Design and Implementation of a Nonlinear Speed

SICE Annual Conference 2010 August 18-21, 2010, The Grand Hotel, Taipei, Taiwan

Design and Implementation of a Nonlinear Speed Controller with Adaptive Backstepping Sliding Mode Technique for an IPMSM Drive System Cheng-Kai Lin and Li-Chen Fu, Fellow, IEEE

Tian-Hua Liu, Senior, IEEE

Department of Electrical Engineering National Taiwan University Taipei, Taiwan [email protected]

Department of Electrical Engineering National Taiwan University of Science and Technology Taipei, Taiwan [email protected]

Abstract-This paper proposes a nonlinear speed controller for an interior permanent magnet synchronous motor (IPMSM) drive system. First, the input-output feedback linearization control is used to derive a linearized model including load torque variations and lumped parameter uncertainties. In order to reduce the influence of load torque disturbances on the linearized model, an external load torque estimator is employed for the load torque estimation. To overcome the lumped parameter uncertainties, an adaptive backstepping approach with adaptation mechanisms is derived. After that, combining with a sliding mode control technique is used to improve the robustness of the speed control system. By using Lyapunov stability theory and Barbalat’s lemma, the proposed control scheme is shown to be globally asymptotically stable. A digital signal processor, TMS320LF2407, is used to implement the proposed control scheme. The experimental results show that the proposed system has fast transient responses, good load disturbance rejection responses, and good tracking responses.

parameter variations. In general, the feedback linearization technique is used to transform a nonlinear dynamic system into a linear one. After that, many well-known control techniques can be applied to the linearized model to complete the controller design. Even though the feedback linearization technique provides a systematic controller design, it requires the full information of the system parameters. In the real world, some parameters of the system cannot be measured precisely, therefore, the complete linearized model cannot be obtained. In addition, this method does not guarantee the robustness under parameter uncertainties or load torque disturbances. In order to overcome the problem, Zhou and Wang proposed an adaptive backstepping speed controller for a PMSM drive system [5]. The adaptive backstepping controller can compensate for slowly varying parameters in the system model. Baik et al. proposed a boundary layer integral sliding mode control technique for a PMSM drive system to improve the robustness and compensate for the influence of quickly varying parameters and disturbances [6]. However, to the authors’ best knowledge, no researchers have actually studied the realization of the adaptive backstepping sliding mode control based on the feedback linearization technique applied to an IPMSM speed control system. This motivated us to deeply study this issue. In addition, by using the DSP to execute the controller, the experimental results show the feasibility of the proposed nonlinear speed control scheme.

Keywords-IPMSM; adaptive backstepping sliding mode

I.

INTRODUCTION

The interior permanent magnet synchronous motor (IPMSM) has been widely used in industry due to its high efficiency, high torque/ampere ratio, and rugged structure. Several nonlinear controllers have been developed for the IPMSM to increase drive systems performance [1]-[6]. For example, Mohamed proposed an adaptive self-tuning maximum torque/ ampere vector controller for an IPMSM drive system [1]. The experimental results are satisfactory. However, the controller is very complicated. As a result, a lot of computation is required for the DSP. Rahman used an adaptive backstepping technique to control an IPMSM. The idea is sound but the realization of the controller is difficult [2]. Shi, Liu, and Chang proposed an adaptive controller design for a sensorless IPMSM drive system with a maximum torque control. The experimental results are good. Unfortunately, the adaptive law and controller are very complicated [3]. Although many different nonlinear controllers have been applied to the IPMSM drives [1]-[3], only a few researchers have implemented the feedback linearization technique to control an IPMSM [4]. Lin, Liu, and Yang proposed a nonlinear position controller with H ∞ control law for an IPMSM with input-output feedback linearization technique. The H ∞ control law used in [4] is a kind of linear control, therefore, the robustness is not guaranteed for heavy

II. MATHEMATICAL MODEL The d-q axis voltage equations of the IPMSM can be expressed as: di vd = rs id + Ld d − Poωrm Lq iq (1) dt diq vq = rs iq + Lq + Poωrm Ld id + Poωrm λm (2) dt where vd is the d-axis voltage, rs is the stator resistance, id is the d-axis current, Ld is the d-axis inductance, d/dt is the differential operator, Po is the pole pair of the motor, rm is the motor speed, Lq is the q-axis inductance, iq is the q-axis current, vq is the q-axis voltage, and m is the flux linkage. The electromagnetic torque can be described as:

- 110 -

Te =

3 Po λmiq + ( Ld − Lq )id iq  2 

(3)

PR0001/10/0000-0110 ¥400 © 2010 SICE

where Te is the electromagnetic torque of the motor. The dynamic equation of the speed is d ω rm 1 (Te − TL − Bm ω rm ) = dt Jm The dynamic equation of the position is d θ rm = ω rm dt

3Po iq λm + 3Po ( Ld − Lq )id iq

ω rm =

2 J mo

−

ωrm Bmo J mo

−

Td J mo

(11)

Then, the system dynamic equation can be rearranged as: x = fˆ ( x) + g1vd + g 2 vq + g3 ΔTd (12)

(4)

where ΔTd = Td − Tˆd ᧨ g 3 = [ 0 0 −1 / J mo ]

T

(5)

and

III. NONLINEAR SPEED CONTROLLER DESIGN

  Po Lq iqωrm ri − sd +

 Ld Ld   fˆ1 ( x) 





r i P ω L i P ω λ s q  fˆ ( x ) = fˆ2 ( x )  =

− o rm d d − − o rm m  Lq Lq Lq

ˆ 

  f3 ( x )  3P i λ + 3P ( L − L )i i ω B Tˆd  o q m o d q d q rm mo − −

J mo J mo  2 J mo 

According to (1)-(5), the dynamic equation of the motor can be expressed as:   r i Po Lqiqωrm − sd +

 Ld Ld

  id 

 0   1/ Ld  Poωrm Ld id rsiq Poωrmλm

 

0  v + 1/ L  v  i = − − − + q q

q 

 d

Lq Lq Lq 

ω rm 

 0    0   

3Poiqλm + 3Po ( Ld − Lq )id iq ωrm Bm TL  − − 

Jm Jm  2J m 

(6)

(13)

Where Ϙ ^ ϙis the symbol of the estimating value. In this paper, we choose the d-axis current and the motor speed as output variables [7],[8]. The aim of the controller is to As you can see, (6) is a nonlinear dynamic equation. As a result, maintain the speed and the d-axis current in the trajectories we the linearization technique, the external load estimator, the want. The new state variables can be defined as᧶ adaptive backstepping technique, and the sliding mode z1 = h1 ( x ) = ωrm (14) controller can be designed as follows. ˆ 3Po λm iq + 3Po ( Ld − Lq )id iq ωrm Bmo Td (15) z2 = L fˆ h1 ( x) = − − A. Input-Output Linearization Technique 2 J mo J mo J mo

z3 = h2 ( x ) = id (16) where, z1 is the motor speed, z2 is the acceleration can be computed by using the estimated load torque, and z3 is the qaxis current. In addition, the dynamic equation of the system can be expressed as: z1 = z2 + Lg 3 h1ΔTd (17)

Letting   Po Lq iqωrm ri − sd +

 Ld Ld



 Poωrm Ld id rs iq Poωrm λm  − − − f ( x) =

Lq Lq Lq





3Po iq λm + 3Po ( Ld − Lq )id iq ωrm Bm TL  − − 

2J m Jm Jm  

(7)

Then, by substituting (7) into (6), we can obtain: x = f ( x ) + g1vd + g 2 vq

iq

g1 = [1 / Ld

ωr 

z3 = L fˆ h2 + Lg1h2 vd

(19)

Lg 1 L fˆ h1 =

T

Lg 2 L fˆ h1 =

0 0]

g 2 = 0 1 / Lq

(18)

and

(8)

and x = id

z1 = z2 = L2fˆ h1 (x) + Lg1Lfˆ h1vd + Lg 2 Lfˆ hv 1 q + Lg 3 Lfˆ h1ΔTd

T

0 

T

L fˆ h2 = −

In fact, the real system includes an unknown external load TL , and the deviations of the lumped parameters. Then, the total uncertainty can be expressed as d ωrm Te = J mo + Bmoω rm + Td dt d ωrm Td = ΔJ m + ΔBmωr + TL dt

(9) L2fˆ h1 =

(10)

where ΔJ m = J m − J mo , ΔBm = Bm − Bmo , and J mo , Bmo are the nominal parameters of the motor, and Td is the disturbance torque. By substituting (9) and (10) into (4), we can obtain

2 J mo Ld 3Po λm + 3Po ( Ld − Lq )id

Bmo J m2 o

1 Ld

3Po ( Ld − Lq )iq ˆ B f1 ( x ) − mo fˆ3 ( x ) J mo 2 J mo

+

(20) (21)

2 J mo Lq

rs id Po Lq iq ωrm + Ld Ld

Lg 3 L fˆ h1 = Lg 1h2 =

3Po ( Ld − Lq )iq

3Po λm + 3Po ( Ld − Lq )id ˆ f 2 ( x) 2 J mo

(22) (23) (24)

(25)

To effectively control the system, the d-q axis voltages can be selected as:

- 111 -

 u1 − L2fˆ h1  vd*  −1 

*  = D ( x)

 vq   u2 − L f h2 

(26)

 Lg1 L fˆ h1 D ( x) =

 Lg1h2

(27)

Where u1 and u2 are new control variables. The D ( x ) can be expressed as: Lg 2 L fˆ h1   0 

Where L1 is the gain of the load estimator. Unfortunately, in the real world, it is difficult to obtain ω rm without causing high frequency noise. To solve the problem, a new variable is defined as xc1 = Tˆd − L1ωrm (36) Then, it is not difficult to obtain  xc1 = Tˆd − L1ω rm 3Po ( Ld − Lq )id iq 

B 1 ˆ 3Po λm = L1  + mo ωrm + Td − iq −  2 2 J mo J J J   mo mo mo

By substituting (27) into (18) and (19), we can obtain : z1 = z2 + Lg 3 h1 ( x)ΔTd

(28)

z2 = u1 + Lg 3 L fˆ h1ΔTd

(29)

z3 = u2 (30) Due to the influence of the external load, the nonlinear system can not match the linear decoupled Brunovski canonical form. As a result, the performance of the system is deteriorated. If we consider the ideal condition, which means that ΔTd = 0 , then it is possible to select linear state feedback control laws as follows: * * * u1 = ωrm + K11 (ω rm − z 2 ) + K12 (ωrm − z1 )

(31)

u2 = K 21 (id* − z3 ) + id*

(32)

* Where ωrm and id* are the motor speed command and d-axis current command, respectively. The desired control performance can be determined by properly selecting the controller gains K11 , K12 , and K 21 .

B. The External Load Torque Estimator In the real world, the motor is used to convert the electrical energy into mechanical energy. As a result, an external load is added to the drive system. For a fixed sampling interval, the external load does not change abruptly. The major reason is the sampling interval is very short. As a result, we can assume that the external load Td is a constant, and its derivative Td is equal to zero for each sampling interval. According to (3) and letting Td = 0 , we can obtain  Bmo ω rm  −

T  = J mo  d 

 0

 3Po ( Ld − Lq )id iq  1   3Po λm  − ω   J mo  rm  + 2 J mo  iq +

2 J mo   T

  d 

 0  0  0   

(33)

and ωrm 

yo = ωrm = [1 0]   Td 

By letting eTd = Td − Tˆd , we can obtain L  eTd = Td − Tˆd = 1 eTd J mo

(38)

Finally, we can find the estimating load is Tˆd = xc1 + L1ω r . The dynamic behaviour can be determined by the gain of the estimator, which is L1 . In addition, the influence of the load disturbances can be effectively compensated for. 

C. Adaptive Backstepping Sliding mode controller Design To overcome the lumped-parameter variation and load estimating error, an adaptive backstepping sliding mode controller is proposed here to achieve maximum torque/ampere control. By suitably adjusting the d-axis current, a maximum torque/ampere characteristic can be achieved. In addition, combining the adaptive backstepping sliding mode controller with the external load torque estimator can compensate for the uncertainties caused by the lumped parameter variations, measuring errors, and external load disturbances. According to (28)-(30), we can obtain

z1 = L fˆ h1 ( x) + Lg 3 h1 ( x)ΔTd

(39)

z2 = L h ( x) + u1 + Lg 3 L fˆ h1 ( x)ΔTd

(40)

z3 = L fˆ h2 ( x) + u2

(41)

where u1 = u1 − L2fˆ h1 ( x)

(42)

2 fˆ 1

u2 = u2 − L fˆ h2 ( x)

(43)

where u1 and u2 are the equivalent control input of the designed adaptive backstepping sliding mode control laws. It is assumed that the lumped parameter variations in (39)-(41) are unknown but constants and are only related to iq2ωrm  iqωrm  id 

(34)

The system of (33)-(34) is observable, and the external load Td can be estimated. Now, the reduced-order load torque estimator can be used as follows[4]: 3Po ( Ld − Lq )id iq 

B 1 ˆ 3Po λm  Tˆd = L1  ω rm + mo ωrm + Td − iq −  2 2 J mo J J J   mo mo mo

(37)

iq  and ωrm . The new linearized model with lumped para-meter

uncertainties can be rewritten as     Lg 3 h1 ( x)ΔTd  z1  L fˆ h1 ( x)  0 0    u1 

 2

 

z2  = L fˆ h1 ( x)  + 1 0  u  + Lg 3 L fˆ h1 ( x)ΔTd + d 2ϕ2 ( x)    2

 z3 

L h ( x)  0 1  d3ϕ3 ( x) 

  fˆ 2 

(35)

where

- 112 -

(44)

d 2ϕ 2 ( x) = [ Δθ1 d3ϕ3 ( x) = [ Δθ 4

iq2ωrm 

 Δθ 2 Δθ 3 ] iq 

ωrm    i  d  Δθ 5 ]

 iqωrm 

(45)

(46)

2

and

θ1 =

θ3 =

3Po ( Ld − Lq ) 2 J mo Ld

, θ2 = −

unknown. Therefore, the estimated lumped parameter uncertainties dˆ2 and dˆ3 are required. The Lyapunov function with sliding manifold information is selected as 1 1 1 1  T 1  T V = e12 + s22 + s32 + d 2 d2 + d3 d3 (57) 2 2 2 2γ 2 2γ 3 Where d = dˆ − d , d = dˆ − d , and γ and γ are adaptive

3Po Bmo λm 3Po λm rs − 2 2 J mo 2 J mo Lq

−λm λm2 − + iq2 2( Ld − Lq ) 4( Ld − Lq ) 2

(47)

Next, we define the tracking error variables between the state variables and the reference commands as follows * e1 = ωrm − z1 (48) e2 = z − z2

(49)

e3 = i − z3

(50)

* 2

* d

and * z2* = k1e1 + ω rm

(51) * 2

where k1 is the closed-loop feedback constant and z is the virtual control variable to stabilise the acceleration error. From (48)-(50), the time derivatives of the tracking error variables are written as * e1 = ω rm − z2 − Lg 3 h1 ( x) ΔTd (52)

e3 = id* − L fˆ h2 ( x ) − u2 − d3ϕ3 ( x)

t

0 t

s3 = e3 + k5  e3 dt 0

2

3

3

(54)

(55) (56)

Where k4 and k5 are the strictly positive constants. In the real world, the lumped parameter uncertainties d 2 and d3 are

γ3

1   T 1   d 2 d 2 + d3 d3 γ2 γ3

( + s ( i − L h ( x) − u

+ s2 −u1 − Lg 3 L fˆ h1 ( x)ΔTd − d 2ϕ 2 ( x) − L2fˆ h1 ( x) 3

* d

fˆ

2

2

− d3ϕ3 ( x) + k5 e3

)

(58)

)

According to (58), adaptive backstepping sliding mode control laws u1 and u2 are proposed as

s  * − k12 e1 − L2fˆ h1 ( x ) + k1e2 + k 2 s2 + k3e1 − dˆ2ϕ 2 ( x ) + η2 sat  2  (59) u1 = ωrm  φ2 

s  u2 = id* − L fˆ h2 ( x) − dˆ3ϕ3 ( x) + k3 s3 + k5 e3 + η3 sat  3   φ3 

(60)

where η2 = η2 + Td ( (1 + k1 ) Lg 3h1 ( x) + Lg 3 L fˆ h1 ( x) ) , and k1 , k2 , k3 , η 2 , and η3 are strictly positive constants, and Td is the upper bound of ΔTd . In (59) and (60), sat (⋅) is defined as  +1, if si > φi

si   sat   =  si / φi , if − φi ≤ si ≤ φi (61)  φi   −1, if s < −φ i = , 2,3 i i  Substituting (59) and (60) into (58), the derivative of the Lyapunov function can be rearranged as V = −k1e12 + e1e2 − k2 s22 − k3 s32 − η3 s3

(

)

− η 2 + Td (1 + k1 ) Lg 3 h1 ( x) + Lg 3 L fˆ h1 ( x) s2

1  

1   + d2  d2T + s2ϕ 2 ( x)  + d3  d3T + s3ϕ3 ( x)  γ γ  2   3 

(

(53)

After that, the sliding surfaces s2 and s3 with the integrated error are chosen as s2 = e2 + k4  e2 dt

3

* + s2 (ωrm − k12 e1 − k1 Lg 3 h1 ( x)ΔTd + k1e2 + k4 e1 )

= −k1e1 − Lg 3 h1 ( x)ΔTd + e2

− L2fˆ h1 ( x) − u1 − Lg 3 L fˆ h1 ( x)ΔTd − d 2ϕ2 ( x)

3

= −k1e12 + e1e2 − e1 ( Lg 3 h1 ( x)ΔTd ) +

Where Δθ1 , Δθ 2 , Δθ 3 , Δθ 4 , and Δθ 5 are the lumped

* e2 = −k12 e1 − k1 Lg 3 h1 ( x)ΔTd + k1e2 + ωrm

2

γ2

2 Po Lq 3P λ Bmo r − o m , θ 4 = − s , θ5 = 2 J mo J mo Lq Ld Ld

parameter variations of θ1 , θ 2 , θ 3 , θ 4 , and θ 5 , respectively. The conventional control laws (31) and (32), applying to (44) can not guarantee the desired performance. Therefore, an adaptive backstepping sliding mode controller is designed for the speed and the d-axis current to track the reference commands as well. In order to achieve maximum torque/ampere control, the relationship of the d-axis current command and the q-axis current was developed in a previous paper, and can be expressed as [9] id* =

2

gains. computing the derivative of the Lyapunov function, one can obtain 1 1   V = e1e1 + s2 s2 + s3 s3 + d2 d2T + d3 d3T

+ ΔTd −e1 Lg 3 h1 ( x) − k1 Lg 3 h1 ( x) s2 − Lg 3 L fˆ h1 ( x) s2

(62)

)

Assume that the lumped parameter uncertainties are varied slowly. Then, the derivative of d 2 and d 3 are zero during a sampling interval. From (62), the following adaptive laws can be obtained as  d = −γ s ϕ T ( x ) (63) 2

2 2

2

 d3 = −γ 3 s3ϕ3T ( x) By substituting (63) and (64) into (62), one can obtain V ≤ −k1e12 + e1e2 − k2 s22 − k3 s32 − η3 s3 − η2 s2 From (65), it is easy to find a new function as

- 113 -

(64) (65)

W (t ) = e T Qe + k3 s32 + η3 s3 + η 2 s2

Where e T = [ e1 e2 ] , and the symmetric matrix Q can be expressed as −1 / 2   k (67) Q= 1 − 1 / 2 k2   By choosing k1 and k2 , the matrix Q can be guaranteed as

1 >0 (68) 4 From (65)-(68), one can obtain the following expression as (69) V ≤ −W (t ) ≤ 0 As you can observe, the derivative of the Lyapunov function is less or equal to zero, and it means that the proposed control system is stable. From (57) and (69), we can obtain that e1 , s2 , s , d , and d are bounded. Then, by integrating (69), we can det Q = k1k2 −

3

2

obtain

IV. EXPERIMENTAL RESULTS

(66)

The IPMSM used in this paper was made by the Shin-Ding Company, typed 130-750MS-ZK-L2. The parameters of the IPMSM are shown in Table I. The sampling intervals of the drive system are: 106 μ s for inverter switching interval, and 1.06ms for speed-control sampling interval. In this paper, a digital signal processor, TMS320LF2407, is used to execute the proposed control algorithm including the input-output feedback linearization technique, the external load estimator, and the adaptive backstepping sliding mode controller. Finally, the DSP computes the d-axis and the q-axis voltages to the inverter. A closed-loop control system with the maximum torque/ampere control is thus achieved. Table I The Parameters of Motor

Po

3

0 ( k1e1 (τ ) − e1(τ )e2 (τ ) + k2 s2 (τ ) + k3s3 (τ ) ) dτ ∞ ∞ ≤  W (τ )dτ ≤ − V (τ )dτ = V (0) − V (∞) < ∞ 0 0 ∞

2

2

rs , Ω Ld Lq

2

(70)

λm

From (69), we can guarantee e1 , s2 , s3 ∈ L2 . Assume that control laws u and u are bounded, therefore, e , s , and s 1

2

1

2

J mo Bmo

3

J mo Bmo

are also bounded. Combining the previous results and using Barbalat’s lemma, the following results can be concluded as lim e1 (t ) = 0 (71) lim s2 (t ) = 0

(72)

lim s3 (t ) = 0

(73)

t →∞ t →∞

From (71)-(73), we can conclude that the closed-loop system is asymptotically stable even if lumped parameter uncertainties, external load disturbances, and estimation errors exist. For the control purpose, the guarantee in (71)-(73) does not imply that the estimated errors d2 and d3 can approach to zero values without satisfying the persistent excitation condition. Substituting (59) and (60) into (42), (43), and (26), we can obtain the control input voltages vd* and vq* , which are shown in Fig. 1.

Z1

* rm u1 = ω − k12e1 − L2fˆ h1 ( x ) − dˆ2ϕ2

k1e1

ω ω

* rm * rm

id*

e1

−

+

Tˆd

L fˆ h1

Z2

*

+ + Z2

e2

−

+

e3

+ − Z3

+ k1e2 + k2 s2 + k4 e1 + η2 sat ( s2 / φ2 )

u2 = id* − L fˆ h2 ( x) − dˆ3ϕ3

+ k3 s3 + k5 e3 + η3 sat ( s3 / φ3 )

s2 s2 s3

s3

iq

u1 u2

L L h L L h 

 0   Lg1h2 2 g1 fˆ 1

2 g 2 fˆ 1

−1

vd*

ωrm

vq*

dˆ2 dˆ3

 d2 = −γ 2 s2ϕ2T  d3 = −γ 3 s3ϕ3T

Fig. 1. Computation of the control input voltages.

id





t →∞

The designed gains for the proposed control scheme are selected as η 2 = 67500 , η3 = 546 , k1 = 13 , k2 = 313 , k3 = 4656 , γ 2 = 0.0163 , and γ 3 = 0.00291 . Several experimental results are shown from comparing the traditional PI controller, the pole placement controller, and the proposed controller in order to verify the effectiveness of the latter. Fig. 2 shows the measured transient response of the proposed controller, the PI controller, and the pole placement controller. As you can observe, the proposed controller has a faster transient response than both the PI controller and the pole placement controller; however, the three kinds of controllers have the same steady-state characteristics under no load disturbances. Fig. 3(a)(b) show the experimental results of the proposed controller and the pole placement controller starting with constant load torque. As you can see, when the motor is started with 1.5 N.m load, the proposed controller has the better performance than the pole placement controller. By using the same controller gains as in Fig. 2, the proposed controller can track the reference speed well as shown in Fig. 3(b). Fig. 4(a)(b) show the experimental results of the proposed controller and the pole placement controller tracking periodic sinusoidal-wave commands. Without using the external load torque estimator, the pole placement controller shows the undesirable dynamic performance tracking the sinusoidal-wave commands as shown in Fig. 4(a); however, the tracking response is obviously improved as shown in Fig. 4(b) by using the proposed control scheme.

- 114 -

ωrm * ωrm

* ωrm

* ω rm

Z2

iq

id

Fig. 2. Comparison of transient responses. * ωrm

ωrm *  rm ω

(b) Fig. 4. Speed response, acceleration response, and d-q axis current responses of periodic sinusoidal-wave command. (a) Conventional control scheme. (b) Proposed control scheme.

Z2

V. CONCLUSIONS iq

In this paper, an adaptive backstepping sliding mode controller has been designed and implemented for an IPMSM speed control system. The proposed system has good transient responses, load disturbance responses, and tracking responses even if the lumped parameter uncertainties, external load disturbances, and load torque estimation errors exist. This paper provides a new direction of applying advanced controller in IPMSM speed control system.

id

(a) * ωrm

ωrm Z2

ACKNOWLEDGMENT * ω rm

This research was supported by the National Science Council of the Republic of China under grant NSC 98-2218-E002-004.

iq

REFERENCES

id

[1]

Tˆd [2] (b) Fig. 3. Speed response, acceleration response, and d-q axis current responses at 500 rpm with 1.5 N.m load. (a) Conventional control scheme. (b) Proposed control scheme.

[3]

[4]

ωrm * ωrm

*  rm ω

[5]

[6]

Z2

[7]

iq

id [8] [9] (a)

- 115 -

Y. A. I. Mohamed and T. K. Lee, “Adaptive self-tuning MTPA vector controller for IPMSM drive system,” IEEE Trans. Energy Conver., vol. 21, no. 3, pp. 636-644, Sep. 2006. M. A. Rahman, D. M. Vilathgamuwa, M. N. Uddin, and K. J. Tseng, “Nonlinear control of interior permanent-magnet synchronous motor,” IEEE Trans. Ind. Appl., vol. 39, no. 2, pp. 408-416, Mar./ Apr. 2003. J. L. Shi, T. H. Liu, and Y. C. Chang, “Adaptive controller design for a sensorless IPMSM drive system with a maximum torque control,” IEE Proc. – Electr. Power Appl., vol. 153, no. 6, pp. 823-833, Nov. 2006. C. K. Lin, T. H. Liu, and S. H. Yang, “Nonlinear position controller design with input-output linearisation technique for an interior permanent magnet synchronous motor control system,” IET Power Electr., vol. 1, no. 1, pp. 14-26, Mar. 2008. J. Zhou and Y. Wang, “Adaptive backstepping speed controller design for a permanent magnet synchronous motor,” IEE Proc. – Electr. Power Appl., vol. 149, no. 2, pp. 165-172, Mar. 2002. I. C. Baik, K. H. Kim, and M. J. Youn, “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique,” IEEE Trans. Control Syst. Technol., vol. 8, no. 1, pp. 47-54, Jan. 2000. J. E. Slotine and W. Li, Applied Nonlinear Control. Prentice-Hall : New Jersey, 1991. A. Isidori, Nonlinear Control Systems. Springer : New York, 1995. Y. A. R. I. Mohamed and T. K. Lee, “Adaptive self-tuning MTPA vector controller for IPMSM drive system,” IEEE Trans. Energy Convers., vol. 21, no. 3, pp. 636-644, Sep. 2006.