld − Lq )id iq + Φ f iq ] + Ω+ L ⎥ J J⎦ ⎣ 2J Utilizing the backstepping design method, we consider the d-q axes currents components id and iq as our virtual control elements and specify its desired behavior, which are called stabilising function in the backstepping design terminology as follows: vq 3p ⎞ 3 pΦ f (Ld − Lq )ed iq − kwew ⎟ + ew − + 2J 2J Lq ⎠ Φf ⎤ Rs L iq + Ω d id + Ω ⎥ Lq Lq Lq ⎥ ⎦ idref = 0 iqref Where The expression (25) found above following control laws: (19) vd = kd Ld ed + Rs i d −ΩLq iq + k w is a positive constant (20) dew 3 pΦ f 3p = eq + (Ld − Lq )ed iq − kwew (21) dt 2J 2J 1 Furthermore the dynamics of the stabilizing errors Park- SVMPWM (ed , eq ) can be computed as: vd vq L de di di di v R d = dref − d = − d = − d + s id −Ω q iq (22) Ld dt dt dt dt Ld Ld = diqref − diq wref id iq ia i b PMSM ic Park w In the controller development in the previous section, it was assumed that all the system parameters are known. However, this assumption is not always true. The flux linkage varies nonlinearly with the temperature rise and, also, with the external fields produced by the stator current due to the nonlinear demagnetization characteristics of the magnets. The winding resistance may vary due to heating. In addition, working condition changes Φf Rs L (23) iq + Ω d i d + Ω Lq Lq Lq Lq To analyze the stability of this system we propose the following Lyapunov function: 1 2 2 2 (e w + e d + e q ) 2 Inverter 5.2 Adaptive case + V2 = Equation (26) sa sb sc Figure3: Simulation scheme of non-adaptive case = dt dt dt 3 pΦ f ⎤ 2(k w J − f ) ⎡ 3p eq + ( Ld − Lq )ed iq − k w ew ⎥ − ⎢ 3 pΦ f ⎣ 2 J 2J ⎦ vq 3 pLd (Ld − Lq )ewi q 2J vq = Substituting (14) and (19) into (16) the dynamics of the tracking error of the velocity becomes: deq requires the 2Lq (kwJ − f ) ⎛ 3 pΦ f ⎞ 3p ⎜ ⎜ 2J eq + 2J (Ld − Lq )ed iq − kwew ⎟ + ⎟ 3 pΦ f ⎝ ⎠ (26) 3 pΦ f Lq ew + Rsiq + ΩLd id + ΩΦf + kq Lqeq 2J With this choice the derivatives of (24) become: & V2 = − k w e w − k d e d − k q e q ≤ 0 (27) Substituting (19) in (18) the derivative of V1 becomes: 2 & V1 = − k w ew (25) 2 2 2 = −k wew − kd ed − kq eq + Its derivative along the solution of (16), is given by: 2 ( fΩ + TL + k w Jew ) = 3 pΦ f E-ISSN: 1817-3195 (24) 4
  5. 5. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org such as load torque and inertia mismatch inevitably impose parametric uncertainties in control system design. Hence, it becomes necessary to account for all these uncertainties in the design of high performance controller. We will see how we efficiently handle these uncertainties through stepby-step adaptive backstepping design and parameter adaptations. In ( ) , we do not know exact value of the load torque TL ; then , it is necessary to estimate them adaptively, and replace them with their estimates ) TL . ( ) ) 2 f ω + T L + k ω Je ω 3 pΦ f So from ( ) and ( ) the following speed error dynamics can be deduced : ) Or i sqref = deω 1 = dt J & & & & V2 = e w e w + ed ed + eq eq + 1 ~~ 1 ~ ~ 1 ~ ~ & & & TL TL + Rs Rs + Φ f Φ f γ1 (28) ⎤ ~ ⎡1 ~ 3p 1 ~⎡1 ~ 1 & & + Rs ⎢ Rs + id ed − iqeq ⎥ + Φ f ⎢ Φ f − eωeq − 2J Ld Lq ⎣γ 3 ⎢γ 2 ⎥ ⎣ ⎦ ⎤ (kω J − f ) 2 1 eq − Ωeq ⎥ ˆ Lq JΦ f ⎥ ⎦ According to the above equation (33) , the control laws are now designed as: 3 pLd ( Ld − Lq )ew i q 2J (34) ˆ ⎞ 2Lq (kwJ − f ) ⎛ 3pΦf 3p ⎜ vq = eq + (Ld − Lq )ediq − kwew ⎟ + ⎟ ˆ 2J 3pΦf ⎜ 2J ⎝ ⎠ ˆ 3pΦf Lq ˆ ˆ ew + Rsiq + ΩLdid + ΩΦf + kqLqeq 2J ˆ v d = k d Ld ed + Rs i d −ΩLq iq + Φf Rs L 2 ⎛f ⎞~ iq + Ω d id + Ω + ⎜ − kω ⎟TL (31) Lq Lq Lq Lq 3 pΦ f ⎝ J ⎠ Let us define a new Lyapunov function the closed loop system as follows: ~ ~2 Φ2 ⎞ ~2 1⎛ 2 2 2 (32) ⎜ e w + e d + e q + TL + R s + f ⎟ V2 = 2⎜ γ1 γ2 γ3 ⎟ ⎠ ⎝ + Consequently (33) is simplified to : 2 2 2 & V 2 = −k w e w − k d e d − k q e q + 2 feq e ⎤ ~ ⎡ 1 ~ 2k ω eq & + TL ⎢ TL − + − ω ⎥+ ˆ ˆ J ⎥ 3 pΦ f 3 pJΦ f ⎢γ 1 ⎦ ⎣ eω , ed , eq , the ~ error TL , the resistance Where the stabilizing errors ~ ~ ~ rotor magnetic flux linkage estimation error Φ f ˆ = Φ f − Φ γ 2 and γ 3 are f (35) ⎤ 1 1 ~ ⎡1 ~ & R s ⎢ Rs + i d ed − i q eq ⎥ + Ld Lq ⎢γ 2 ⎥ ⎣ ⎦ ˆ estimation error R s ( R s = R s − R s and the ~ (Φ γ3 2 q q ˆ Φ f ⎤ ~ ⎡ 1 ~ 2kωeq 2 feq e ⎤ Rs L & + − ω⎥ iq + Ω d id + Ω ⎥ + TL ⎢ TL − ˆ ˆ Lq Lq Lq ⎥ 3 pΦ f 3 pJΦ f J ⎥ ⎢γ 1 ⎣ ⎦ ⎦ diq load torque estimation 2 d d Lq ⎤ ⎡ v R 3p ed ⎢kd ed − d + s − Ω iq + (Ld − Lq )ewiq ⎥ 2J Ld Ld Ld ⎣ ⎦ ˆf ⎡ 2(k J − f ) ⎛ 3 pΦ ⎜ + eq ⎢kqeq + w eq + ˆ 3 pΦ f ⎜ 2J ⎢ ⎝ ⎣ ˆ vq 3p ⎞ 3 pΦ f (Ld − Lq )ed iq − kwew ⎟ + ew − + 2J 2J Lq ⎠ = − = dt dt dt ⎤ 2(kwJ − f ) ⎡3 pΦ f 3p eq + (Ld − Lq )ediq − kwew ⎥ − ⎢ 3 pΦ f ⎣ 2J 2J ⎦ vq (33) = −k e − k e − k e + ⎛ ~ 3 pΦ f ⎞ 3p ⎜ − TL + eq + ( Ld − Lq )ed iq − kω Jeω ⎟ ⎜ ⎟ 2 2 ⎝ ⎠ diqref γ2 2 w w ) ~ (29) where TL = TL − TL As a result, the d-q currents errors dynamics can be rewritten as : Lq de d di v R (30) = − d = − d + s id − Ω iq dt dt Ld Ld Ld deq E-ISSN: 1817-3195 f ⎤ (k J − f ) 2 1 3p ~ ⎡1 ~ & Φf ⎢ Φf − eω eq − ω e q − Ωe q ⎥ ˆ 2J Lq JΦ f ⎢γ 3 ⎥ ⎣ ⎦ ) are all included. And γ 1 , an adaptation gain. Tracking the The parameter adaptation laws are then chosen as : derivative of (32) and substing (29) , (30) and (31) into this derivative we can obtain: 5
  6. 6. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org ⎡ 2k ω e q 2 feq e ⎤ ~ & TL = γ 1 ⎢ − + ω⎥ ˆ ˆ J ⎥ ⎢ 3 pΦ f 3 pJΦ f ⎣ ⎦ ⎡ 1 ⎤ 1 ~ & Rs = γ 2 ⎢− id ed + iq eq ⎥ Lq ⎢ Ld ⎥ ⎣ ⎦ E-ISSN: 1817-3195 6 (36) 4 Iabc (A ) 2 (37) 0 -2 -4 -6 ⎡3p ⎤ (k J − f ) 2 1 ~ & eq + Ωeq ⎥ (38) Φ f = γ 3 ⎢ eω eq + ω ˆ Lq JΦ f ⎢ 2J ⎥ ⎣ ⎦ Figure3: iabc currents without uncertainties Then (35) can be rewritten as follows 200 & V2 = −k e − k e − k e ≤ 0 2 d d 2 q q (39) Define the following equation 2 2 2 Q (t ) = k w e w + k d e d + k q e q ≥ 0 (40) vq v d SVMPWM Equation (34) Equats (36), (37) and (38) id iq 0.3 0.35 0.4 0 0 0.05 0.1 0.15 time(s) 0.2 0.25 0.3 0.35 0.4 Figure4: speed response without uncertainties 14 ia Inverter 0.25 50 Tload T 12 i b PMSM 10 ic T, TL(N.m ) Park -1 0.15 time(s) 0.2 0.1 100 Furthermore, by using LaSalle Yoshizawa’s theorem [11], its can be shown that Q(t) tends to zero as t →∞, ew, ed and eq will converge to zero as. Consequently, the proposed controller is stable and robust, despite the parameter uncertainties. sa sb sc 0.05 150 w re f , w (ra d / s ) 2 w w 0 Park w 8 6 4 Figure4: simulation scheme of an adaptive case 2 6. SIMULATION RESULTS 0 0 6.1 PMSM parameter’s 0.15 time(s)0.2 0.25 0.3 0.35 0.4 0.18 phid phiq 0.16 value 0.14 300 v 3000 tr/s to 150 Hz 14.2 N.ms 0.4578 Ω 4 3.34 mH 3.58 mH 0.001469 kg.m2s 0.0003035 Nm/Rad/s 0.171 0.12 phidq(w b) Maximal voltage of food Maximal speed Nominal Torque ;Tn Rs Number of pair poles :p Ld Lq The moment of inertia J Coefficient of friction viscous f Flux of linquage Фf 0.1 Figure5: Torque response without uncertainties TABLE I : PARAMETERS OF PMSM parameter 0.05 0.1 0.08 0.06 0.04 0.02 0 time(s) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 figure6: d-q axis flux without uncertainties 6.2 Results For a trajectory wref=200 rad/s at 0s, =100rad/s at t=0.15s, =200rad/s at t=0.25s, the following figures ( 1 to 7) show the performance of the input output linearization control. 6 0.4
  7. 7. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org E-ISSN: 1817-3195 14 12 iq Id Iq 12 id iq TL 10 10 8 8 Idq(A ) 6 6 4 4 2 2 id 0 0 -2 0 0.05 Time(s) 0.15 0.2 0.1 0.25 0.3 0.35 -2 0 0.4 Figure7: d-q axis currents response without uncertainties 0.1 0.2 0.3 0.4 0.5 0.6 figure10: d-q axis currents with step load and without uncertainties. The figures below (figures 8, 9 and 10) shows the evolution of speed, electromagnetic torque and d-q axis currents in the presence of a disturbance load torque. The figures 10 to 13 show the robustness of adaptive backstepping controller compared with that of non-adaptive case and feedback input-output linearization. 200 200 w 150 100 w (ra d / s ) w (ra d /s ) a n d T L (N . m ) 150 50 TL 50 0 -50 time(s) 0 0.05 0.1 0.15 0.2 100 0.25 0.3 0.35 0 0.4 time(s) 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure11: speed response of the proposed adaptive controller with step load (+12Nm at 0.2s , +8Nm at 0.4s) Figure8: response speed with step load at t= 0.2s 12 iq 10 8 200 6 adaptive control case non adaptive control case 150 2 S peed 4 id 100 0 50 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 Figure9: electromagnetic torque with step load time(s) 0.1 0.2 0.3 0.4 0.5 0.6 Figure12: speed response of the non-adaptive and adaptive controller with step load 7
  8. 8. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org REFRENCES: 250 [1] Guy GRELLET & Guy CLERC « Actionneurs électriques » Eyrolles–November 1996 [2] Melicio, R; Mendes, VMF; Catalao, JPS “Modeling, Control and Simulation of FullPower Converter Wind Turbines Equipped with Permanent Magnet Synchronous Generator” vol 5 (issue 2) pg 397-408 MarApr 2010 [3] Gholamian, S. Asghar; Ardebili, M.; Abbaszadeh, K “Selecting and Construction of High Power Density Double-Sided Axial Flux Slotted Permanent Magnet Motors for Electric Vehicles” vol 4 (issue 3) pg 470-476 Mai-Jun 2009 [4] Muyeen, S. M.; Takahashi, R.; Murata, T.; Tamura, J. “Miscellaneous Operations of Variable Speed Wind Turbine Driven Permanent Magnet Synchronous Generator” vol 3 (issue 5) pg 912-921 Sep-Oct 2008 [5] K.Paponpen and M.Konghirum “ Speed Sensorless Control Of PMSM using An Improved Sliding Mode Observer With Sigmoid function” ECTI- Vol5.NO1 February 2007. [6] S.Hunemorder, M.Bierhoff, F.W.Fuchs ‘’ Drive with PMSM and voltage source inverter for wind power application’’ - IEEE. [7] J. Thomas K. René A.A. Melkebeek “Direct Torque Of Permanent Magnet Synchronous Motors- An Overview” 3RD IEEE – April 2006. [8] Shahnazari, M.; Vahedi, A. “Improved Dynamic Average Modeling of Synchronous Machine with Diode-Rectified Output” Vol 4 ‘issue6) pg 1248-1258 Nov-Dec 2009 [9] Youngju Lee, Y.B. Shtessel, “Comparison of a feedback linearization controller and sliding mode controllers for a permanent magnet stepper motor," ssst, pp.258, 28th Southeastern Symposium on System Theory (SSST '96), 19960 [10] A.Lagrioui, H.Mahmoudi « Contrôle Non linéaire en vitesse et en courant de la machine synchrone à aimant permanent » ICEED’07Tunisia [11] A. Lagrioui, H.Mahmoudi « Nonlinear Tracking Speed Control for the PMSM using an Adaptive Backstepping Method » ICEED’08-Tunisia 200 S peed 150 Feedback linearization Adaptive Control case non-adaptive control case 100 50 0 -50 E-ISSN: 1817-3195 time(s) 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure13: speed response of the input-output linearization- non-adaptive and adaptive controller with variation of parameters ( Rs = RS 0 ± ∆RS ) 250 Non adaptive case Adaptive case 200 input output linearization Speed 150 100 50 0 Time(s) 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure14: speed response of the input-output linearization- non-adaptive and adaptive controller with variation of parameters ( Φ f = Φ f ± ∆Φ f ) at t=0.3 s 0 6. CONCLUSION: A method of nonlinear backstepping control has been proposed and used for the control of a PMSM. The simulation results show, with a good choice of control parameters, good performance obtained with the proposed control as compared with the non-linear control by state feedback. From the speed tracking simulations results, we can find that the nonlinear adaptive backstepping controllers have excellent performance. The direct axis current id is always forced to zero in order to orient all the linkage flux in the d-axis and to achieve maximum torque per ampere. The q-axis current will approach a constant when the actual motor speed achieves reference speed. The show simulation results, fast response without overshoot and robust performance to parameter variations and disturbances throughout the system 8
  9. 9. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN Journal of Theoretical and Applied Information Technology 15th July 2011. Vol. 29 No.1 © 2005 - 2011 JATIT & LLS. All rights reserved. ISSN: 1992-8645 www.jatit.org [12] A. Lagrioui, Hassan Mahmoudi “Modeling and simulation of Direct Torque Control applied to a Permanent Magnet Synchronous Motor” IREMOS- Journal – August 2010. [13] A. Lagrioui, Hassan Mahmoudi “Current and Speed Control for the PMSM Using a Sliding Mode Control” 2010 IEEE 16th International Symposium for Design and Technology in Electronic Packaging (SIITME). [14] A. Lagrioui, Hassan Mahmoudi “Speed and Current Control for the PMSM Using a Luenberger” The 2nd International Conference on Multimedia Computing and Systems (ICMCS'11)- Ouarzazate, Morocco [15] D M. Rajashekar, I.V.Venu Gopala swamy, T.Anil kumar “Modeling and Simulation of Discontinuous Current Mode Inverter Fed Permanent Magnet Synchronous Motor Drive” JATIT-Vol 24. No. 2 – 2011 [16] B.Belabbes, M.K.Fellah, A.Lousdad, A.Maroufel, A.Massoum ‘’ Speed Control by backstepping with non linear observer of a PMSM’’ Acta Electrotechnica et Informatica No:4 Vol:6 -2006 9 E-ISSN: 1817-3195

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