# Nonlinear Speed Control of Switched Reluctance Motor Drives Taking into Account Mutual Inductance

Authors: M. Alrifai; M. Zribi; R. Krishnan; M. Rayan
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

A speed control algorithm is proposed for variable speed switched reluctance
motor (SRM) drives taking into account the effects of mutual inductances. The control
scheme adopts two-phase excitation; exciting two adjacent phases can overcome the problems associated with single-phase excitation such as large torque ripple, increased acoustic noise, and rotor shaft fatigues. The effects of mutual coupling between two adjacent phases and their contribution to the generated electromagnetic torque are considered in the design of the proposed control scheme for the motor. The proposed controller guarantees the convergence of the currents and the rotor speed of the motor to their desired values. Simulation results are given to illustrate the developed theory; the simulation studies show that the proposed controller works well. Moreover, the simulation results indicate that the proposed controller is robust to changes in the parameters of the motor and to changes in the load torque.


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## Body

## 1. Introduction

Switched
reluctance motors (SRMs) have gained  increasing popularity in recent years due
to their simplicity of construction, low cost, and simplicity of the associated
unipolar power converters. Moreover, SRMs can produce high torque at low
speeds. These characteristics, combined with advanced power electronic devices,
and the availability of high-speed processors make these motors attractive for
many applications not only for low performance applications such as fans and
hand-tools but also for high dynamic performance applications such as electric
vehicles and motors used for aerospace applications.The switched reluctance motor is a doubly salient
brushless motor, with concentrated copper windings on the stator poles, and no
windings or magnets on the rotor poles. Both stator and rotor cores are
constructed from iron laminations. Each phase is wound on diametrically
opposite stator poles. Figure1 shows the cross-section of an SRM with eight
stator poles and six rotor poles (a typical 8/6 SRM).
Excitation of a phase, that is, excitation of a pair of diametrically opposite
stator poles, causes the nearest pair of rotor poles to align with the excited
stator poles. This produces torque regardless of the direction of the current
in the phase winding. Therefore, only unipolar currents are required in stator
phases, and sequential excitation based on rotor position causes the rotor to
rotate and align its poles with the excited stator poles. A shaft position
encoder is often used to provide the rotor position information. The reader can
refer to [1–3] and the references therein
for more details on SRMs.Figure 1
The cross-section of an8/6
SRM.Both spatial and magnetic nonlinearities are inherent
characteristics of the SRM. As a result, the dynamic equations of the SRM are
nonlinear and time varying. High performance can hardly be achieved by using
conventional linear controllers because linearizing the system dynamics around
an operating point and designing a linear controller is generally not
sufficient to achieve the high dynamical performance required for high
performance drives. Therefore, it is necessary to take the system
nonlinearities into account and design feedback control laws that compensate
for the system nonlinearities and parameter uncertainties.Many researchers have focused on the control of
switched reluctance motors. Optimal current profiling [4] and torque distribution
functions [5–9] were proposed in
the literature to minimize torque ripples during commutation. In [5, 6], the basic idea was to
distribute the desired torque to two adjacent phases during commutation using
proposed torque distribution functions. Linearly decreasing the outgoing phase
and increasing the incoming phase during the commutation interval was proposed
in [7] to obtain a
high performance torque controller. In [8], a torque distribution function that minimizes the
rates of change of currents over the commutation interval was proposed. In some
of these 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 inductance and the possibility of two-phase
excitation were mentioned in [9–14], but no active control scheme was suggested to
overcome the effects of mutual inductances. A novel dynamic model for the SRM
which is suitable for two-phase excitation control studies and which takes into
account the mutual inductance was proposed in [15]. In that paper, a torque
distribution function which reduces the rates of change of currents and which
compensates for the effects of mutual coupling was presented; a control scheme
which is based on linearizing and decoupling the SRM model was also proposed.Due to the developments in nonlinear control theory,
many nonlinear control techniques have been developed for nonlinear systems in
the last three decades or so. Among these techniques are feedback linearization
control, sliding mode control, adaptive control, optimal control neural
control, fuzzy logic control, and backstepping control. Although some of these
nonlinear control techniques have been known for some time now, it is only
recently that they are being applied to SRM drives.Some of the speed controllers for the SRM reported in
the literature are open loop in nature. These controllers do not achieve high
dynamic performance which rivals that of DC and AC motor-based drives. The main
purpose of this research is to investigate the development of a nonlinear
control scheme that can achieve high dynamical performance as required for
speed regulation of an SRM while taking into account mutual inductances.In this paper, a nonlinear speed controller is
proposed for the SRM drive. The mathematical model proposed in [15] for the SRM is adopted. The
adopted model is suitable for two-phase excitation and takes into account the
effects of mutual inductance between two adjacent phases. Simulation results
indicate that the proposed controller works well and it is robust to changes in
the parameters of the system and to changes in the load torque.The paper is organized as follows. The model of the
SRM which takes into account mutual inductance is presented in Section2. In
Section 3, a nonlinear feedback controller is proposed for the SRM. Simulation
results are presented and discussed in Section 4. Finally, some concluding
remarks are given in Section 5.


## 2. Modeling of the Switched Reluctance Motor Including Mutual Inductance

Using a reliable mathematical model is essential for
properly evaluating the SRM performances when different control schemes are
used. Both spatial and magnetic nonlinearities are inherent characteristics of
the SRM and, as a result  the motor parameters, are functions of the
rotor position and current. The model introduced by [15], which takes into account
mutual inductance, is adopted in this paper. In many linear drive applications
[16, 17], the SRM is operating in
the magnetically linear region where the parameters of the system are expressed
as functions of the rotor position only. In this paper, magnetic saturation is
not considered and the analysis of the SRM is performed assuming that the motor
operates in the linear magnetic region. The extension of the proposed control
algorithm to include the nonlinear operation region can be achieved by
expressing the inductances in terms of the rotor position and the phase
currents.


### 2.1. The Parameters of the SRM

The parameters of the SRM, such as self-inductances,
mutual inductances, and the rates of change of the inductances with respect to
the rotor position, are usually obtained analytically by the finite element
analysis (FEA) method and they are usually verified experimentally by
measurements [18–24]. The SRM
parameters obtained by the FEA method and experimentally verified in [15] are adopted in this paper;
some small changes due to curve fitting are made. The SRM used is operating in
the linear magnetic region when the load torque is less than 0.2 Nm or when the
phase current is less than 1.2 amp. Therefore,
motor parameters are obtained at phase currents equal to 1.2 amp.The rates of change of the self- and mutual
inductances, with respect to rotor position, are defined as the self-torque
functions and the mutual torque functions, respectively. They are denoted as
follows:(1)gk(θ)=∂Lk(θ)∂θfork=a,b,c,d,gjk(θ)=∂Mjk(θ)∂θforjk=ab,bc,cd,da,
where Lk is the phase
inductance of phase k, Mjk is the mutual
inductance between phase j and phase k, and θ is the rotor
position.The self-inductances and the mutual inductances of the
prototype SRM are shown in Figure2. From the geometry of the prototype SRM
with 8 stator and 6 rotor poles, it is noticed that all the parameters have a
period of 60° and there is a
phase shift of 15° between
adjacent phases. The mutual inductance between the two adjacent phases is not
greater than 6.4% of the related
self-inductances at any position. The mutual inductance between the two
nonadjacent phases is negligible as they are less than 0.07% of the self-inductances.The self- and mutual inductances of the used8/6
SRM.
(a)(b)By examining the properties of the self-torque
functions and the torque functions from the plots in Figure2, it can be
deduced that two adjacent phases can generate the desired output torque during
a period of 15° degrees because
the polarities of the self-torque functions are equal during that interval.
This interval is defined as the excitation region. There are four different
excitation region in a period and only two phases are useful in an excitation
region. In addition, it is noticed that the mutual coupling between them is not
negligible [15]. In
Figures 3 and 4, the considered torque functions in each region are redrawn
from Figure 2. The mutual torque functions are magnified ten times for ease of
comparison. The waveforms of the corresponding self-torque functions in four
regions at the same torque are equal. However, the waveforms of the mutual
torque functions in the first region when Te is positive and
the third region when Te is negative are
different from the waveforms of the mutual torque functions in other excitation
regions because only the mutual coupling between phases d and a
is additive.The self-and mutual torque functions for positive
torque.
(a)(b)(c)(d)The self- and mutual torque functions for
negative torque.
(a)(b)(c)(d)


### 2.2. Dynamic Equations for Two-Phase Excitation of the SRM

To include the effects of mutual coupling, two phases
are considered simultaneously. 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 two-phase equations as proposed
in [15], and they are
written as follows:(2)dixdt=−a1ix−a2ωix+a3iy+a4ωiy+a0(vx−MxyLyvy),diydt=−b1iy−b2ωiy+b3ix+b4ωix+b0(vy−MxyLxvx),dωdt=−BJω+1JTewith(3)Te=12gxix2+12gyiy2+gxyixiy=c2Jix2+c3Jiy2+c4Jixiy,where the subscript x
and
y are the phases
in consideration and the set (x,y) is one
of (a,b), (b,c), (c,d), and (d,a).
Either phase x or phase y is leading and
the other phase is following according to the direction of rotation. The phase
voltages for phase x and phase y are denoted as vx and vy, respectively. The phase currents in phase x and phase y are denoted as ix and iy, respectively. The speed of the rotor is denoted as ω, θ is the rotor
position, J is the rotor
inertia, B is the damping
factor, and
Te is the motor
torque.The parameters of the motor
are such that(4)c1=BJ,c2=12Jgx,c3=12Jgy,c4=1Jgxy,D=LxLy−Mxy2,a0=1DLy,a1=a0Rs,a2=a0(gx−MxyLygxy),a3=a0MxyRsLy,a4=a0(−gxy+MxyLygy),b0=1DLx,b1=b0Rs,b2=b0(gy−MxyLxgxy),b3=b0MxyRsLx,b4=b0(−gxy+MxyLxgx),
where Rs is the winding
phase resistance. It should be mentioned that most of the motor parameters are
implicitly dependent on θ since dθ/dt=ω.Letux and uy be such
that(5)ux=vx−MxyLyvy,uy=vy−MxyLxvx.Note that using ux and uy, given in (5), one can obtain the phase voltages vx and vy such
that(6)vx=LxLyDux+LxMxyDuy,vy=LyMxyDux+LxLyDuy.
Also let the state variables x1, x2, and x3 be such that x1=ix, x2=iy, and x3=ω. Then, the model of the SRM motor, given by (2) and (3),
can be written as(7)x˙1=−a1x1−a2x3x1+a3x2+a4x3x2+a0ux,x˙2=−b1x2−b2x3x2+b3x1+b4x3x1+b0uy,x˙3=−c1x3+c2x12+c3x22+c4x1x2.
The objective of the paper is to design a control
scheme such that the motor speed is driven to a desired constant speed ωd.Defineixd, iyd, and ωd as the desired
values of the phase currents ix, iy and the speed ω, respectively. Note that ωd is constant.The motor torqueTe, expressed by (3), can be indirectly controlled to a
desired value by controlling phase currents. Torque
distribution functions which reduce the rate of change of phase currents and
compensate for the mutual coupling were proposed in [15]. In that paper, the authors
suggested an approach to calculate the desired phase currents using self- and
mutual torque functions given by (1), this approach is adopted in this
paper and the equations used to calculate the desired phase currents are as
follows:(8)ixd=2gxTegx2+gy2±2gxygxgy,iyd=2gyTegx2+gy2±2gxygxgy.Note that in (8), the  “±”
term should be interpreted as follows: a plus sign is used when the desired
output torque Te is positive and
minus sign is used when Te is negative.It should be noted that the desired values should
satisfy the equation(9)−c1ωd+c2ixd2+c3iyd2+c4ixdiyd=0.Equation (9) can be easily
obtained from the dynamic model given in (7).Define the following errors:(10)e1(t)=x1(t)−ixd(t),e2(t)=x2(t)−iyd(t),e3(t)=x3(t)−ωd.
Hence, using (7), (9), and (10), the model of the SRM
motor can be written as(11)e˙1=−a1(e1+ixd)−a2(e3+ωd)(e1+ixd)+a3(e2+iyd)+a4(e3+ωd)(e2+iyd)−dixd+a0ux,e˙2=−b1(e2+iyd)−b2(e3+ωd)(e2+iyd)+b3(e1+ixd)+b4(e3+ωd)(e1+ixd)−diyd+b0uy,e˙3=−c1(e3+ωd)+c2(e1+ixd)2+c3(e2+iyd)2+c4(e1+ixd)(e2+iyd)with dixd=dixd/dt,diyd=diyd/dt,anddωd/dt=0,
or
(12)e˙1=−a1e1−a1ixd−a2e1e3−a2ixde3−a2ωde1−a2ixdωd+a3e2+a3iyd+a4e2e3+a4iyde3+a4ωde2+a4iydωd−dixd+a0uxe˙2=−b1e2−b1iyd−b2e2e3−b2iyde3−b2ωde2−b2iydωd+b3e1+b3ixd+b4e1e3+b4ixde3+b4ωde1+b4ixdωd−diyd+b0uye˙3=−c1e3−c1ωd+c2e12+c2ixd2+2c2ixde1+c3e22+c3iyd2+2c3iyde2+c4e1e2+c4iyde1+c4ixde2+c4ixdiyd=−c1e3+c2e12+2c2ixde1+c3e22+2c3iyde2+c4e1e2+c4iyde1+c4ixde2.


## 2.1. The Parameters of the SRM

The parameters of the SRM, such as self-inductances,
mutual inductances, and the rates of change of the inductances with respect to
the rotor position, are usually obtained analytically by the finite element
analysis (FEA) method and they are usually verified experimentally by
measurements [18–24]. The SRM
parameters obtained by the FEA method and experimentally verified in [15] are adopted in this paper;
some small changes due to curve fitting are made. The SRM used is operating in
the linear magnetic region when the load torque is less than 0.2 Nm or when the
phase current is less than 1.2 amp. Therefore,
motor parameters are obtained at phase currents equal to 1.2 amp.The rates of change of the self- and mutual
inductances, with respect to rotor position, are defined as the self-torque
functions and the mutual torque functions, respectively. They are denoted as
follows:(1)gk(θ)=∂Lk(θ)∂θfork=a,b,c,d,gjk(θ)=∂Mjk(θ)∂θforjk=ab,bc,cd,da,
where Lk is the phase
inductance of phase k, Mjk is the mutual
inductance between phase j and phase k, and θ is the rotor
position.The self-inductances and the mutual inductances of the
prototype SRM are shown in Figure2. From the geometry of the prototype SRM
with 8 stator and 6 rotor poles, it is noticed that all the parameters have a
period of 60° and there is a
phase shift of 15° between
adjacent phases. The mutual inductance between the two adjacent phases is not
greater than 6.4% of the related
self-inductances at any position. The mutual inductance between the two
nonadjacent phases is negligible as they are less than 0.07% of the self-inductances.The self- and mutual inductances of the used8/6
SRM.
(a)(b)By examining the properties of the self-torque
functions and the torque functions from the plots in Figure2, it can be
deduced that two adjacent phases can generate the desired output torque during
a period of 15° degrees because
the polarities of the self-torque functions are equal during that interval.
This interval is defined as the excitation region. There are four different
excitation region in a period and only two phases are useful in an excitation
region. In addition, it is noticed that the mutual coupling between them is not
negligible [15]. In
Figures 3 and 4, the considered torque functions in each region are redrawn
from Figure 2. The mutual torque functions are magnified ten times for ease of
comparison. The waveforms of the corresponding self-torque functions in four
regions at the same torque are equal. However, the waveforms of the mutual
torque functions in the first region when Te is positive and
the third region when Te is negative are
different from the waveforms of the mutual torque functions in other excitation
regions because only the mutual coupling between phases d and a
is additive.The self-and mutual torque functions for positive
torque.
(a)(b)(c)(d)The self- and mutual torque functions for
negative torque.
(a)(b)(c)(d)


## 2.2. Dynamic Equations for Two-Phase Excitation of the SRM

To include the effects of mutual coupling, two phases
are considered simultaneously. 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 two-phase equations as proposed
in [15], and they are
written as follows:(2)dixdt=−a1ix−a2ωix+a3iy+a4ωiy+a0(vx−MxyLyvy),diydt=−b1iy−b2ωiy+b3ix+b4ωix+b0(vy−MxyLxvx),dωdt=−BJω+1JTewith(3)Te=12gxix2+12gyiy2+gxyixiy=c2Jix2+c3Jiy2+c4Jixiy,where the subscript x
and
y are the phases
in consideration and the set (x,y) is one
of (a,b), (b,c), (c,d), and (d,a).
Either phase x or phase y is leading and
the other phase is following according to the direction of rotation. The phase
voltages for phase x and phase y are denoted as vx and vy, respectively. The phase currents in phase x and phase y are denoted as ix and iy, respectively. The speed of the rotor is denoted as ω, θ is the rotor
position, J is the rotor
inertia, B is the damping
factor, and
Te is the motor
torque.The parameters of the motor
are such that(4)c1=BJ,c2=12Jgx,c3=12Jgy,c4=1Jgxy,D=LxLy−Mxy2,a0=1DLy,a1=a0Rs,a2=a0(gx−MxyLygxy),a3=a0MxyRsLy,a4=a0(−gxy+MxyLygy),b0=1DLx,b1=b0Rs,b2=b0(gy−MxyLxgxy),b3=b0MxyRsLx,b4=b0(−gxy+MxyLxgx),
where Rs is the winding
phase resistance. It should be mentioned that most of the motor parameters are
implicitly dependent on θ since dθ/dt=ω.Letux and uy be such
that(5)ux=vx−MxyLyvy,uy=vy−MxyLxvx.Note that using ux and uy, given in (5), one can obtain the phase voltages vx and vy such
that(6)vx=LxLyDux+LxMxyDuy,vy=LyMxyDux+LxLyDuy.
Also let the state variables x1, x2, and x3 be such that x1=ix, x2=iy, and x3=ω. Then, the model of the SRM motor, given by (2) and (3),
can be written as(7)x˙1=−a1x1−a2x3x1+a3x2+a4x3x2+a0ux,x˙2=−b1x2−b2x3x2+b3x1+b4x3x1+b0uy,x˙3=−c1x3+c2x12+c3x22+c4x1x2.
The objective of the paper is to design a control
scheme such that the motor speed is driven to a desired constant speed ωd.Defineixd, iyd, and ωd as the desired
values of the phase currents ix, iy and the speed ω, respectively. Note that ωd is constant.The motor torqueTe, expressed by (3), can be indirectly controlled to a
desired value by controlling phase currents. Torque
distribution functions which reduce the rate of change of phase currents and
compensate for the mutual coupling were proposed in [15]. In that paper, the authors
suggested an approach to calculate the desired phase currents using self- and
mutual torque functions given by (1), this approach is adopted in this
paper and the equations used to calculate the desired phase currents are as
follows:(8)ixd=2gxTegx2+gy2±2gxygxgy,iyd=2gyTegx2+gy2±2gxygxgy.Note that in (8), the  “±”
term should be interpreted as follows: a plus sign is used when the desired
output torque Te is positive and
minus sign is used when Te is negative.It should be noted that the desired values should
satisfy the equation(9)−c1ωd+c2ixd2+c3iyd2+c4ixdiyd=0.Equation (9) can be easily
obtained from the dynamic model given in (7).Define the following errors:(10)e1(t)=x1(t)−ixd(t),e2(t)=x2(t)−iyd(t),e3(t)=x3(t)−ωd.
Hence, using (7), (9), and (10), the model of the SRM
motor can be written as(11)e˙1=−a1(e1+ixd)−a2(e3+ωd)(e1+ixd)+a3(e2+iyd)+a4(e3+ωd)(e2+iyd)−dixd+a0ux,e˙2=−b1(e2+iyd)−b2(e3+ωd)(e2+iyd)+b3(e1+ixd)+b4(e3+ωd)(e1+ixd)−diyd+b0uy,e˙3=−c1(e3+ωd)+c2(e1+ixd)2+c3(e2+iyd)2+c4(e1+ixd)(e2+iyd)with dixd=dixd/dt,diyd=diyd/dt,anddωd/dt=0,
or
(12)e˙1=−a1e1−a1ixd−a2e1e3−a2ixde3−a2ωde1−a2ixdωd+a3e2+a3iyd+a4e2e3+a4iyde3+a4ωde2+a4iydωd−dixd+a0uxe˙2=−b1e2−b1iyd−b2e2e3−b2iyde3−b2ωde2−b2iydωd+b3e1+b3ixd+b4e1e3+b4ixde3+b4ωde1+b4ixdωd−diyd+b0uye˙3=−c1e3−c1ωd+c2e12+c2ixd2+2c2ixde1+c3e22+c3iyd2+2c3iyde2+c4e1e2+c4iyde1+c4ixde2+c4ixdiyd=−c1e3+c2e12+2c2ixde1+c3e22+2c3iyde2+c4e1e2+c4iyde1+c4ixde2.


## 3. Design of the Proposed Speed Controller

This section proposes a speed controller for the SRM
modeled in the previous section.Letk1, k2, W1, and W2 be positive
scalars; these scalars are design parameters.Proposition 1.
The controllersux and uy such
that
(13)ux=1a0(a1ixd+a2e1e3+a2ixde3+a2ωde1+a2ixdωd−a3e2−a3iyd−a4e2e3−a4iyde3+dixd−a4ωde2−a4iydωd−c2e1e3−2c2ixde3−c4iyde3−12c4e2e3−k1e1−W1signe1−k3e2e3),uy=1b0(b1iyd+b2e2e3+b2iyde3+b2ωde2+b2iydωd−b3e1−b3ixd−b4e1e3−b4ixde3+diyd−b4ωde1−b4ixdωd−c3e2e3−2c3iyde3−c4ixde3−12c4e1e3−k2e2−W2signe2−k3e1e3)guarantee the asympyotic
stability of the closed loop system.Proof.
Consider the following Lyapunov function candidate:
(14)V1=12e12+12e22+12e32; taking the deivation of V1 with respect to time and using (12) and (13), one obtains that
(15)V˙1=e1e˙1+e2e˙2+e3e˙3=−a1e12−a1ixde1−a2e12e3−a2ixde1e3−a2ωde12−a2ixdωde1−dixde1+a3e1e2+a3iyde1+a4e1e2e3+a4iyde1e3+a4ωde1e2+a4iydωde1+a0e1ux−b1e22−b1iyde2−b2e22e3−b2iyde2e3−b2ωde22−b2iydωde2−diyde2+b3e1e2+b3ixde2+b4e1e2e3+b4ixde2e3+b4ωde1e2+b4ixdωde2+b0e2uy−c1e32+c2e12e3+2c2ixde1e3+c3e22e3+2c3iyde2e3+c4e1e2e3+c4iyde1e3+c4ixde2e3=−a1e12−b1e22−c1e32−k1e12−W1e1sign(e1)−k2e22−W2e2sign(e2). Since V1 is positive
definite and V˙1 is negative
definite, it can be concluded that the errors converge asymptotically to zero
and hence the closed loop system is asymptotically stable. Therefore, the
currents ix, iy and the speed ω converge to
their desired values.Remark 1.
Note
that in the controllers given by (13), the sign(e1) and sign(e2) terms are
added. The addition of these two terms enable the controllers to change
structures. Variable structure control is capable of making a control system
robust with respect to system parametric uncertainties and external disturbances [25, 26].


## 4. Simulation Results

Simulations of
the proposed controller when it is applied to the SRM drive are carried out
using MATLAB. The parameters of the motor used for the simulation studies are
given in Table1. The excitation
angles (θon and θoff) are kept
fixed throughout the simulation studies at 0° and 30°, respectively (note that 0° and 60° correspond to
the aligned and the unaligned positions). Two phases are allowed to be excited
at one time.Table 1
Parameters of the SRM.
ParameterValueNumber of phases4Number of stator poles8Number of rotor poles6Stator pole arc16°Rotor pole arc18°Phase resistance1.6ΩRated torque0.4 NmThe proposed control scheme needs the measurements of
all the states for implementation purposes. The control scheme decides which
phases are to be excited at any given instant of time based on rotor position
information. High-resolution position encoders are often used to provide the
rotor information, and quadrature encoding is used to boost the resolution
(pulses per revolution). These encoders are not sensitive to noise. Hall effect
sensors are often used to sense the motor phase currents.For simulation studies, no precautions are needed to
increase the drive EMI noise immunity. However, precautions should be taken in
real life to protect the power inverter and increase the gate drive circuit EMI
noise immunity.The results of the simulations are presented in the
following subsections.


### 4.1. Performance of the System When the Developed Controller Is Used

The control law described by (13) is applied
to the SRM system described in Section 2. Figure 5 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 Nm. The speed
response of the motor when it is allowed 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 Nm is depicted
in Figure 6. It can be seen from the figures that the motor speed converges to
the desired speed with a very small steady-state error. It should be mentioned
that the ripples in the speed response is due to the sequential switching
between the phases and it is not caused by the controller. For completeness,
the SRM phase currents waveforms are shown in Figure 7.Figure 5
Speed response
of the SRM drive.Figure 6
Speed response
of the SRM drive.Figure 7
Phase currents
waveform of the SRM drive.


### 4.2. Performance of the System When PI Controller Is Used

The PI control law(16)u=kPωer+kI∫ωerdtis used for simulation studies.
The PI controller is applied to the SRM drive system described in Section 2.
Because the state variables of the SRM drive are highly nonlinearly coupled,
the conventional pole-placement method is not suitable in designing a PI
controller for the SRM. The gains kP and kI are tuned for
low-speed and low-load torque operations using trial-and-error method. Figure 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 Nm. It can be
seen from the figure that the motor speed converges to the desired speed with
some overshoot and a small steady state error. The simulation results show that
the proposed nonlinear controller yields a better
performance than the typical PI controller. Moreover, as discussed in [27], the PI controller requires
manual tuning. The PI controller needs to be retuned when the speed command
changes. It should be mentioned that the gains used for simulation studies are
obtained after several trials.Figure 8
Speed response
of the SRM drive when the PI controller is used.


### 4.3. Robustness of the Proposed Control Scheme

Simulation studies are undertaken to test the
robustness of the proposed controllers to changes in the parameters of the SRM
and changes in the load torque. Changes in the phase resistanceR, the rotor inertia J, and 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 Nm. Step
changes in one of the parameters of the motor occur at time t=0.06s and at time t=0.14s.Figures9–11  show the motor responses when there are
changes in the parameters of the SRM system. Figure 9 shows the response of the
motor when the phase resistance is increased by 100% of its original
value and then decreased by 20% of its original
value. Figure 10  shows the response of the motor when the rotor inertia is
varied by up to 10 times its
original value. Figure 11  shows the response of the motor when the damping
factor B is varied by up
to 10 times its
original value. It is clear from the figures that the speed responses of the
motor converge to the desired speed even when the parameters of the system
change drastically. Hence, it can be concluded that the simulation results show
that the proposed controller is robust to changes in the parameters of the SRM
system.Figure 9
Speed response
of the SRM drive with changes inR.Figure 10
Speed response
of the SRM drive with changes inJ.Figure 11
Speed response
of the SRM drive with changes inB.It is desirable for high-performance applications that
the proposed control scheme be robust to variations in the load torque.
Simulation studies are carried out to demonstrate the robustness of the
proposed controller to changes in the load torque. The motor is commanded to
accelerate from rest to a desired speed of100 rad/s. Figures
12–14 show the motor responses when the load torque takes on the values of 0.05, 0.1, and 0.15 Nm,
respectively. Figure 12 shows the motor response when the load torque is 0.05 Nm. Figure 13
shows the motor response when the load torque equals 0.1 Nm. Figure 14
shows the motor response when the load torque equals 0.15 Nm. It can be
seen that the speed response converges to its desired value for the different
values of the load torque. Figure 15  depicts the motor response when the load
torque changes during the motion of the rotor. The load torque is originally 0.15 Nm, then it
briefly becomes 0.4 Nm, and then it
changes back to its original value. It can be seen from the figure that the
motor speed response shows a small dip in the speed when the load is suddenly
changed, however the controller is able to keep the motor speed very close to
the desired speed.Figure 12
Speed response
of the SRM drive with load torque0.05 Nm.Figure 13
Speed response
of the SRM drive with the load torque0.1Nm.Figure 14
Speed response
of the SRM drive with load torque0.15Nm.Figure 15
Speed response
of the SRM drive with changes in the load torque.Therefore, it can be concluded from the simulation
studies that the proposed nonlinear controller is robust to changes in the
parameters of the SRM system and to changes in the load torque.For comparison purposes, the output torque at low
speed of10 rad/s (around 100 rpm) is shown
in Figure 16 when the mutual inductance is neglected; the output torque is shown
in Figure 17  when the mutual inductance is considered. It can be seen from these
figures that the proposed control scheme has to some extent reduced the torque
ripples when the mutual inductance is taken into account. It should be noted
that the overall system performance is determined by the characteristics of the
machine, the power converter, and the controller. Torque ripples at very low
speed are an important factor in characterizing position control. However, for
speed control, small torque ripples at high speeds are usually accepted given
that the motor speed follows the desired speed.Figure 16
Output torque
of the SRM drive with mutual inductance neglected (load torque0.2Nm
and speed
10rad/s
).Figure 17
Output torque
of the SRM drive with mutual inductance considered (load torque0.2Nm
and speed
10rad/s
).


## 4.1. Performance of the System When the Developed Controller Is Used

The control law described by (13) is applied
to the SRM system described in Section 2. Figure 5 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 Nm. The speed
response of the motor when it is allowed 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 Nm is depicted
in Figure 6. It can be seen from the figures that the motor speed converges to
the desired speed with a very small steady-state error. It should be mentioned
that the ripples in the speed response is due to the sequential switching
between the phases and it is not caused by the controller. For completeness,
the SRM phase currents waveforms are shown in Figure 7.Figure 5
Speed response
of the SRM drive.Figure 6
Speed response
of the SRM drive.Figure 7
Phase currents
waveform of the SRM drive.


## 4.2. Performance of the System When PI Controller Is Used

The PI control law(16)u=kPωer+kI∫ωerdtis used for simulation studies.
The PI controller is applied to the SRM drive system described in Section 2.
Because the state variables of the SRM drive are highly nonlinearly coupled,
the conventional pole-placement method is not suitable in designing a PI
controller for the SRM. The gains kP and kI are tuned for
low-speed and low-load torque operations using trial-and-error method. Figure 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 Nm. It can be
seen from the figure that the motor speed converges to the desired speed with
some overshoot and a small steady state error. The simulation results show that
the proposed nonlinear controller yields a better
performance than the typical PI controller. Moreover, as discussed in [27], the PI controller requires
manual tuning. The PI controller needs to be retuned when the speed command
changes. It should be mentioned that the gains used for simulation studies are
obtained after several trials.Figure 8
Speed response
of the SRM drive when the PI controller is used.


## 4.3. Robustness of the Proposed Control Scheme

Simulation studies are undertaken to test the
robustness of the proposed controllers to changes in the parameters of the SRM
and changes in the load torque. Changes in the phase resistanceR, the rotor inertia J, and 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 Nm. Step
changes in one of the parameters of the motor occur at time t=0.06s and at time t=0.14s.Figures9–11  show the motor responses when there are
changes in the parameters of the SRM system. Figure 9 shows the response of the
motor when the phase resistance is increased by 100% of its original
value and then decreased by 20% of its original
value. Figure 10  shows the response of the motor when the rotor inertia is
varied by up to 10 times its
original value. Figure 11  shows the response of the motor when the damping
factor B is varied by up
to 10 times its
original value. It is clear from the figures that the speed responses of the
motor converge to the desired speed even when the parameters of the system
change drastically. Hence, it can be concluded that the simulation results show
that the proposed controller is robust to changes in the parameters of the SRM
system.Figure 9
Speed response
of the SRM drive with changes inR.Figure 10
Speed response
of the SRM drive with changes inJ.Figure 11
Speed response
of the SRM drive with changes inB.It is desirable for high-performance applications that
the proposed control scheme be robust to variations in the load torque.
Simulation studies are carried out to demonstrate the robustness of the
proposed controller to changes in the load torque. The motor is commanded to
accelerate from rest to a desired speed of100 rad/s. Figures
12–14 show the motor responses when the load torque takes on the values of 0.05, 0.1, and 0.15 Nm,
respectively. Figure 12 shows the motor response when the load torque is 0.05 Nm. Figure 13
shows the motor response when the load torque equals 0.1 Nm. Figure 14
shows the motor response when the load torque equals 0.15 Nm. It can be
seen that the speed response converges to its desired value for the different
values of the load torque. Figure 15  depicts the motor response when the load
torque changes during the motion of the rotor. The load torque is originally 0.15 Nm, then it
briefly becomes 0.4 Nm, and then it
changes back to its original value. It can be seen from the figure that the
motor speed response shows a small dip in the speed when the load is suddenly
changed, however the controller is able to keep the motor speed very close to
the desired speed.Figure 12
Speed response
of the SRM drive with load torque0.05 Nm.Figure 13
Speed response
of the SRM drive with the load torque0.1Nm.Figure 14
Speed response
of the SRM drive with load torque0.15Nm.Figure 15
Speed response
of the SRM drive with changes in the load torque.Therefore, it can be concluded from the simulation
studies that the proposed nonlinear controller is robust to changes in the
parameters of the SRM system and to changes in the load torque.For comparison purposes, the output torque at low
speed of10 rad/s (around 100 rpm) is shown
in Figure 16 when the mutual inductance is neglected; the output torque is shown
in Figure 17  when the mutual inductance is considered. It can be seen from these
figures that the proposed control scheme has to some extent reduced the torque
ripples when the mutual inductance is taken into account. It should be noted
that the overall system performance is determined by the characteristics of the
machine, the power converter, and the controller. Torque ripples at very low
speed are an important factor in characterizing position control. However, for
speed control, small torque ripples at high speeds are usually accepted given
that the motor speed follows the desired speed.Figure 16
Output torque
of the SRM drive with mutual inductance neglected (load torque0.2Nm
and speed
10rad/s
).Figure 17
Output torque
of the SRM drive with mutual inductance considered (load torque0.2Nm
and speed
10rad/s
).


## 5. Conclusion

The design of a
nonlinear controller for speed regulation applications for an SRM has been
presented in this paper. An SRM model which includes the effects of mutual
inductances and which is suitable for two-phase excitation has been adopted.
Experimentally verified SRM parameters which takes mutual effects into account
were used for the simulation studies.Simulation results show that the proposed nonlinear
control scheme works well. The controller reduces the torque ripples and
compensates the effect of mutual inductance. Moreover, the simulation results
indicate that the proposed control scheme is robust to changes in the
parameters of the system; and robust to variations in the load torque.Future work will address the extension of the proposed
control algorithm to include the nonlinear operational region, and the
implementation of the proposed control scheme using a DSP-based digital
controller board and a hardware setup.



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*Source: 491625-2008-03-25.xml*