optimal state feedback control of brushless direct-current motor
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8/3/2019 Optimal State Feedback Control of Brushless Direct-current Motor
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Wang et al . / J Zhejiang Univ Sci A 2007 8(12):1889-1893 1890
Mathematical modelThe dynamic equations of a brushless DC motor
may be expressed as
a
2
a L
a e
d,
d
d,
375 d
,
iu u E iR L
t
GD nT T
t
E K n
− ∆ = + +
− =
=
(1)
where R is the inner resistance of motor, T L the load
torque of motor, GD2
the moment of inertia of motor rotor (in N·m
2), ∆u the voltage drop of the thyristor,
K e the back EMF coefficient. Torque T a and back
electromotive force (EMF) E a, using the concept of
average torque and average back electromotive force.
Due to the bigger ripple (Li et al ., 2004; Wang et al .,
2005), the back EMF and torque have
a T ,T K I = (2)
where I is the current of every phase, K T the torque
coefficient of motor.Taking Laplace transform of Eqs.(1) and (2)
yields the following relationships
a
a T
2
a L
a e
( ) ( ) ( ) ( ) ( ),
( ) ( ),
( ) ( ) ( ),375
( ) ( ),
U s U s E s RI s LsI s
T s K I s
GDT s T s sN s
E s K N s
− ∆ = + +
=
− =
=
(3)
where ∆U (s) is small to be neglected, then open loop
transfer function of the system may be expressed as
2a l m l
l m
( ) 1/( )( ) ( 1/ ) ( )
E s TT T aU s s s T T s s a
= = ⋅+ +
, (4)
where T l= L/ R is the effective electrical time constant,
T m= RGD2/(375 K e K T) is the mechanical time constant,
a=1/T l.
Allowing for a zero-order hold (sample-hold) in
the discrete-time domain representation, the Z trans-
form of the above input-output representation may be
expressed as (Kwo, 1992)
21l
2
m e
1 1
l
1 1
m e
2
( )(1 )
( ) ( )
[( 1 e ) (1 e e ) ]
(1 )(1 e )
, (5)
aT aT aT
aT
T N z a z Z
U z T K s s a
T aT aT z z
T K z z
Az B
z Fz D
−
− − − − −
− − −
= −
+
− + + − −=
− −
+=
+ +
where T is the sample period.
State and output equations at the k th sampled
interval are
1 1
2 2
1
2
( 1) ( )0 1( )
( 1) ( )
( ) ( ),
( )( ) [1 0] ( ),
( )
x k x k Au k
x k x k D F B AF
k u k
x k n k k
x k
+ = + + − − −
= +
= =
Gx H
Cx
(6)
where
1
2 1
( ) ( ),
( ) ( 1) ( ),
x k n k
x k x k Au k
=
= + −(7)
A/D
D/A
-
PWM
PWMPWM
PWMPWM
C
Kalman filter
H1
+
Adaptive controller
Optimal state feed back
matrix K
×
u(k )
DSP
H3
H2
PWM1
PWM6
74ls14Drive
circuit B
A
CAP3
n(k )
+
uo(k )
n(k )
n(k )
+
CAP2
CAP1
× u*(k )
Fig.1 Function block diagram of brushless DC motor drive system
ˆ( )n k u*(k )
. . .
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Wang et al . / J Zhejiang Univ Sci A 2007 8(12):1889-1893 1891
G , H , C are all constant matrices for brushless DC
motor. From Eq.(5), the input-output difference
equation may be obtained:
( ) ( 1) ( 2)
( 1) ( 2).
n k F n k D n k
A u k B u k
+ ⋅ − + ⋅ −
= ⋅ − + ⋅ −(8)
STATE ESTIMATION USING KALMAN FILTER
In order to realize the optimal feedback control
using state feedback, the state vector x (k ) needs to be
estimated (Rodriguez and Emadi, 2006). Using Kal-
man filter theory, the algorithm for the derivationˆ( )k x can be obtained as
[ ]{ }
ˆ ˆ( ) ( 1) ( 1) ( )
ˆ( ) ( 1) ( 1) ,
k k u k k
n k k u k
= − + − + ⋅
− − + −
x Gx H K
C Gx H (9)
where ˆ( )k x is the estimated state vector, and G , H , C
are constant matrices known for a brushless DC motor
drive system. Choose Q, R as the weight matrices, and
p(0) the initial matrix, the recurrence algorithms of
Kalman filter gain matrix K (k ) can be calculatedoff-line with
T T
T T 1
( ) ( 1) ,
( ) ( ) [ ( ) ] ,
( ) [ ( ) ] ( ),
k k
k k k
k k k
−
= − +
= +
= −
P GP G HQH
K P C CP C R
P I K C P
(10)
where P (k ) is an estimation error covariance matrix,
and ( )k P is a prediction matrix of the error covari-
ance.
OPTIMAL STATE FEEDBACK CONTROL BY
LYAPUNOV STABILITY CRITERION
The state equation of the system is expressed as
( 1) ( ) ( ).k k u k + = + x Gx H (11)
The control u(k ) is defined as
( ) ( ).u k k = − Kx (12)
The design objective is to find the feedback
matrix K that will bring the state x (k ) from any initial
state x (0) to the equilibrium state x =0 in some optimalsense.
The basic assumption is that the system of
Eq.(11) is asymptotically stable with u(k )=0. This
guarantees that given a positive-definite real sym-
metric matrix Q, there exists a positive-definite real
symmetric matrix P such that
T .− = −G PG P Q (13)
The Lyapunov function is defined as
V [ x (k )]= x T(k ) Px (k ), (14)
and
T
[ ( )] [ ( 1)],
[ ( )] ( ) ( ).
V k V k
V k k k
∆ = +
− = −
x x
x x Qx (15)
As an optimal control problem, we find an op-
timal control uo(k ) which minimizes the performance
index
J =∆V [ x (k )] (16)
in the instant k .Since ∆V [ x (k )] represents the discrete rate of
change of V [ x (k )], we can define V [ x (k )]. So the
minimization of the performance index of Eq.(16)
will carry a physical meaning in optimal control. For
instance,∆V [ x (k )] may represent the rate of change of
speed, distance, or energy along the trajectory x (k ).
For the general form of V [ x (k )] given in Eq.(14),
we write
T T
T T T
[ ( )] ( 1) ( 1) ( ) ( )
( )( ) ( ) ( ) ( ) ( ).
V k k k k k
k k k k
∆ = + + −
= − − −
x x Px x Px
x G HK P G HK x x Px
(17)
By the state equation Eq.(11), the last equation can be
written as
T T T T
T T T T T
[ ( )] ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ).
V k k k u k k
k u k u k u k k k
∆ = +
+ + −
x x G PGx H PGx
x G PH H PH x Px
(18)
For minimum ∆V [ x (k )] with respect to u(k )
[ ( )]
0.( )
V k
u k
∂∆
=∂
x
(19)
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Wang et al . / J Zhejiang Univ Sci A 2007 8(12):1889-1893 1892
Substituting Eq.(18) into Eq.(19) leads to
T T
2 ( ) 2 ( ) 0.k u k + = H PGx H PH (20)
Therefore, the optimal control is
T 1 T( ) ( ) ( ).u k k
−= − H PH H PGx (21)
Given Q= I (identity matrix) with V [ x ]= x T(k ) Px (k )=0
and P , the optimal control uo(k ) can be obtained.
ADAPTIVE CONTROL TO COMPENSATE
EFFECTS OF NOISE AND DISTURBANCE
If a system is subject to process disturbance and
measurement noise, which are from power supple and
load disturbance, the equations for the system can be
expressed as
( ) ( 1) ( 1) ( 1),
( ) ( ) ( ),
k k u k k
k k k
= − + − + −
= +
x Gx H ξ
n Cx ε(22)
where ξ (k ), ε(k ) are Gaussian noises. Since the output
error due to the above noise and disturbance cannot be
eliminated by optimal feedback control, use of anadaptive algorithm is needed.
The output expectation of the system due to an
arbitrary input u(k ) may be expressed as
( ) ( 1) ( 2) [ ( 1) ( )]
[ ( 2) ( )], (23)
n k Fn k Dn k A u k u k
B u k u k
= − − − − + − + ∆
+ − + ∆
where ∆u(k ) is the input error, due to the existence of
the noise.
The output error is
ˆ( ) ( ) ( ) ( ) ( ).n k n k n k A u k B u k ∆ = − = ∆ + ∆ (24)
So an indirect measurement value of ∆u(k ) is
3
( )( ) ( ).
n k u k K n k
A B
∆∆ = = ∆
+(25)
Using an exponential smoothing filter
ˆ ˆ( ) (1 ) ( 1) ( ),n k n k n k β β ∆ = − ∆ − + ∆ (26)
where ˆ( )n k ∆ is the filtered value of ∆n(k ), and β
(0< β <1) is the filter coefficient. The following equa-
tion may be used as an adaptive corrector:
o o
3 ˆ( ) ( 1) ( ).u k u k K n k = − − ∆ (27)
SIMULATION EXPERIMENTS
Technique data of experimented brushless DC
motor
Technique data of the experimented brushless
DC motor are: rated power P =1.57 kW, rated speed
n=1500 rpm, rated current I =4.5 A, moment of iner-
tia-including load J =16 kg/cm2.
Algorithm
Step 1: Calculate coefficients A, B, F , D and
matrices G , H , C .
Step 2: Calculate the optimal feedback matrix K
and control signal u*(k ).
Step 3: Calculate the adaptive signal uo(k ).
Simulation results
Simulation results are given in Fig.2.
In Fig.2, curve 1 corresponds to the dynamic
response of system under the rated load disturbance
without adaptive control, while curve 2 corresponds
to that with adaptive control. The method of optimal
feedback is seen to provide faster step response of
output shaft velocity. To further reduce the sensitivity
to load disturbances, adaptive torque correct methods
were applied in conjunction with optimal feedback control methods.
Fig.2 Dynamic response of system underload disturbance
Rated torque disturbance
1600
S p e e d ( r p m )
1400
1000
800
600
1200
200
400
00 0.05 0.10 0.15 0.20
1 Without adaptive control
2 With adaptive control
Time (s)
1500
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