Aug 18, 1993 - Abstract: A new control method for the robust position control of a brushless DC (BLDC) motor using the adaptive load torque observer is pre-.
Robust digital position control of brushless DC motor with adaptive load torque observer J.-S. KO J.-H. Lee M.-J. Youn
Indexing term: Adaptiue load torque Observer, BmWess DC motor, Robust position control
1
Abstract: A new control method for the robust position control of a brushless DC (BLDC) motor using the adaptive load torque observer is presented. It is shown that the augmented state variable feedback can be applied to the BLDC motor system approximately linearised using the fieldorientation method. To overcome the problem of an unknown parameter or a parameter variation such as a flux linkage, an adaptive mechanism is employed. As a result, the robustness can be obtained without affecting the overall system response. The load disturbance is detected by the adaptive 0-observer of an unknown and inaccessible input, and is compensated by feedforwarding the equivalent current having the fast response.
List of symbols
r, P L,, Lq L& L,, L ,
= stator resistance = number of poles = d and q axes stator inductances, respectively = stator leakage inductance = d and q axes magnetising inductances,
respectively = flux linkage of permanent magnet 0, = rotor angular velocity = q axis stator voltage ab, aqs = d and q axes stator currents, respectively = q axis current command ih, ,i = three-phase current commands = motor torque = estimated load torque TL * y, y, y, = rotor position, estimated rotor position, An
5.
ks ,
x, f, 2,
a,,& E e
h
and reference rotor position, respectively = state space vector, augmented state space vector, and state space vector of observer, respectively =system parameter vectors of known and unknown coefficients, respectively = estimated output error = rotor position error = sampling interval
8 IEE, 1994 Papa 9858B (PI), received Und January and in revised form 18th August 1993 The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusongdong, Yusong-gu, Taejon 305-701, Korca I E E Proc.-Electr. Power Appl., Vol. 141, No. 2,March I994
Introduction
DC motors have gradually been replaced by BLDC motors since industrial applications require more powerful actuators in small sizes. The advantage of using a BLDC motor is that it can be controlled to have speedtorque characteristics similar to those of a permanent magnet DC motor. In addition, the BLDC motor has low inertia, a large power-to-volume ratio, and low noise as compared with a permanent magnet DC servo motor having the same output rating [l, 23. But its disadvantages are the high cost and a more complex controller caused by the nonlinear characteristics [3,4]. The PI controller is usually employed in a BLDC motor control, which is simple in realisation but has difficulty obtaining a sufficiently high performance in the tracking application. It is, however, known that the tracking controller problem using a state variable feedback can be simply solved by augmenting the state variables using the output error [SI. Even though this method is more complex than a PI controller, it is more efficient to obtain the control gain using the optimal control theory than the trial-anderror method for the PI controller. Recently, the direct drive motor system has been used for the fast position controller because it has no backlash. However, the load torque directly imposed by the motor shaft must be quickly removed to obtain robustness in the position controller. Therefore, the BLDC motor is the best choice of robust control system under large torque variation. For the unknown and inaccessible inputs, the observer was studied by Meditch and Hostetter [61]. Reference 7 considers the application of the load torque observer, where a load torque is considered as an unknown and inaccessible input in a DC servo motor system with a P controller. This load torque observer needs the measured current value which is too noisy to be used in either a digital controller or a digital observer with an inverter. The fixed gain observer is not enough to overcome the parameter uncertainty of the BLDC system. In this paper, the augmented state variable feedback controller based on the linear quadratic control (LQC) is introduced to the position control of a BLDC motor having a sinusoidal back EMF and linearised by the field-orientation. In particular, to obtain the robustness against the load variation, the load torque is estimated by a dead beat estimator excluding the current using the Luenberger observer and the control signal is compensated by the feedforward terms. Furthermore, this observer needs to sense only one variable, i.e., the position value having no noisy term. Its gains are simply realised by Ackerman’s rule. Since the flux linkage is not exactly known for a load torque observer, there is a problem of 63
uncertainty. Therefore, a model reference adaptive observer is considered to overcome the problem of the unknown parameter. A detailed comparison between the two system responses has been made. It is shown that the load disturbance can be reduced without affecting the overall system performance under all operating conditions if the proposed algorithm is employed. 2
Modelling of BLDC motor
2.1 Nonlinear model
Generally, a small horsepower BLDC motor used for a position control is the same as a permanent magnet synchronous machine. The stator is constructed by a threephase Y-connection without the neutral and the rotor is made by the permanent magnets. Since each phase has a phase angle difference of 120",the summation of all three phase currents becomes zero. The system equations in a d - q model can be expressed as follows [3] iqs =
.
rs L* - - iqa - - o,id,
L,
L,
1 V, - Am +o,
4
L,
(1)
3
Control algorithm
3.1 Position controller The reference is a step value as in a tracking servo problem [lo]. The dynamic equation of a given system can be expressed as follows i ( t ) = Ax(t)
+h(t)
(10)
y = cx(t) (11) where the dimensions of the matrices A , b and c are n x n, n x 1, and 1 x n, respectively. Usually, a linear quadratic controller is used to solve the regulator problem resulting in a state variable feedback. Applying it to a servo problem, another control variable iic is needed as follows
+
u(t) = - Lx(t) ii&) (12) where L and ii&) are a feedback gain matrix and a compensation input, respectively, and in the case of a regulator, iic(t)= 0. It is, however, difficult to find the value of ii,(t) for the servo problem. Therefore, a new state is defined for the tracking controller as
(13)
i=y-y,
where y, is the rotor position reference [ll]. It is then evident from eqns. 10, 11, and 13 that the open loop tracking system is governed by a state equation of the form as follows
=
2 2 Linearised model By means of a field-oriented control, it is possible to make ib become zero [8,9]. Therefore, the system equations of a BLDC motor model can be described as
AZ + bu - (y)y,
where Z = [ x
2]*.
If (A,
6) is controllable,then
lim (y - y,) = 0 1-CO
by the control input of and the torque equation is expressed as
= k,ips
where
k, =
(;),Im
U
= -kx
- klz
(17)
where k is a 1 x n vector and k , is a scalar [SI. It is well known that this controller is able to cancel the steady state error caused by the unmeasurable or inaccessible disturbance. Also, if this closed-loop system is asymptotically stable, the overall system is robust to the system parameter variation or the feedback gain variation [ll]. The augmented system for the position control of a BLDC motor is expressed as follows
Since the current control is employed in a position control, the system model expressing the position dynamics becomes eqn. 7 and the rotor position dynamics becomes jJ = 0, (9) For the implementation of the field-orientation, each threephase current control command must be generated separately. This command can be obtained by converting the controller current command based on the rotor reference frame to the stator reference frame. Using the current regulated PWM(CRPWM) scheme, the threephase current commands ,i isc , and ,i are tracked by the actual three-phase current. 64
y = (0 1 0)Z
(19)
For this system, the rank of a controllability matrix is 3. Therefore the augmented control system is controllable IEE Proc.-Ekctr. Power Appl., Vol. 141, No. 2. March 1994
and the steady state value of z becomes zero by the control input given in the form of
steady state position error is controlled by the controller given as follows
u(t) = -h (20) where 6 is a 1 x 3 vector. The block diagram of an augmented state variable feedback control system is shown in Fig. 1.
u(k) = - Kx(k) (26) However, a large feedback gain is needed for the fast reduction of an error caused by the disturbance, which results in a very large current command. Therefore, a new control algorithm is required to reduce the influence of the disturbance at the transient state without affecting the predefined overall system performance. If the load torque TL is known, a equivalent current command iW2 can be expressed, from eqn. 8, as
T2g+l+,
(27)
TL = kt iqc2
1
kl
Fig. 1
Block diagram of uugmented state wriable feedbuck controller
3.2 Digital control and proposed algorithm The equation of the discrete time invariant system can be obtained from eqns. 10 and 11 under the following conditions (a) Simultaneous sampling of input and output (synchronisedA/D and D/A converter) (b) Negligible conversion time (c) Constant during the sampling period (d)Constant sampling interval
Thus the discrete system equation of (18) without disturbance and reference can be written as follows [l2]
x(kh + h) = @x(kh)+ ru(kh) y(kh) = cx(kh) where
(21) (22)
iw2 = - TL kt
Therefore the equivalent q axis current can be calculated to compensate the load torque. A fast compensation can be obtained by feeding forward an equivalent q axis current command for the load torque to the output controller as shown in Fig. 2. The control input ipc consists
estimated Iwd toraue
Fig. 2
of the position controller output iqcl and the equivalent current igc2 generated by dividing the estimated load torque with a motor torque constant k,. Therefore the control input U can be obtained as follows
.
. +
U E 1qc = I gcl
(29) However, if the load torque is an unmeasurable or inaccessible disturbance, it is difficult to obtain this value. Therefore the derivation of a load torque by an observer is required.
@ = 2 h
I
O
i,
Feedforward compensatorfor loud torque disturbme
o\
igc2
3.3 Load torque observer Generally, to estimate a state, it is required to know all the inputs given to the system. But in the real system, there are many cases where some of the inputs are unknown or inaccessible. Almost all these inputs are the internal or external disturbances or, some other case, the unknown error from an output measurement [6, 131. For a given system of eqns. 10 and 11, the observer is known as follows
From this equation, the state feedback controller gain can be obtained by the optimal control law minimising the performance index such that N-l
J = lim
1 {b:Qbk+ u:Ruk}
N-rm i=o
(23)
where the weighting matrix Q and R are defined as diag [ q l , qZ2q3J and 1, respectively. The steady state solution of a LQC can be obtained by solving the following equations as [12] s = @ - rKTs@- TK
+ GY + HU w = Ly + M Z
2 = FZ
0
+ Q + KTRK
(24)
+ rTd-)-lrTs@ (25) where s and R + T’sT are the positive definites. Then the
(30)
(31) where F, G, H , L, and M are the real constant matrices and there exists T such that w(t) = TX(t). Two types of the observer are available when u(t) is unknown and inaccessible [SI. Firstly, a @observer is available under the condition that u(t) is constant (ti = 0)and there exists T such that w(t) =
T(T))
Secondly, a K-observer is suggested under the condition that u(t) = uo u,t + . . . uktk and there exists
+
+
K = -(R
IEE Proc.-Electr. Power Appl., Vol. Ill,No. 2,March I994
(33) 65
For simplicity, a 0-observer is selected and the augmented system is given as follows (34)
Therefore, the load torque observer can be expressed as i0(k
+ 1) = @io(k) + fdk) + N
k ) - Hk))
fik) = ti,,(k)
(41)
The observability matrix WObecomes
Y
)o:(,
=(c
(35)
To guarantee the existence of a 0-observer, the observability matrix must have a full rank 16, 141. An application method to the case where the measurement contains a constant but an unknown bias error has been introduced by Bryson and Luenberger [13]. Thus, TL can be considered as an unknown input in eqn. 34 and assumed to be a constant. From eqns. 10 and 11, an observer for the load torque estimation is obtained as
TL= 0
(36)
and the system equation can be expressed as
(37) where L is a 3 x 1 gain matrix. 3.4 Implementation of load torque observer The discrete form of eqn. 36 is represented as
+
TL(k 1) = T'(k)
(38)
A load torque observer is obtained considering TL as an unknown input. From eqns. 21, 22 and 38, the discrete load torque observer can be obtained by z transformation as follows [12]
+ TL(k + 1)
( ( fik 5 ( k + 1) 1)) = (ai a,
0
(:)
0 1 a4 az)( (5(k)) #k) + b, i,(k) 0
1
TL(k)
where 11, I,, and I , are the elements of L, and a, = e-(B/JM
(i)
w,= t@ = (ala30+a3 a,
(40)
1 11 a2a,+2a
n",
3
(42)
Since the rank of a matrix WOis 3, the discrete load torque is observable. To guarantee the less time required calculating the load torque rather than the overall system response time and to compensate the load torque at transient state, a dead beat observer is desirable. The real load torque is quickly estimated by this observer and the position error can be compensated by the feedforward term using this estimated load torque even at the transient state. The dead beat observer is simply obtained by using the pole assignment technique where the desired poles of this system are all chosen to be at the origin in the z-domain. Therefore the characteristicequation of the dead beat load torque observer becomes 2" = 0 (43) where n is the order of the system [121. It then follows from the Cayley-Hamilton theorem that the system matrix of the closed loop system satisfies the following equation as
q = 0 (44) where CP, = 6 - L.2 and L is a feedback gain. This strategy has the property that it will drive all the states to zero in at most n steps. Since the settling time is n . h, the design parameter is determined only by the sampling interval h, which is a unique characteristicsof a discrete sampled data system. On the contrary, the control input U is large and is increased as h is decreased. It is therefore difficult to obtain the desired characteristics by the bounded input, and they are sensitive to the measurement error in the fast response observer. The full state load torque observer can be implemented using only one variable, the position error, instead of using both the position and the speed errors. Also, this observer is not influenced by the noise caused by the A/D conversion of the speed. The feedback gain L can be obtained by by pole placement using the Ackermann formula [lo]. If the given system is reachable, a gain L can be represented as follows
L=P(@)W,'[O 0 ... 11 (45) Because of the dead beat controller, the characteristicsof the load torque observer in this system is
P(@) = 8,
(46)
3.5 Proposed adaptive load torque observer Even though the observer feedback gain is obtained by using the nominal parameter value, there is a certain variation or uncertainty of the parameter such as the flux linkage. To overcome this problem, an adaptive observer is considered. As shown in Fig. 3, the difference between the reference model and the adjustable system can be reduced by an adaptive mechanism [lS]. In this system, the reference model is the real plant with the augmented state variable feedback controller. In a similar way, the adaptive load torque observer is considered as an adjust66
IEE I4oc.-Electr. Power Appl, Vol. 141, No. 2, March 1994
able system. From eqn. 39, a discrete motor equation can be written as follows
+ y(k) + b2 i,(k) + 1) = a,o,(k - 1 ) + bli,,(k - 1)
y(k + 1 ) = a3w,(k)
(47)
o,(k
(48)
Y,(k
+ 1) = b , iJk) + a3b,i,(k - 1) = y(k + 1 ) - A,y(k) - A z y ( k - 1)
- A,y(k - 2) (54) Then, the new quasi-output and E can be expressed as
model w t p u t
reference model
adaptation rate vector, respectively. The model output Y, is assumed to be the difference between the real plant output and past output as follows
+ 1) = 6, i,(k) + a36,i,(k - 1 ) Y(k + 1) - Ym(k+ 1 ) = (6, - b,)i,(k) + a3(6, - b&(k - 1 )
Y(k
E system output
(55)
E
(56)
Therefore, the gradient can be obtained as
aE -a& - iqc(k) Fig. 3 Basic configuration of model reference adoptiw scheme (MRAS)
(57)
-'E - i,(k - 1 )
a&
Using a simple relationship between the speed and the rotor position of 1 o,(k) = - {y(k)- y(k - 1)) h
(49)
an autoregressive moving average (ARMA) model of this BLDC motor system can be obtained from eqns. 47 and 48 as follows y(k + 1 ) = y(k) +%y(k h
A discrete form of eqn. 53 can be expressed simply by using the Euler method as follows (59)
where a, and uz are the elements of the vector a. The resultant block diagram of the model reference adaptive observer is shown in Fig. 4. During the operation, the
- 1 ) - y y ( k - 2)
+ b, iJk) + u3 b,i,(k
TL
I y(k) :y,;rotor
- 1)
posilim
= [ j l ( k ) y ( k - 1 ) y(k-2)]
where A,, A , , A , , B,, and B , are 1, (a,a,)/h, -(a,a,)/h, b, , and a, b,, respectively. Since b, and b, are not exact values, the system equation given by eqn. 49 can be expressed as y ( k + 1 ) = cP, @ 1 + cP2 Q , where*- means an estimated value and
(51)
8 , = CAk) Ak - 1) Ak - 211 @,=CA, 8, = M k )
%=C&
41' u(k - 91 &I A,
The order of the estimated matrix can be reduced by employing the definition of the new quasi output as follows Y(k + 1 )
Cv(k + 1) - @,el) = 8, 6, (52) The MIT rule is employed to adaptate the real values of B, and B, as
de = -aE dt
grad:
(53)
where E, 0, and U are the difference between the model and real outputs, adjustable parameter vector and IEE Proc.-Electr. Power Appl., Vol. 141, No. 2, March 1994
Fig. 4
Configurationof model reference adoptive torque obserwr
quasi output Y from eqn. 55 is continuously compared to the model output Y, in eqn. 54. The obtained error E is used in an adaptive mechanism which adjusts the parameters in the primary controller so that the process response becomes equal to that of the reference model. This means that the estimated torque of the adaptive load torque observer tracks well the real load torque. In this system, the reference model output for an adaptive observer is assumed to be the rotor position of a BLDC motor. 4
Configuration of the overall systems
The block diagram of the proposed controller is shown in Fig. 5 where the controller is composed of two parts. The first part is the digital controller consisting of the state feedback controller with an optimal gain and the adaptive load torque observer. This observer is realised by a dead beat control gain with a parameter adaptation mechanism. The load disturbance is compensated by the 67
is the power control unit of a field orientation which proposed algorithms are depicted in Figs. 7-10. In an augmented state feedback case, the feedback gain is includes the position and current sensors. The position obtained by a linear optimal control theory under the controller is composed of the aygmented state feedback defined in eqns. 14, 15 and 17. For the realisation of the .
Fig. 6
Block diagram of proposed digital position control system
augmented state z(k + 1) defined in eqn. 13, the discrete form of this state is approximately obtained by using a trapezoidal rule as follows
m i
where e(k) = y(k) - y,. The power converter is controlled by the field orientation method, which is composed of a 2- to 3-phase converter and a CRPWM. The CRPWM makes the three-phase real currents track the three-phase current commands by the on-off control with a hysteresis band gap P,91. 6
read goal position
augmented state variable feedback
Simulation results
U =IqCl
The parameters of a BLDC motor used in this simulation and experiment are given as follows Power
4OOW
Rated torque
1.438 N m
Inertia
1.3132 x lo-* kg m2
Stator resistance
6.2 R 1.35 ms
The proposed digital position controller is implemented as shown in Fig. 5. The CRPWM is employed as a power converter and the hysteresis band gap is chosen as 0.05 A. Also the sampling time h is determined as 0.1 ms. After some trial and error, the weighting matrix is selected as follows Q=diag[lO-'
lo3 lo6] R = 1
As a result, the optimal gain matrix becomes k = c0.2501 34.82 857.11 by solving the Riccati equation and the dead beat observer gain becomes L = [22468 3 -65241 from eqns. 45 and 46. In the adaptation mechanism, the adaptation rates 0 1 ~and a2 are obtained as 0.81 and 0.41, respectively, by trial and error. The control flow chart of 68
t I
output
adaptive load torque observer
n l
I
fixed gam loud torque observer
Mechanical time contant 2.89 ms Electrical time contant
stop ?
Fig. 6
Flow chart of control program using adaptive load torque
obserwr
conditions that there is no overshoot and the settling time is about 0.1 s. And a step load torque of 1 N m is considered in this system to illustrate the overall performance. This torque disturbance is chosen to be about half the rated torque. Furthermore, the parameter variation is also considered to be -20%. As shown in Fig 7, the step response of the rotor position is sensitive to the change of the load torque T- . This change of the load torque creates the position error about 0.02 rad as illustrated in Fig. 9a. Also, the negative peak value of iqc is almost 0.7 A in the conventional scheme as shown in Fig. 7. This unwanted command is caused by a nonexact parameter value. I E E hoc.-Electr. Power Appl., Vol. 141, No. 2, March 1994
However, in the proposed scheme shown in Fig. 8, the position error is gradually reduced and the negative peak value of igc is about 0.4A. This rotor position error is decreased to 0.01 rad under all operating conditions. This result comes from the adaptive load torque observer with the adaptive mechanism given by eqn. 53 and the feedforward compensation. The q phase current command iF shown in Fig. 8 shows this compensation effect. The phase a stator current is illustrated in Fig. 10. The results shown in Figs. 7-10 demonstrate the effectiveness of the adaptive torque observer, fast adaptation, and good control performance. The adaptive observer provides better damping than the conventional fmed gain observer.
,$ .1.
6
Conclusions
A load torque observer using the model reference adaptive system is employed to obtain the better perfomance from the BLDC motor in a position control. Also the augmented state variable feedback is employed in the digital control system with an optimal gain. Considering the load torque as the unknown and inaccessible input,
toad torque, Nm
12
I
I
2
3
08
4
5
time. s
0
a
e
e
04
r
"0
1
3 time. s
2
5
4
1
2
3
4
5
1ime.s b I
0.-0.1' 0
'
'
1
"
2
'
'
3
'
'
I
'
5
time, s
Fig. 9
Rotor position errors
a coovcntional s c h e (deadbeat obacrver) b proposcd s c h e (adaptive obacncr)
Fig. 7 Rotor position and q phase current command of conventional scheme (20% parameter miation) 2-
load torque, Nm
time. s
L
0
:
02
5a
r -04 -1U
-0.1
0
2
3
3 time. s b
5
time. s
Fig. 8
0
Rotor position and q phase current command of proposed scheme (20% parameter variation)
Fig. 10
Phase on stator currents of two controllers
a conventionalscheme b proposed sehwnc
IEE Proc.-Electr. Power Appl. Vol. I l l , No. 2, March I994
69
.
the robust digital position control system is implemented based on observer theory. The system response between the fixed gain observer and the adaptive observer has been compared. The load torque compensator based on the adaptive observer and the feedforward can be used to cancel out the steady state and the transient position error due to the external disturbances such as various friction and a load torque. In this proposed scheme, the overall system response is not affected by the disturbance compensator, and the rotor position error caused by the nonexact parameter is gradually decreased. The total control system can be realised by a digital controller where the gain is obtained in z-domain using optimal theory.
7
References
less DC motors’ (Sogo Electronics Publishing Co. Tokyo, 1984). pp. 96-100 5 PORTER, B., and BRADSHAW, A.: ‘Design of linear multivariable continuous-time trackinc svstems’.. Int. J. Svst. Sci... 1974.. 5.. f121 , ,. pp. 1155-1164 6 MEDITCH, J.S., and HOSTETTER, G.H.: ‘Observer for systems with unknown and inaccessible inmts’. Inc. J. Control... 1974.19.131 ,.,,, pp. 473-480 7 OHISHI, K., NAKAO, M., OHNISHI, K., and MIYACHI, K.: I
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‘Microprosser-controlled D C motor for load-insensitive position servo system’, I E E E Trans., 1987, IE34, (I), pp. 44-49 8 NOVOTNY, D.W., and LORENZ, R.D.: ‘Introduction to field orientation and high performancc AC drives’. IAS-Tutorial Course, 1986 9 BOSE, B.K.: ‘Power electronics and AC drives’ (McGraw-Hill, 1986) 10 TZAFESTAS. S.G.: ‘Aoolied - control’ (North-Holland. New .. diaital York, 1985), pp. 51-53 11 DAVISON, E.J.: ‘The output control of linear time-invariant multi-
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12
1 ELECTRO-CRAFT CORPORATION USA: ‘DC motors speed controls servo system’, 1972 2 KUSKO, A., and PEERAN, S.M.: ‘Brushless DC motors using unsymmetrical field magnetization’, I E E E Trans., 1987, IA-23, (2). pp. 319-326 3 KRAUSE, P.C.: ‘Analysis of electric machinery’ (Mdiraw-Hill, 1984) 4 KENJO, T., and NAGAMORI, S.: ‘Permanent magnet and brush-
I E E Proc-Electr. Power Appl., Vol. 141, No.2, March I994
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