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Speed control of switched reluctance motors taking into account mutual inductances and magnetic saturation effects
Energy Conversion and Management 51 (2010) 1287–1297
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Speed control of switched reluctance motors taking into account mutual inductances and magnetic saturation effects M. Alrifai a,*, M. Zribi a, M. Rayan a, R. Krishnan b a b
Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Center for Rapid Transit Systems, Electrical and Computer Engineering Department, Virginia Tech University, 461 Durham Hall, Blacksburg, VA 24061-011, USA
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 13 June 2009 Accepted 9 January 2010
Keywords: Switched reluctance motor Mutual inductances Magnetic saturation Feedback linearization Sliding mode control Speed control Two-phase excitation
a b s t r a c t This paper deals with the speed control of switched reluctance motor (SRM) drives taking into account the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core. To overcome the problems commonly associated with single-phase excitation, a nonlinear SRM model, which is suitable for two-phase excitation and which takes into account the effects of mutual inductances between two adjacent phases and the magnetic saturation effects, is considered in the design of the proposed controllers. A feedback linearization control scheme and a sliding mode control scheme are designed for this motor drive. The proposed controllers guarantee the convergence of the phase currents and the rotor speed of the motor to their desired values. Simulation results indicate that the proposed controllers work well and that they are robust to changes in the parameters of the system and to changes in the load torque. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Switched reluctance motor drives have been receiving renewed attention as attractive candidates for many applications. For example, SRMs are widely used in high performance servo applications, such as the aerospace industry, and other industrial applications, such as electric vehicles and robotics. Combining the unique features of an SRM with its relatively simple and efficient power converter leads to a variable speed motor drive which users prefer in many applications to AC or DC motor drives. The main advantages of an SRM drive are: (1) its simplicity and its low-cost machine construction; this is due to the absence of rotor windings and permanent magnets, (2) the simplicity of the associated unipolar converter, (3) its fault tolerant operation, (4) its rugged behavior and large torque output over a very wide speed range. On the other hand, torque ripples, acoustic noise and rotor position sensing requirements are the main disadvantages of the SRM drive. The switched reluctance motor is a doubly salient machine with independent phase windings on the stator and a solid laminated rotor. The stator windings (on diametrically opposite poles) are connected in series to form one phase of the motor. Fig. 1 shows a four phase switched reluctance motor with eight stator poles and six rotor poles (a typical 8/6 SRM). When a stator phase is * Corresponding author. Tel.: +965 9875751. E-mail address: [email protected] (M. Alrifai). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.01.004
energized (i.e., a pair of diametrically opposite stator poles are excited) the most adjacent rotor pole-pair is attracted towards the energized stator in order to minimize the reluctance of the magnetic path. This excitation produces a torque regardless of the direction of the current in the phase winding. By energizing consecutive phases in succession, it is possible to develop a constant torque in either direction of rotation. The reader can refer to references [1,2] and the references therein for more details on SRMs. Usually, SRM drives operate in the magnetic-flux-saturation region so that high torque-to-mass ratios can be realized. Hence, magnetic saturation is a very important key to the high-performance operations of SRM drives. High performances can hardly be achieved by using conventional linear controllers, because linearizing the system dynamics around an operating point and designing linear controllers is generally not sufficient to achieve the high dynamical performances required for high performance drives. Moreover, the nonlinearity arising from the high saturation of the magnetic characteristics complicates the design of the control algorithms for such motors. Thus, the modeling of the nonlinear magnetic characteristics and the establishment of good SRM drive models are crucial for the design, performance prediction, and control of SRM drives [3–5]. Therefore, it is necessary to take the system nonlinearities into account and design feedback control laws that compensate for these nonlinearities and also compensate for the uncertainties on the parameters of the system.
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Fig. 1. The cross-section of an 8/6 SRM.
Several nonlinear control techniques have been developed for the control of electrical motor drives. For example, the feedback linearization control technique, the sliding mode control, adaptive control, optimal control, neural control, fuzzy logic have been used to control motor drives. Some of these techniques have been applied to SRM drives. Control of the sum of the squares of the phase currents to minimize torque ripples was proposed in [6]; also, the authors applied a sliding mode controller to the speed control loop to compensate for the low frequency oscillations on the torque output. Although the mutual inductance effects were included in [6], the modeling and simulation of the SRM were accomplished for operations in the linear region of its magnetic characteristics. In reference [7], an approach to reduce the torque ripple while controlling the speed of an SRM was presented. A neural network is used to estimate the generated torque of the SRM; this network is trained off line using data acquired from both the linear region and the saturation region. The main drawback of this approach is the use of a motor data obtained using the finite element method to train the neural network torque estimator. In addition, this method does not consider the effects of mutual inductances between the phases during commutations. In order to obtain high dynamical performance from an SRM drive, torque distribution functions were proposed in the literature to minimize torque ripples during commutation. In Refs. [8–11], the basic idea was to distribute the desired torque to two adjacent phases during commutation using a specified torque distribution functions. The researchers in [12] proposed a linear decrease of the outgoing phase and a linear increase of the incoming phase during the commutation interval to obtain a high performance drive. In [13], a torque distribution function that minimizes the rates of change of currents over the commutation interval was proposed. In some of the above mentioned control schemes, an ideal inductance profile was assumed. However, none of these controllers considered the effects of mutual inductances during commutation. The effects of mutual inductances and the possibility of twophase excitation were mentioned in [14–18]. However, no control schemes were suggested to overcome the effects of mutual inductances. A novel dynamic model for the SRM which is suitable for twophase excitation control studies and which takes into account the effects of mutual inductances was proposed in [19]. A torque distribution function which reduces the rates of change of currents and which compensates for the effects of mutual couplings and the magnetic saturation was also presented. In addition, a control scheme which is based on linearizing and decoupling the mathematical nonlinear SRM model was proposed. The main objective of this paper is to investigate the development of nonlinear control schemes that can achieve high dynamical performances for the speed regulation of SRM drives taking into ac-
count the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core. A model which is suitable for two-phase excitation and which takes into account the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core is adopted in this paper. Two nonlinear speed controllers, which take into account the coupling and the nonlinearity in the current loop, are proposed for the SRM drive. Simulation results indicate that the proposed controllers work well and that they are robust to changes in the parameters of the system and to changes in the load torque. Moreover, the proposed control schemes are compared to a linearizing PI current controller motivated by the work [19] which was based on linearizing and decoupling the dynamics of the currents in the mathematical nonlinear SRM model. The paper is organized as follows. The model of the SRM is presented in Section 2. A feedback linearization controller and a sliding mode controller are proposed in Sections 3 and 4, respectively. Simulation results of the closed loop systems are presented and discussed in Section 5. In Section 6, comparisons of the proposed controllers with the results obtained from a linearizing PI current controller motivated by the work in [19] are provided. Finally, some concluding remarks are given in Section 7. 2. Model of the SRM including mutual inductances and magnetic saturation effects Since mutual inductances between two adjacent phases and the magnetic saturation of the iron core have important effects on the performances of the SRM drive and on the torque ripples, a reliable mathematical model which includes these effects has to be adopted to properly evaluate the SRM performances when different control techniques are used. Both spatial and magnetic nonlinearities are inherent characteristics of the SRM; as a result the motor parameters are functions of the rotor position and the phase currents. However, in many linear drive applications, for example [20,21], the SRM is operating in the magnetically linear region where the parameters of the system are expressed as functions of the rotor position only. Bae [19] developed a model which takes into account mutual inductances and the magnetic saturation effects. In this paper, magnetic saturation is considered and the analysis of the SRM is performed assuming that the motor operates in either the linear or the nonlinear magnetic regions. To include the effects of operating in the nonlinear magnetic operating region, the parameters of the system are expressed in terms of both the rotor position and the phase currents. 2.1. The parameters of the SRM In this paper, we adopt the SRM model developed by Bae [19]. The SRM parameters, such as self and mutual inductances for different rotor positions and phase currents, are obtained analytically by using the finite element method. The SRM parameters are experimentally verified in [19]. In Ref. [22], the SRM parameters were obtained when the phase currents are equal to 1.2 A, while assuming that the motor operates in the linear magnetic region. If the phase currents exceed 1.2 A, the magnetic circuit of the prototype SRM becomes saturated when the rotor moves towards the aligned position. Therefore, the inductances are no longer determined only by the rotor position. Thus, the general form of the inductances at any operating condition with respect to the rotor position and phase currents are represented as follows:
Lx ¼ Lx ðix ; hÞ Ly ¼ Ly ðiy ; hÞ M xy ¼ M xy ðix ; iy ; hÞ
ð1Þ
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0.09
I =1.2 Amp x
0.08
1.6
Mutual inductance Mxy (H)
Self inductance Lx (H)
x 10−3
1.8
I x =1.2 Amp
0.07 0.06 0.05 I =2.0 Amp x
0.04 0.03
1.4
1.2
1 I =2.0 Amp x
0.8
0.6
0.02
0.4
0.01 0
20
40
0
60
20
40
60
Rotor position (degrees)
Rotor position (degrees)
Fig. 2. The self and mutual inductances at various current levels.
where Lx and Ly are the self inductances, Mxy is the mutual inductance between phase x and phase y; ix and iy are the actual phase currents for phase x and phase y, respectively. The self inductance Lx and the mutual inductance Mxy at various phase current levels are shown in Fig. 2. The torque functions proposed in [19] are adopted. They are denoted as follows:
@K x ðix ; hÞ @h @K y ðiy ; hÞ Gy ðiy ; hÞ ¼ @h @K xy ðix ; iy ; hÞ Gxy ðix ; iy ; hÞ ¼ @h
Gx ðix ; hÞ ¼
ð2Þ
where Gx and Gy are the self torque functions and Gxy is the mutual torque function. Also, the functions K x ðix ; hÞ; K y ðiy ; hÞ and K xy ðix ; iy ; hÞ are arbitrary functions determined by solving the co-energy function [19]. The self and mutual torque functions are shown in Fig. 3. It should be noted that at the same phase current levels, these parameters have the same properties. The rates of change of self and mutual inductances with respect to rotor position are shown in Fig. 4. Note that the torque functions are less affected by the
phase currents than the rates of change of inductances with respect to the rotor position. Also, it should be noted that the variations in the parameters with respect to the mutual inductances are quite small and can be neglected. 2.2. The dynamic model of the SRM Two-phases excitation is considered to include the effects of mutual couplings. A four-phase system can be considered as a two-phase system because only one excitation region is considered at one time. Thus, the four phases equations are reduced to twophase equations as proposed in [19], and they are written as follows:
dix ¼ p1 ix p2 xix þ p3 iy þ p4 xiy þ p5 dt diy ¼ p6 iy p7 xiy þ p8 ix þ p9 xix þ p10 dt dx B 1 ¼ x þ ðT e T L Þ J J dt
vx
M xy Ly
vy
M xy
with
x 10−3
0.3
8
I =1.2 amp
(N.m/A )
x
2
2
Self torque function G (N.m/A )
0.2
6 4
x
Mutual torque function G
xy
0.1
0
I x =2.0 amp
−0.1
−0.2
−0.3
2 0 −2 −4 −6 −8
−0.4 0
20
40
60
0
Rotor position(degrees) Fig. 3. The self and mutual torque functions.
20
40
Rotor position(degrees)
!
vy
60
Lx
vx
ð3Þ
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0.3
8
0.2
6
x 10-3 I =1.2 Amp
∂ M (i ,0,θ) / ∂θ (H/rad)
x
∂ L (i , θ) / ∂θ (H/rad)
0.1
0 I x =2.0 Amp
x x
xy x
−0.1
−0.2
−0.3
4 2 I x =2.0 Amp 0 −2 −4 −6
I =1.2 Amp x
−0.4
−8 0
20
40
60
0
20
Rotor position (degrees)
40
60
Rotor position (degrees)
Fig. 4. The rates of change of self and mutual inductances with respect to rotor position.
Te ¼
1 2 1 2 2 2 Gx i þ Gy i þ Gxy ix iy ¼ p12 Jix þ p13 Jiy þ p14 Jix iy 2 x 2 y
ð4Þ
where the subscripts x are y are the phases under consideration. Note that the set (x, y) can take on the value (a, b), (b, c), (c, d), and (d, a). Also, note that, either phase x or phase y is leading and the other phase is following depending on the direction of rotation. The currents ix and iy are the currents in phase x and phase y. The speed of the rotor is denoted as x; the angle h is the rotor position, J is the rotor inertia, B is the damping factor, T e is the motor torque. Denote the winding phase resistance by R. The parameters of the motor can be written as,
@Lx ix ; @ix
Lx ¼ Lx þ p5 ¼
1 D
Ly ;
Ly ¼ Ly þ
p10 ¼
1 D
@Ly iy ; @iy
D ¼ Lx Ly M 2xy ;
Lx
p11 ¼
p2 ¼ p5
B ; J
p12 ¼
1 Gx ; 2J
p13 ¼
1 Gy ; 2J
p14 ¼
u1 ¼ v x u2 ¼
Ly
vy
1 Gxy J
ð6Þ
Note that by using u1 and u2 given in (5) and (6), one can easily obtain the phase voltages v x and v y such that:
vx ¼ vy ¼
Lx Ly D
u1 þ
Ly Mxy D
Lx Mxy
u1 þ
D Lx Ly D
u2 ð7Þ u2
ð8Þ
x_ 3 ¼ p11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Note that in model (8), the load torque T L is taken to be zero. However, the effects of the load torque on the performance of the system are studied in the simulations section. The objective of the paper is to design nonlinear control schemes such that the motor speed is driven to a desired constant speed xd . In this regards, define ixd ; iyd and xd as the desired values of the phase currents ix , iy and the speed x, respectively.
2
ð5Þ
M xy vx þ vy Lx
x_ 2 ¼ p6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 þ p10 u2
2
p11 xd þ p12 ixd þ p13 iyd þ p14 ixd iyd ¼ 0
Define the state variables x1 , x2 and x3 such that x1 ¼ ix , x2 ¼ iy and x3 ¼ x. Also, let the controllers u1 and u2 be such that:
Mxy
x_ 1 ¼ p1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 x2 þ p5 u1
Remark 1. It is clear from the third equation of the dynamic model given in (8) that the desired values of the phase currents and the motor speed should satisfy the equation:
!
@Lx M xy @M xy M xy R ; p3 ¼ p5 ; @h Ly @h Ly ! Mxy @Ly @Mxy ; p4 ¼ p5 @h Ly @h @Ly Mxy @M xy M xy R p6 ¼ p10 R; p7 ¼ p10 ; ; p8 ¼ p10 @h Lx @h Lx M xy @Lx @Mxy p9 ¼ p10 : @h Lx @h
p1 ¼ p5 R;
Using the definitions of the parameters of the motor given above and using Eqs. (5) and (6), the model of the SRM motor given by (3) and (4) can be written as,
ð9Þ
The motor torque T e given by (4) is a function of the phase currents. Therefore, this torque T e is indirectly controlled to a desired value by controlling the phase currents. Torque distribution functions which reduce the rate of changes of the phase currents, compensate for the mutual couplings between two adjacent phases and consider the magnetic saturation of the core were proposed in [19]; an approach to calculate the desired phase currents using the self and mutual torque functions given by (2) is suggested. The approach in [19] is adopted in this paper and the following equations are used to calculate the desired phase currents:
ixd iyd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gx T e ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 2 Gx þ Gy 2Gxy Gx Gy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gy T e ¼ pffiffiffiffiffiffiffiffiffiffiffi G2x þ G2y 2Gxy Gx Gy
ð10Þ
Note that in Eq. (10), the ‘‘±” term should be interpreted as follows: a plus sign is used when the desired output torque T e is positive and the minus sign is used when the torque T e is negative. To facilitate the design of control schemes for the SRM motor, we propose to use the following transformation:
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z1 ¼ x3 xd z2 ¼ p11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2
ð11Þ
z3 ¼ ðx1 ixd Þ2 þ ðx2 iyd Þ2
Remark 2. The choice of the states z1 ðtÞ; z2 ðtÞ and z3 ðtÞ in (11) guarantees that when z1 ðtÞ; z2 ðtÞ and z3 ðtÞ converge to zero then the states x1 ðtÞ; x2 ðtÞ and x3 ðtÞ converge to their desired values, respectively. The dynamic model of the SRM motor given by Eq. (8) can be transformed into the following model through the use of the transformation given in (11) as follows:
z_ 1 ¼ z2 1 z_ 2 ¼ f1 þ u _z3 ¼ f2 þ u 2
ð12Þ
1 and u 2 are defined as follows: where the functions f1 ; f 2 ; u
f1 ¼ p11 ðp11 x3 þ
p12 x21
þ
p13 x22
It can be easily checked that since a1 ; a2 and a3 are positive scalars, the matrix Ac1 is a stable matrix. The solution of Eq. (21) is zðtÞ ¼ expðAc tÞzð0Þ. Since Ac is a stable matrix, then zðtÞ converges to zero as t ! 1. h Therefore, since z(t) converges to zero as t ! 1, then the speed of the motor and the phase currents of the motor will converge to their desired values, respectively as t ! 1 because of transformation (11). Remark 4. The controller given by (19) and (20) can be written in the original coordinates by using transformation (11) and (18). Therefore,
1 ¼ f1 a1 ðx3 xd Þ a2 ðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Þ ð23Þ u 2 ¼ f2 a3 ðx1 ixd Þ2 a3 ðx2 iyd Þ2 u
where f1 and f2 are given by (13) and (14). Then, the controllers u1 and u2 are such:
þ p14 x1 x2 Þ
ð24Þ
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð25Þ
þ ð2p12 x1 þ p14 x2 Þðp1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 x2 Þ þ ð2p13 x2 þ p14 x1 Þðp6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 Þ d ixd f2 ¼ 2ðx1 ixd Þ p1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 dt d iyd þ 2ðx2 iyd Þ p6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 dt 1 ¼ ð2p12 x1 þ p14 x2 Þp5 u1 þ ð2p13 x2 þ p14 x1 Þp10 u2 u 2 ¼ 2ðx1 ixd Þp5 u1 þ 2ðx2 iyd Þp10 u2 : u
ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ
4. Design of a sliding mode controller for the SRM Let b1 ; c1 ; c2 and W 1 ; W 2 be positive scalars. Since, the SRM has two inputs, one has to design two sliding surfaces. Let the first sliding surface r1 be such that,
r1 ¼ z2 þ b1 z1 Also, choose the second sliding surface
ð26Þ
r2 such that:
r2 ¼ z3 Remark 3. The design of the controllers will be accomplished using the transformed dynamic model given by (12), then the 2 will be transformed back into the original 1 and u controllers u coordinates using the following state dependent transformation:
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð18Þ
In the following two sections, two nonlinear control schemes are designed for the switched reluctance motor.
3. Design of a feedback linearization controller for the SRM
Proposition 2. The sliding mode controller,
1 ¼ f1 b1 z2 c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ u 2 ¼ f2 c2 z3 W 2 signðz3 Þ u
Proof. Differentiating Eq. (26) with respect to time and using (12), it follows that,
ð20Þ
when applied to the transformed motor system model given by (12) guarantees the asymptotic convergence of the states zðtÞ ¼ ½z1 ðtÞ z2 ðtÞ z3 ðtÞT to zero as t tends to infinity. Proof. The closed loop system when the controller (19) and (20) is applied to the system (12) is such:
ð21Þ
where,
2
0 6 Ac ¼ 4 a1 0
1 a2 0
0
3
7 0 5 a3
ð30Þ
r_ 1 ¼ f1 þ u1 þ b1 z2 ð19Þ
z_ ¼ Ac z
ð29Þ
1 by its value from (28), it follows that, Substituting u
Proposition 1. The feedback linearization controller:
1 ¼ f1 a1 z1 a2 z2 u 2 ¼ f2 a3 z3 u
ð28Þ
when applied to the SRM motor model given by (12), ensures that the states z1 ðtÞ; z2 ðtÞ and z3 ðtÞ converge to zero as t tends to infinity.
r_ 1 ¼ z_ 2 þ b1 z_ 1 ¼ f1 þ u1 þ b1 z2 Let a1 ; a2 and a3 be positive scalars.
ð27Þ
ð22Þ
¼ f1 þ b1 z2 f1 b1 z2 c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ ¼ c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ ¼ c1 r1 W 1 signðr1 Þ
ð31Þ
Also, differentiating Eq. (27) with respect to time, using (12) and 2 by its value from (29), it follows that, substituting u
r_ 2 ¼ f2 þ u2 ¼ c2 z3 W 2 signðz3 Þ ¼ c2 r2 W 2 signðr2 Þ
ð32Þ
The dynamics (31) and (32) guarantees that ri r_ i < 0 for (i ¼ 1; 2). Also, it can be checked that the trajectories associated with the discontinuous dynamics (31) and (32) exhibit a finite time reachability to zero from any given initial condition provided that the scalars c1 ; c2 ; W 1 and W 2 are chosen to be strictly positive. Therefore, we are guaranteed to reach r1 ¼ 0 and r2 ¼ 0 in a finite time.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
Since r1 is driven to zero in finite time, the variable z1 ðtÞ is governed after such finite amount of time by the first order differential equation z_ 1 ðtÞ þ b1 z1 ðtÞ ¼ 0. Thus z1 ðtÞ will asymptotically converge to 0 as t ! 1 because b1 is a positive scalar. Hence, z2 ðtÞ will also converge to zero as t ! 1. In addition, since r2 is driven to zero in finite time, the variable z3 ðtÞ will converge to zero in finite time.
speed (rad/s)
1292
Therefore, it can be concluded that the sliding mode controller (28) and (29) guarantees the asymptotic convergence of z1 ðtÞ; z2 ðtÞ and z3 ðtÞ to zero as t ! 1. h Thus, the controller (28) and (29) guarantees the asymptotic convergence of x1 ðtÞ; x2 ðtÞ and x3 ðtÞ to their desired values as t ! 1. Remark 5. The controller (28) and (29) can be written in the original coordinates by using the transformation (11) and (18). Therefore,
1 ¼ f1 ðb1 þ c1 Þðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Þ u b1 c1 ðx3 xd Þ W 1 signðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 þ b1 ðx3 xd ÞÞ
ð33Þ
2 ¼ f2 c2 ðx1 ixd Þ2 Þ c2 ðx2 iyd Þ2 W 2 signððx1 ixd Þ2 u þ ðx2 iyd Þ2 Þ
ð34Þ
where f1 and f2 are given by (13) and (14). Thus, the controllers u1 and u2 are such:
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð35Þ
5. Simulation results
100 90 80 70 60 50 40 30 20 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (s) Fig. 6. Speed response (using FLC) of the SRM drive with changes in the motor reference speed.
5.1. Performance of the system using the feedback linearization controller The performance of the SRM system when the feedback linearization controller (FLC) is used is simulated using the Matlab software. The control law described by (23)–(25) is applied to the SRM system given by (3). Fig. 5 shows the speed response when the motor accelerates from rest to a desired speed of 100 rad/s, with a load torque of 0.15 N m. Fig. 6 shows the speed response of the motor with step changes in the reference speed. First, the motor is commanded to accelerate from rest to a speed of 100 rad/s, then to a speed of 150 rad/s and finally to a speed of 200 rad/s with a load torque of 0.2 N m. It can be seen from the figures that the motor speed converges to the desired speed with a very small steady state error. Note that the steady state error could be made to go to zero by tuning the gains of the controller. It should be mentioned that the ripples in the speed response are due to the sequential switching between the phases and they are not caused by the controller. The SRM phase currents are shown in Fig. 7. 5.2. Performance of the system using the sliding mode controller The performance of the SRM system when the sliding mode controller (SMC) is used is simulated. The control law described by (33)–(35) is applied to the SRM system given by (3). Fig. 8 shows the speed response when the motor is commanded to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.15 N m. The speed response of the motor with step changes in the reference speed is depicted in Fig. 9. First, the speed response of the motor is commanded to accelerate from rest to a speed of 100 rad/s, then to a speed of 150 rad/s and finally to a speed of 200 rad/s with a load torque of 0.2 N m. It can be seen from the figures that the motor speed converges to the desired speed. It should be mentioned that the ripples in the speed response are due to the sequential switching between the phases and they are not caused by the controller. The SRM phase currents are shown in Fig. 10.
phase currents (Amp)
speed (rad/s)
The SRM system is simulated using the MATLAB software [23]. The SRM model discussed in Section 2 is adopted; the model takes into account the effects of mutual inductances coupling and the magnetic saturation. The simulations are completed on a fourphase motor which has eight stator poles and six rotor poles. The stator pole arc is 16°. The rotor pole arc is 18°. The phase resistance is 1:6 X and the rated torque of the motor is 0.4 N m. The excitation angles (hon and hoff ) are kept fixed throughout the simulation studies at 0° and 30°, respectively (note that 0° corresponds to the aligned position; and 60° corresponds to the unaligned position). Note that two phases are allowed to be excited at one time. Simulations are performed when the proposed feedback linearization and sliding mode controllers are applied to the SRM system. The corresponding results are presented in the following subsections.
200 180 160 140 120 100 80 60 40 20
1.5 I d
I
Ia
Ic
b
I
d
1
0.5
0 0
0.05
0.1
0.15
0.2
0.25
Time (s) Fig. 5. Speed response (using FLC) of the SRM drive with fixed xd .
0.3
0.062
0.064
0.066
0.068
0.07
0.072
Time (s) Fig. 7. Phase currents (using FLC) waveforms of the SRM drive.
0.074
1293
120
120
100
100
speed (rad/s)
speed (rad/s)
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
80 60 40 20
80 60 R=2x1.6 ohm
40 20 0
0 −20
R=1x1.6 ohm
R=1x1.6 ohm
−20 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
0.05
0.1
0.15
Time (sec)
speed (rad/s)
speed (rad/s)
100 90 80 70 60 50 40 30 20 10
0.15
0.2
0.25
0.35
0.4
0.45
j=10x0.001
j=1x0.001
0 0.1
0.3
Fig. 12. Speed response (using SMC) of the SRM drive with changes in R.
200 180 160 140 120 100 80 60 40 20 0 0.05
0.25
Time (s)
Fig. 8. Speed response (using SMC) of the SRM drive.
0
0.2
0.3
0.35
0.4
j=1x0.001
0.05
0.1
0.15
0.2
0.25
0.3
0.45
Time (s)
Time (s)
Fig. 13. Speed response (using FLC) of the SRM drive with changes in J.
Fig. 9. Speed response (using SMC) of the SRM drive with changes in the motor reference speed.
120
Phase currents(Amp)
I
d
Ia
Ic
Ib
I
speed (rad/s)
100
1.5
d
1
0.5
80 60 j=10x0.001
40 20
j=1x0.001
j=1x0.001
0 −20 0
0 0.076
0.078
0.08
0.082
0.084
0.086
0.088
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (sec)
0.09
Time (s)
Fig. 14. Speed response (using SMC) of the SRM drive with changes in J.
100 90 80 70 60 50 40 30 20 10
speed (rad/s)
speed (rad/s)
Fig. 10. Phase currents (using SMC) waveforms of the SRM drive.
R=2x1.6 ohm
R=1x1.6 ohm
R=1x1.6 ohm
100 90 80 70 60 50 40 30 20 10
B=10x1e−5
0 0
0.05
0.1
0.15
0.2
0.25
0.3
Time (s)
B=1x1e−5
B=1x1e−5
0.05
0.1
0.15
0.2
0.25
0.3
Time (s) Fig. 15. Speed response (using FLC) of the SRM drive with changes in B.
Fig. 11. Speed response (using FLC) of the SRM drive with changes in R.
5.3. Robustness of the proposed control schemes The robustness of the proposed control schemes to changes in the parameters of the SRM and to changes in the load torque are tested through simulation studies. The effects of the changes in the phase resistance R, the rotor inertia J, the damping factor B are investigated through simulations. The simulations are carried
out by step changing one parameter at one time while keeping all the other parameters unchanged. The motor is commanded to accelerate from rest to a desired speed of 100 rad/s with a load torque of 0.15 N m. The motor responses when there are changes in the parameters of the SRM system are depicted in Figs. 11–16. Figs. 11 and 12 show the responses of the motor using the FLC and the SMC, respectively when the phase resistance is increased by 100% of its original value and then it is changed back to its original value.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297 120
120
100
100
80
speed (rad/s)
speed (rad/s)
1294
60 40
B=10x1e−5
20
B=1x1e−5
B=1x1e−5
80 60 40 T =0.1Nm l
20
0 −20 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.4
0
0.05
0.1
0.15
Time (sec)
100
120
80
100
60 40 T l =0.05 Nm
0
0.3
0.35
0.4
80 60 40 T =0.15 Nm l
20 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45 −20
Time (sec)
0
100 90 80 70 60 50 40 30 20 10 0
T l =0.05 Nm
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Fig. 21. Speed response (using FLC) of the SRM drive with load torque T L ¼ 0:15 N m.
0.35
110 100 90 80 70 60 50 40 30 20 10
Time (s)
T =0.15Nm l
0
Fig. 18. Speed response (using SMC) of the SRM drive with load torque T L ¼ 0:05 N m.
0.05
Time (sec)
speed (rad/s)
Fig. 17. Speed response (using FLC) of the SRM drive with load torque T L ¼ 0:05 N m.
speed (rad/s)
0.25
Fig. 20. Speed response (using SMC) of the SRM drive with the load torque T L ¼ 0:1 N m.
speed (rad/s)
speed (rad/s)
Fig. 16. Speed response (using SMC) of the SRM drive with changes in B.
20
0.2
Time (sec)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (s) Fig. 22. Speed response (using SMC) of the SRM drive with load torque T L ¼ 0:15 N m.
80
100
60
speed (rad/s)
speed (rad/s)
100
40 T =0.1Nm l
20 0
80 60 T l =2x0.2 Nm
40 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
T l =1x0.2 Nm
T =1x0.2 Nm l
0.45 0
Time (sec) Fig. 19. Speed response (using FLC) of the SRM drive with the load torque T L ¼ 0:1 N m.
Figs. 13 and 14 show the responses of the motor using the FLC and the SMC, respectively when the rotor inertia is varied by up to 10 times its original value. Figs. 15 and 16 show the responses of the motor using the FLC and the SMC, respectively when the damping factor B is varied by 10 times its original value. It is clear from these figures that the speed responses of the motor converge to the de-
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec) Fig. 23. Speed response (using FLC) of the SRM drive with changes in the load torque.
sired speed even when the parameters of the system change. Hence, it can be concluded that the simulation results show that the proposed controllers are robust to changes in the parameters of the SRM system.
1295
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.2
0.3
0.2 0.15 0.1 0.05 0 0.12
0.14
0.16
0.18
0.2
Time (s) Fig. 27. Output torque (using FLC) of the SRM drive with mutual inductance considered (T L ¼ 0:2 N m and x ¼ 10 rad=s).
Torque with mutual (Nm)
For high performance applications, it is desirable that the proposed control schemes reduce the torque ripples in the output torque. For comparison purposes, the output torque at low speeds (around 100 rpm) using the FLC and the SMC, respectively is shown in Figs. 25 and 26 when the mutual inductances are neglected. The output torque is shown in Figs. 27 and 28 using the FLC and the SMC, respectively when the mutual inductances are considered. It can be seen from these figures that the proposed control schemes have to some extent reduced the torque ripples when the mutual
speed (rad/s)
0.18
0.25
0.1
0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
Time (s) T =2x0.2Nm l T l =1x0.2Nm
0
0.05
Fig. 28. Output torque (using SMC) of the SRM drive with mutual inductance considered (T L ¼ 0:2 N m and x ¼ 10 rad=s). T =1x0.2Nm l
0.1
0.15
0.2
0.25
0.3
Time (sec) Fig. 24. Speed response (using SMC) of the SRM drive with changes in the load torque.
Torque without mutual (Nm)
0.16
Fig. 26. Output torque (using SMC) of the SRM drive with mutual inductance neglected (T L ¼ 0:2 N m and x ¼ 10 rad=s).
5.4. Output torque with and without mutual inductance
100 90 80 70 60 50 40 30 20 10
0.14
Time (s)
Torque with mutual (Nm)
In addition, simulation studies are carried out to demonstrate the robustness of the proposed controllers to changes in the load torque. The motor is commanded to accelerate from rest to a desired speed of 100 rad/s. Figs. 17–22 show the motor responses when the load torque takes on the values of 0.05, 0.1 and 0.15 N m, respectively. Figs. 17 and 18 show the motor responses using the FLC and the SMC, respectively when the load torque is 0.05 N m. Figs. 19 and 20 show the motor responses using the FLC and the SMC, respectively when the load torque equals 0.1 N m. Figs. 21 and 22 show the motor responses using the FLC and the SMC, respectively when the load torque equals 0.15 N m. It can be seen that for all the cases, the speed response converges to its desired value with small steady state errors. Figs. 23 and 24 depict the motor responses using the FLC and the SMC, respectively when the load torque changes during the motion of the rotor. The load torque is originally 0.2 N m, then it briefly becomes 0.4 N m, then it changes back to its original value. It can be seen from the figures that the motor speed responses show mall dips in the speed when the load is suddenly changed, however the controllers are able to keep the motor speed very close to the desired speed. Therefore, it can be concluded from the simulation studies that the proposed controllers are robust to changes in the parameters of the SRM system and to changes in the load torque.
Torque without mutual (Nm)
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
Time (s) Fig. 25. Output torque (using FLC) of the SRM drive with mutual inductance neglected (T L ¼ 0:2 N m and x ¼ 10 rad=s).
inductances are taken into account. Also, it can be seen from these figures that the proposed sliding mode controller gave better results than the feedback linearization controller. 6. Comparisons of the proposed controllers with previous work The control of SRM drives which adopts two-phase excitation, and takes into account both the mutual inductance effects between two adjacent phases and the magnetic saturation effects was tackled for the first time in the literature in [19]. It was shown that the mutual inductance between two adjacent phases partly contributes to generate the electromagnetic torque and introduces coupling between adjacent phases in the current control loop. The dynamics of the current loop are coupled and highly nonlinear due to the mutual inductance between two adjacent phases and the time varying nature of the inductance profile. Bae [19] proposed a linearizing PI controller to linearize and decouple the dynamics of the current loop. We will compare our proposed controllers, which take into account the coupling and the nonlinearity in the current loop, with a linearizing PI current controller motivated by the work in [19].
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Recall that the dynamic model of the SRM system given by (3) is such that:
dix ¼ p1 ix p2 xix þ p3 iy þ p4 xiy þ p5 u1 dt diy ¼ p6 iy p7 xiy þ p8 ix þ p9 xix þ p10 u2 dt dx B 1 ¼ x þ ðT e T L Þ J J dt
ð36Þ
To speed up the speed response, the controllers given by (39) and (40) are modified so that terms related to the error in the speed are included in the controllers. Therefore, the controllers u1 and u2 are now given as follows:
u1 ¼
1 d ixd ðp xix p3 iy p4 xiy þ þ p1 ixd m1 ex m2 p5 2 dt Z t Z t ex ðrÞdr m5 ex m6 ex ðrÞdrÞ 0
ð44Þ
0
with 2
2
T e ¼ p12 Jix þ p13 Jiy þ p14 Jix iy u1 ¼
vx
M xy Ly
!
vy
and u2 ¼
ð37Þ
vy
Mxy Lx
vx
ex ¼ ix ixd ey ¼ iy iyd
ð38Þ
e x ¼ xy xd Bae [19] proposed a PI controller which linearizes and decouples the phase currents. Motivated by his approach, a linearizing PI current controller is derived below. Let m1 ; m2 ; m3 and m4 be positive scalars. Define the controllers u1 and u2 such that:
1 d ixd ðp xix p3 iy p4 xiy þ þ p1 ixd m1 ex m2 p5 2 dt Z t ex ðrÞdrÞ
ð39Þ
0
u2 ¼
d iyd 1 ðp xiy p8 ix p9 xix þ þ p6 iyd m3 ey m4 p10 7 dt Z t ey ðrÞdrÞ
ð40Þ
Z
t
ex ðrÞdr
ð41Þ
ey ðrÞdr
ð42Þ
speed (rad/s)
The application of the controller given by (39) and (40) into the SRM model given by (36) results in the following closed loop dynamics:
0
e_ x ¼ ðp6 þ m3 Þey m4
Z
t
100 90 80 70 60 50 40 30 20 10 0 0
0
0.1
0.2
ð43Þ
Clearly, Eqs. (41) and (42) imply that ex and ey converge to zero asymptotically as t tends to infinity since p1 ; p6 ; m1 ; m2 ; m3 and m4 are positive scalars. Therefore, the phase currents ix and iy converge to their desired values ixd and iyd , respectively. Recalling that the desired values of the phase currents and the motor speed 2 2 should satisfy the equation: p11 xd þ p12 ixd þ p13 iyd þ p14 ixd iyd ¼ 0, it can be deduced from (43) and the fact that T L ¼ 0, that the motor speed converges to it desired constant value xd as t tends to infinity. The performance of the SRM system when the current linearizing PI controller is used is simulated. The control laws described by (39) and (40) is applied to the SRM system given by (36). Fig. 29 shows the speed response when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.2 N m. It is clear from the simulation results that the controller given by Eqs. (39) and (40) is a bit slow in forcing the speed of the motor to converge to its desired speed.
0.3
0.4
0.5
0.6
0.7
Time (s) Fig. 29. Speed response (using the linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
120 100
speed (rad/s)
dx B 1 ¼ x þ ðT e T L Þ J J dt
ð45Þ
0
where m5 ; m6 ; m7 and m8 are positive scalars. The simulation results of he SRM system given by (36) using the FLC, the SMC and the controllers (44) and (45) are depicted in Figs. 30–32. Fig. 30 shows the speed response when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.2 N m. It is clear from this figure that the improved current linearizing PI controller is able to force the speed to converge to its desired value. However, the response exhibits some ripples. Fig. 31 shows the speed responses when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, while using the FLC, the SMC and the improved current linearizing PI controller. This figure clearly shows that the FLC and SMC controllers gave better performances than the improved current linearizing PI controller. Note that the response when using the FLC controller is a bit faster than that of the SMC controller . However, the FLC controlled response has a small steady state error. Fig. 32 shows the speed responses when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s when the load torque changes during the motion of the rotor. The load torque is origi-
0
e_ x ¼ ðp1 þ m1 Þex m2
d iyd 1 þ p6 iyd m3 ey m4 ðp xiy p8 ix p9 xix þ p10 7 dt Z t Z t ey ðrÞdr m7 ex m8 ex ðrÞdrÞ 0
Define the errors ex and ey ; ew such that:
u1 ¼
u2 ¼
80 60 40 20 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s) Fig. 30. Speed response (using the modified linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
100 90 PI Controller
speed (rad/s)
80
SMC Controller FLC Controller
70 60 50 40 30 20 10 0 0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
100 PI Controller SMC Controller FLC Controller
speed (rad/s)
60 T =2x0.2Nm l
40 T l =1x0.2Nm
T l =1x0.2Nm
20
0 0
0.1
0.2
0.3
high performance characteristics of the proposed sliding mode controller, and that the sliding mode control scheme is very robust to variations in the parameters of the motor and to variations in the loads of the system. The proposed controllers are able to reduce the torque ripples to some extent and compensates for the effects of both mutual inductances and magnetic saturation of the core. Moreover, the proposed schemes are compared to a current linearizing PI controller proposed in the literature. Simulations studies indicate that the proposed schemes outperform the linearizing PI controller. Future research will address the implementation of the proposed control schemes on a hardware setup. References
Fig. 31. Speed responses (using the FLC, the SMC and the modified linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
80
1297
0.4
0.5
0.6
Time (s) Fig. 32. Speed responses (using the FLC, the SMC and the modified linearizing PI controller) of the SRM drive with changes in the load torque T L .
nally 0.2 N m, then it briefly becomes 0.4 N m, then it changes back to its original value. This figure clearly shows that the proposed controllers are more robust to changes in the load torque than the current linearizing PI controller. Therefore, it can be concluded that the proposed controllers gave better performances than the current linearizing PI controller, and that the SMC controller gave the best performance among the three controllers. 7. Conclusion In this paper, a feedback linearization controller and a sliding mode controller are proposed for speed regulation applications for switched reluctance motors. An SRM model which includes the effects of mutual inductances and the effects of the magnetic saturation of the core is adopted for the design of the control schemes and for simulation studies; the adopted model is also suitable for two-phase excitation. The proposed controllers, which take into account the coupling and the nonlinearity in the current loop, ensure the asymptotic convergence of the speed of the motor and the phase currents to their desired values. Simulation results indicate that the proposed nonlinear control schemes works well. In addition, the simulation results reveal the
[1] Miller TJE. Switched reluctance motors and their control. Clarendon Press; 1993. [2] Krishnan R. Switched reluctance motor drives. CRC Press LLC; 2001. [3] Mese E. A rotor position estimator for switched reluctance motors using CMAC. Energy Convers Manage 2003;44:1229–45. [4] Daldaban F, Ustkoyuncu N, Guney K. Phase inductance estimation for switched reluctance motor using adaptive neuro-fuzzy inference system. Energy Convers Manage 2006;47:485–93. [5] Xue XD, Cheng KWE, Ho SL. Correlation of modeling techniques and power factor for switched reluctance machines drives. IEE Proc Electr Power Appl 2005;152(3):710–22. [6] Inanc N, Ozbulur V. Torque ripple minimization of a switched reluctance motor by using continuous sliding mode control technique. Electr Power Syst Res 2003;66(3):241–51. [7] Hajatipour M, Farrokhi M. Adaptive intelligent speed control of switched reluctance motors with torque ripple reduction. Energy Convers Manage 2008;49:1028–38. [8] Schramm DS, Williams BW, Gaeen TC. Torque ripple reduction of switched reluctance motors by phase current optimal profiling. In: Power electronics specialist conference, PESC92; 1992, p. 857–60. [9] Hung JY. Torque ripple reduction of variable reluctance motors. Mechatronics 1994;4(8):785–98. [10] Husain L, Ehsani M. Torque ripple minimization in switched reluctance motor drives by PWM current control. IEEE Trans Power Electron 1996;11(1):83–8. [11] Ilic-Spong M, Miller TJE, Macminn SR, Thorp JS. Instantaneous torque control of electric motor drives. IEEE Trans Power Electron 1987;2(1):55–61. [12] Wallace RS, Taylor DG. A balanced commutator for switched reluctance motors to reduce torque ripple. IEEE Trans Power Electron 1992;7(4):617–26. [13] Kim CH, Ha IJ. A new approach to feedback-linearizing control of variable reluctance motors for direct drive applications. IEEE Trans Control Syst Technol 1996;4(4):348–62. [14] Pillay P, Liu Y, Cai W, Sebastian T. Multiphase operation of switched reluctance motor drives. In: IEEE industry applications conference, IAS’97, vol. 1, 1997, p. 310–7. [15] Moreiara JC, Lipo TA. Simulation of a four phase switched reluctance motor including the effects of mutual coupling. Electr Power Compo Syst 1989;16(4):281–99. [16] De Paula PP, Da Silva WM, Cardoso JR, Nabeta SI. Assessment of the influences of the mutual inductances on switched reluctance machines performance. In: Proceedings of the IEEE international electric machines and drives conference, IEMDC’03, vol. 3, 2003, p. 1732–8. [17] Krishnamurthy M, Fahimi B, Edrington CS. On the measurement of mutual inductance for a switched reluctance machine. In: Proceedings of the 37th IEEE power electronics specialists conference, PESC’06, 2006, p. 1–7. [18] Ahn JW, Oh SG, Moon JW, Hwang YM. A three-phase switched reluctance motor with two-phase excitation. IEEE Trans Ind Appl 1999;35(5):1067–75. [19] Bae HK. Control of switched reluctance motors considering mutual inductance. Ph.D. Thesis Dissertation, Virginia Polytechnic Institute and State University; 2000. [20] Bae HK, Lee BS, Vijayraghavau P, Krishnan R. A linear switched reluctance motor converter and control. IEEE Trans Ind Appl 2000;36(5):1351–9. [21] Lee BS, K Bae H, Vijayraghavan P, Krishnan R. Design of a linear switched reluctance machine. IEEE Trans Ind Appl 2000;36(6):1571–80. [22] Alrifai M, Zribi M, Krishnan R, Rayan M. Nonlinear speed control of switched reluctance motor drives taking into account mutual inductance. J Control Sci Eng 2008;11 [Article ID 491625]. [23] The MathWorks, Inc.; 1984–2009.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Speed control of switched reluctance motors taking into account mutual inductances and magnetic saturation effects M. Alrifai a,*, M. Zribi a, M. Rayan a, R. Krishnan b a b
Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Center for Rapid Transit Systems, Electrical and Computer Engineering Department, Virginia Tech University, 461 Durham Hall, Blacksburg, VA 24061-011, USA
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 13 June 2009 Accepted 9 January 2010
Keywords: Switched reluctance motor Mutual inductances Magnetic saturation Feedback linearization Sliding mode control Speed control Two-phase excitation
a b s t r a c t This paper deals with the speed control of switched reluctance motor (SRM) drives taking into account the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core. To overcome the problems commonly associated with single-phase excitation, a nonlinear SRM model, which is suitable for two-phase excitation and which takes into account the effects of mutual inductances between two adjacent phases and the magnetic saturation effects, is considered in the design of the proposed controllers. A feedback linearization control scheme and a sliding mode control scheme are designed for this motor drive. The proposed controllers guarantee the convergence of the phase currents and the rotor speed of the motor to their desired values. Simulation results indicate that the proposed controllers work well and that they are robust to changes in the parameters of the system and to changes in the load torque. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Switched reluctance motor drives have been receiving renewed attention as attractive candidates for many applications. For example, SRMs are widely used in high performance servo applications, such as the aerospace industry, and other industrial applications, such as electric vehicles and robotics. Combining the unique features of an SRM with its relatively simple and efficient power converter leads to a variable speed motor drive which users prefer in many applications to AC or DC motor drives. The main advantages of an SRM drive are: (1) its simplicity and its low-cost machine construction; this is due to the absence of rotor windings and permanent magnets, (2) the simplicity of the associated unipolar converter, (3) its fault tolerant operation, (4) its rugged behavior and large torque output over a very wide speed range. On the other hand, torque ripples, acoustic noise and rotor position sensing requirements are the main disadvantages of the SRM drive. The switched reluctance motor is a doubly salient machine with independent phase windings on the stator and a solid laminated rotor. The stator windings (on diametrically opposite poles) are connected in series to form one phase of the motor. Fig. 1 shows a four phase switched reluctance motor with eight stator poles and six rotor poles (a typical 8/6 SRM). When a stator phase is * Corresponding author. Tel.: +965 9875751. E-mail address: [email protected] (M. Alrifai). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.01.004
energized (i.e., a pair of diametrically opposite stator poles are excited) the most adjacent rotor pole-pair is attracted towards the energized stator in order to minimize the reluctance of the magnetic path. This excitation produces a torque regardless of the direction of the current in the phase winding. By energizing consecutive phases in succession, it is possible to develop a constant torque in either direction of rotation. The reader can refer to references [1,2] and the references therein for more details on SRMs. Usually, SRM drives operate in the magnetic-flux-saturation region so that high torque-to-mass ratios can be realized. Hence, magnetic saturation is a very important key to the high-performance operations of SRM drives. High performances can hardly be achieved by using conventional linear controllers, because linearizing the system dynamics around an operating point and designing linear controllers is generally not sufficient to achieve the high dynamical performances required for high performance drives. Moreover, the nonlinearity arising from the high saturation of the magnetic characteristics complicates the design of the control algorithms for such motors. Thus, the modeling of the nonlinear magnetic characteristics and the establishment of good SRM drive models are crucial for the design, performance prediction, and control of SRM drives [3–5]. Therefore, it is necessary to take the system nonlinearities into account and design feedback control laws that compensate for these nonlinearities and also compensate for the uncertainties on the parameters of the system.
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M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
Fig. 1. The cross-section of an 8/6 SRM.
Several nonlinear control techniques have been developed for the control of electrical motor drives. For example, the feedback linearization control technique, the sliding mode control, adaptive control, optimal control, neural control, fuzzy logic have been used to control motor drives. Some of these techniques have been applied to SRM drives. Control of the sum of the squares of the phase currents to minimize torque ripples was proposed in [6]; also, the authors applied a sliding mode controller to the speed control loop to compensate for the low frequency oscillations on the torque output. Although the mutual inductance effects were included in [6], the modeling and simulation of the SRM were accomplished for operations in the linear region of its magnetic characteristics. In reference [7], an approach to reduce the torque ripple while controlling the speed of an SRM was presented. A neural network is used to estimate the generated torque of the SRM; this network is trained off line using data acquired from both the linear region and the saturation region. The main drawback of this approach is the use of a motor data obtained using the finite element method to train the neural network torque estimator. In addition, this method does not consider the effects of mutual inductances between the phases during commutations. In order to obtain high dynamical performance from an SRM drive, torque distribution functions were proposed in the literature to minimize torque ripples during commutation. In Refs. [8–11], the basic idea was to distribute the desired torque to two adjacent phases during commutation using a specified torque distribution functions. The researchers in [12] proposed a linear decrease of the outgoing phase and a linear increase of the incoming phase during the commutation interval to obtain a high performance drive. In [13], a torque distribution function that minimizes the rates of change of currents over the commutation interval was proposed. In some of the above mentioned control schemes, an ideal inductance profile was assumed. However, none of these controllers considered the effects of mutual inductances during commutation. The effects of mutual inductances and the possibility of twophase excitation were mentioned in [14–18]. However, no control schemes were suggested to overcome the effects of mutual inductances. A novel dynamic model for the SRM which is suitable for twophase excitation control studies and which takes into account the effects of mutual inductances was proposed in [19]. A torque distribution function which reduces the rates of change of currents and which compensates for the effects of mutual couplings and the magnetic saturation was also presented. In addition, a control scheme which is based on linearizing and decoupling the mathematical nonlinear SRM model was proposed. The main objective of this paper is to investigate the development of nonlinear control schemes that can achieve high dynamical performances for the speed regulation of SRM drives taking into ac-
count the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core. A model which is suitable for two-phase excitation and which takes into account the effects of the mutual inductances between two adjacent phases and the effects of the magnetic saturation of the core is adopted in this paper. Two nonlinear speed controllers, which take into account the coupling and the nonlinearity in the current loop, are proposed for the SRM drive. Simulation results indicate that the proposed controllers work well and that they are robust to changes in the parameters of the system and to changes in the load torque. Moreover, the proposed control schemes are compared to a linearizing PI current controller motivated by the work [19] which was based on linearizing and decoupling the dynamics of the currents in the mathematical nonlinear SRM model. The paper is organized as follows. The model of the SRM is presented in Section 2. A feedback linearization controller and a sliding mode controller are proposed in Sections 3 and 4, respectively. Simulation results of the closed loop systems are presented and discussed in Section 5. In Section 6, comparisons of the proposed controllers with the results obtained from a linearizing PI current controller motivated by the work in [19] are provided. Finally, some concluding remarks are given in Section 7. 2. Model of the SRM including mutual inductances and magnetic saturation effects Since mutual inductances between two adjacent phases and the magnetic saturation of the iron core have important effects on the performances of the SRM drive and on the torque ripples, a reliable mathematical model which includes these effects has to be adopted to properly evaluate the SRM performances when different control techniques are used. Both spatial and magnetic nonlinearities are inherent characteristics of the SRM; as a result the motor parameters are functions of the rotor position and the phase currents. However, in many linear drive applications, for example [20,21], the SRM is operating in the magnetically linear region where the parameters of the system are expressed as functions of the rotor position only. Bae [19] developed a model which takes into account mutual inductances and the magnetic saturation effects. In this paper, magnetic saturation is considered and the analysis of the SRM is performed assuming that the motor operates in either the linear or the nonlinear magnetic regions. To include the effects of operating in the nonlinear magnetic operating region, the parameters of the system are expressed in terms of both the rotor position and the phase currents. 2.1. The parameters of the SRM In this paper, we adopt the SRM model developed by Bae [19]. The SRM parameters, such as self and mutual inductances for different rotor positions and phase currents, are obtained analytically by using the finite element method. The SRM parameters are experimentally verified in [19]. In Ref. [22], the SRM parameters were obtained when the phase currents are equal to 1.2 A, while assuming that the motor operates in the linear magnetic region. If the phase currents exceed 1.2 A, the magnetic circuit of the prototype SRM becomes saturated when the rotor moves towards the aligned position. Therefore, the inductances are no longer determined only by the rotor position. Thus, the general form of the inductances at any operating condition with respect to the rotor position and phase currents are represented as follows:
Lx ¼ Lx ðix ; hÞ Ly ¼ Ly ðiy ; hÞ M xy ¼ M xy ðix ; iy ; hÞ
ð1Þ
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0.09
I =1.2 Amp x
0.08
1.6
Mutual inductance Mxy (H)
Self inductance Lx (H)
x 10−3
1.8
I x =1.2 Amp
0.07 0.06 0.05 I =2.0 Amp x
0.04 0.03
1.4
1.2
1 I =2.0 Amp x
0.8
0.6
0.02
0.4
0.01 0
20
40
0
60
20
40
60
Rotor position (degrees)
Rotor position (degrees)
Fig. 2. The self and mutual inductances at various current levels.
where Lx and Ly are the self inductances, Mxy is the mutual inductance between phase x and phase y; ix and iy are the actual phase currents for phase x and phase y, respectively. The self inductance Lx and the mutual inductance Mxy at various phase current levels are shown in Fig. 2. The torque functions proposed in [19] are adopted. They are denoted as follows:
@K x ðix ; hÞ @h @K y ðiy ; hÞ Gy ðiy ; hÞ ¼ @h @K xy ðix ; iy ; hÞ Gxy ðix ; iy ; hÞ ¼ @h
Gx ðix ; hÞ ¼
ð2Þ
where Gx and Gy are the self torque functions and Gxy is the mutual torque function. Also, the functions K x ðix ; hÞ; K y ðiy ; hÞ and K xy ðix ; iy ; hÞ are arbitrary functions determined by solving the co-energy function [19]. The self and mutual torque functions are shown in Fig. 3. It should be noted that at the same phase current levels, these parameters have the same properties. The rates of change of self and mutual inductances with respect to rotor position are shown in Fig. 4. Note that the torque functions are less affected by the
phase currents than the rates of change of inductances with respect to the rotor position. Also, it should be noted that the variations in the parameters with respect to the mutual inductances are quite small and can be neglected. 2.2. The dynamic model of the SRM Two-phases excitation is considered to include the effects of mutual couplings. A four-phase system can be considered as a two-phase system because only one excitation region is considered at one time. Thus, the four phases equations are reduced to twophase equations as proposed in [19], and they are written as follows:
dix ¼ p1 ix p2 xix þ p3 iy þ p4 xiy þ p5 dt diy ¼ p6 iy p7 xiy þ p8 ix þ p9 xix þ p10 dt dx B 1 ¼ x þ ðT e T L Þ J J dt
vx
M xy Ly
vy
M xy
with
x 10−3
0.3
8
I =1.2 amp
(N.m/A )
x
2
2
Self torque function G (N.m/A )
0.2
6 4
x
Mutual torque function G
xy
0.1
0
I x =2.0 amp
−0.1
−0.2
−0.3
2 0 −2 −4 −6 −8
−0.4 0
20
40
60
0
Rotor position(degrees) Fig. 3. The self and mutual torque functions.
20
40
Rotor position(degrees)
!
vy
60
Lx
vx
ð3Þ
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0.3
8
0.2
6
x 10-3 I =1.2 Amp
∂ M (i ,0,θ) / ∂θ (H/rad)
x
∂ L (i , θ) / ∂θ (H/rad)
0.1
0 I x =2.0 Amp
x x
xy x
−0.1
−0.2
−0.3
4 2 I x =2.0 Amp 0 −2 −4 −6
I =1.2 Amp x
−0.4
−8 0
20
40
60
0
20
Rotor position (degrees)
40
60
Rotor position (degrees)
Fig. 4. The rates of change of self and mutual inductances with respect to rotor position.
Te ¼
1 2 1 2 2 2 Gx i þ Gy i þ Gxy ix iy ¼ p12 Jix þ p13 Jiy þ p14 Jix iy 2 x 2 y
ð4Þ
where the subscripts x are y are the phases under consideration. Note that the set (x, y) can take on the value (a, b), (b, c), (c, d), and (d, a). Also, note that, either phase x or phase y is leading and the other phase is following depending on the direction of rotation. The currents ix and iy are the currents in phase x and phase y. The speed of the rotor is denoted as x; the angle h is the rotor position, J is the rotor inertia, B is the damping factor, T e is the motor torque. Denote the winding phase resistance by R. The parameters of the motor can be written as,
@Lx ix ; @ix
Lx ¼ Lx þ p5 ¼
1 D
Ly ;
Ly ¼ Ly þ
p10 ¼
1 D
@Ly iy ; @iy
D ¼ Lx Ly M 2xy ;
Lx
p11 ¼
p2 ¼ p5
B ; J
p12 ¼
1 Gx ; 2J
p13 ¼
1 Gy ; 2J
p14 ¼
u1 ¼ v x u2 ¼
Ly
vy
1 Gxy J
ð6Þ
Note that by using u1 and u2 given in (5) and (6), one can easily obtain the phase voltages v x and v y such that:
vx ¼ vy ¼
Lx Ly D
u1 þ
Ly Mxy D
Lx Mxy
u1 þ
D Lx Ly D
u2 ð7Þ u2
ð8Þ
x_ 3 ¼ p11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Note that in model (8), the load torque T L is taken to be zero. However, the effects of the load torque on the performance of the system are studied in the simulations section. The objective of the paper is to design nonlinear control schemes such that the motor speed is driven to a desired constant speed xd . In this regards, define ixd ; iyd and xd as the desired values of the phase currents ix , iy and the speed x, respectively.
2
ð5Þ
M xy vx þ vy Lx
x_ 2 ¼ p6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 þ p10 u2
2
p11 xd þ p12 ixd þ p13 iyd þ p14 ixd iyd ¼ 0
Define the state variables x1 , x2 and x3 such that x1 ¼ ix , x2 ¼ iy and x3 ¼ x. Also, let the controllers u1 and u2 be such that:
Mxy
x_ 1 ¼ p1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 x2 þ p5 u1
Remark 1. It is clear from the third equation of the dynamic model given in (8) that the desired values of the phase currents and the motor speed should satisfy the equation:
!
@Lx M xy @M xy M xy R ; p3 ¼ p5 ; @h Ly @h Ly ! Mxy @Ly @Mxy ; p4 ¼ p5 @h Ly @h @Ly Mxy @M xy M xy R p6 ¼ p10 R; p7 ¼ p10 ; ; p8 ¼ p10 @h Lx @h Lx M xy @Lx @Mxy p9 ¼ p10 : @h Lx @h
p1 ¼ p5 R;
Using the definitions of the parameters of the motor given above and using Eqs. (5) and (6), the model of the SRM motor given by (3) and (4) can be written as,
ð9Þ
The motor torque T e given by (4) is a function of the phase currents. Therefore, this torque T e is indirectly controlled to a desired value by controlling the phase currents. Torque distribution functions which reduce the rate of changes of the phase currents, compensate for the mutual couplings between two adjacent phases and consider the magnetic saturation of the core were proposed in [19]; an approach to calculate the desired phase currents using the self and mutual torque functions given by (2) is suggested. The approach in [19] is adopted in this paper and the following equations are used to calculate the desired phase currents:
ixd iyd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gx T e ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 2 Gx þ Gy 2Gxy Gx Gy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gy T e ¼ pffiffiffiffiffiffiffiffiffiffiffi G2x þ G2y 2Gxy Gx Gy
ð10Þ
Note that in Eq. (10), the ‘‘±” term should be interpreted as follows: a plus sign is used when the desired output torque T e is positive and the minus sign is used when the torque T e is negative. To facilitate the design of control schemes for the SRM motor, we propose to use the following transformation:
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z1 ¼ x3 xd z2 ¼ p11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2
ð11Þ
z3 ¼ ðx1 ixd Þ2 þ ðx2 iyd Þ2
Remark 2. The choice of the states z1 ðtÞ; z2 ðtÞ and z3 ðtÞ in (11) guarantees that when z1 ðtÞ; z2 ðtÞ and z3 ðtÞ converge to zero then the states x1 ðtÞ; x2 ðtÞ and x3 ðtÞ converge to their desired values, respectively. The dynamic model of the SRM motor given by Eq. (8) can be transformed into the following model through the use of the transformation given in (11) as follows:
z_ 1 ¼ z2 1 z_ 2 ¼ f1 þ u _z3 ¼ f2 þ u 2
ð12Þ
1 and u 2 are defined as follows: where the functions f1 ; f 2 ; u
f1 ¼ p11 ðp11 x3 þ
p12 x21
þ
p13 x22
It can be easily checked that since a1 ; a2 and a3 are positive scalars, the matrix Ac1 is a stable matrix. The solution of Eq. (21) is zðtÞ ¼ expðAc tÞzð0Þ. Since Ac is a stable matrix, then zðtÞ converges to zero as t ! 1. h Therefore, since z(t) converges to zero as t ! 1, then the speed of the motor and the phase currents of the motor will converge to their desired values, respectively as t ! 1 because of transformation (11). Remark 4. The controller given by (19) and (20) can be written in the original coordinates by using transformation (11) and (18). Therefore,
1 ¼ f1 a1 ðx3 xd Þ a2 ðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Þ ð23Þ u 2 ¼ f2 a3 ðx1 ixd Þ2 a3 ðx2 iyd Þ2 u
where f1 and f2 are given by (13) and (14). Then, the controllers u1 and u2 are such:
þ p14 x1 x2 Þ
ð24Þ
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð25Þ
þ ð2p12 x1 þ p14 x2 Þðp1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 x2 Þ þ ð2p13 x2 þ p14 x1 Þðp6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 Þ d ixd f2 ¼ 2ðx1 ixd Þ p1 x1 p2 x3 x1 þ p3 x2 þ p4 x3 dt d iyd þ 2ðx2 iyd Þ p6 x2 p7 x3 x2 þ p8 x1 þ p9 x3 x1 dt 1 ¼ ð2p12 x1 þ p14 x2 Þp5 u1 þ ð2p13 x2 þ p14 x1 Þp10 u2 u 2 ¼ 2ðx1 ixd Þp5 u1 þ 2ðx2 iyd Þp10 u2 : u
ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ
4. Design of a sliding mode controller for the SRM Let b1 ; c1 ; c2 and W 1 ; W 2 be positive scalars. Since, the SRM has two inputs, one has to design two sliding surfaces. Let the first sliding surface r1 be such that,
r1 ¼ z2 þ b1 z1 Also, choose the second sliding surface
ð26Þ
r2 such that:
r2 ¼ z3 Remark 3. The design of the controllers will be accomplished using the transformed dynamic model given by (12), then the 2 will be transformed back into the original 1 and u controllers u coordinates using the following state dependent transformation:
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð18Þ
In the following two sections, two nonlinear control schemes are designed for the switched reluctance motor.
3. Design of a feedback linearization controller for the SRM
Proposition 2. The sliding mode controller,
1 ¼ f1 b1 z2 c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ u 2 ¼ f2 c2 z3 W 2 signðz3 Þ u
Proof. Differentiating Eq. (26) with respect to time and using (12), it follows that,
ð20Þ
when applied to the transformed motor system model given by (12) guarantees the asymptotic convergence of the states zðtÞ ¼ ½z1 ðtÞ z2 ðtÞ z3 ðtÞT to zero as t tends to infinity. Proof. The closed loop system when the controller (19) and (20) is applied to the system (12) is such:
ð21Þ
where,
2
0 6 Ac ¼ 4 a1 0
1 a2 0
0
3
7 0 5 a3
ð30Þ
r_ 1 ¼ f1 þ u1 þ b1 z2 ð19Þ
z_ ¼ Ac z
ð29Þ
1 by its value from (28), it follows that, Substituting u
Proposition 1. The feedback linearization controller:
1 ¼ f1 a1 z1 a2 z2 u 2 ¼ f2 a3 z3 u
ð28Þ
when applied to the SRM motor model given by (12), ensures that the states z1 ðtÞ; z2 ðtÞ and z3 ðtÞ converge to zero as t tends to infinity.
r_ 1 ¼ z_ 2 þ b1 z_ 1 ¼ f1 þ u1 þ b1 z2 Let a1 ; a2 and a3 be positive scalars.
ð27Þ
ð22Þ
¼ f1 þ b1 z2 f1 b1 z2 c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ ¼ c1 ðz2 þ b1 z1 Þ W 1 signðz2 þ b1 z1 Þ ¼ c1 r1 W 1 signðr1 Þ
ð31Þ
Also, differentiating Eq. (27) with respect to time, using (12) and 2 by its value from (29), it follows that, substituting u
r_ 2 ¼ f2 þ u2 ¼ c2 z3 W 2 signðz3 Þ ¼ c2 r2 W 2 signðr2 Þ
ð32Þ
The dynamics (31) and (32) guarantees that ri r_ i < 0 for (i ¼ 1; 2). Also, it can be checked that the trajectories associated with the discontinuous dynamics (31) and (32) exhibit a finite time reachability to zero from any given initial condition provided that the scalars c1 ; c2 ; W 1 and W 2 are chosen to be strictly positive. Therefore, we are guaranteed to reach r1 ¼ 0 and r2 ¼ 0 in a finite time.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
Since r1 is driven to zero in finite time, the variable z1 ðtÞ is governed after such finite amount of time by the first order differential equation z_ 1 ðtÞ þ b1 z1 ðtÞ ¼ 0. Thus z1 ðtÞ will asymptotically converge to 0 as t ! 1 because b1 is a positive scalar. Hence, z2 ðtÞ will also converge to zero as t ! 1. In addition, since r2 is driven to zero in finite time, the variable z3 ðtÞ will converge to zero in finite time.
speed (rad/s)
1292
Therefore, it can be concluded that the sliding mode controller (28) and (29) guarantees the asymptotic convergence of z1 ðtÞ; z2 ðtÞ and z3 ðtÞ to zero as t ! 1. h Thus, the controller (28) and (29) guarantees the asymptotic convergence of x1 ðtÞ; x2 ðtÞ and x3 ðtÞ to their desired values as t ! 1. Remark 5. The controller (28) and (29) can be written in the original coordinates by using the transformation (11) and (18). Therefore,
1 ¼ f1 ðb1 þ c1 Þðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 Þ u b1 c1 ðx3 xd Þ W 1 signðp11 x3 þ p12 x21 þ p13 x22 þ p14 x1 x2 þ b1 ðx3 xd ÞÞ
ð33Þ
2 ¼ f2 c2 ðx1 ixd Þ2 Þ c2 ðx2 iyd Þ2 W 2 signððx1 ixd Þ2 u þ ðx2 iyd Þ2 Þ
ð34Þ
where f1 and f2 are given by (13) and (14). Thus, the controllers u1 and u2 are such:
u1 u2
¼
ð2p12 x1 þ p14 x2 Þp5
ð2p13 x2 þ p14 x1 Þp10
2ðx1 ixd Þp5
2ðx2 iyd Þp10
1
1 u 2 u
ð35Þ
5. Simulation results
100 90 80 70 60 50 40 30 20 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (s) Fig. 6. Speed response (using FLC) of the SRM drive with changes in the motor reference speed.
5.1. Performance of the system using the feedback linearization controller The performance of the SRM system when the feedback linearization controller (FLC) is used is simulated using the Matlab software. The control law described by (23)–(25) is applied to the SRM system given by (3). Fig. 5 shows the speed response when the motor accelerates from rest to a desired speed of 100 rad/s, with a load torque of 0.15 N m. Fig. 6 shows the speed response of the motor with step changes in the reference speed. First, the motor is commanded to accelerate from rest to a speed of 100 rad/s, then to a speed of 150 rad/s and finally to a speed of 200 rad/s with a load torque of 0.2 N m. It can be seen from the figures that the motor speed converges to the desired speed with a very small steady state error. Note that the steady state error could be made to go to zero by tuning the gains of the controller. It should be mentioned that the ripples in the speed response are due to the sequential switching between the phases and they are not caused by the controller. The SRM phase currents are shown in Fig. 7. 5.2. Performance of the system using the sliding mode controller The performance of the SRM system when the sliding mode controller (SMC) is used is simulated. The control law described by (33)–(35) is applied to the SRM system given by (3). Fig. 8 shows the speed response when the motor is commanded to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.15 N m. The speed response of the motor with step changes in the reference speed is depicted in Fig. 9. First, the speed response of the motor is commanded to accelerate from rest to a speed of 100 rad/s, then to a speed of 150 rad/s and finally to a speed of 200 rad/s with a load torque of 0.2 N m. It can be seen from the figures that the motor speed converges to the desired speed. It should be mentioned that the ripples in the speed response are due to the sequential switching between the phases and they are not caused by the controller. The SRM phase currents are shown in Fig. 10.
phase currents (Amp)
speed (rad/s)
The SRM system is simulated using the MATLAB software [23]. The SRM model discussed in Section 2 is adopted; the model takes into account the effects of mutual inductances coupling and the magnetic saturation. The simulations are completed on a fourphase motor which has eight stator poles and six rotor poles. The stator pole arc is 16°. The rotor pole arc is 18°. The phase resistance is 1:6 X and the rated torque of the motor is 0.4 N m. The excitation angles (hon and hoff ) are kept fixed throughout the simulation studies at 0° and 30°, respectively (note that 0° corresponds to the aligned position; and 60° corresponds to the unaligned position). Note that two phases are allowed to be excited at one time. Simulations are performed when the proposed feedback linearization and sliding mode controllers are applied to the SRM system. The corresponding results are presented in the following subsections.
200 180 160 140 120 100 80 60 40 20
1.5 I d
I
Ia
Ic
b
I
d
1
0.5
0 0
0.05
0.1
0.15
0.2
0.25
Time (s) Fig. 5. Speed response (using FLC) of the SRM drive with fixed xd .
0.3
0.062
0.064
0.066
0.068
0.07
0.072
Time (s) Fig. 7. Phase currents (using FLC) waveforms of the SRM drive.
0.074
1293
120
120
100
100
speed (rad/s)
speed (rad/s)
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
80 60 40 20
80 60 R=2x1.6 ohm
40 20 0
0 −20
R=1x1.6 ohm
R=1x1.6 ohm
−20 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
0.05
0.1
0.15
Time (sec)
speed (rad/s)
speed (rad/s)
100 90 80 70 60 50 40 30 20 10
0.15
0.2
0.25
0.35
0.4
0.45
j=10x0.001
j=1x0.001
0 0.1
0.3
Fig. 12. Speed response (using SMC) of the SRM drive with changes in R.
200 180 160 140 120 100 80 60 40 20 0 0.05
0.25
Time (s)
Fig. 8. Speed response (using SMC) of the SRM drive.
0
0.2
0.3
0.35
0.4
j=1x0.001
0.05
0.1
0.15
0.2
0.25
0.3
0.45
Time (s)
Time (s)
Fig. 13. Speed response (using FLC) of the SRM drive with changes in J.
Fig. 9. Speed response (using SMC) of the SRM drive with changes in the motor reference speed.
120
Phase currents(Amp)
I
d
Ia
Ic
Ib
I
speed (rad/s)
100
1.5
d
1
0.5
80 60 j=10x0.001
40 20
j=1x0.001
j=1x0.001
0 −20 0
0 0.076
0.078
0.08
0.082
0.084
0.086
0.088
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (sec)
0.09
Time (s)
Fig. 14. Speed response (using SMC) of the SRM drive with changes in J.
100 90 80 70 60 50 40 30 20 10
speed (rad/s)
speed (rad/s)
Fig. 10. Phase currents (using SMC) waveforms of the SRM drive.
R=2x1.6 ohm
R=1x1.6 ohm
R=1x1.6 ohm
100 90 80 70 60 50 40 30 20 10
B=10x1e−5
0 0
0.05
0.1
0.15
0.2
0.25
0.3
Time (s)
B=1x1e−5
B=1x1e−5
0.05
0.1
0.15
0.2
0.25
0.3
Time (s) Fig. 15. Speed response (using FLC) of the SRM drive with changes in B.
Fig. 11. Speed response (using FLC) of the SRM drive with changes in R.
5.3. Robustness of the proposed control schemes The robustness of the proposed control schemes to changes in the parameters of the SRM and to changes in the load torque are tested through simulation studies. The effects of the changes in the phase resistance R, the rotor inertia J, the damping factor B are investigated through simulations. The simulations are carried
out by step changing one parameter at one time while keeping all the other parameters unchanged. The motor is commanded to accelerate from rest to a desired speed of 100 rad/s with a load torque of 0.15 N m. The motor responses when there are changes in the parameters of the SRM system are depicted in Figs. 11–16. Figs. 11 and 12 show the responses of the motor using the FLC and the SMC, respectively when the phase resistance is increased by 100% of its original value and then it is changed back to its original value.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297 120
120
100
100
80
speed (rad/s)
speed (rad/s)
1294
60 40
B=10x1e−5
20
B=1x1e−5
B=1x1e−5
80 60 40 T =0.1Nm l
20
0 −20 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.4
0
0.05
0.1
0.15
Time (sec)
100
120
80
100
60 40 T l =0.05 Nm
0
0.3
0.35
0.4
80 60 40 T =0.15 Nm l
20 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45 −20
Time (sec)
0
100 90 80 70 60 50 40 30 20 10 0
T l =0.05 Nm
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Fig. 21. Speed response (using FLC) of the SRM drive with load torque T L ¼ 0:15 N m.
0.35
110 100 90 80 70 60 50 40 30 20 10
Time (s)
T =0.15Nm l
0
Fig. 18. Speed response (using SMC) of the SRM drive with load torque T L ¼ 0:05 N m.
0.05
Time (sec)
speed (rad/s)
Fig. 17. Speed response (using FLC) of the SRM drive with load torque T L ¼ 0:05 N m.
speed (rad/s)
0.25
Fig. 20. Speed response (using SMC) of the SRM drive with the load torque T L ¼ 0:1 N m.
speed (rad/s)
speed (rad/s)
Fig. 16. Speed response (using SMC) of the SRM drive with changes in B.
20
0.2
Time (sec)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (s) Fig. 22. Speed response (using SMC) of the SRM drive with load torque T L ¼ 0:15 N m.
80
100
60
speed (rad/s)
speed (rad/s)
100
40 T =0.1Nm l
20 0
80 60 T l =2x0.2 Nm
40 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
T l =1x0.2 Nm
T =1x0.2 Nm l
0.45 0
Time (sec) Fig. 19. Speed response (using FLC) of the SRM drive with the load torque T L ¼ 0:1 N m.
Figs. 13 and 14 show the responses of the motor using the FLC and the SMC, respectively when the rotor inertia is varied by up to 10 times its original value. Figs. 15 and 16 show the responses of the motor using the FLC and the SMC, respectively when the damping factor B is varied by 10 times its original value. It is clear from these figures that the speed responses of the motor converge to the de-
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec) Fig. 23. Speed response (using FLC) of the SRM drive with changes in the load torque.
sired speed even when the parameters of the system change. Hence, it can be concluded that the simulation results show that the proposed controllers are robust to changes in the parameters of the SRM system.
1295
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.2
0.3
0.2 0.15 0.1 0.05 0 0.12
0.14
0.16
0.18
0.2
Time (s) Fig. 27. Output torque (using FLC) of the SRM drive with mutual inductance considered (T L ¼ 0:2 N m and x ¼ 10 rad=s).
Torque with mutual (Nm)
For high performance applications, it is desirable that the proposed control schemes reduce the torque ripples in the output torque. For comparison purposes, the output torque at low speeds (around 100 rpm) using the FLC and the SMC, respectively is shown in Figs. 25 and 26 when the mutual inductances are neglected. The output torque is shown in Figs. 27 and 28 using the FLC and the SMC, respectively when the mutual inductances are considered. It can be seen from these figures that the proposed control schemes have to some extent reduced the torque ripples when the mutual
speed (rad/s)
0.18
0.25
0.1
0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
Time (s) T =2x0.2Nm l T l =1x0.2Nm
0
0.05
Fig. 28. Output torque (using SMC) of the SRM drive with mutual inductance considered (T L ¼ 0:2 N m and x ¼ 10 rad=s). T =1x0.2Nm l
0.1
0.15
0.2
0.25
0.3
Time (sec) Fig. 24. Speed response (using SMC) of the SRM drive with changes in the load torque.
Torque without mutual (Nm)
0.16
Fig. 26. Output torque (using SMC) of the SRM drive with mutual inductance neglected (T L ¼ 0:2 N m and x ¼ 10 rad=s).
5.4. Output torque with and without mutual inductance
100 90 80 70 60 50 40 30 20 10
0.14
Time (s)
Torque with mutual (Nm)
In addition, simulation studies are carried out to demonstrate the robustness of the proposed controllers to changes in the load torque. The motor is commanded to accelerate from rest to a desired speed of 100 rad/s. Figs. 17–22 show the motor responses when the load torque takes on the values of 0.05, 0.1 and 0.15 N m, respectively. Figs. 17 and 18 show the motor responses using the FLC and the SMC, respectively when the load torque is 0.05 N m. Figs. 19 and 20 show the motor responses using the FLC and the SMC, respectively when the load torque equals 0.1 N m. Figs. 21 and 22 show the motor responses using the FLC and the SMC, respectively when the load torque equals 0.15 N m. It can be seen that for all the cases, the speed response converges to its desired value with small steady state errors. Figs. 23 and 24 depict the motor responses using the FLC and the SMC, respectively when the load torque changes during the motion of the rotor. The load torque is originally 0.2 N m, then it briefly becomes 0.4 N m, then it changes back to its original value. It can be seen from the figures that the motor speed responses show mall dips in the speed when the load is suddenly changed, however the controllers are able to keep the motor speed very close to the desired speed. Therefore, it can be concluded from the simulation studies that the proposed controllers are robust to changes in the parameters of the SRM system and to changes in the load torque.
Torque without mutual (Nm)
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
Time (s) Fig. 25. Output torque (using FLC) of the SRM drive with mutual inductance neglected (T L ¼ 0:2 N m and x ¼ 10 rad=s).
inductances are taken into account. Also, it can be seen from these figures that the proposed sliding mode controller gave better results than the feedback linearization controller. 6. Comparisons of the proposed controllers with previous work The control of SRM drives which adopts two-phase excitation, and takes into account both the mutual inductance effects between two adjacent phases and the magnetic saturation effects was tackled for the first time in the literature in [19]. It was shown that the mutual inductance between two adjacent phases partly contributes to generate the electromagnetic torque and introduces coupling between adjacent phases in the current control loop. The dynamics of the current loop are coupled and highly nonlinear due to the mutual inductance between two adjacent phases and the time varying nature of the inductance profile. Bae [19] proposed a linearizing PI controller to linearize and decouple the dynamics of the current loop. We will compare our proposed controllers, which take into account the coupling and the nonlinearity in the current loop, with a linearizing PI current controller motivated by the work in [19].
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Recall that the dynamic model of the SRM system given by (3) is such that:
dix ¼ p1 ix p2 xix þ p3 iy þ p4 xiy þ p5 u1 dt diy ¼ p6 iy p7 xiy þ p8 ix þ p9 xix þ p10 u2 dt dx B 1 ¼ x þ ðT e T L Þ J J dt
ð36Þ
To speed up the speed response, the controllers given by (39) and (40) are modified so that terms related to the error in the speed are included in the controllers. Therefore, the controllers u1 and u2 are now given as follows:
u1 ¼
1 d ixd ðp xix p3 iy p4 xiy þ þ p1 ixd m1 ex m2 p5 2 dt Z t Z t ex ðrÞdr m5 ex m6 ex ðrÞdrÞ 0
ð44Þ
0
with 2
2
T e ¼ p12 Jix þ p13 Jiy þ p14 Jix iy u1 ¼
vx
M xy Ly
!
vy
and u2 ¼
ð37Þ
vy
Mxy Lx
vx
ex ¼ ix ixd ey ¼ iy iyd
ð38Þ
e x ¼ xy xd Bae [19] proposed a PI controller which linearizes and decouples the phase currents. Motivated by his approach, a linearizing PI current controller is derived below. Let m1 ; m2 ; m3 and m4 be positive scalars. Define the controllers u1 and u2 such that:
1 d ixd ðp xix p3 iy p4 xiy þ þ p1 ixd m1 ex m2 p5 2 dt Z t ex ðrÞdrÞ
ð39Þ
0
u2 ¼
d iyd 1 ðp xiy p8 ix p9 xix þ þ p6 iyd m3 ey m4 p10 7 dt Z t ey ðrÞdrÞ
ð40Þ
Z
t
ex ðrÞdr
ð41Þ
ey ðrÞdr
ð42Þ
speed (rad/s)
The application of the controller given by (39) and (40) into the SRM model given by (36) results in the following closed loop dynamics:
0
e_ x ¼ ðp6 þ m3 Þey m4
Z
t
100 90 80 70 60 50 40 30 20 10 0 0
0
0.1
0.2
ð43Þ
Clearly, Eqs. (41) and (42) imply that ex and ey converge to zero asymptotically as t tends to infinity since p1 ; p6 ; m1 ; m2 ; m3 and m4 are positive scalars. Therefore, the phase currents ix and iy converge to their desired values ixd and iyd , respectively. Recalling that the desired values of the phase currents and the motor speed 2 2 should satisfy the equation: p11 xd þ p12 ixd þ p13 iyd þ p14 ixd iyd ¼ 0, it can be deduced from (43) and the fact that T L ¼ 0, that the motor speed converges to it desired constant value xd as t tends to infinity. The performance of the SRM system when the current linearizing PI controller is used is simulated. The control laws described by (39) and (40) is applied to the SRM system given by (36). Fig. 29 shows the speed response when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.2 N m. It is clear from the simulation results that the controller given by Eqs. (39) and (40) is a bit slow in forcing the speed of the motor to converge to its desired speed.
0.3
0.4
0.5
0.6
0.7
Time (s) Fig. 29. Speed response (using the linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
120 100
speed (rad/s)
dx B 1 ¼ x þ ðT e T L Þ J J dt
ð45Þ
0
where m5 ; m6 ; m7 and m8 are positive scalars. The simulation results of he SRM system given by (36) using the FLC, the SMC and the controllers (44) and (45) are depicted in Figs. 30–32. Fig. 30 shows the speed response when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, with a load torque of 0.2 N m. It is clear from this figure that the improved current linearizing PI controller is able to force the speed to converge to its desired value. However, the response exhibits some ripples. Fig. 31 shows the speed responses when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s, while using the FLC, the SMC and the improved current linearizing PI controller. This figure clearly shows that the FLC and SMC controllers gave better performances than the improved current linearizing PI controller. Note that the response when using the FLC controller is a bit faster than that of the SMC controller . However, the FLC controlled response has a small steady state error. Fig. 32 shows the speed responses when the motor is allowed to accelerate from rest to a desired speed of 100 rad/s when the load torque changes during the motion of the rotor. The load torque is origi-
0
e_ x ¼ ðp1 þ m1 Þex m2
d iyd 1 þ p6 iyd m3 ey m4 ðp xiy p8 ix p9 xix þ p10 7 dt Z t Z t ey ðrÞdr m7 ex m8 ex ðrÞdrÞ 0
Define the errors ex and ey ; ew such that:
u1 ¼
u2 ¼
80 60 40 20 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s) Fig. 30. Speed response (using the modified linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
M. Alrifai et al. / Energy Conversion and Management 51 (2010) 1287–1297
100 90 PI Controller
speed (rad/s)
80
SMC Controller FLC Controller
70 60 50 40 30 20 10 0 0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
100 PI Controller SMC Controller FLC Controller
speed (rad/s)
60 T =2x0.2Nm l
40 T l =1x0.2Nm
T l =1x0.2Nm
20
0 0
0.1
0.2
0.3
high performance characteristics of the proposed sliding mode controller, and that the sliding mode control scheme is very robust to variations in the parameters of the motor and to variations in the loads of the system. The proposed controllers are able to reduce the torque ripples to some extent and compensates for the effects of both mutual inductances and magnetic saturation of the core. Moreover, the proposed schemes are compared to a current linearizing PI controller proposed in the literature. Simulations studies indicate that the proposed schemes outperform the linearizing PI controller. Future research will address the implementation of the proposed control schemes on a hardware setup. References
Fig. 31. Speed responses (using the FLC, the SMC and the modified linearizing PI controller) of the SRM drive with load torque T L ¼ 0:2 N m.
80
1297
0.4
0.5
0.6
Time (s) Fig. 32. Speed responses (using the FLC, the SMC and the modified linearizing PI controller) of the SRM drive with changes in the load torque T L .
nally 0.2 N m, then it briefly becomes 0.4 N m, then it changes back to its original value. This figure clearly shows that the proposed controllers are more robust to changes in the load torque than the current linearizing PI controller. Therefore, it can be concluded that the proposed controllers gave better performances than the current linearizing PI controller, and that the SMC controller gave the best performance among the three controllers. 7. Conclusion In this paper, a feedback linearization controller and a sliding mode controller are proposed for speed regulation applications for switched reluctance motors. An SRM model which includes the effects of mutual inductances and the effects of the magnetic saturation of the core is adopted for the design of the control schemes and for simulation studies; the adopted model is also suitable for two-phase excitation. The proposed controllers, which take into account the coupling and the nonlinearity in the current loop, ensure the asymptotic convergence of the speed of the motor and the phase currents to their desired values. Simulation results indicate that the proposed nonlinear control schemes works well. In addition, the simulation results reveal the
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