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 )

 . . .

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 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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