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Design and implementation of a grey sliding mode controller for synchronous reluctance motor drive
ARTICLE IN PRESS
Control Engineering Practice 12 (2004) 155–163
Design and implementation of a grey sliding mode controller for synchronous reluctance motor drive Huann-Keng Chianga,*, Chih-Huang Tsengb a
Department of Electrical Engineering, National Yunlin University of Science and Technology, 123 University Road, Section 3, Electrical Engineering, Touliu Yunlin 640, Taiwan, ROC b Graduate School of Engineering Science and Technology, National Yunlin University of Science and Technology, ROC Received 8 November 2001; accepted 30 January 2003
Abstract A synchronous reluctance motor (SynRM) driven by a grey sliding mode controller is presented. The mathematical model for the SynRM includes core loss. We develop a maximum power efficiency control strategy. The grey sliding mode controller is used to reject the uncertain bounded disturbances and parameter variations. The grey prediction can forecast and reject the disturbances and parameter variations. The grey sliding mode controller is implemented using dSPACE DS1102 processor board. This system has a fast response and a good disturbance rejection capability. Simulation and experimental results show that the proposed controller is valid for the SynRM. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Synchronous reluctance motor; Core loss; Sliding mode control; Grey prediction; Maximum power efficiency control
1. Introduction The Synchronous Reluctance Motor (SynRM) has a mechanically simple and robust structure. It can rotate at high speeds in high temperature environments. The SynRM has been frequently discussed in the field of high performance ac motor drives (Xu, Xu, Lipo, & Novotny, 1991; Boldea, 1996; Jovanovic & Betz, 1999). The rotor circuit is opened such that the flux linkage of the SynRM is directly proportional to the stator currents. Hence, the torque of a SynRM can be controlled by adjusting the stator current. In general, we only consider the copper loss in motor modeling analysis (Betz, 1992; Jovanovic & Betz, 1999). The core losses of hysteresis and eddy currents are neglected. Core loss makes the torque decrease in acceleration and increase during braking (Xu et al., 1991; El-Antably & Lipo, 1997; Lee, Kang, & Sul, 1999; Sharf-Eldin, Dunnigan, Fletche, & Williams, 1999). Core loss also decreases the efficiency, at high switching *Corresponding author. Tel.: +886-5-5342601 ext 4247; fax: +8865-5312065. E-mail address: [email protected] (H.-K. Chiang).
frequencies, of pulse width modulation. Therefore, the SynRM model including core loss is considered in this paper. One of the popular methods about robust control is the so-called variable structure control (Itkis 1976; Utkin, Guldner, & Shi, 1999). It has been proven as an effective and robust control technology in SynRM (Liu & Lin, 1996; Sharf-Eldin et al., 1999; Shyu, Lai, & Tsai, 2000). The variable structure control can offer fast dynamic response, insensitivity to parameter variations and external disturbances rejection. However, this theory still has the problem of state catching and chattering. We used integral variable structure strategy to solve the catching state problem, and a boundary layer to reduce the chattering phenomenon. The grey theory proposed by Deng (1982) had been successfully employed in motor control systems (Tsai & Lu, 2000; Lu & Tsai, 2000). However, there are no reports regarding the use of the grey theory in the SynRM control systems. The grey prediction method only requires a few sampled data to develop the grey model and to forecast the future. In this paper, we first apply the grey sliding mode controller in SynRM
0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0967-0661(03)00019-4
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
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including the core loss for the maximum power efficiency control (MPEC) strategy. SynRM modeling, including the core loss, is discussed in Section 2. In Section 3, the sliding mode control is used to reject uncertain bounded disturbances and system parameter variations. We propose the grey sliding mode controller to forecast the state in Section 4. The grey predictor makes the state into the boundary layer when it exceeds the allowable deviation range and reduces the chattering phenomenon. In Section 5, we use the MPEC strategy for the SynRM. In Section 6, the simulation and experimental results are presented to validate the proposed grey sliding mode controller. Our conclusions are presented in Section 7.
2. Modeling of the SynRM Fig. 1 shows the d–q equivalent circuits of a SynRM in a synchronously rotating rotor reference frame, including core losses. The voltage equations of the SynRM are represented as (Xu et al., 1991; Kang & Sul, 1998) didt Vd ¼ Rs ids or Lq iqt þ Ld ; ð1Þ dt diqt ; ð2Þ dt where Vd and Vq are the direct axis and quadrature axis terminal voltages, ids and iqs the direct axis and quadrature axis terminal currents, idt and iqt the direct axis and quadrature axis torque producing currents, Ld and Lq the direct axis and quadrature axis magnetizing inductances, Rs and Rc the stator resistance and core loss per phase and or is the speed of the rotor. The electromagnetic torque Te and motor dynamic equation are stated as Vq ¼ Rs iqs þ or Ld idt þ Lq
Te ¼ 32PðLd Lq Þidt iqt ;
ð3Þ
dor þ Bm or ; ð4Þ dt where TL is torque of load, Jm the inertia moment of the rotor, Bm the viscous friction coefficient and P the number of pole pairs. The torque producing currents idt and iqt differ from the stator currents ids and iqs ; respectively. The terminal currents ids and iqs can be measured but the torque Te TL ¼ Jm
ids +
Vd
−
Rs
− ω r Lq iqt idt + − idc Rc Ld
iqs
Rs
+
Vq
Rc
iqt ω+ r Ld−idt iqc Lq
−
Fig. 1. Equivalent circuit of SynRM including core loss.
producing currents idt and iqt must be calculated. The core loss resistance Rc is difficult to measure in the transition state. Hence, we ignore the inductance transition voltage in the steady state, and the relationship between stator currents and torque currents are represented as 1 ids ¼ idt ðor Lq iqt Þ; ð5Þ Rc iqs ¼ iqt þ
1 ðor Ld idt Þ: Rc
ð6Þ
Substituting (5) and (6) into (1) and (2), we obtain the voltage equations as Rs didt ; ð7Þ Vd ¼ Rs idt or 1 þ Lq iqt þ Ld Rc dt diqt Rs Vq ¼ Rs iqt þ or 1 þ : ð8Þ Ld idt þ Lq Rc dt
3. The sliding mode control Defining the velocity error eðtÞ ¼ or oref ; oref is the velocity command. Assume that the time required to change the velocity command is much longer than the velocity response time (i:e: doref =dt ¼ 0). The velocity error differential equation of SynRM can be expressed as follows: deðtÞ Bm 1 1 ¼ or þ Te þ TL dt Jm Jm Jm * * L ðtÞ; ¼ aeðtÞ * þ ao * ref þ buðtÞ þ dT ð9Þ where a*
Bm ¼ a0 þ Da; Jm
1 b* ¼ b0 þ Db; Jm 1 ¼ d0 þ Dd; d* Jm u Te : The subscript index ‘‘o’’ and ‘‘D’’ symbol express the system parameter nominal and variation values, respectively. To have complete robustness, the sliding function S is combined with the integration of the state (Liu & Lin, 1996; Utkin et al., 1999) as Z t S ¼ eðtÞ þ c eðtÞ dt; c > 0: ð10Þ N
If we want the system to be stable, the velocity error integration and the velocity error must be reversed. Therefore, the sliding line S ¼ 0 must be designed in the second and fourth quadrants. The larger c value of (10) shows that the system has a faster dynamic response.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
The sliding mode controller has different dynamic responses, depending on the initial condition of the integrator in the sliding line. In other words, the system dynamic response can be controlled by selecting the initial condition of the integrator. At t ¼ 0; Eq. (10) can be expressed as S ¼ eð0Þ þ cIð0Þ;
ð11Þ
where eð0Þ is the initial condition of the error e and Ið0Þ is the initial condition of the integrator which is defined as Z 0 Ið0Þ ¼ eðtÞ dt: ð12Þ N
We choose the initial condition as oref or Ið0Þ ¼ : ð13Þ c The sliding mode control satisfies the hitting condition in the sliding surface and no reaching time when t ¼ 0; and e-0 for t-N:
157
where the parameters a and b are called the development coefficient and the grey input, respectively. The whitening equation corresponding to the grey differential Eq. (19) is dxð1Þ m þ axð1Þ ð20Þ m ¼ b: dt The parameters a and b can be solved by means of the least-square method as follows: " # a ð21Þ ¼ ðBT BÞ1 BT YN ; b where 2
zð1Þ m ð2Þ
6 ð1Þ 6 zm ð3Þ B¼6 6 ^ 4 zð1Þ m ðnÞ
1
3
7 17 7; ^7 5 1
T ð0Þ ð0Þ YN ¼ ½xð0Þ m ð2Þ; xm ð3Þ; y; xm ðnÞ :
The prediction value of the GM model with respect to the data sequence xð1Þ m is given by b ak b ð0Þ ð22Þ x# ð1Þ ðk þ 1Þ ¼ x ð1Þ þ ; e m m a a
4. Design of the grey sliding mode controller 4.1. The grey prediction (Lu & Tsai, 2000; Wong & Chen, 1998) Let xð0Þ be the original data sequence ð0Þ
x
ð0Þ
ð0Þ
ð0Þ
¼ fx ð1Þ; x ð2Þ; y; x ðnÞg;
ð14Þ
where n is the sampling size of the recorded data. The original data sequence xð0Þ may include positive and negative data simultaneously. Hence, we adopt the mapping generating operation method (MGO) to map the original sequence xð0Þ into the non-negative sequence xð0Þ m ð1Þ ð1Þ ð1Þ xð1Þ m ¼ fxm ð1Þ; xm ð2Þ; y; xm ðnÞg:
ð15Þ
The relation of xð0Þ and xð0Þ m are represented as follows: ð0Þ xð0Þ m ðkÞ ¼ a þ bx ðkÞ;
ð16Þ
where a and b are the arbitrary positive constants. We obtain the first-order accumulated generating operation (AGO) sequence as ( ) k X ð0Þ ð1Þ ð0Þ ð1Þ ð0Þ xm ¼ fxm ð1Þ; xm ð2Þ; y; xm ðnÞg ¼ xm ðiÞ : ð17Þ i¼1
zð1Þ m
Let to xð1Þ m :
be the following MEAN generating operation
ð1Þ 1 ð1Þ zð1Þ m ðkÞ ¼ 2ðxm ðk 1Þ þ xm ðkÞÞ;
k ¼ 2; 3; y; n:
ð18Þ
We form the following GM(1,1) first-order differential equation as dxð1Þ m þ azð1Þ m ¼ b; dt
ð19Þ
where L denotes the predicted value. Next, we take the inverse accumulated generating operation (IAGO) on # ð0Þ x# ð1Þ m ðk þ 1Þ; the forecasting value of x m ðk þ 1Þ can be expressed as b ð0Þ ð0Þ x# m ðk þ 1Þ ¼ xm ð1Þ ðeak eaðk1Þ Þ: ð23Þ a Similarly, the inverse mapping generating operation (IMGO) should be applied to x# ð0Þ m for obtaining the prediction value of the original data sequence which is defined as 1 b ak ð0Þ ð0Þ aðk1Þ xm ð1Þ ðe x# ðk þ 1Þ ¼ e Þ a : ð24Þ b a The total procedure of the grey predictor is shown in Fig. 2. 4.2. The grey sliding mode controller In the variable structure control system, sliding mode guarantees the robustness of the system. The chattering phenomenon by the switching control makes the system to be influenced by the external disturbance and parameter variations. The integral variable structure control with a boundary layer can reduce the chattering phenomenon for the disturbances and parameter variations. We combine the grey predictor and sliding mode technique to reduce the sensitivity for the disturbances and the parameter variations. We need not know the
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
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x (0) (k )
MGO
∧ (1 )
x m( 1 ) ( k )
x m( 0 ) ( k )
AGO
∧ (0)
IAGO
GM(1,1)
∧ (0)
x m ( k + 1)
x m ( k + 1)
x
( k + 1)
IMGO
Fig. 2. Operation of the grey predictor.
x1s < − w
x2
x2
x2
x2 s < −w x1
x1s > w
x2 s > w x1s < − w (a)
x1s ≤ w
x2s > w
x1s > w
x1 x2 s < −w
s=0
x2 s ≤ w
x1
s=0
s =0 (b)
(c)
Fig. 3. Boundary layer form: (a) the boundary layer jx1 Sj exceeds the region of boundary width w; (b) the boundary layer jx2 Sj exceeds the region of boundary width w; (c) the boundary layer jx1 Sj and jx2 Sj inside the region of boundary width w:
uncertain bounded disturbances and parameter variations. The sliding function is estimated to reject the disturbances and the parameter variations. The grey predictor design is represented as the following: First, we design a boundary layer with a boundary # x1 and x2 are the width w in the phase plane. The S; S; sliding function, prediction sliding function, motor angle and motor velocity, respectively. The boundary layer form is shown in Fig. 3. The grey predictor forecasts the movement of the state in the following conditions: (a) The positive step prediction: jx1 S j > w or jx2 S j > w and one of the following four conditions is satisfied # þ 1Þ SðkÞ > 0 if SðkÞ > 0 and Sðk # þ 1Þ > 0; 1. Sðk # # 2. Sðk þ 1Þ SðkÞo0 if SðkÞo0 and Sðk þ 1Þo0; # þ 1Þo0; 3. SðkÞ > 0 and Sðk # 4. SðkÞo0 and Sðk þ 1Þ > 0: (b) The negative step prediction: jx1 Sj > w or jx2 S j > w and one of the following two conditions is satisfied # þ 1Þ SðkÞo0 if SðkÞ > 0 and Sðk # þ 1Þ > 0; 1. Sðk # # þ 1Þo0: 2. Sðk þ 1Þ SðkÞ > 0 if SðkÞo0 and Sðk The foregoing forecast action is shown in Fig. 4. We add the positive step prediction control to prompt the state into the boundary layer when the estimation state far away. Similarly, we add the negative step predition control when the state outside the boundary layer and the estimation state go toward the boundary. Finally, the state will be toward the origin and stays in the neighborhood of the origin by the integral variable structure control. For avoiding violent tremble, the control input uðtÞ (the electromagnetic torque Te ) can be defined as uðtÞ ¼ u0eq ðtÞ þ un ðtÞ;
ð25Þ
x2
x2
1 3 2
1
x1
x1
4
2
S =0
(a)
S =0
(b)
Fig. 4. Grey predictor forecast conditions for the state point move direction ( ) and the grey forecast move direction (yy) using (a) the positive step prediction and (b) the negative step prediction.
where u0eq ðtÞ is used to control the overall behavior of the system and un ðtÞ is used to reject disturbances and to suppress parameter uncertainties. To satisfy the equivalent control concept, Eqs. (9) and (25) are substituted into S’ ¼ 0; we get 1 u0eq ¼ ½ða0 þ cÞe þ a0 oref Þ
b
ð26Þ
and ( x2 ¼
# un ¼ x1 S x2 S; s>0 0
for positive or negative step prediction; otherwise; ð27Þ
where x1 represents the reciprocal of time constant of the system dynamic response when the state trajectory is into the sliding line, x2 represents the gain of the positive or negative step prediction and S# represents the forecast value of the sliding function S by the grey predictor. The
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
grey sliding mode controller satisfies the hitting condition at the sliding line and no reaching time when t ¼ 0; and the error of e in the boundary layer w when t-N:
159
5. The MPEC strategy
the model including core loss resistance to maximize the power efficiency. The losses concentrated in the stator and the rotor due to the flux ripple are assumed to be negligible. From (5) and (6), we know ids aidt and iqs aiqt : Hence, the total power losses are represented as
A comparative study of different control methods of the SynRM was already reported (Betz, 1992; Betz, Lagerquist, & Jovanovic, 1993). The performances of the ideal motor model are analyzed for different control techniques. In practice, the forementioned current angle cannot satisfy the characteristics of the control strategies for the motor including the core loss. In this paper, we derive the minimal power losses control strategy from
Ptotal ¼ Pcopper þ Pcore 3 ðor Ld Þ2 Rs 2 2 Rs þ ¼ þ 2 ðor Ld Þ idt 2 Rc Rc ( ) ðor Lq Þ2 Rs 3 2 Rs þ þ þ 2 ðor Lq Þ2 iqt 2 Rc Rc 3 2Rs þ or ðLd Lq Þ idt iqt : 2 Rc
ω ref +
ωr
∆ωr
Te*
IVSC speed controller
-
MPEC core comp. & optimal current angle
+
+
iqt* -
i qt
idt
ωr
V I S G V gating B P T W M
Vd* Vα*
i dt* -
PI current controller
2φ / 2φ
Vq* Vβ* ids
core compensation
ð28Þ
SynRM
ias ibs encoder
3φ / 2φ
θ
i qs
o
d dt
Fig. 5. Experiment structure of SynRM
0 600
velocity (rpm)
velocity error (rad/s)
-10 500
C=8 C=6
400
C=4
300 200
-20 -30 -40 -50
100
-60
0
-70
C= 8
0
0.5
1
(a)
1.5
2
0
2
4
6
(b)
time (sec)
C= 4
C= 6
8
10
12
14
16
integral velocity error (rad)
Fig. 6. Simulation results for the speed command 600 rpm with different sliding lines c: (a) speed responses and (b) phase plane trajectories.
900
0
800
-10 800 rpm
velocity error (rad/s)
velocity (rpm)
700 600 600 rpm
500 400 400 rpm
300
-20 -30 -40
-60
200
-70
100
-80
0
(a)
0
0.5
1
time (sec)
1.5
2
-90
(b)
400 rpm
-50
600 rpm
800 rpm 0
5
10
15
integral velocity error (rad)
Fig. 7. Simulation results for the different speed commands with the same sliding line c ¼ 6: (a) speed responses and (b) phase plane trajectories.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
160
10
700
0
500
velocity error (rad/s)
velocity (rpm)
600 C= 8 C= 6
400 C= 4
300 200
-20 -30 -40 -50
100 0
-10
-60 0
0.5
1
(a)
1.5
C= 8
-70 -2
2
0
2
(b)
time (sec)
4
C= 4
C= 6
6
8
10
12
14
integral velocity error (rad)
Fig. 8. Experimental results for the 600 rpm reference speed command with different sliding lines c: (a) speed responses and (b) phase plane trajectories.
1000
10
900
0
800
-10 -20
velocity error
velocity (rpm)
800 r.p.m
700 600 600 r.p.m
500 400 400 r.p.m
-30
-50
300
-60
200
-70
100
-80
0
(a)
0
0.5
1
1.5
2
time (sec)
400 r.p.m
-40
(b)
600 r.p.m
800 r.p.m
-90 -2
0
2
4
6
8
10
12
integral velocity error
4
4
3
3
2
2
1
1
ias (A)
ias (A)
Fig. 9. Experimental results for the different speed commands with the same sliding line c ¼ 6: (a) speed responses and (b) phase plane trajectories.
0
0
-1
-1
-2
-2
-3
-3 -4
-4 2.75
2.8
2.85
2.9
2.95
3
0.5
0.52
0.54
8
6
6
4
4
i-beta (A)
i-beta (A)
8
2 0
-2 -4
-6
-6
(a)
-4
-2
0
2
i-alpha (A)
0.6
0.62
0
-4
-6
0.58
2
-2
-8 -8
0.56
time (sec)
time (sec)
4
6
8
-8 -8
(b)
-6
-4
-2
0
2
4
6
8
i-alpha (A)
Fig. 10. Experimental results: the stator current ias and the ia and ib currents in phase plane with speed command 600 rpm and c ¼ 6 by using (a) sign() function and (b) proposed controller.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
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¼ sgnðTe Þjopt idt : iqt
If the core losses are ignored, i.e. Rc -N; the optimal current ratio of (29) is jopt ¼ 71: The current angle is jopt ¼ 7p=4: From the electromagnetic torque (3) and optimal current ratio (29), we can obtain the command torque current as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi T e ; ð30Þ idt ¼ 3 PðL L d q Þ jopt 2
6. The simulation and experimental results 6.1. Simulation results The proposed system, shown in Fig. 5, has verified the global behavior and the robustness of the sliding mode control with grey prediction. We simulate it by using the Simulink of Matlab. For simplifying the simulation process, the parameters of inductance and all resistances
900
900
800
800
700
700
600
600
500 400
500 400
300
300
200
200
100
100
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
0
0.5
1
1.5
2
time (sec)
3
3.5
4
4.5
5
5
3
4
2
3
1
2
ia (A)
ia (A)
2.5
time (sec)
4
0
1
-1
0
-2
-1
-3
-2
-4
ð31Þ
By combing (5), (6), (30) and (31), we can have the torque current loop.
velocity (rpm)
velocity (rpm)
Let the ratio of d-axis and q-axis currents be j ¼ iqt =idt and assume the torque command as constant on one sampling time, i.e., idt iqt is constant. Let dPtotal =dj ¼ 0; the optimal current ratio jopt can be obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rs R2c þ Rc ðor Ld Þ2 þ Rs ðor Ld Þ2 : ð29Þ jopt ¼ 7 Rs R2c þ Rc ðor Ld Þ2 þ Rs ðor Lq Þ2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
5
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
time (sec) 6
2 5 1.5 4
torque (Nt-m)
torque (Nt-m)
1 0.5 0
0
-1
(a)
2
1
-0.5
-1.5 0
3
0.5
1
1.5
2
2.5
time (sec)
3
3.5
4
4.5
-1
5
(b)
0
0.5
1
1.5
2
2.5
time (sec)
Fig. 11. Experimental results: the speed regulation, the stator current ia ; and electromagnetic torque Te by using (a) sat( ) function and (b) proposed controller.
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162
are assumed constant in the sampling period. The motor parameters are shown in Appendix. The initial conditions of the integrator are set in the sliding line with different constants c: Fig. 6 shows the simulation results of the MPEC with core losses compensation. Fig. 6(a) shows that the dynamic response has a faster transient response when a larger constant c is chosen. The phase plane trajectories of Fig. 6(b) follow the sliding line toward the origin with different sliding line. The different low speed responses are shown in Fig. 7(a). In Fig. 7(b), the phase plane trajectories follow the sliding line toward the origin.
The initial conditions of the integrator are set in the sliding line and adding the machine load 0.2 N-m at the beginning of the experimentation. Fig. 8 shows the experimental results of speed responses and phase plane trajectories. The dynamic response has a faster transient speed response when a larger constant c is chosen as shown in Fig. 8(a). From Fig. 8(b), the phase plant trajectories follow the sliding line toward the origin and stay in the neighborhood of the origin. The different speed responses in the low speed region are shown in Fig 9(a). In Fig. 9(b), the phase plane trajectories follow the sliding line toward the origin and stay in the neighborhood of the origin.
6.2. Experimental results The dSPACE DS1102 control board is adopted to act as the digital system speed control. The SynRM is driven using IGBTs by a three-phase voltage space vector pulsewidth modulation (VSVPWM) inverter. The sampling period of control rules is set as 200 ms.
Table 1 Parameters of SynRM (0.37 kW) Rs 4.2 O Jm 0.00076 kg m2
Rc 50 O (f ¼ 60 Hz) Bm 0.00012 N-m s
400
Lq 0.181 H
900
700
450
Ld 0.328 H P 1
800
600
700
350
250 200 150
velocity (rpm)
velocity (rpm)
velocity (rpm)
500 300
400 300
600 500 400 300
200 200
100 100
50 0
0
0.5
1
1.5
2
2.5
0
3
100
0
0.5
1
time (sec)
1.5
2
2.5
0
3
0
0.5
1
time (sec)
4
2
2.5
3
2
2.5
3
2
2.5
3
9
7
4.5
1.5
time (sec)
8
6
7
3.5 5
6
2.5 2 1.5
4
torque (nt-m)
torque (nt-m)
torque (nt-m)
3
3 2
5 4 3 2
1 1
1
0.5 0
0 -0.5
0
0.5
1
1.5
2
2.5
-1
3
0 -1 0
0.5
1
time (sec)
1.5
2
2.5
3
0
0.5
1
time (sec)
6
6
4
4
2
2
1.5
time (sec) 6 4 2
i (A)
i (A)
i (A)
0 0
0
-2
-2
-2
-4
-4
-4
-6
(a)
0
0.5
1
1.5
time (sec)
2
2.5
-6
3
(b)
-6 -8 0
0.5
1
1.5
time (sec)
2
2.5
3
(c)
0
0.5
1
1.5
time (sec)
Fig. 12. The speed, electromagnetic torque and a-phase line current responses with a 0.2 N-m machine load is added at the beginning: (a) the speed command 400 rpm and a 1.7 N-m disturbance is added at 0.9 s; (b) the speed command 600 rpm and a 1.6 N-m disturbance is added at 0.9 s; and (c) the speed command 800 rpm and a 0.9 N-m disturbance is added at 1.1 s.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
In order to reduce the chattering phenomenon, a sign() function and a boundary layer with grey prediction are used as shown in Fig. 10. Fig. 10(a) shows that the steady-state a-phase current ias has a seriously larger harmonic and the stator current trajectory in ia ; ib phase plane is largely varied by using the sign() function. From Fig. 10(b), by the proposed controller, the a-phase current ias is very close to a sinusoidal waveform and the stator current trajectory ia; ib in the phase plane is smooth. Therefore, the chattering phenomenon in the integral variable structure controllers with a boundary layer by grey prediction has a better performance than using the sign() function. In Fig. 11, the reference speed commands are changed from 0 to 600 rpm, 600 to 800 rpm, and 800 to 400 rpm by using the sat() function and the proposed controller, respectively. The speed regulation is satisfied for the proposed controller. It has better transient response than using the sat() function especially from 800 to 400 rpm. For the external load disturbance regulation, the dynamic responses with different speed commands are shown in Fig. 12. The robust control performance of the proposed controller in the command tracking and load disturbance regulation is obvious.
7. Conclusions A complete model development and analysis for the grey sliding mode speed control of synchronous reluctance motor including the core loss is first presented in this paper. The mathematical model of synchronous reluctance motor includes the core loss. We develop a maximum power efficiency control strategy. The grey sliding mode controller is used to reject the load disturbance and parameter variations. We need not know the uncertain bounded disturbances and parameter variations. The grey prediction can forecast the system state variations and force the state into the boundary layer. The boundary layer gets a smooth current trajectory and reduces the chattering phenomenon. Finally, we employ the simulations and experiments to validate the proposed method.
163
Appendix The parameters of SynRM are shown in Table 1.
References Betz, R. E. (1992). Theoretical aspects of control of synchronous reluctance machines. IEE Proceedings B, 139(4), 355–364. Betz, R. E., Lagerquist, R., & Jovanovic, M. (1993). Control of synchronous reluctance machines. IEEE Transactions on Industry Applications, 29(6), 1110–1121. Boldea, R. E. (1996). Reluctance synchronous machines and drives. Oxford: Clarendon Press. Deng, J. L. (1982). Control problems of grey system. System and Control Letters, 1(5), 288–294. El-Antably, T. M. A, & Lipo, T. A. (1997). A new control strategy for optimum-efficiency operation of a synchronous reluctance motor. IEEE Transaction on Industry Applications, 33(5), 1146–1153. Itkis, U. (1976). Control system of variable structure. New York: Wiley. Jovanovic, M. G., & Betz, R. E. (1999). Effects of uncompensated vector control on synchronous reluctance motor performance. IEEE Transactions on Energy Conversion, 14(3), 532–537. Kang, S. J., & Sul, S. K. (1998). Highly dynamic torque control of synchronous reluctance motor. IEEE Transactions on Power Electronics, 13(4), 793–798. Lee, H. D., Kang, S. J., & Sul, S. K. (1999). Efficiency-optimized direct torque control of synchronous reluctance motor using feedback linearization. IEEE Transactions on Industrial Electronics, 46(1), 192–198. Liu, T. H., & Lin, M. T. (1996). A fuzzy sliding-mode controller design for a synchronous reluctance motor drive. IEEE Transactions on Aerospace and Electronic Systems, 32(3), 1065–1076. Lu, H. C., & Tsai, C. H. (2000). Grey-fuzzy implementation of direct torque control of induction machines. Electric Machines and Power Systems, 28, 1127–1139. Sharf-Eldin, T., Dunnigan, M., Fletche, J. E., & Williams, B. W. (1999). Nonlinear robust control of a vector-controlled synchronous reluctance machine. IEEE Transactions on Power Electronics, 14(6), 1111–1121. Shyu, K. K., Lai, C. K., & Tsai, Y. W. (2000). Optimal position control of synchronous reluctance motor via totally invariant variable structure control. IEE Proceeding-Control Theory and Applications, 147(1), 28–36. Tsai, C. H., & Lu, H. C. (2000). Design and implementation of a DSPbased grey-fuzzy controller for induction motor drive. Electric Machines and Power Systems, 28, 373–384. Utkin, V. I., Guldner, J., & Shi, J. (1999). Sliding mode control in electromechanical systems. London: Taylor & Francis. Wong, C. C., & Chen, J. Y. (1998). A hybrid grey PID controller design through fuzzy gain. The Journal of Grey System, 1, 13–29. Xu, L., Xu, X., Lipo, T. A., & Novotny, D. W. (1991). Vector control of a synchronous reluctance motor including saturation and iron loss. IEEE Transactions on Industry Applications, 27(5), 977–985.
Control Engineering Practice 12 (2004) 155–163
Design and implementation of a grey sliding mode controller for synchronous reluctance motor drive Huann-Keng Chianga,*, Chih-Huang Tsengb a
Department of Electrical Engineering, National Yunlin University of Science and Technology, 123 University Road, Section 3, Electrical Engineering, Touliu Yunlin 640, Taiwan, ROC b Graduate School of Engineering Science and Technology, National Yunlin University of Science and Technology, ROC Received 8 November 2001; accepted 30 January 2003
Abstract A synchronous reluctance motor (SynRM) driven by a grey sliding mode controller is presented. The mathematical model for the SynRM includes core loss. We develop a maximum power efficiency control strategy. The grey sliding mode controller is used to reject the uncertain bounded disturbances and parameter variations. The grey prediction can forecast and reject the disturbances and parameter variations. The grey sliding mode controller is implemented using dSPACE DS1102 processor board. This system has a fast response and a good disturbance rejection capability. Simulation and experimental results show that the proposed controller is valid for the SynRM. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Synchronous reluctance motor; Core loss; Sliding mode control; Grey prediction; Maximum power efficiency control
1. Introduction The Synchronous Reluctance Motor (SynRM) has a mechanically simple and robust structure. It can rotate at high speeds in high temperature environments. The SynRM has been frequently discussed in the field of high performance ac motor drives (Xu, Xu, Lipo, & Novotny, 1991; Boldea, 1996; Jovanovic & Betz, 1999). The rotor circuit is opened such that the flux linkage of the SynRM is directly proportional to the stator currents. Hence, the torque of a SynRM can be controlled by adjusting the stator current. In general, we only consider the copper loss in motor modeling analysis (Betz, 1992; Jovanovic & Betz, 1999). The core losses of hysteresis and eddy currents are neglected. Core loss makes the torque decrease in acceleration and increase during braking (Xu et al., 1991; El-Antably & Lipo, 1997; Lee, Kang, & Sul, 1999; Sharf-Eldin, Dunnigan, Fletche, & Williams, 1999). Core loss also decreases the efficiency, at high switching *Corresponding author. Tel.: +886-5-5342601 ext 4247; fax: +8865-5312065. E-mail address: [email protected] (H.-K. Chiang).
frequencies, of pulse width modulation. Therefore, the SynRM model including core loss is considered in this paper. One of the popular methods about robust control is the so-called variable structure control (Itkis 1976; Utkin, Guldner, & Shi, 1999). It has been proven as an effective and robust control technology in SynRM (Liu & Lin, 1996; Sharf-Eldin et al., 1999; Shyu, Lai, & Tsai, 2000). The variable structure control can offer fast dynamic response, insensitivity to parameter variations and external disturbances rejection. However, this theory still has the problem of state catching and chattering. We used integral variable structure strategy to solve the catching state problem, and a boundary layer to reduce the chattering phenomenon. The grey theory proposed by Deng (1982) had been successfully employed in motor control systems (Tsai & Lu, 2000; Lu & Tsai, 2000). However, there are no reports regarding the use of the grey theory in the SynRM control systems. The grey prediction method only requires a few sampled data to develop the grey model and to forecast the future. In this paper, we first apply the grey sliding mode controller in SynRM
0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0967-0661(03)00019-4
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including the core loss for the maximum power efficiency control (MPEC) strategy. SynRM modeling, including the core loss, is discussed in Section 2. In Section 3, the sliding mode control is used to reject uncertain bounded disturbances and system parameter variations. We propose the grey sliding mode controller to forecast the state in Section 4. The grey predictor makes the state into the boundary layer when it exceeds the allowable deviation range and reduces the chattering phenomenon. In Section 5, we use the MPEC strategy for the SynRM. In Section 6, the simulation and experimental results are presented to validate the proposed grey sliding mode controller. Our conclusions are presented in Section 7.
2. Modeling of the SynRM Fig. 1 shows the d–q equivalent circuits of a SynRM in a synchronously rotating rotor reference frame, including core losses. The voltage equations of the SynRM are represented as (Xu et al., 1991; Kang & Sul, 1998) didt Vd ¼ Rs ids or Lq iqt þ Ld ; ð1Þ dt diqt ; ð2Þ dt where Vd and Vq are the direct axis and quadrature axis terminal voltages, ids and iqs the direct axis and quadrature axis terminal currents, idt and iqt the direct axis and quadrature axis torque producing currents, Ld and Lq the direct axis and quadrature axis magnetizing inductances, Rs and Rc the stator resistance and core loss per phase and or is the speed of the rotor. The electromagnetic torque Te and motor dynamic equation are stated as Vq ¼ Rs iqs þ or Ld idt þ Lq
Te ¼ 32PðLd Lq Þidt iqt ;
ð3Þ
dor þ Bm or ; ð4Þ dt where TL is torque of load, Jm the inertia moment of the rotor, Bm the viscous friction coefficient and P the number of pole pairs. The torque producing currents idt and iqt differ from the stator currents ids and iqs ; respectively. The terminal currents ids and iqs can be measured but the torque Te TL ¼ Jm
ids +
Vd
−
Rs
− ω r Lq iqt idt + − idc Rc Ld
iqs
Rs
+
Vq
Rc
iqt ω+ r Ld−idt iqc Lq
−
Fig. 1. Equivalent circuit of SynRM including core loss.
producing currents idt and iqt must be calculated. The core loss resistance Rc is difficult to measure in the transition state. Hence, we ignore the inductance transition voltage in the steady state, and the relationship between stator currents and torque currents are represented as 1 ids ¼ idt ðor Lq iqt Þ; ð5Þ Rc iqs ¼ iqt þ
1 ðor Ld idt Þ: Rc
ð6Þ
Substituting (5) and (6) into (1) and (2), we obtain the voltage equations as Rs didt ; ð7Þ Vd ¼ Rs idt or 1 þ Lq iqt þ Ld Rc dt diqt Rs Vq ¼ Rs iqt þ or 1 þ : ð8Þ Ld idt þ Lq Rc dt
3. The sliding mode control Defining the velocity error eðtÞ ¼ or oref ; oref is the velocity command. Assume that the time required to change the velocity command is much longer than the velocity response time (i:e: doref =dt ¼ 0). The velocity error differential equation of SynRM can be expressed as follows: deðtÞ Bm 1 1 ¼ or þ Te þ TL dt Jm Jm Jm * * L ðtÞ; ¼ aeðtÞ * þ ao * ref þ buðtÞ þ dT ð9Þ where a*
Bm ¼ a0 þ Da; Jm
1 b* ¼ b0 þ Db; Jm 1 ¼ d0 þ Dd; d* Jm u Te : The subscript index ‘‘o’’ and ‘‘D’’ symbol express the system parameter nominal and variation values, respectively. To have complete robustness, the sliding function S is combined with the integration of the state (Liu & Lin, 1996; Utkin et al., 1999) as Z t S ¼ eðtÞ þ c eðtÞ dt; c > 0: ð10Þ N
If we want the system to be stable, the velocity error integration and the velocity error must be reversed. Therefore, the sliding line S ¼ 0 must be designed in the second and fourth quadrants. The larger c value of (10) shows that the system has a faster dynamic response.
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The sliding mode controller has different dynamic responses, depending on the initial condition of the integrator in the sliding line. In other words, the system dynamic response can be controlled by selecting the initial condition of the integrator. At t ¼ 0; Eq. (10) can be expressed as S ¼ eð0Þ þ cIð0Þ;
ð11Þ
where eð0Þ is the initial condition of the error e and Ið0Þ is the initial condition of the integrator which is defined as Z 0 Ið0Þ ¼ eðtÞ dt: ð12Þ N
We choose the initial condition as oref or Ið0Þ ¼ : ð13Þ c The sliding mode control satisfies the hitting condition in the sliding surface and no reaching time when t ¼ 0; and e-0 for t-N:
157
where the parameters a and b are called the development coefficient and the grey input, respectively. The whitening equation corresponding to the grey differential Eq. (19) is dxð1Þ m þ axð1Þ ð20Þ m ¼ b: dt The parameters a and b can be solved by means of the least-square method as follows: " # a ð21Þ ¼ ðBT BÞ1 BT YN ; b where 2
zð1Þ m ð2Þ
6 ð1Þ 6 zm ð3Þ B¼6 6 ^ 4 zð1Þ m ðnÞ
1
3
7 17 7; ^7 5 1
T ð0Þ ð0Þ YN ¼ ½xð0Þ m ð2Þ; xm ð3Þ; y; xm ðnÞ :
The prediction value of the GM model with respect to the data sequence xð1Þ m is given by b ak b ð0Þ ð22Þ x# ð1Þ ðk þ 1Þ ¼ x ð1Þ þ ; e m m a a
4. Design of the grey sliding mode controller 4.1. The grey prediction (Lu & Tsai, 2000; Wong & Chen, 1998) Let xð0Þ be the original data sequence ð0Þ
x
ð0Þ
ð0Þ
ð0Þ
¼ fx ð1Þ; x ð2Þ; y; x ðnÞg;
ð14Þ
where n is the sampling size of the recorded data. The original data sequence xð0Þ may include positive and negative data simultaneously. Hence, we adopt the mapping generating operation method (MGO) to map the original sequence xð0Þ into the non-negative sequence xð0Þ m ð1Þ ð1Þ ð1Þ xð1Þ m ¼ fxm ð1Þ; xm ð2Þ; y; xm ðnÞg:
ð15Þ
The relation of xð0Þ and xð0Þ m are represented as follows: ð0Þ xð0Þ m ðkÞ ¼ a þ bx ðkÞ;
ð16Þ
where a and b are the arbitrary positive constants. We obtain the first-order accumulated generating operation (AGO) sequence as ( ) k X ð0Þ ð1Þ ð0Þ ð1Þ ð0Þ xm ¼ fxm ð1Þ; xm ð2Þ; y; xm ðnÞg ¼ xm ðiÞ : ð17Þ i¼1
zð1Þ m
Let to xð1Þ m :
be the following MEAN generating operation
ð1Þ 1 ð1Þ zð1Þ m ðkÞ ¼ 2ðxm ðk 1Þ þ xm ðkÞÞ;
k ¼ 2; 3; y; n:
ð18Þ
We form the following GM(1,1) first-order differential equation as dxð1Þ m þ azð1Þ m ¼ b; dt
ð19Þ
where L denotes the predicted value. Next, we take the inverse accumulated generating operation (IAGO) on # ð0Þ x# ð1Þ m ðk þ 1Þ; the forecasting value of x m ðk þ 1Þ can be expressed as b ð0Þ ð0Þ x# m ðk þ 1Þ ¼ xm ð1Þ ðeak eaðk1Þ Þ: ð23Þ a Similarly, the inverse mapping generating operation (IMGO) should be applied to x# ð0Þ m for obtaining the prediction value of the original data sequence which is defined as 1 b ak ð0Þ ð0Þ aðk1Þ xm ð1Þ ðe x# ðk þ 1Þ ¼ e Þ a : ð24Þ b a The total procedure of the grey predictor is shown in Fig. 2. 4.2. The grey sliding mode controller In the variable structure control system, sliding mode guarantees the robustness of the system. The chattering phenomenon by the switching control makes the system to be influenced by the external disturbance and parameter variations. The integral variable structure control with a boundary layer can reduce the chattering phenomenon for the disturbances and parameter variations. We combine the grey predictor and sliding mode technique to reduce the sensitivity for the disturbances and the parameter variations. We need not know the
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x (0) (k )
MGO
∧ (1 )
x m( 1 ) ( k )
x m( 0 ) ( k )
AGO
∧ (0)
IAGO
GM(1,1)
∧ (0)
x m ( k + 1)
x m ( k + 1)
x
( k + 1)
IMGO
Fig. 2. Operation of the grey predictor.
x1s < − w
x2
x2
x2
x2 s < −w x1
x1s > w
x2 s > w x1s < − w (a)
x1s ≤ w
x2s > w
x1s > w
x1 x2 s < −w
s=0
x2 s ≤ w
x1
s=0
s =0 (b)
(c)
Fig. 3. Boundary layer form: (a) the boundary layer jx1 Sj exceeds the region of boundary width w; (b) the boundary layer jx2 Sj exceeds the region of boundary width w; (c) the boundary layer jx1 Sj and jx2 Sj inside the region of boundary width w:
uncertain bounded disturbances and parameter variations. The sliding function is estimated to reject the disturbances and the parameter variations. The grey predictor design is represented as the following: First, we design a boundary layer with a boundary # x1 and x2 are the width w in the phase plane. The S; S; sliding function, prediction sliding function, motor angle and motor velocity, respectively. The boundary layer form is shown in Fig. 3. The grey predictor forecasts the movement of the state in the following conditions: (a) The positive step prediction: jx1 S j > w or jx2 S j > w and one of the following four conditions is satisfied # þ 1Þ SðkÞ > 0 if SðkÞ > 0 and Sðk # þ 1Þ > 0; 1. Sðk # # 2. Sðk þ 1Þ SðkÞo0 if SðkÞo0 and Sðk þ 1Þo0; # þ 1Þo0; 3. SðkÞ > 0 and Sðk # 4. SðkÞo0 and Sðk þ 1Þ > 0: (b) The negative step prediction: jx1 Sj > w or jx2 S j > w and one of the following two conditions is satisfied # þ 1Þ SðkÞo0 if SðkÞ > 0 and Sðk # þ 1Þ > 0; 1. Sðk # # þ 1Þo0: 2. Sðk þ 1Þ SðkÞ > 0 if SðkÞo0 and Sðk The foregoing forecast action is shown in Fig. 4. We add the positive step prediction control to prompt the state into the boundary layer when the estimation state far away. Similarly, we add the negative step predition control when the state outside the boundary layer and the estimation state go toward the boundary. Finally, the state will be toward the origin and stays in the neighborhood of the origin by the integral variable structure control. For avoiding violent tremble, the control input uðtÞ (the electromagnetic torque Te ) can be defined as uðtÞ ¼ u0eq ðtÞ þ un ðtÞ;
ð25Þ
x2
x2
1 3 2
1
x1
x1
4
2
S =0
(a)
S =0
(b)
Fig. 4. Grey predictor forecast conditions for the state point move direction ( ) and the grey forecast move direction (yy) using (a) the positive step prediction and (b) the negative step prediction.
where u0eq ðtÞ is used to control the overall behavior of the system and un ðtÞ is used to reject disturbances and to suppress parameter uncertainties. To satisfy the equivalent control concept, Eqs. (9) and (25) are substituted into S’ ¼ 0; we get 1 u0eq ¼ ½ða0 þ cÞe þ a0 oref Þ
b
ð26Þ
and ( x2 ¼
# un ¼ x1 S x2 S; s>0 0
for positive or negative step prediction; otherwise; ð27Þ
where x1 represents the reciprocal of time constant of the system dynamic response when the state trajectory is into the sliding line, x2 represents the gain of the positive or negative step prediction and S# represents the forecast value of the sliding function S by the grey predictor. The
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grey sliding mode controller satisfies the hitting condition at the sliding line and no reaching time when t ¼ 0; and the error of e in the boundary layer w when t-N:
159
5. The MPEC strategy
the model including core loss resistance to maximize the power efficiency. The losses concentrated in the stator and the rotor due to the flux ripple are assumed to be negligible. From (5) and (6), we know ids aidt and iqs aiqt : Hence, the total power losses are represented as
A comparative study of different control methods of the SynRM was already reported (Betz, 1992; Betz, Lagerquist, & Jovanovic, 1993). The performances of the ideal motor model are analyzed for different control techniques. In practice, the forementioned current angle cannot satisfy the characteristics of the control strategies for the motor including the core loss. In this paper, we derive the minimal power losses control strategy from
Ptotal ¼ Pcopper þ Pcore 3 ðor Ld Þ2 Rs 2 2 Rs þ ¼ þ 2 ðor Ld Þ idt 2 Rc Rc ( ) ðor Lq Þ2 Rs 3 2 Rs þ þ þ 2 ðor Lq Þ2 iqt 2 Rc Rc 3 2Rs þ or ðLd Lq Þ idt iqt : 2 Rc
ω ref +
ωr
∆ωr
Te*
IVSC speed controller
-
MPEC core comp. & optimal current angle
+
+
iqt* -
i qt
idt
ωr
V I S G V gating B P T W M
Vd* Vα*
i dt* -
PI current controller
2φ / 2φ
Vq* Vβ* ids
core compensation
ð28Þ
SynRM
ias ibs encoder
3φ / 2φ
θ
i qs
o
d dt
Fig. 5. Experiment structure of SynRM
0 600
velocity (rpm)
velocity error (rad/s)
-10 500
C=8 C=6
400
C=4
300 200
-20 -30 -40 -50
100
-60
0
-70
C= 8
0
0.5
1
(a)
1.5
2
0
2
4
6
(b)
time (sec)
C= 4
C= 6
8
10
12
14
16
integral velocity error (rad)
Fig. 6. Simulation results for the speed command 600 rpm with different sliding lines c: (a) speed responses and (b) phase plane trajectories.
900
0
800
-10 800 rpm
velocity error (rad/s)
velocity (rpm)
700 600 600 rpm
500 400 400 rpm
300
-20 -30 -40
-60
200
-70
100
-80
0
(a)
0
0.5
1
time (sec)
1.5
2
-90
(b)
400 rpm
-50
600 rpm
800 rpm 0
5
10
15
integral velocity error (rad)
Fig. 7. Simulation results for the different speed commands with the same sliding line c ¼ 6: (a) speed responses and (b) phase plane trajectories.
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160
10
700
0
500
velocity error (rad/s)
velocity (rpm)
600 C= 8 C= 6
400 C= 4
300 200
-20 -30 -40 -50
100 0
-10
-60 0
0.5
1
(a)
1.5
C= 8
-70 -2
2
0
2
(b)
time (sec)
4
C= 4
C= 6
6
8
10
12
14
integral velocity error (rad)
Fig. 8. Experimental results for the 600 rpm reference speed command with different sliding lines c: (a) speed responses and (b) phase plane trajectories.
1000
10
900
0
800
-10 -20
velocity error
velocity (rpm)
800 r.p.m
700 600 600 r.p.m
500 400 400 r.p.m
-30
-50
300
-60
200
-70
100
-80
0
(a)
0
0.5
1
1.5
2
time (sec)
400 r.p.m
-40
(b)
600 r.p.m
800 r.p.m
-90 -2
0
2
4
6
8
10
12
integral velocity error
4
4
3
3
2
2
1
1
ias (A)
ias (A)
Fig. 9. Experimental results for the different speed commands with the same sliding line c ¼ 6: (a) speed responses and (b) phase plane trajectories.
0
0
-1
-1
-2
-2
-3
-3 -4
-4 2.75
2.8
2.85
2.9
2.95
3
0.5
0.52
0.54
8
6
6
4
4
i-beta (A)
i-beta (A)
8
2 0
-2 -4
-6
-6
(a)
-4
-2
0
2
i-alpha (A)
0.6
0.62
0
-4
-6
0.58
2
-2
-8 -8
0.56
time (sec)
time (sec)
4
6
8
-8 -8
(b)
-6
-4
-2
0
2
4
6
8
i-alpha (A)
Fig. 10. Experimental results: the stator current ias and the ia and ib currents in phase plane with speed command 600 rpm and c ¼ 6 by using (a) sign() function and (b) proposed controller.
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161
¼ sgnðTe Þjopt idt : iqt
If the core losses are ignored, i.e. Rc -N; the optimal current ratio of (29) is jopt ¼ 71: The current angle is jopt ¼ 7p=4: From the electromagnetic torque (3) and optimal current ratio (29), we can obtain the command torque current as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi T e ; ð30Þ idt ¼ 3 PðL L d q Þ jopt 2
6. The simulation and experimental results 6.1. Simulation results The proposed system, shown in Fig. 5, has verified the global behavior and the robustness of the sliding mode control with grey prediction. We simulate it by using the Simulink of Matlab. For simplifying the simulation process, the parameters of inductance and all resistances
900
900
800
800
700
700
600
600
500 400
500 400
300
300
200
200
100
100
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
0
0.5
1
1.5
2
time (sec)
3
3.5
4
4.5
5
5
3
4
2
3
1
2
ia (A)
ia (A)
2.5
time (sec)
4
0
1
-1
0
-2
-1
-3
-2
-4
ð31Þ
By combing (5), (6), (30) and (31), we can have the torque current loop.
velocity (rpm)
velocity (rpm)
Let the ratio of d-axis and q-axis currents be j ¼ iqt =idt and assume the torque command as constant on one sampling time, i.e., idt iqt is constant. Let dPtotal =dj ¼ 0; the optimal current ratio jopt can be obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rs R2c þ Rc ðor Ld Þ2 þ Rs ðor Ld Þ2 : ð29Þ jopt ¼ 7 Rs R2c þ Rc ðor Ld Þ2 þ Rs ðor Lq Þ2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
5
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
time (sec) 6
2 5 1.5 4
torque (Nt-m)
torque (Nt-m)
1 0.5 0
0
-1
(a)
2
1
-0.5
-1.5 0
3
0.5
1
1.5
2
2.5
time (sec)
3
3.5
4
4.5
-1
5
(b)
0
0.5
1
1.5
2
2.5
time (sec)
Fig. 11. Experimental results: the speed regulation, the stator current ia ; and electromagnetic torque Te by using (a) sat( ) function and (b) proposed controller.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
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are assumed constant in the sampling period. The motor parameters are shown in Appendix. The initial conditions of the integrator are set in the sliding line with different constants c: Fig. 6 shows the simulation results of the MPEC with core losses compensation. Fig. 6(a) shows that the dynamic response has a faster transient response when a larger constant c is chosen. The phase plane trajectories of Fig. 6(b) follow the sliding line toward the origin with different sliding line. The different low speed responses are shown in Fig. 7(a). In Fig. 7(b), the phase plane trajectories follow the sliding line toward the origin.
The initial conditions of the integrator are set in the sliding line and adding the machine load 0.2 N-m at the beginning of the experimentation. Fig. 8 shows the experimental results of speed responses and phase plane trajectories. The dynamic response has a faster transient speed response when a larger constant c is chosen as shown in Fig. 8(a). From Fig. 8(b), the phase plant trajectories follow the sliding line toward the origin and stay in the neighborhood of the origin. The different speed responses in the low speed region are shown in Fig 9(a). In Fig. 9(b), the phase plane trajectories follow the sliding line toward the origin and stay in the neighborhood of the origin.
6.2. Experimental results The dSPACE DS1102 control board is adopted to act as the digital system speed control. The SynRM is driven using IGBTs by a three-phase voltage space vector pulsewidth modulation (VSVPWM) inverter. The sampling period of control rules is set as 200 ms.
Table 1 Parameters of SynRM (0.37 kW) Rs 4.2 O Jm 0.00076 kg m2
Rc 50 O (f ¼ 60 Hz) Bm 0.00012 N-m s
400
Lq 0.181 H
900
700
450
Ld 0.328 H P 1
800
600
700
350
250 200 150
velocity (rpm)
velocity (rpm)
velocity (rpm)
500 300
400 300
600 500 400 300
200 200
100 100
50 0
0
0.5
1
1.5
2
2.5
0
3
100
0
0.5
1
time (sec)
1.5
2
2.5
0
3
0
0.5
1
time (sec)
4
2
2.5
3
2
2.5
3
2
2.5
3
9
7
4.5
1.5
time (sec)
8
6
7
3.5 5
6
2.5 2 1.5
4
torque (nt-m)
torque (nt-m)
torque (nt-m)
3
3 2
5 4 3 2
1 1
1
0.5 0
0 -0.5
0
0.5
1
1.5
2
2.5
-1
3
0 -1 0
0.5
1
time (sec)
1.5
2
2.5
3
0
0.5
1
time (sec)
6
6
4
4
2
2
1.5
time (sec) 6 4 2
i (A)
i (A)
i (A)
0 0
0
-2
-2
-2
-4
-4
-4
-6
(a)
0
0.5
1
1.5
time (sec)
2
2.5
-6
3
(b)
-6 -8 0
0.5
1
1.5
time (sec)
2
2.5
3
(c)
0
0.5
1
1.5
time (sec)
Fig. 12. The speed, electromagnetic torque and a-phase line current responses with a 0.2 N-m machine load is added at the beginning: (a) the speed command 400 rpm and a 1.7 N-m disturbance is added at 0.9 s; (b) the speed command 600 rpm and a 1.6 N-m disturbance is added at 0.9 s; and (c) the speed command 800 rpm and a 0.9 N-m disturbance is added at 1.1 s.
ARTICLE IN PRESS H.-K. Chiang, C.-H. Tseng / Control Engineering Practice 12 (2004) 155–163
In order to reduce the chattering phenomenon, a sign() function and a boundary layer with grey prediction are used as shown in Fig. 10. Fig. 10(a) shows that the steady-state a-phase current ias has a seriously larger harmonic and the stator current trajectory in ia ; ib phase plane is largely varied by using the sign() function. From Fig. 10(b), by the proposed controller, the a-phase current ias is very close to a sinusoidal waveform and the stator current trajectory ia; ib in the phase plane is smooth. Therefore, the chattering phenomenon in the integral variable structure controllers with a boundary layer by grey prediction has a better performance than using the sign() function. In Fig. 11, the reference speed commands are changed from 0 to 600 rpm, 600 to 800 rpm, and 800 to 400 rpm by using the sat() function and the proposed controller, respectively. The speed regulation is satisfied for the proposed controller. It has better transient response than using the sat() function especially from 800 to 400 rpm. For the external load disturbance regulation, the dynamic responses with different speed commands are shown in Fig. 12. The robust control performance of the proposed controller in the command tracking and load disturbance regulation is obvious.
7. Conclusions A complete model development and analysis for the grey sliding mode speed control of synchronous reluctance motor including the core loss is first presented in this paper. The mathematical model of synchronous reluctance motor includes the core loss. We develop a maximum power efficiency control strategy. The grey sliding mode controller is used to reject the load disturbance and parameter variations. We need not know the uncertain bounded disturbances and parameter variations. The grey prediction can forecast the system state variations and force the state into the boundary layer. The boundary layer gets a smooth current trajectory and reduces the chattering phenomenon. Finally, we employ the simulations and experiments to validate the proposed method.
163
Appendix The parameters of SynRM are shown in Table 1.
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