1 INTRODUCTION Brushless DC motor (BLDCM) DC motor not only has good speed performance, but also has the advantages of AC servo motor such as simple structure, reliable operation and so on, so the servo system in many high performances has been applied widely [1]. Brushless DC motor servo system is a multivariable, nonlinear, strong coupling and time-varying system. Traditional PID control cannot satisfy the requirement of high precision for controlling motor. Thus, advanced control strategies, such as fuzzy control, neural network, variable structure control and self-adaptation [2-7], have been widely applied in Brushless DC motor servo system. Input structure control is the key problem of predictive functional control, which solves the irregular input control problem in predictive control of other models. Besides input structure control has good tracking ability and strong robustness. At present, the predictive functional control, a control strategy that is able to provide better velocity servo control for brushless DC motor, has been applied in motor servo system and control of induction motor. 2 PREDICTIVE FUNCTIONAL CONTROL

ALGORITHM In the predictive functional control (PFC), the input structure control is the key factor that influences system performance. The new control function added can be expressed as a linear combination of several known primary functions {un}:

1( ) ( ) ( )

D

n nn

u k i k u iμ=

+ = 0,..., 1i H= − (1)

Where, ( )u k i+ is controlled variable at time k i+ , ( )n kμ is a linear combination weighted coefficient of

This work is supported by National Nature Science Foundation under

Grant 61104005.

primary function that needs optimal calculation every time it is applied; ( )nu i represents the primary function value at

time si iT= , sT is the sampling period; D represents the number of primary functions; H is the time domain while predicting optimal calculation being applied. For the selected primary function, through off-line calculation in advance, response function ( )ng i of controlled plant will be obtained under its effect, while primary function is a step function 1u .The reference trajectory is taken as a first-order exponential function form, it can be expressed by:

0

0 0 01

( ) ( ) ( ) [ ( ) ( )]N

j ir j

jy k i c k c k i c k y kγ

=

+ = + − − (2)

Where ( )ry k i+ expresses the value of reference

trajectory at time k i+ ; 0 ( )c k is the value of tracking set at time k; ( )y k represents actual system output value at time k; is attenuation coefficient, generally expressed by 3 /s rT Teγ −= ; rT represents expected closed loop response time of the reference trajectory; The prediction model using the discrete state space model is given by:

( ) ( 1) ( 1)( ) ( )

m m

m m

X k A X k B u ky k CX k

= − + −= (3)

Where 1nmX M ×∈ is the state vector of prediction model;

1 1my M ×∈ is s the output of prediction model;

1 1u M ×∈ is the

controlled input of system; n nA M ×∈ , 1nB M ×∈ and 1 nC M ×∈

represent the coefficient matrices of prediction model state

equations, respectively. After calculation, it can be

expressed by

( ) ( ) ( ) ( )im my k i CA X k k g iμ+ = + (4)

Brushless DC motor speed control based on predictive functional control

Deying Gu, Jingquan Zhang, Jingxiao Gu School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

E-mail: gdy0335@163.com Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65201, U.S.A

E-mail: jgy46@mail.missouri.edu

Abstract: Because the DC motor is characterized by multi-variables, nonlinearity, strong coupling and time-varying as well as tradition control method strongly depends on the mathematical model of motor, predictive functional control strategy is proposed to control speed of brushless direct-current motor in the paper. The current-loop adopts PI controller while speed-loop uses predictive functional controller. Simulation result is shown that predictive functional control is characterized by small on-line calculations, simple control and stronger anti- interference, along with the robustness, and good static and dynamic performance, thus the control performance of system is enhanced in general. Key Words: Brushless DC motor, Predictive Functional Control, Speed Control, DSP

3456978-1-4799-7016-2/15/$31.00 c©2015 IEEE

Because brushless DC motor servo is a nonlinear, strongly coupling and time-varying control system, there is some degree of error between prediction model output ym(k) and actual output y(k ). Self-compensator uses the difference between the predicted model output and actual output value of the past several times to fit into a polynomial of order N after filtered then it is written as follows:

1 1

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) (5)

mD N

i im n n m i

n i

y k i y k i e k i

CA X k k g i y k y k e k jμ= =

+ = + + +

= + + − +

Where e(k) is the error at time k; y(k) is the actual output at time k; ( )my k is the predictive output at time k; ( )ie k is the polynomial coefficient. In the prediction function, the objective of optimization is to calculate formula (1) and get the coefficients 1 2, ,..., Bμ μ μ of the primary function, so that the predictive output y(k+i) of controlled object in optimal time domain closes to reference trajectory value as much as possible. In the online optimal algorithm of PFC, quadratic form performance index [25-27] is mostly used. It is expressed by: '

2

1

min ( ) [ ( ) ( ) ( )] (6)n

i ii

P k k g h Q k hμ=

= − +

Where[ ] [ ]1 2 1 2( ) ( ), ( ),..., ( ) ( ) ( ), ( ),..., ( )D i i i D ik k k k g h g h g h g hμ μ μ μ= = ;

0

0 01

( ) (1 )( ( ) ( )) ( ) ( ) ( )i i

Nh hj

i j i mj

Q k h c k y k c k h C A I X kγ=

+ = − − + − − ;

As a result, the controlled output of predictive functional control at time k is expressed by:

0

0 0 01

( ) ( ( ) ( )) ( ) ( ) (7)N

j j m mj

u k k c k y k k c k k X k=

= − + ∗ +

3 BRUSHLESS DC MOTOR SPEED CONTROL SYSTEM Brushless DC motor speed control system uses speed

and current double closed-loop control. The speed loop adopted in predictive functional control while current loop uses PI regulator in order to limit the maximal current of system and guarantee the stable running of the system. The transfer function of PI controller for the current loop is /ci p iG K K s= + , where rI is the current reference value and Ω is the motor mechanical angular velocity. The structure diagram of current loop is shown in figure 1.

( )sΩ( )dU s ( )I s ( )E s PI

rI

Fig. 1 The current loop structure of Brushless DC Motor

The speed loop can enhance the ability of anti-load disturbance and restrain the fluctuation of speed and at the same time guarantee the system to achieve better static and dynamic tracking performance; Speed loop controller design has tight connection with the servo system stability

and performance at high running speed. The structure diagram of Brushless DC motor with speed loop is shown in Figure 2. The transfer function of entire current loop (including transfer function of motor speed loop) is taken as the controlled object ( )iG s .

1

1/1

RTs+

m

RTs

1

eC( )sΩ( )dU s ( )E s PI

ru I=( )cG s( )rG s

( )pG s

( )fG s

sy ry

δ

Fig. 2 The speed loop structure of Brushless DC Motor

The transfer function of controlled object in speed loop is expressed by:

2 2 2

4 2 3 2 2

21( )( ) (2 )

p p i ii

e m p m m p a p p i i a i

K Rs K KRs K RG s

C T K s K T K r K s K K Kr s K s+ +

=+ + + + + +

8

Where 2ar R= is the winding resistance, Tm mechanical time constant, Cg is the back EMF coefficient, and Kp, Ki are proportional constant and integral constant of PI controller, respectively. For asymptotically stable objects with higher order, through fitting and simplifying, second order model can be obtained as follows.

2 4 2

23.15( )1.32 10 0.028 1i

KG sas bs c s s−= =

+ + × + + (9)

Where parameters (K,T, ) represents steady-state gain, time constant and lag time of system.

4 SIMULATION OF SYSTEM Parameters of Brushless DC motor contain: rated voltage is 24VDC; rated current is 6A; output power is 80W; The rated speed and rated torque are 2000r/min and 0.4N · m, respectively; pole pairs is 2; armature resistance is 1.5 ; the armature winding inductance is 0.004mH, mutual inductance is 7.05mH, mechanical time constant is 0.028s; the electromagnetic time constant is 0.0047s;the back EMF coefficient is 0.0432Vs/rad and moment of inertias 1.766e-5Kgm2. Brushless DC motor speed model (the equation No.8) is transformed using Z transform and then discretized to get difference equation, while the sampling period is 0.01s.

( ) 1.8236 ( 1) 0.9435 ( 2)0.018 ( 1) 0.0444 ( 2)

y k y k y ku k u k

= − − −+ − + −

10

Set the sampling period of velocity loop Ts=0.001s and determine factors: A=1, B=1, C=2.379. Set response time of closed-loop of reference trajectory Tr=0.005s so that the attenuation coefficient of reference path is calculated and get the results: 3 / 0.55s rT Teγ −= = , (0,1)γ ∈ . Given a step function with the amplitude ( 1=1500rad/s ) as primary function, fitting point h=0.075 and coefficients A=1, B=1 and C=2.379, process response of primary functions

2015 27th Chinese Control and Decision Conference (CCDC) 3457

g(0.0075) and B=(g(0.0075)g(0.0075)T)-1g(0.0075) are obtained. Based on the parameters above, coefficients in formula No.7 are expressed by k0=0.115, kj=0.0001 and km=0.4388. Furthermore, Setting the initial value of state vector Xm(k) of prediction mode is zero. According to equation No.7, the given signal current PI control loop U (k) is achieved. Based on design experience, the parameters of PI controller of current loop are set as Kp=0.528, Ki=0.0014. When it starts, given speed is 1000rad/s with 3 N m⋅ load torque. At time 0.02s, change the given speed to 1500r/min, while at time 0.035s, reduce the given speed to 1000r/min. The simulation is shown in Figure 3(a). When it starts, given speed is 1000rad/s with 3 N m⋅ load torque. At time 0.03s, the load torque increases to 5 N m⋅ .The simulation is shown in Figure 3(b). When it starts, given speed is 1000rad/s with 3 N m⋅ load torque. At time 0.03s, the load torque increases suddenly. The comparison simulation chart of prediction function and PI is shown in fig 3 (c). When it starts, given speed is 1000rad/s with 4 N m⋅ load torque. Then add random disturbance into it. The comparison simulation chart of prediction function and PI control is shown in Figure 3 (d).

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

200

400

600

800

1000

1200

1400

1600

t/s

n•r/

min

)

PFC

PI

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600

700

800

900

1000

1100

t/s

n(r/

min

)

(b)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

100

200

300

400

500

600

700

800

900

1000

1100

t/s

n•r/

min

)

PFC

PI

(c)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

200

400

600

800

1000

1200

t/s

n(r/

min

)

PFC

PI

(d)

Fig.3 Output waveform of BLDCM by using PFC in different simulation. From the simulation results, it can be concluded that based on predictive functional control to achieve speed loop control, at different given speed, the system can show nice tracking performance. Thus, there are several advantages such as small overshoot, low torque ripple and small starting current. Besides, zero-error in steady state and ability to restrain interference signal are also excellent properties. 5 CONCLUSION Predictive functional control has strong anti-interference ability and robustness, which the steady-state accuracy and dynamic performance superior to the conventional PID control. The predictive functional control method can provide unbiased tracking to speed Co set by the step function. It can eliminate interference effectively, which has strong robustness, and then control performance is improved. Brushless DC motor speed servo system based on the predictive functional control has a good dynamic and static performance. It can also solve problems of brushless DC motor speed servo system such as time-varying parameters, anti-interference, rapid tracking and high control accuracy requirements. Therefore, predictive functional control is able to improve the overall control of the brushless DC motor speed servo system and will be applied extensively.

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3458 2015 27th Chinese Control and Decision Conference (CCDC)