Real-time nonlinear adaptive backstepping speed control for a PM synchronous motor

ARTICLE IN PRESS

Control Engineering Practice 13 (2005) 1259–1269 www.elsevier.com/locate/conengprac

Real-time nonlinear adaptive backstepping speed control for a PM synchronous motor Jianguo Zhoua, Youyi Wangb, a

Seagate Technology International, Science Park Drive, 118249, Singapore School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

b

Received 26 July 2001; accepted 14 November 2004 Available online 7 January 2005

Abstract A real-time nonlinear adaptive speed control scheme based on backstepping control technique is proposed for a permanent magnet synchronous motor. In the controller design, the input–output feedback linearization is firstly used to compensate the nonlinearities in the nominal system. Then, adaptive backstepping control approach is adopted to derive the control scheme, which is robust to the parameter uncertainties and load torque disturbance. Simulation and real-time experimental results clearly show that the proposed control scheme can track the speed reference signal generated by a reference model successfully under parameter uncertainties and load torque disturbance without singularity and overparameterization. r 2004 Published by Elsevier Ltd. Keywords: Nonlinear control; Adaptive control; Permanent magnet synchronous motor; Robustness; Uncertainty; Backstepping

1. Introduction Permanent magnet synchronous motors (PMSM) are receiving increased attention for electric drive applications due to their high power density, large torque to inertia ratio and high efficiency over other kinds of motors such as DC motors or induction motors (Leonhard, 1995). But the dynamic model of a PMSM is highly nonlinear because of the coupling between the motor speed and the electrical quantities, such as the d–q axis currents. The model parameters such as the stator resistance and the friction coefficient may also not be exactly known. Even worse, the load torque is always unknown. All these factors make controller design for a PMSM difficult when high speed and high precision are required in the real application. In order to deal with this problem, sliding-mode variable structure control approach has been used due to its favorable advantages such as insensitive to Corresponding author. Tel.: +65 67904537; fax: +65 67920415.

E-mail address: [email protected] (Y. Wang). 0967-0661/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.conengprac.2004.11.007

parameter uncertainties and external disturbances and only the bounds of the uncertainties are needed in design procedure (Slotine & Li, 1991; Hung, Gao, & Hung, 1993; Zhang & Panda, 1999). The robustness of this control is guaranteed but the worst drawback is the chattering, which limits its application. Furthermore, the bounds of the uncertainties are often chosen much larger than the real bounds in order to ensure the robustness. This may cause much higher control effort. A robust speed control of PMSM using boundary layer integral sliding mode control technique (Baik, Kim, & Youn, 2000) was presented to reduce the chattering phenomenon but this would cause the steady-state error (Zhang & Panda, 1999). Feedback linearization control (Isidori, 1995) has been thoroughly studied in the last 20 years, by which the original nonlinear model can be transformed into a linear model through proper coordinate transformation. Thus, almost all the well developed linear control techniques might be applied. But when parameter uncertainties and unknown disturbance are taken into account, this approach may not be applicable because it

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J. Zhou, Y. Wang / Control Engineering Practice 13 (2005) 1259–1269

is based on the exact cancellation of the nonlinearity. In (Kim, Baik, Chung, & Youn, 1997), a robust speed control of brushless DC using adaptive input–output linearization technique was developed to deal with the uncertainties and load disturbance, in which the Popov’s hyperstability theory and positivity concept were used to derive the adaptation law. Backstepping control is a newly developed technique to the control of uncertain nonlinear systems, particularly those systems that do not satisfy matching conditions (Krstic, Kanellakopoulos, & Kokotovic, 1995). The most appealing point of it is to use the virtual control variable to make the original high order system to be simple enough thus the final control outputs can be derived step by step through suitable Lyapunov functions. A nonlinear torque controller for an induction motor was designed based on adaptive backstepping approach, in which overparameterization may occur (Shieh & Shyu, 1999). In that paper, there are only two uncertainties DRr and DRs ; however in the adaptation laws, there are totally 5 uncertainty parameters. Thus the problem of overparameterization happened. A speed controller based on backstepping was developed for induction motors, but there was a possible singularity problem (Tan & Chang, 1999). The singularity problem is from the estimation of uncertainty R2 ; whose estimated values R^ 2 appears as the denominator of the control output ðu1d Þ: It is obvious that when the error signals ei ; i ¼ 1; 2; 3 tend to zero, R^ 2 may tend to zero. When R^ 2 tends to zero, the control output u1d becomes quite large. This is not acceptable in the practical application because the control efforts have a limit. In this paper, input–output feedback linearization method is firstly used to simplify the uncertain nonlinear model of a PMSM under the parameter uncertainties and load torque disturbance. Then, a reference model is used to give the desired transient performance in order to get the error model and thus make the derivation of the control scheme easily. Finally nonlinear adaptive speed controller based on nonlinear adaptive backstepping control technique is derived step by step. It has no singularity and overparameterization. The resulted control scheme can track the reference signal quite well under parameter uncertainties and load torque disturbance. The paper is organized as follows. Section 2 introduces the nonlinear model of the system. Feedback linearization control for a PMSM is given in Section 3. The main idea of the proposed nonlinear adaptive speed control scheme without singularity and overparameterization is presented in Section 4. The stability analysis is given through Lyapunov function and Barbalat’s lemma in Section 5. Simulation and experimental results are given in Section 6. Section 7 shows some conclusions.

2. Mathematical model of surface-mounted PMSM motor The following assumptions are made in the derivation of the mathematical model of a PMSM: (Pillay & Krishnan, 1988)

 

Saturation and iron losses are neglected although it can be taken into account by parameter changes; the back emf is sinusoidal.

The model of a typical surface-mounted PMSM can be described in the well known (d–q) frame through the Park transformation as follows: the stator d, q equations in the rotor frame are expressed as follows: (Pillay & Krishnan, 1988; Krause, 1995) ud ¼ Rid þ pld  olq ; uq ¼ Riq þ plq þ old ;

ð1Þ

where lq ¼ Lq iq ;

(2)

ld ¼ Ld id þ f;

(3)

f is the magnet flux linkage. For a uniform airgap surfaced-mounted PMSM motor, Ld ¼ Lq : Thus the dynamic model of a surface-mounted PMSM can be described as follows: did R 1 ¼  id þ Poiq þ ud ; L L dt diq R Pf 1 o þ uq ; ¼  iq  Poid  L L L dt do 3Pf B TL ¼ iq  o  ; dt 2J J J

ð4Þ

where id and iq are the d–q axis currents, ud and uq are the d–q axis voltages, R is the stator resistance, L is the stator inductor, P is the pole pairs, J is the rotor moment of inertia, B is the viscous friction coefficient, T L is the load torque. Remark 1. From (4), it is obvious that the dynamic model of a PMSM is highly nonlinear due to the coupling between the speed and the electrical currents of d–q axis. Meanwhile, the parameters such as resistance may vary due to heating during operation and the load torque is also possible to be changed. Thus, both the nonlinearity and uncertainties as well as the load torque disturbance have to be taken into account in the design procedures of high performance speed controller.

ARTICLE IN PRESS J. Zhou, Y. Wang / Control Engineering Practice 13 (2005) 1259–1269

3. Input–output feedback linearization control for PMSM The compact form of (4) can be written as follows: x_ ¼ fðxÞ þ g1 ðxÞud þ g2 ðxÞuq ; T

(5) T

where x ¼ ½id iq o ; g1 ðxÞ ¼ ½1=L 0 0 ; ½0 1=L 0T ; 2 3  RL id þ Poiq 6 R 7 7  i  Poid  Pf fðxÞ ¼ 6 L o 5: 4 L q 3Pf TL B 2J iq  J o  J

g2 ðxÞ ¼

The notation Lf hðxÞ; Rn ! R is adopted for Lie’s derivative of a function hðxÞ along the direction of vector field fðxÞ and iteratively L2f hðxÞ ¼ Lf ðLf hðxÞÞ in this paper. The details of Lie’s derivative and feedback linearization control can be found in (Isidori, 1995). In order to make the actual speed follow the desired one and avoid the zero dynamic, o and id are chosen as the control outputs. Then the new variable are defined as follows: y1 ¼ h1 ðxÞ ¼ o; _ ¼ Lf h1 ðxÞ; y2 ¼ o y3 ¼ h2 ðxÞ ¼ id ;

ð6Þ

where y1 ; y2 and y3 are the speed, the acceleration and the direct axis current respectively. Then the new state equation in the new state coordinates can be written as

Remark 2. Because this method is based on the exact cancellation of the nonlinear items in the system, the drawbacks of this control technique are very obvious. If the parameters or the load torque are not exactly known, the nonlinearities in the system cannot be thoroughly cancelled. This will consequently cause errors in the cancellation, and thus cause the speed error. Also, this method requires the direct measurement of acceleration signal or computation from the position or speed signal which often suffers from noise. These problems drive us to search a more effective way to design the speed controller for the PMSM motor. In the next section, feedback linearization technique is used to partially simplify the system and obtain the suitable form for backstepping control design.

4. Nonlinear adaptive backstepping controller From the previous section, it is found that exact input–output feedback control techniques might not be able to work well when parameter uncertainties and load disturbances exist. Adaptive backstepping is a recently developed control method for nonlinear system with unknown but constant uncertainties. The most appealing feature of adaptive backstepping technique is that the final controller as well as the adaptation laws can be derived systematically step by step. This can be clearly seen from the following procedures. 4.1. Control objective

y_ 1 ¼ y2 ; y_ 2 ¼ L2f h1 ðxÞ þ Lg1 Lf h1 ud þ Lg2 Lf h1 uq ; y_ 3 ¼ Lf h2 ðxÞ þ Lg1 h2 ud þ Lg2 h2 uq :

1261

ð7Þ

The definitions of Lie’s derivatives used in (7) can be found in appendix. Using the following nonlinear state feedback for (7): " 0 # " 2 # ud Lf h1 ðxÞ þ Lg1 Lf h1 ud þ Lg2 Lf h1 uq ¼ (8) u0q Lf h2 ðxÞ þ Lg1 h2 ud þ Lg2 h2 uq

The control objective is to design a asymptotically stable speed controller for the PMSM (4) to make the mechanical speed and d-axis follow the reference signals satisfactorily. It is assumed that the stator resistance R, the friction coefficient B, and the load torque T L are the unknown but constant parameters in the control system as follows: R ¼ RN þ DR;

gives the closed-loop system as y_ 1 ¼ y2 ; y_ 2 ¼ u0d ; y_ 3 ¼ u0q :

B ¼ BN þ DB; T L ¼ T LN þ DT L ; ð9Þ

Now, the standard pole-placement technique can be adopted to design the state feedback control output as follows: _ ref Þ þ o € ref ; u0d ¼ k1 ðy1  oref Þ  k2 ðy2  o 0 u ¼ k3 ðy3  idref Þ þ i_dref ; q

ð10Þ

where oref and idref are the commands for the speed and the d-axis current, respectively.

ð11Þ

where RN ; BN and T LN are the known nominal values of the stator resistance, the friction coefficient and the load torque respectively, DR; DB and DT L are the differences between the real values and the nominal values of the stator resistance, the friction coefficient and the load torque respectively. 4.2. Nonlinear adaptive backstepping controller design Under the uncertainties mentioned above, a nonlinear adaptive backstepping speed controller can be designed

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systematically based on suitable Lyapunov function and adaptation laws. The compact form of the system (4) with uncertainties can be written as follows: ¯ þ DfðxÞ þ g1 ðxÞud þ g2 ðxÞuq ; x_ ¼ fðxÞ where x ¼ ½id iq o; g1 ðxÞ ¼ ½1=L 0 0T ; ½0 1=L 0T ; 2 3  RLN id þ Poiq 6 7 Pf 6 R 7 ¯ fðxÞ ¼ 6  LN iq  Poid  L o 7; 4 5 3Pf BN T LN i  o  2J q J J 2 3 DR  L id 6 7 DR 6 7 DfðxÞ ¼ 6  L iq 7: 4 5 DT L  DB J o J

(12) g2 ðxÞ ¼

e ¼ ½e1 e2 e3 T

z1 ¼ h1 ðxÞ ¼ o;

¼ ½z1  zm1 ; z2  zm2 ; z3  zm3 T

z2 ¼ Lf¯ h1 ðxÞ; z3 ¼ h2 ðxÞ ¼ id :

ð13Þ

The state-space equations of system (12) can be written as z_1 ¼ z2 þ LDf h1 ; z_2 ¼

þ Lg1 Lf¯ h1 ud þ Lg2 Lf¯ h1 uq þ LDf Lf¯ h1 ;

z_3 ¼ Lf¯ h2 þ Lg1 h2 ud þ Lg2 h2 uq þ LDf h2 :

ð14Þ

In (Shieh & Shyu, 1999), overparameterization problem occurred because of duplicate definition of uncertainties. Instead of defining three vector uncertainties as described in (Shieh & Shyu, 1999), three scalar new uncertain parameters are clearly defined in order to avoid this overparameterization as y1 ¼ 

DB DT L DR ; y2 ¼  : ; y3 ¼  J L J

Then system (14) can be written as follows: z_1 ¼ z2 þ y1 z1 þ y2 ; BN BN 3Pf i q y3 ; oy1  y2 þ z_2 ¼ L2f¯ h1 þ u¯ d  2J J J z_3 ¼ Lf¯ h2 þ u¯ q þ id y3 ; where " # ud uq

" ¼

0 Lg1 h2

Lg2 Lf¯ h1 0

#"

ud uq

ð15Þ

# :

ð17Þ

where zm1 ; zm2 and zm3 are the reference states, km1 ; km2 and km3 are the designed gains, oref and idref are the speed and d-axis current command. Using the reference model, we can easily evaluate the performance of the system because the tracking problem could be changed to a regulation problem. Define the error variables between the dynamic model (15) of PMSM and the reference model (17) as

Step 1: Define the new variable as follows:

L2f¯ h1

system (Shieh & Shyu, 1999). 2 3 2 32 3 0 1 0 zm1 z_m1 6 7 6 76 7 6 z_m2 7 ¼ 6 km1 km2 6 7 0 7 4 5 4 54 zm2 5 0 0 km3 zm3 z_m3 2 3 0 0 " # 6 7 oref 0 7 þ6 ; 4 km1 5 i dref 0 km3

(16)

Now, a reference model below is used to assign the desired output dynamic behavior for the nonlinear

and use the following transformation: " # u~ d e U¼ u~ q " # u¯ d þ km1 zm1 þ km2 zm2  km1 oref ¼ : u¯ q þ km3 zm3  km3 idref

ð18Þ

ð19Þ

Then the differential equations of the errors are given as follows: e_1 ¼ e2 þ y1 e1 þ y1 zm1 þ y2 ; BN BN 3Pf i q y3 ; oy1  y2 þ e_2 ¼ L2f¯ h1 þ u~ d  2J J J e_3 ¼ Lf¯ h2 þ u~ q þ id y3 :

ð20Þ

Step 2: For the first equation of (20), if y1 and y2 are known, then e2 can be taken as the new control input. The controller a ¼ k1 e1  y1 e1  y1 zm1  y2 can make the first equation stable by a Lyapunov function V 1 ¼ 1 2 2 e1 easily. But actually e2 is not the real control. Furthermore, parameter uncertainties and load disturbance also exist in the system, and they should be considered in the design procedures. The uncertain parameter errors are defined as follows: y~ 1 ¼ y1  y^ 1 ; y~ 2 ¼ y2  y^ 2 ; y~ 3 ¼ y3  y^ 3 ;

ð21Þ

where y^ 1 ; y^ 2 and y^ 3 are the estimations of y1 ; y2 and y3 ; respectively, y~ 1 ; y~ 2 and y~ 3 are the estimation errors between the unknown real values and their estimations, respectively.

ARTICLE IN PRESS J. Zhou, Y. Wang / Control Engineering Practice 13 (2005) 1259–1269

Define the new virtual control a for e2 as a ¼ k1 e¯ 1  y^ 1 e1  y^ 1 zm1  y^ 2 :

(22)

Then define new error variables as e¯ 1 ¼ e1 ; e¯ 2 ¼ e2  a; e¯ 3 ¼ e3 :

(23)

The derivatives of the new error variables in (23) are written as e_¯ 1 ¼ k1 e¯ 1 þ e¯ 2 þ y~ 1 e1 þ y~ 1 zm1 þ y~ 2 ; BN ^ BN ^ oðy1 þ y~ 1 Þ  ðy2 þ y~ 2 Þ e_¯ 2 ¼ L2f¯ h1 þ u~ d  J J 3Pf ^ da iq ðy3 þ y~ 3 Þ  ; þ 2J dt e_¯ 3 ¼ Lf¯ h2 þ ðy^ 3 þ y~ 3 Þid þ u~ q ;

(24)

da _ _ _ ¼ k1 e_¯ 1  y^ 1 e¯ 1  y^ 1 e_¯ 1  y^ 2  y^ 1 zm1  y^ 1 z_m1 : (25) dt The nonlinear adaptive controller and the adaptation laws can be easily designed through suitable Lyapunov candidate. Step 3: Define the final Lyapunov candidate V as

u~ d ¼  k2 e¯ 2  e¯ 1  L2f¯ h1 þ

BN ^ BN ^ o y1 þ y2 J J

3Pf ^ iq y3  k1 ðk1 e¯ 1 þ e¯ 2 Þ 2J _ _ _  y^ 1 e¯ 1  y^ 2 þ k1 e¯ 1 y^ 1  e¯ 2 y^ 1  y^ 1 zm1  y^ 1 z_m1 ;



Parameter adaptation laws of y^ 1 ; y^ 2 and y^ 3 are given as

BN _^ o¯e2 þ k1 e¯ 1 e¯ 2 þ y^ 1 e¯ 1 e¯ 2 þ e¯ 1 zm1 y1 ¼ g1 e¯ 21  J  þ k1 zm1 e¯ 2 þ y^ 1 zm1 e¯ 2 ;

 BN _ e¯ 2 þ k1 e¯ 2 þ y^ 1 e¯ 2 ; y^ 2 ¼ g2 e¯ 1  J

 3Pf _ iq e¯ 2 þ id e¯ 3 : y^ 3 ¼ g3 2J

ð28Þ

Thus the derivative the Lyapunov candidate becomes as V_ ¼ ðk1 e¯ 21 þ k2 e¯ 22 þ k3 e¯ 23 Þ ¼ ¯eT K¯eo0; ð26Þ

where g1 ; g2 and g3 are constants. Considering the error dynamic equation (24), the derivative of the Lyapunov candidate in (26) becomes as follows: 1 _ 1 _ 1 _ V_ ¼ e¯ 1 e_¯ 1 þ e¯ 2 e_¯ 2 þ e¯ 3 e_¯ 3 þ y~ 1 y~ 1 þ y~ 2 y~ 2 þ y~ 3 y~ 3 g1 g2 g3

B N o¯e2 þ k1 e¯ 1 e¯ 2 þ y^ 1 e¯ 1 e¯ 2 ¼  k1 e¯ 21 þ y~ 1 e¯ 21  J  1 _^ ^  y1 þ e¯ 1 zm1 þ k1 e¯ 2 zm2 þ y1 zm1 e¯ 2 g1

 BN 1 _^ ~ ^ þ y2 e¯ 1  e¯ 2 þ k1 e¯ 2 þ y1 e¯ 2  y2 g2 J

 3Pf 1 _ iq e¯ 2 þ id e¯ 3  y^ 3 þ y~ 3 2J g3

BN ^ BN ^ 3Pf ^ i q y3 þ e¯ 2 e1 þ L2f¯ h1  o y1  y2 þ 2J J J _ _ þ u~ d þ k1 ðk1 e¯ 1 þ e¯ 2 Þ þ y^ 1 e¯ 1 þ y^ 2 i _  k1 e¯ 1 y^ 1 þ e¯ 2 y^ 1 þ y^ 1 zm1 þ y^ 1 z_m1 þ e¯ 3 ½Lf¯ h2 þ id y^ 3 þ u~ q :

items in the fourth and the fifth brackets equal to k2 e¯ 2 and k3 e¯ 3 ; respectively. Then the following results can be obtained. The control output u~ d and u~ q are designed as

u~ q ¼ k3 e¯ 3  Lf¯ h2  id y^ 3 :

where

1 1 1 1 ~2 y V ¼ e¯ 21 þ e¯ 22 þ e¯ 23 þ 2 2 2 2g1 1 1 ~2 1 ~2 y þ y þ 2g2 2 2g3 3

1263

(29)

where K ¼ dig½k1 k2 k3 ; ki 40; i ¼ 1; 2; 3; and V_ ¼ 0 iff e¯ ¼ 0: It follows that V in (26) is Lyapunov function of (24). The final control inputs ud and uq can be easily derived through (16), (19) and (28). Remark 3. It should be noted that there is no singularity and overparameterization in the proposed control scheme. There are three uncertainties DR; DB and DT L : From (28), it is clear that each uncertainty is estimated by a unique adaptation law. Thus, there is no overparameterization, which occurred in (Shieh & Shyu, 1999). Remark 4. Furthermore, it can be easily proved that the coordinate transformation (16) is invertible, and there is no estimated uncertainty in the denominator of any item in (28). Therefore it can be concluded that the proposed nonlinear adaptive backstepping speed controller is more acceptable in the practical real-time applications.

5. Stability analysis ð27Þ

To ensure V_ p0; the simplest way is to make the items in the first, the second and the third square brackets in (27) equal to zero to cancel the uncertainties. Then make the

In this section, stability of the proposed control scheme and parameter adaptation laws are to be analyzed. Considering the fact from V_ o0; it follows that all the errors e¯ 1 ; e¯ 2 and e¯ 3 are bounded.

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diagram of the nonlinear controller and adaptation laws is given in Fig. 2. The parameters of the PMSM motor under our experiment is shown in Table 1. The d-axis current command is set as zero and the speed trajectory reference is given by the reference model (17). In this paper, the reference model (17) is adopted to make the PMSM follow the command signals in a specified manner. Pole placement technique is used to determine the gains. Actually, from Eq. (17), there are two decoupled subsystems, one is a second-order mechanical subsystem for motor speed o: The other is a first-order electrical subsystem for d-axis current id : For the mechanical subsystem, the gains km1 and km2 can be chosen according to the expected transient performance. In this paper, we would like the motor speed reach the step command within 0.4 s without any overshot. The poles of the speed reference are assigned as 11:5 j52:7 by choosing km1 ¼ 160; km2 ¼ 23: For the electrical subsystem, we would like d-axis current reach the step command within 0.1 s without any overshot. The pole of the d-axis current reference dynamics is assigned as 50 by choosing km3 ¼ 50: For the nonlinear controllers, k1 ; k2 and k3 may be chosen for the nominal system, which means they are obtained by forcing the uncertainties y1 ; y2 and y3 to be zero. Therefore, the error dynamics (20) of the control system becomes as

Define the following new function: MðtÞ ¼ e¯ T K¯eX0: Now, integrating (29) and gives that Z t ~ MðtÞ dt ¼ V ð¯eð0Þ; yð0ÞÞ  V ðtÞ: 0

Since V ðtÞX0; it gives that Z t ~ MðtÞ dtpV ð¯eð0Þ; yð0ÞÞo1; lim t!1

0

for all t. Through Barbalat’s Lemma (Slotine & Li, 1991), it can be obtained that MðtÞ ! 0 as t ! 1: Therefore it can be concluded that all the error variables are bounded and converge to zero asymptotically. Further, since the motor speed o 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.

6. Simulation and experiment results The overall configuration of the control system for a PMSM is shown in Fig. 1. The main circuit is a voltagesource inverter using six Semikron IGBT driven by a IGBT driver and the inverter is realized by space vector PWM (SVPWM) method. The real-time algorithms of the controller and the adaptation laws are realized by TMS320C31 processor based DSAPCE DS1102 DSP card through C language programming. The phase currents and motor speed are measured by Hall-effect current transducers and encoder with resolution of 2048 line. The sampling rate is chosen as 160 us. The block

e_1 ¼ e2 ; e_2 ¼ ðk1 þ k2 Þe2  k1 k2 e1 ; e_3 ¼ k3 e3 :

From Eq. (30), it is very clear that the controller gains k1 ; k2 and k3 are the poles of the system error dynamics. In this paper, we chose k1 ¼ 1500; k2 ¼ 250; k3 ¼ 100 to

PERSONAL COMPUTER

DC POWER SUPPLY

ω ref idref

Nonlinear Backstepping Control Algorithm

ud

ð30Þ

+

Vdc

uα

d, q

uq

α, β

uβ

IGBT DRIVER (SKHI60)

SVPWM Algorithm

PWM 3-PHASE IGBT INVERTER

Θ

id d,q d, q

iα α , β

iq

iβ

α, β

ω

S

a, b, c

ADC1 ADC2

Θ

ia ib

PMSM MOTOR

ENCODER

DSPACE DSP DS1102 BOARD

Fig. 1. Overall diagram of the proposed control system of PMSM.

LOAD

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L f h1

α = −k1e1 − θˆ1e1 − θˆ1 z m1 − θˆ2

ω ref I dref

e1

zm1 Reference z m 2 model

z m3

-

- e2 - e3

B B 3 Pφ ˆ u~d = −k2e2 − e1 − L2f h1 + N ωθˆ1 + N θˆ2 − iqθ 3 − k1 ( −k1e1 + e2 ) J J 2J . . . . ˆ ˆ ˆ ˆ ˆ ˆ − θ1e1 − θ 2 + k1e1θ1 − e2θ1 − θ1zm1 − θ1zm1 CONTROL u~ = −k e − L h − i θˆ q

3 3

f 2

d 3

θˆ1

θˆ2

id SVPWM

ω

PMSM

OUTPUTS

iq

θˆ1

θˆ2

θˆ3

. B θˆ1 = γ 1 (e12 − N ωe2 + k1e1e2 + θˆ1e1e2 + e1zm1 + k1zm1e2 + θˆ1zm1e2 ) J . B θˆ2 = γ 2 (e1 − N e2 + k1e2 + θˆ1e2 ) J ADAPTATION . 3 Pφ LAWS θˆ3 = γ 3 ( iqe2 + id e3 ) 2J Fig. 2. Block diagram of the nonlinear controller and adaptation laws.

Table 1 PMSM motor parameters Power Rated torque Rotor inertia Torque constant Inductance Rated speed Resistance Viscous constant

1.49 KW 2.2 Nm 0:0000765 kg m2 0.29 Nm/A 0.0094 H 7500 rpm 1:6 O 0.000004 Nm/rpm

place the poles of the speed error dynamics as 1500; 250 and the d-axis current error dynamics as 100; respectively. Therefore, under the nominal system parameter, the speed command oref can be successfully followed. It should be noted that theoretically, the system can be stabilized by any positive gains k1 ; k2 and k3 : For the gains of the adaptation laws, from Eq. (28), there is no any limitation on the gains of g1 ; g2 and g3 except that they are positive constants. However, the choices of the estimation gains have fundamental influence on the convergence behavior of the estimator (28). Poor choice may leads to oscillatory and slower convergence. In (Slotine & Li, 1991), It is stated that for a multi-parameter estimation case, the convergence rate of the estimated parameters is not as simple as a singleparameter estimation case. In this paper, it is very difficult to determine the estimation gains by poleplacement technique because the parameter estimator dynamics suitable for pole-placement is not easy to be obtained. Thus, we choose the gains of the adaptation laws by trial-and-error method to make the estimation

converge within 0.2 s without large oscillation. The gains of the adaptation laws are given as g1 ¼ 0:0000002; g2 ¼ 0:0000001 and g3 ¼ 0:0000001: 6.1. Simulation results The simulation results of the pure input–output feedback nonlinear controller (10) are shown in Figs. 3 and 4. Case 1: Pure feedback linearization controller is applied for the nominal system, i.e. DR ¼ DB ¼ DT L ¼ 0: The speed reference changed from 0 to 100 rad/s at t ¼ 0 s: Then at t ¼ 3 s; the reference is increased to 200 rad/s. Fig. 3 shows obviously that the system speed can satisfactorily follow the reference signal without any steady error. But when there are parameter uncertainties and load torque, the exact cancellation based feedback linearization controller cannot satisfy the speed tracking requirement, which can be seen in Fig. 4. In this case, there is a large steady-state error of speed when a step load T L ¼ 2:2 Nm is added at t ¼ 1:5 s: Case 2: Figs. 5–8 give the simulation results of the proposed control scheme (16), (19) and (28) with adaptation laws (28). Fig. 5 gives the simulation results of the proposed control scheme of the nominal system. The results are as perfect as Fig. 3. In Fig. 6, in order to test the robustness of the controller to the change of the system parameters and the load disturbance, the motor is started with the stator resistance and friction ¯ and B ¼ 2B ¯ and T L ¼ 0 Nm: coefficient as R ¼ 2R Then at t ¼ 1:5 s; a step rated load torque T L ¼ 2:2 Nm is added suddenly and at t ¼ 4:5 s; the load torque is unloaded. Fig. 6 shows that compared with the results in Fig. 4, the proposed adaptive backstepping controller

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2.5

250

d, q axis current (A)

Speed (rad/s)

d axis current

150

100 Speed reference 50

0 0

q axis current

2

200

1.5 1 0.5 0

Speed response

1

2

3 Time (sec)

4

5

0

6

1

2

3 Timer (sec)

4

5

6

Fig. 3. Simulation results of nominal system without adaptation laws. (Left) Speed reference and actual speed response. (Right) d, q axis current response.

250

12 q axis current 10

d, q axis current (A)

Speed (rad/s)

200

150

100

d axis current

8 6 4

Speed reference 50

0 0

2

Speed response

0 1

2

3 Time (sec)

4

5

6

0

1

2

3 Time (sec)

4

5

6

Fig. 4. Simulation results without adaptation laws with R ¼ 2RN and step load T L ¼ 2:2 Nm at t ¼ 1:5 s: (Left) Speed reference and actual speed response. (Right) d, q axis current response.

250

2 q axis current d axis current 1.5

d, q axis current (A)

Speed (rad/s)

200

150

100 Speed reference 50

1

0.5

Speed response 0

0 0

1

2

3 Time (sec)

4

5

6

0

1

2

3 Time (sec)

4

5

6

Fig. 5. Simulation results of nominal system with proposed adaptive control scheme. (Left) Speed reference and actual speed response. (Right) d, q axis current response.

can track the reference speed successfully although there is a small speed drop when the load is added. After a very short period (less than 200 ms), the speed recovers to follow the reference speed. Further more, the d-axis current id is well decoupled from motor speed and is regulated to zero quite well. The estimation results of parameter y1 and y2 are shown in Fig. 7. Fig. 8 gives the estimation of y3 and the speed tracking error under parameter uncertainties and load disturbance of the proposed control scheme. It shows that the proposed

controller can track thereference signal satisfactorily even under parameter uncertainties and load torque disturbance. It should be noted that the estimated values of y1 ; y2 and y3 are not the actual values of y1 ; y2 and y3 : 6.2. Experimental results In order to further evaluate the effectiveness of the proposed control scheme compared with pure feedback linearization controller, some real-time experiments are

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12 q axis current 10

d axis current

d, q axis current (A)

Speed (rad/s)

200

150

100 Speed reference 50

Speed response

0 0

1

2

3 Time (sec)

4

5

8 6 4 2 0 –2 0

6

1

2

3 Time (sec)

4

5

6

Fig. 6. Simulation results of the proposed control scheme with R ¼ 2RN and step load T L ¼ 2:2 Nm at t ¼ 1:5 s: (Left) Speed reference and actual speed response. (Right) d, q axis current response.

5

0.1

0

0 Esimation 2

Estimation 1

-5 -10 -15

-0.1

-20 -25 -30

0

1

2

3 Time (sec)

4

5

6

-0.2

0

1

2

3 Time (sec)

4

5

6

4

5

6

Fig. 7. Parameter estimations. (Left) y1 ; (right) y2 :

0.05

8 6

0

4 Speed error (rad/s)

Estimation 3

-0.05 -0.1 -0.15 -0.2

2 0 -2 -2 -2

-0.25

-2

-0.3 0

1

2

3 Time (sec)

4

5

6

0

1

2

3 Time (sec)

Fig. 8. Estimation of y3 (left) and speed error of proposed control scheme (right).

carried out. Figs. 9 and 10 show the experimental results of the two controllers under no parameter uncertainties and load disturbances, respectively. It can be seen that both controller can track the reference speed signal very well. However, when there are uncertainties and load disturbances, the performance are quite different. The pure feedback linearization controller has large error under uncertainty R ¼ 2RN and load torque disturbance at t ¼ 1:15 s; shown in Fig. 11. Fig. 12 gives the dynamic

responses of the proposed nonlinear backstepping adaptive control scheme when the motor is started under uncertainty R ¼ 2RN and the full load torque disturbance. Fig. 13 shows the performance of the proposed scheme when the load torque is unloaded at t ¼ 1:2 s: It is evident that the proposed controller has improved the dynamic and steady-state performance significantly. From these simulation and real-time experiments, it can be concluded that the proposed nonlinear adaptive

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Fig. 9. Experimental results of pure feedback linearization controller under no parameter uncertainties and load disturbance. (a) Speed reference and actual speed response. (b) d, q axis current.

Fig. 11. Experimental results of pure feedback linearization controller under uncertainties ðR ¼ 2RN Þ and load disturbance (T L ¼ 1:7 Nm at t ¼ 1:15 s). (a) Speed reference and actual speed response. (b) d, q axis current.

Fig. 10. Experimental results of the proposed controller scheme under no parameter uncertainties and load disturbance. (a) Speed reference and actual speed response. (b) d, q axis current.

Fig. 12. Experimental results of the proposed controller scheme under uncertainties ðR ¼ 2RN Þ and load disturbance ðT L ¼ 1:7 NmÞ: (a) Speed reference and actual speed response. (b) d, q axis current.

backstepping speed controller is quite effective under parameter variations and load torque disturbance without singularity and overparameterization.

uncertainties considered in the system are the stator resistance, the friction coefficient and the load torque, which are the most concerned variations in a electric drive system. Then the parameter adaptation laws and the final control laws are derived systematically step by step. The speed tracking errors between the reference model and the actual speed of PMSM converge to zero asymptotically, and the robustness against the parameter uncertainties and load torque disturbance is achieved. Furthermore, the resulted control scheme has no singularity and overparameterization. Simulation and DSP-based real-time experiments have been carried out to verify the validity of the proposed control scheme.

7. Conclusion In this paper, based on adaptive backstepping approach, a nonlinear adaptive speed controller is designed for a permanent magnet synchronous motor under parameter uncertainties and the load torque disturbance. Feedback linearization technique is used to simplify the nonlinear system and a suitable error model with related to the reference model is derived. The

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Lg1 Lf¯ h1 ¼ 0; 3Pf ; 2JL RN Lf¯ h2 ¼  id þ Poiq ; L 1 Lg1 h2 ¼ ; L Lg2 h2 ¼ 0; DB DT L o ; LDf h1 ¼  J



J  BN DB DT L 3Pf DR o  iq ; LDf Lf¯ h1 ¼   þ J 2J L J J DR id : LDf h2 ¼  L Lg2 Lf¯ h1 ¼

Fig. 13. Experimental results of the proposed controller scheme under uncertainties (R ¼ 2RN ) and load disturbance (T L ¼ 1:7 Nm till t ¼ 1:2 s). (a) Speed reference and actual speed response. (b) d, q axis current.

Appendix Lie’s derivatives used in this paper are given as follows: 3Pf B TL iq  o  Lf h1 ¼ ; 2J J J

 3Pf R Pf  iq  Poid  o L2f h1 ¼ 2J L L

 B 3Pf B TL  iq  o  ; J 2J J J Lg1 Lf h1 ¼ 0; 3Pf ; Lg2 Lf h1 ¼ 2JL R Lf h2 ¼  id þ Poiq ; L 3Pf BN T LN iq  o ; Lf¯ h1 ¼ 2J J J

 3Pf RN Pf  o L2f¯ h1 ¼ iq  Poid  2J L L

 BN 3Pf BN T LN  iq  o ; 2J J J J

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