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The Discrete-Time Sliding Mode Torque Control of a Brushless DC Motor
CopyriSU1
THE DISCRETE·TIME SLIDING MODE TORQUE CONTROL OF A BRUSHLESS DC l\10TOR K. JEZERNIK and M. LUT AR
University of Maribor, Faculty of Technical Sciences Smetanova 17.6200 Maribor, Slovenia Abstract. When designing the control of a multiphase ac motor its model is usually transformed from the natural reference frame into the equivalent two-phase reference frame . In most cases this simplifies the model and consecutively the control scheme . Since the magnetic field of a brusWess de motor is not sinusoidally distributed. this transformauon In common doesn't simplify the model. This paper presents an approach that leads to a relatively simple model of a brusWess de motor in the rotor d-q reference frame whic~ enables the use of some knO\\l1 methods in the design process of the control. Based on this model, the discrete-time sliding mode torque control of a brusWess de motor is proposed. Simulation results are presented to verify the theoretical study.
Ke) Words. BrusWess de motor; discrete-time sliding mode torque control ; d-q transformation
1. INTRODUCTION According to their structure, brushless de motors (BLDC) belong to the class of ae machines. They are similar to the permanent-magnet synchronous machines (pMSM). The difference is in back emf waveforms. In order to accomplish sinusoidal waveforms by PMSM. stator windings have to be properly formed to compensate the rotor magnet field distribution. This makes PMSM to have more weight at the same nominal power. Because of considerable torque chanering, the conventional control ofBLDC is not suitable for high-performance servo drives [1] . This fact implies the idea that a control similar to the control of PMSM should be performed. The model of BLDC in the natural three-phase stator reference frame is represented by a system of equations, which is Dot suitable for control design. In case of PMSM, this model is transformed into the equivalent two-phase rotor reference frame, known as d-q frame. The result is a more simple system of equations, appropriate for the control design. In the case of BLDC, the appropriate model shall be used. Otherwise, this transformation doesn't simplify the model. because the expressions become more complicated and unsurveyable. In this paper, the approach which leads to the model of the machine with arbitrary back emf waveforms in d-q
frame will be presented and the discrete-time sliding mode torque control based on this model will be proposed. The simulation of this kind of drive is made and results are included.
2. MODEL OF BLDC In this paper the three-phase BLDC with two pole pairs (p=2) is modelled in the natural frame . All windings of the same phase are represented by a single equivalent winding and all of the parameters are appropriately changed. In this way the representation in electric frame is achieved. Electrical and mechanical quantities are related by: B,=p·B.
(2 .1)
=p.
(2.2)
(J),
(J).
The position of winding axes is shown in Fig. 1. Axes in the transformed d-q frame are positioned so that the d-axis coincides with the magnetic axis of the equivalent rotor magnet and the q-axis is perpendicular to the d-axis following it. The current direction in stator winding is determined so that the magnetic field it produces has the same direction as the appunenant axis. Voltage is determined so that it acts in the same sense as the current. The electromechanical torque T, is oriented so that it sup-
n-1l7
ports posItive rotation. and the load torque T, acts in the opposite direction. The transformation from the d-q frame into the natural frame is:
c=!f
- sine e, ) l/hj cos{ e, -~r.13) -sin{ e, -2 nI3 ) 1/J2 cos(
e)
[ cos{e,+2n/3)
-sin{e,+2nI3)
(2 .3)
l/h
and the inverse transformation is :
cJ=/f
[
cos( e,)
cos( e, - 2 n13)
-sin(e,)
-since, -2nI3) -sin{e,+2nI3) (2 .4)
I/J2
cos(e, +2nI
3)]
I/h
1/J2
\
\
b-axu ' \
" ,
\,
c~,
/ )~~ \ I ' '-, / / ~' ../
\\
I
~
~
', '
\~.
,
Due to the saliency of the pole structure (nonuniform air gap) the inductances vary with the rotor position as described in [2) . In order to get the expressions appropriate for transformation into the d-q frame, the simplified cosinusoidal variations are assumed : L.J, =L .(I+KP .cos(2 · e, +2 · '" )) ""''' •
(2 .6)
L IY =L" =L +m )) , . . .(I-K ,cos(2·e +", '"r'.J Y'y
(2 .7)
,
e = d)."" = d)."". de, =K .F(e)UJ , dt det dl • , t
~ ';. \.. -~- \
..T, \
\
\8.
'
•
I
(2.8)
•
n
___
~
__
where F(e,) can be "ntten as Fourier series : •
F( e, )= -
/ i
.~ .
....,
,
(2 .5)
'
.~ . ' 'eb d-ax.. .'
\'
..
dt
Back emf due to the flux linkage established by the permanent magnet as viewed from the stator phase winding is described by:
\ Q-UI5
,
U=R ' i+~' (L(e ).i)+e
M
•
\
currents are related at any instant of time by:
..~ / "
-'
--: .
•
Fig 1. Model of BLDC \\;th equivalent \\;ndings in electric reference frame
I,.0 K
l •• J
.sin((2 ·/ + I) · e,)
(2.9)
and KJ = I. In the continuation this function ,,;11 be called the form function . With PMSM there exists only K J , so the back emf has the sinusoidal waveform. With BLDC it is trapezoidal. so higher harmonics are present. Now, it is possible to transfonn (2.5) into the d-q frame . The result is the below equation: (2 .10)
Voltage equations Fig 2 shows the equivalent circuit of the voltage source invener fed BLDC. Each substitutional winding is represented by equivalent resistance R., inductance L. and back emf e,. It is assumed that the windings are symmetrical, that the magnetic system is linear, that large stator currents can be tolerated without significant demagnetization of the pennanent magnet, and that all of the circulating rotor currents are negligibly small. Voltages and
To simplify the expressions, following vinual mductivities are introduced: L.~L,-L.
(2 . 11)
L.=LK .1 ~ l , ·L, -K.4.ooor
(2 .12)
L,=K, ·L, +K. ·L.
(2 . 13)
Matrices in (2.10) are now: R
0 0]
o
0 R
R .. = 0 R 0 [
(2.14)
j
L ... -
-(L.. -L1 ) -J2. L,·sin (30J 0 (L.+L1 ) 0 J2.L,.cos(30J [ -+,.L,.sin(30J 7·L,.cos(30J 0
(2.15)
Fig. 2. Equivalent circuit ofBLDC with invencr
n-118
L ___
o
L.+LI
=
0 L.-LI [ i·L,.cos(3B,) -~·L ·sin(3B) .,;:'
.
* .L, .COs(3B,)] *·L,.sin(3B,) LL +2·L",
i.e.
T = P.(iT.d..t.(B.) +L1T. dL(B,) , dB, 2 dB,
.i)
(2 .21)
(2.16)
or: The transfonned back emfvector is eXl'ressed as:
F(e) + l- ' 1' T ' dL(e')'J T, =p' ( I'T" KT , 2 de,- ' 1
(2 .22)
(2.17)
where fonn functions F,(e,) and F.(e,) are:
F,( B,) = L -(KOi _1 + KOi.,)-sin( 6/ · e,)
/.,
F.( e.)= 1+
i:( -1)' (K'i-I - K~.,) · cos( 6/ · e,) /.,
(2 .18)
KT is called the torque constant. It is numerically equal to the back emf constant K• . The torque equation in the d-q frame can be obtained by the transfonnation of equation (2.22) or directly from (2.20); the result is :
(2 .19)
Torque equations The electromagnetic torque is produced by the interaction of the poles of the permanent magnet rotor, and the poles resulting from the rotating air gap mrnf established by currents flo\\ing in stator \\indings In machines of this kind. torque contributions arc obtained from both the pennanent magnet flux lmkage change and inductance variations. The expression for the developed torque is obtained from : -r
J
The rotational perfonnance of the motor depends on the developed electromagnetic torque T,. the load torque r;. mechanical characteristics of inertia J and friction B (only viscous friction is taken into account, other types are neglected) : J. dfJ) .. =T -T.-B .fJ) dt ' I •
dH~(i . e)
,=
Mechanical equation
(2.20)
de
+ ()o----{
R .......- ....
Fig. 3. Block scheme ofBLDC in rotor d-q frame
n-U9
(2 .24)
3. TORQUE CONTROL
J
The objective of torque control is to get a controllable electromagnetic torque independent of rotor position. The equation (2 .23) is reduced to a SImple relationsrup if one of the currents is equal to zero at any instant of time. It is simple to show that the c~ent j should be held at zero . In this case the
•
torque depends only on the current
i.
and rotor po-
sition B, : (3 .1)
T,=p ' V2 rI'KT ·F• (B, ).;,
Trus equation is similar to that of PMSM with the exception that there is no form function F. (B,). The relationsrup between the developed torque and the control vanable i, should be constant T=K ·i , ..,
(3 .2)
This requirement is satisfied if the actual current changes \-vith the rotor position in a manner that compensates the form function F. (B,) :
T = \2 {I .p.KT ·F~ (e, )· F-'(P )·i., " •
(3 .3)
#
ess.:· base d con trol cannot sau ~f,' this require::1ent, which introduces chattering of the currents and develc;:>ed torque. The S\\itching point in this case extends to the neighbourhood to which the motion of system trajectory' is bound This neighbourhood depends on time. needed to perform the control algorithm (discretization period -r) and on magrutudesof control inputs (voltages in this case) . While the dlscretization period cannot be shorter because of hmited process capability of available hardware. the idea is to modulate the control inputs in the manner that the neighbourhood of S\\itching points will be reduced. By pulse-v.idth modulation (PWM) of inverter output signals it is possible to generate appropriate average values of control voltages in one period 1, which drives the system trajectory' exactly to the S\\itching point at the end of the period. The control algorithm should be as simple as possible in order to keep the discretization period in the same range . While all of the values "ill be considered in sample instants k (t = K· T). the differentia) equations are replaced \\ith difference equations. The current derivative is replaced "ith the first difference :
According to the S\\ilChing nature of the inverter, sliding mode control [3] is the appropriate solution for the torque control of BLDC. The basic idea of trus control principle is appropriate S\\itching between the structures of the system. Thus, the system is forced to the desired behaviour, wruch is independent of it's 0\\11 dynamics . To explain this princIple, the second order system is suitable, because it can be presented in the phase plane. In this plane, the sliding line is chosen. When the trajectory of the system reaches this line, the switching between the states of the system takes place. This forces the trajectory to slide along the sliding line into the origin of the phase plane. In the case of the torque control of BLDC, the control problem disintegrates into two separate control problems with switching surfaces reduced to points: (3.4)
~ i( k + 1) - i( k ) di - -+ -= dt
•
(4.1 )
As mentioned before. the control should drive the trajectory from an arbitrary' point to the switching point 0'=0 in one period 1:. which means that the difference could be wrinen as
(4.2) By using PWM to modulate the voltages and by the substitution of differential with difference equations an error was introduced. It forces the trajectories to different motion as planned. Consequently, the difference (4.2) needs to be corrected by factor K. and the following conditions are satisfied in each step: (4 .3)
(4.4)
4 .. DISCRETE-TIME IMPLEMENTATION OF SLIDING MODE CONTROL
The meaning of the first condition is that the switching function er should change the sign in each step, thus making average values of actUal currents to be as close to their references as possible. The second condition implies that the trajectory will reach the neighbourhood of the switching point and remain in it; this is the condition for the existence of the sliding mode. So, the factor X, is limited to the lowest value of X, El by the fim. and to the
The drawback of sliding mode control is the infinite switching frequency for control inputs. Microproc-
highest value of X, =2 by the second condition. To make the safety region as big as possible, the value
(7
•
= , -( = i ••
•
-F-'(B ).j 41 I
.,
Control inputs are voltages
(3.5)
=0 11.
and "• .
TI-120
KI = 3/2 was chosen. Now, voltage equation (2.10)
can be wrinen as:
3· (L~ -L
1)
.((k)+(R
3·(L~ -LI))./ (k)+
2· r ·
2·r·
+(L~ + L1 )· tiI, (k ).j,,(k) +e, (tiI, (k ),8, (k)) (4 .6)
The implementation involves a problem with measurement and computing, which takes one period 1: .
So, to satisfy equations (4.5) and (4 .6), values in the moment of response on the invener should be used. While they are not available in the beginning of the computati~nal interval, the prediction of these values should take place. As velocity changes in one period are negligibly small, it is simple to predict values of velocity and position on the base of measured values from the beginning of the interval . The problems are currents, which have to change their values significantly in each period. But, on the base of actual differences in previous sampling instants, the differences in the instantaneous step can be predicted and so can be values of currents. Inspection of the actual behaviour of currents can also be used to properly determine expressions (4 .5) and (4 .6), if
·,,-r-
(a)
(a)
.. -- ~I-
-- -:'.
-..
-~.
. .. - -
(b)
(b)
(c)
(c)
Fig. 4. Free acceleration orBLDC drive with discretizcd conventional sliding-mode torque control algorithm at i~ =3A :
Fig. S. Free acceleration orBLDC drive with proposed sliding-mode torque control algorithm at i~=3A: (a) .....developed torque (b) .....sbaft velocity (c) ..... c:urrent of equivalent phase a
(a) ...•.developed torque (b) .....shaft velocity
(c) .••••c:urrent of equivalent phase a
n -121
exact values of motor parameters are not available.
is then better filtered by mechanical propenies of the machine.
5. CONCLUSION 6. REFERENCES The proposed control algorithm is up to now verified by simulation; results are presented. It is evident that results are much better than those obtained by the discretized conventional sliding mode control algorithm. The advantage of the derived method is that it is appropriate for any permanent magnet ac machine with arbitral)' back emf waveform. In fact, the back emf of any PMSM differs from the sinusoidal , and the back emf of any BLDC differs from the trapezoidal. The advantage of the proposed algorithm, suitable for microprocessor realization. is in the Significantly reduced current and torque chattering at increased frequencies . This chattering
D-l22
[1) Carlson, 1. Cros, M . Lajoie-Mazenc. "The Anal~tical Detennination of the Characteristics of Permanent Magnet Brushless dc Drives", EPE Firenze, pp. 468-471, 1991 [2] Wallace, R. Spee, "The Effects of Motor Parameters on the Performance of Brushless DC Drives", IEEE Transactions on Power Electronics, Vol. 5, No. 1, pp. 2-8, January 1990 [3] Venkataramanan. "Sliding Mode Control of Power Conveners", California Institute of Technology, Pasadena. California. 1986
THE DISCRETE·TIME SLIDING MODE TORQUE CONTROL OF A BRUSHLESS DC l\10TOR K. JEZERNIK and M. LUT AR
University of Maribor, Faculty of Technical Sciences Smetanova 17.6200 Maribor, Slovenia Abstract. When designing the control of a multiphase ac motor its model is usually transformed from the natural reference frame into the equivalent two-phase reference frame . In most cases this simplifies the model and consecutively the control scheme . Since the magnetic field of a brusWess de motor is not sinusoidally distributed. this transformauon In common doesn't simplify the model. This paper presents an approach that leads to a relatively simple model of a brusWess de motor in the rotor d-q reference frame whic~ enables the use of some knO\\l1 methods in the design process of the control. Based on this model, the discrete-time sliding mode torque control of a brusWess de motor is proposed. Simulation results are presented to verify the theoretical study.
Ke) Words. BrusWess de motor; discrete-time sliding mode torque control ; d-q transformation
1. INTRODUCTION According to their structure, brushless de motors (BLDC) belong to the class of ae machines. They are similar to the permanent-magnet synchronous machines (pMSM). The difference is in back emf waveforms. In order to accomplish sinusoidal waveforms by PMSM. stator windings have to be properly formed to compensate the rotor magnet field distribution. This makes PMSM to have more weight at the same nominal power. Because of considerable torque chanering, the conventional control ofBLDC is not suitable for high-performance servo drives [1] . This fact implies the idea that a control similar to the control of PMSM should be performed. The model of BLDC in the natural three-phase stator reference frame is represented by a system of equations, which is Dot suitable for control design. In case of PMSM, this model is transformed into the equivalent two-phase rotor reference frame, known as d-q frame. The result is a more simple system of equations, appropriate for the control design. In the case of BLDC, the appropriate model shall be used. Otherwise, this transformation doesn't simplify the model. because the expressions become more complicated and unsurveyable. In this paper, the approach which leads to the model of the machine with arbitrary back emf waveforms in d-q
frame will be presented and the discrete-time sliding mode torque control based on this model will be proposed. The simulation of this kind of drive is made and results are included.
2. MODEL OF BLDC In this paper the three-phase BLDC with two pole pairs (p=2) is modelled in the natural frame . All windings of the same phase are represented by a single equivalent winding and all of the parameters are appropriately changed. In this way the representation in electric frame is achieved. Electrical and mechanical quantities are related by: B,=p·B.
(2 .1)
=p.
(2.2)
(J),
(J).
The position of winding axes is shown in Fig. 1. Axes in the transformed d-q frame are positioned so that the d-axis coincides with the magnetic axis of the equivalent rotor magnet and the q-axis is perpendicular to the d-axis following it. The current direction in stator winding is determined so that the magnetic field it produces has the same direction as the appunenant axis. Voltage is determined so that it acts in the same sense as the current. The electromechanical torque T, is oriented so that it sup-
n-1l7
ports posItive rotation. and the load torque T, acts in the opposite direction. The transformation from the d-q frame into the natural frame is:
c=!f
- sine e, ) l/hj cos{ e, -~r.13) -sin{ e, -2 nI3 ) 1/J2 cos(
e)
[ cos{e,+2n/3)
-sin{e,+2nI3)
(2 .3)
l/h
and the inverse transformation is :
cJ=/f
[
cos( e,)
cos( e, - 2 n13)
-sin(e,)
-since, -2nI3) -sin{e,+2nI3) (2 .4)
I/J2
cos(e, +2nI
3)]
I/h
1/J2
\
\
b-axu ' \
" ,
\,
c~,
/ )~~ \ I ' '-, / / ~' ../
\\
I
~
~
', '
\~.
,
Due to the saliency of the pole structure (nonuniform air gap) the inductances vary with the rotor position as described in [2) . In order to get the expressions appropriate for transformation into the d-q frame, the simplified cosinusoidal variations are assumed : L.J, =L .(I+KP .cos(2 · e, +2 · '" )) ""''' •
(2 .6)
L IY =L" =L +m )) , . . .(I-K ,cos(2·e +", '"r'.J Y'y
(2 .7)
,
e = d)."" = d)."". de, =K .F(e)UJ , dt det dl • , t
~ ';. \.. -~- \
..T, \
\
\8.
'
•
I
(2.8)
•
n
___
~
__
where F(e,) can be "ntten as Fourier series : •
F( e, )= -
/ i
.~ .
....,
,
(2 .5)
'
.~ . ' 'eb d-ax.. .'
\'
..
dt
Back emf due to the flux linkage established by the permanent magnet as viewed from the stator phase winding is described by:
\ Q-UI5
,
U=R ' i+~' (L(e ).i)+e
M
•
\
currents are related at any instant of time by:
..~ / "
-'
--: .
•
Fig 1. Model of BLDC \\;th equivalent \\;ndings in electric reference frame
I,.0 K
l •• J
.sin((2 ·/ + I) · e,)
(2.9)
and KJ = I. In the continuation this function ,,;11 be called the form function . With PMSM there exists only K J , so the back emf has the sinusoidal waveform. With BLDC it is trapezoidal. so higher harmonics are present. Now, it is possible to transfonn (2.5) into the d-q frame . The result is the below equation: (2 .10)
Voltage equations Fig 2 shows the equivalent circuit of the voltage source invener fed BLDC. Each substitutional winding is represented by equivalent resistance R., inductance L. and back emf e,. It is assumed that the windings are symmetrical, that the magnetic system is linear, that large stator currents can be tolerated without significant demagnetization of the pennanent magnet, and that all of the circulating rotor currents are negligibly small. Voltages and
To simplify the expressions, following vinual mductivities are introduced: L.~L,-L.
(2 . 11)
L.=LK .1 ~ l , ·L, -K.4.ooor
(2 .12)
L,=K, ·L, +K. ·L.
(2 . 13)
Matrices in (2.10) are now: R
0 0]
o
0 R
R .. = 0 R 0 [
(2.14)
j
L ... -
-(L.. -L1 ) -J2. L,·sin (30J 0 (L.+L1 ) 0 J2.L,.cos(30J [ -+,.L,.sin(30J 7·L,.cos(30J 0
(2.15)
Fig. 2. Equivalent circuit ofBLDC with invencr
n-118
L ___
o
L.+LI
=
0 L.-LI [ i·L,.cos(3B,) -~·L ·sin(3B) .,;:'
.
* .L, .COs(3B,)] *·L,.sin(3B,) LL +2·L",
i.e.
T = P.(iT.d..t.(B.) +L1T. dL(B,) , dB, 2 dB,
.i)
(2 .21)
(2.16)
or: The transfonned back emfvector is eXl'ressed as:
F(e) + l- ' 1' T ' dL(e')'J T, =p' ( I'T" KT , 2 de,- ' 1
(2 .22)
(2.17)
where fonn functions F,(e,) and F.(e,) are:
F,( B,) = L -(KOi _1 + KOi.,)-sin( 6/ · e,)
/.,
F.( e.)= 1+
i:( -1)' (K'i-I - K~.,) · cos( 6/ · e,) /.,
(2 .18)
KT is called the torque constant. It is numerically equal to the back emf constant K• . The torque equation in the d-q frame can be obtained by the transfonnation of equation (2.22) or directly from (2.20); the result is :
(2 .19)
Torque equations The electromagnetic torque is produced by the interaction of the poles of the permanent magnet rotor, and the poles resulting from the rotating air gap mrnf established by currents flo\\ing in stator \\indings In machines of this kind. torque contributions arc obtained from both the pennanent magnet flux lmkage change and inductance variations. The expression for the developed torque is obtained from : -r
J
The rotational perfonnance of the motor depends on the developed electromagnetic torque T,. the load torque r;. mechanical characteristics of inertia J and friction B (only viscous friction is taken into account, other types are neglected) : J. dfJ) .. =T -T.-B .fJ) dt ' I •
dH~(i . e)
,=
Mechanical equation
(2.20)
de
+ ()o----{
R .......- ....
Fig. 3. Block scheme ofBLDC in rotor d-q frame
n-U9
(2 .24)
3. TORQUE CONTROL
J
The objective of torque control is to get a controllable electromagnetic torque independent of rotor position. The equation (2 .23) is reduced to a SImple relationsrup if one of the currents is equal to zero at any instant of time. It is simple to show that the c~ent j should be held at zero . In this case the
•
torque depends only on the current
i.
and rotor po-
sition B, : (3 .1)
T,=p ' V2 rI'KT ·F• (B, ).;,
Trus equation is similar to that of PMSM with the exception that there is no form function F. (B,). The relationsrup between the developed torque and the control vanable i, should be constant T=K ·i , ..,
(3 .2)
This requirement is satisfied if the actual current changes \-vith the rotor position in a manner that compensates the form function F. (B,) :
T = \2 {I .p.KT ·F~ (e, )· F-'(P )·i., " •
(3 .3)
#
ess.:· base d con trol cannot sau ~f,' this require::1ent, which introduces chattering of the currents and develc;:>ed torque. The S\\itching point in this case extends to the neighbourhood to which the motion of system trajectory' is bound This neighbourhood depends on time. needed to perform the control algorithm (discretization period -r) and on magrutudesof control inputs (voltages in this case) . While the dlscretization period cannot be shorter because of hmited process capability of available hardware. the idea is to modulate the control inputs in the manner that the neighbourhood of S\\itching points will be reduced. By pulse-v.idth modulation (PWM) of inverter output signals it is possible to generate appropriate average values of control voltages in one period 1, which drives the system trajectory' exactly to the S\\itching point at the end of the period. The control algorithm should be as simple as possible in order to keep the discretization period in the same range . While all of the values "ill be considered in sample instants k (t = K· T). the differentia) equations are replaced \\ith difference equations. The current derivative is replaced "ith the first difference :
According to the S\\ilChing nature of the inverter, sliding mode control [3] is the appropriate solution for the torque control of BLDC. The basic idea of trus control principle is appropriate S\\itching between the structures of the system. Thus, the system is forced to the desired behaviour, wruch is independent of it's 0\\11 dynamics . To explain this princIple, the second order system is suitable, because it can be presented in the phase plane. In this plane, the sliding line is chosen. When the trajectory of the system reaches this line, the switching between the states of the system takes place. This forces the trajectory to slide along the sliding line into the origin of the phase plane. In the case of the torque control of BLDC, the control problem disintegrates into two separate control problems with switching surfaces reduced to points: (3.4)
~ i( k + 1) - i( k ) di - -+ -= dt
•
(4.1 )
As mentioned before. the control should drive the trajectory from an arbitrary' point to the switching point 0'=0 in one period 1:. which means that the difference could be wrinen as
(4.2) By using PWM to modulate the voltages and by the substitution of differential with difference equations an error was introduced. It forces the trajectories to different motion as planned. Consequently, the difference (4.2) needs to be corrected by factor K. and the following conditions are satisfied in each step: (4 .3)
(4.4)
4 .. DISCRETE-TIME IMPLEMENTATION OF SLIDING MODE CONTROL
The meaning of the first condition is that the switching function er should change the sign in each step, thus making average values of actUal currents to be as close to their references as possible. The second condition implies that the trajectory will reach the neighbourhood of the switching point and remain in it; this is the condition for the existence of the sliding mode. So, the factor X, is limited to the lowest value of X, El by the fim. and to the
The drawback of sliding mode control is the infinite switching frequency for control inputs. Microproc-
highest value of X, =2 by the second condition. To make the safety region as big as possible, the value
(7
•
= , -( = i ••
•
-F-'(B ).j 41 I
.,
Control inputs are voltages
(3.5)
=0 11.
and "• .
TI-120
KI = 3/2 was chosen. Now, voltage equation (2.10)
can be wrinen as:
3· (L~ -L
1)
.((k)+(R
3·(L~ -LI))./ (k)+
2· r ·
2·r·
+(L~ + L1 )· tiI, (k ).j,,(k) +e, (tiI, (k ),8, (k)) (4 .6)
The implementation involves a problem with measurement and computing, which takes one period 1: .
So, to satisfy equations (4.5) and (4 .6), values in the moment of response on the invener should be used. While they are not available in the beginning of the computati~nal interval, the prediction of these values should take place. As velocity changes in one period are negligibly small, it is simple to predict values of velocity and position on the base of measured values from the beginning of the interval . The problems are currents, which have to change their values significantly in each period. But, on the base of actual differences in previous sampling instants, the differences in the instantaneous step can be predicted and so can be values of currents. Inspection of the actual behaviour of currents can also be used to properly determine expressions (4 .5) and (4 .6), if
·,,-r-
(a)
(a)
.. -- ~I-
-- -:'.
-..
-~.
. .. - -
(b)
(b)
(c)
(c)
Fig. 4. Free acceleration orBLDC drive with discretizcd conventional sliding-mode torque control algorithm at i~ =3A :
Fig. S. Free acceleration orBLDC drive with proposed sliding-mode torque control algorithm at i~=3A: (a) .....developed torque (b) .....sbaft velocity (c) ..... c:urrent of equivalent phase a
(a) ...•.developed torque (b) .....shaft velocity
(c) .••••c:urrent of equivalent phase a
n -121
exact values of motor parameters are not available.
is then better filtered by mechanical propenies of the machine.
5. CONCLUSION 6. REFERENCES The proposed control algorithm is up to now verified by simulation; results are presented. It is evident that results are much better than those obtained by the discretized conventional sliding mode control algorithm. The advantage of the derived method is that it is appropriate for any permanent magnet ac machine with arbitral)' back emf waveform. In fact, the back emf of any PMSM differs from the sinusoidal , and the back emf of any BLDC differs from the trapezoidal. The advantage of the proposed algorithm, suitable for microprocessor realization. is in the Significantly reduced current and torque chattering at increased frequencies . This chattering
D-l22
[1) Carlson, 1. Cros, M . Lajoie-Mazenc. "The Anal~tical Detennination of the Characteristics of Permanent Magnet Brushless dc Drives", EPE Firenze, pp. 468-471, 1991 [2] Wallace, R. Spee, "The Effects of Motor Parameters on the Performance of Brushless DC Drives", IEEE Transactions on Power Electronics, Vol. 5, No. 1, pp. 2-8, January 1990 [3] Venkataramanan. "Sliding Mode Control of Power Conveners", California Institute of Technology, Pasadena. California. 1986