Current Control Sliding Mode PDF

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Sliding Mode Speed Control of Brushless DC Motor Using

Pulse-Width-Modulated Current Regulator

C.-Y. Chen, Member, IEEE, W.-C. Chan, T.-C. Ou, S.-H. Yu, and T.-W. Liu

AbstractThis paper proposes a sliding mode controller

(SMC) design for a Brushless DC Motor (BLDCM) to achieve

high-performance speed control. Based on Pulse-Width-

Modulation (PWM) technique, a PWM-based controller in the

inner current loop is first developed to perform a fast current

control. A sliding mode speed controller is then designed in its

outer control loop to enhance the speed control. Finally, the

effectiveness of the proposed control scheme under the load

disturbances and parameter uncertainties is verified by

simulation results and comparison with proportional-integral

control (PIC) approach.

I. INTRODUCTION

EC

th

ENTLY, DC motors have been gradually replaced by

e BLDCM since the industrial applications require

more powerful actuators in small size. Elimination ofbrushes and commutators can solve the problem associated

with contacts and give improved reliability and enhances life.

Therefore, the BLDC motor is becoming popular in various

applications because of its high efficiently, high power

factor, high torque, simple control, and lower maintenance.

However, replacement of a DC motor by a BLDC motor

cause the higher demands on a control algorithm, based on

the use of 3 Hall sensors, to commutate the phase current of

the BLDC motor [1]. However, the leakage inductance

causes the stator currents to take a finite time to rise and fall,

thus distorting the ideal waveform into a trapezoidal shape.

This effect has the tendency of introducing torque ripple at

the current transitions. These commutation torque ripples

will produce noise and degrade speed-control characteristics,especially at low speed. Therefore, there are numerous

current control approaches adopted to diminish the torque

pulsation by generating specialized current reference [2-8].

In [2], a direct torque control of BLDCM was proposed to

directly control the torque and stator flux by selecting the

stator voltage vector according to the switching table.

However, DTC has some drawbacks during BLDCM

operation, such as large ripples in torque and flux at low

speeds. Furthermore, there are some instantaneous torque

control was proposed in [4], based on sliding mode control

in d-q reference frame. However, this vector control is

usually effective for the permanent magnet synchronous

motors (PMSM) since three-phase conducting circuit is

adopted. For two-phase conducting BLDC drives,

pulse-width-modulated controllers are extensively applied

to produce the switch logics for the inverters to enforce the

stator phase currents to be as close to the reference

commands [5-9]. Then, in the outer control loop, the speed

controller provides the reference current command to

achieve the speed control. In general, three hall-effect

sensors are usually used as position sensors for the current

commutation points that occur at every 60 electrical degrees.

Therefore, a relatively low cost drive can be reached when

compared to a PMSM drive with expensive high-resolutionrevolver. Consequently, this simple controller design based

on PWM techniques is sensitive to parameter variations and

load disturbances.

Sliding-mode control (SMC) has been found in many

applications in recent years because it can offer many good

properties such as insensitivity to parameter variations,external disturbance rejection, and fast dynamic response [10].

Due to inherent switching characteristic within power converter,

the advantages of SMC are naturally obtained a lot of attraction

in the speed and position control for AC motor. Recently,

several solutions that combine SMC with PWM current

regulator have been considered to achieve the

high-performance speed control of BLDCM [7, 8]. However,

chattering occurs when the control input switches

discontinuously across the boundary, and it is undesirable

because it involves high control activity and may excite

high-frequency dynamics [10]. The chattering phenomenon

can be easily eliminated by the boundary layer technique.

This paper considers the cascade application of a sliding

mode speed controller and a PWM current regulator for a

BLDC drive operating in the two-phase conducting mode, to

achieve the high-performance speed control. In the inner

control loop, the PWM current regulator is first realized. The

control effort of the sliding mode speed controller is then

directly fed through PWM current closed-loop control to

produce an adequate switching signal for driving the

inverter of the BLDC motor. Finally, the effectiveness of the

proposed control scheme under load disturbance andparameter uncertainties will be validated by simulation

results.

Manuscript received January 15, 2009. This work was financially

supported by the National Science Council, Republic of China (Taiwan) for

under grant NSC 97-2221-E-230 -018.

C.-Y. Chen and T.-W Liu (Graduate student) are with the department of

electrical engineering, Cheng Shiu University, No. 840, Cheng-Ching Road,

Niaosong Township, Kaohsiung County 83347, Taiwan (phone:

886-7-7310606; fax: 886-7-733-7390; e-mail: [email protected]).II. DYNAMIC DESCRIPTION FOR BLDCMOTOR

R

S.-H. Yu and W.-C. Chan (Graduate student) are with the department of

electrical engineering, National Sun Yat-Sen University, 70, Lien-hai Rd.,

Kaohsiung 80424, TaiwanA permanent-magnet brushless motor with trapezoidal

flux distribution is considered in this paper. Assuming threeT.-C. Ou is with Institute of Nuclear Energy Research, Atomic Energy

Council, Taoyuan, Taiwan, (e-mail: [email protected])

2009 IEEE/ASME International Conference on Advanced Intelligent MechatronicsSuntec Convention and Exhibition CenterSingapore, July 14-17, 2009

978-1-4244-2853-3/09/$25.00 2009 IEEE 1395


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symmetric phases circuit (see Fig. 1), its dynamics model for

phase-voltage equation can be described as below [1]

1

1

1

0 0 0 0

0 0 0 0

0 0 0 0

as s as as as

bs s bs bs bs

cs s cs cs cs

V R i L i ed

V R i L idt

V R i L i

e

e

(1)

whereRsis the stator resistance per phase and L1=Ls-M;Ls

and Mrepresent the phase inductance and mutual inductance,respectively; ias(Vas), ibs(Vbs) and ics(Vcs) denote the

three-phase currents (voltages) into the motor. The

instantaneous induced EMF for phase a, b and c are all

assumed to be trapezoidal and can be respectively expressed

asFig. 2 Waveforms of back-EMF, phase current, and developing torque.

as as r e me f K (2)

bs bs r e me f K (3)

cs cs r e me f K (4)

where the functionfas,fbs, andfcshave the same shape as eas,

ebs, ecs, with a maximum magnitude of 1;Keis the fluxlinkage constant. The electromagnetic torque is given by

[ ]e as as bs bs cs csT e i e i e i / m (5)The equation of motion for a simple system with inertiaJ,

friction coefficientB, and load torque TLis

me

dT J B T

dt

m L (6)

and the electrical rotor r and mechanical speed m are

related by

2( )Pr m . (7)

Fig. 3. Integration scheme combined sliding mode speed control with PWM

current regulator.

wherePis the number of motor poles. 120voltage-sourceoperation is selectively applied in the inverter stage such that

there is only current conduction path for each switching, as

shown in Fig. 2. It is noted that the phase-voltage equations

are identical to the armature-voltage equations of a DC

motor.

III. SLIDING MODE SPEED CONTROLLER DESIGN

This paper adopts two stages control schematic to achieve

the high-performance speed control, as shown in Fig. 3. The

inner current loops enforce current command, and the outerspeed loop enforces the speed command. The current

command signal is obtained from the speed-error signal

through a sliding mode control under the conditions of load

disturbances and parameter uncertainties. Combining with

respective Hall position signal and steering circuit, the

PWM current regulator can provides the fastest phase

current response according to the current reference, by

means of instantaneous action. PWM current regulator is

simple in form of PI controller [1]. Here, only the speed

controller design will be discussed in below.

After the PWM current controller has been applied in the

inner-loop (See Fig. 3), the current gain can be simply

modeled as unity if the delay due to the PWM carrier

frequency is negligible. According to (5) and (6), the

equivalent electromechanical dynamic equation for each

switching can be considered as

Fig. 1. The electrical equivalent circuit of BLDCM.

2 e m m l K I J B T (8)

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1 1( 1mb a f b b U whereIrepresents the phase current for each commutation,i.e., ias, ibs,ics. Furthermore, the dynamic equation (8) can be

rewritten in the following form:

)

(19)

m ma f b I (9)Based on the conditions of max min/ 1b b and

) (1 (1 1 / )b b 1

where a=B/J; b=2Ke/J; and f=Tl/J. Taking into

consideration load disturbances and parameter uncertainties,

the previous equation can be expressed as:

m n m na a f f b u (10)

where u=I; an (fn) and a (f) represent the known

nominal and uncertainty values of the parameters asandf,

respectively. b denotes control gain and its value is

bounded within the region of bminbbmax.

Define the speed tracking error to be the difference

between the speed of the desired trajectory md and the

motor speed m , i.e., e md m

0

. Next, define a stable

speed sliding surface as

(11)

0

( ) ( ) ( ) 0t

s t e t K e d whereKis the positive design parameter. Then, Taking the

derivative of the above sliding surface with respect to time

yields

(12)0s e K e

Substituting (10) into (12) gives

( )md n m ns a a f f K e b u . (13)

Canceling the unexpected terms and embedding the desired

terms, the ideal equivalent input control law is specified as

( )md n m nb u a a f f K e (14)

Then, the best approximation equivalent control input can be

expressed as:

*m n m nb u U a f K e

(15)

A control law satisfies the hitting condition can be derived as

1 sgn( )u b b u s

(16)

wheremin maxb b b

, is the switching gain, and

is the sign function.sgn( )Let us consider the Lyapunov function candidate for the

speed control as V s and take its time derivative as:2 / 2

] [ ( )md n m nV s a a f f K e b u (17)

Substituting (15) and (16) into (17) leads to

1(1 ) 0mV b a f b b U s

(18)

if is chosen to meet the following condition:

(1 )

, (19) can be rewritten as:

1( (1 1 /mb a f U ) )

(20)

When V is clearly positive definite and V is negative

definite, s(t) asymptotically converges to zero; i.e.,

as . When the speed sliding mode occurs

on the sliding surface (11), i.e., . Therefore,

the relationship between s(t) and e in (12), considering

as , gives e t as ,i.e.,

asymptotic speed trajectory tracking if the gainKis chosen

to be strictly positive.

( ) 0s t t

( ) ( ) 0s t s t

( ) 0s t t ( ) 0

t

Finally, the speed sliding mode controller is summarized as:

1 * sgn( )m n m nu b a f K e s

(21)

Furthermore, in order to reduce the inherent chatteringphenomenon of the sliding mode control, the control input

according to the boundary layer approach can be modified as

below:

1 * ( / )m n m nu b a f K e sat s

(22)

where ( )sat is the saturation function and is the

boundary layer thickness for the speed control.

IV. SIMULATION RESULTS

In order to evaluate the performance of the proposed

controller designs mentioned above, some different cases of

load disturbances and parameter uncertainties are

considered in the simulation studies and compared withproportional-Integral control approach. Considering a

BLDC90 from TECO Electro Devices Co. and its relative

parameters are given in Table I, the design gains of SMC

controller is chosen to be K=0.5, and the parameter

uncertainties of a and f are considered to be 100%

variation with respect to the corresponding nominal

parameter values, listed in Table I.

Figure 4 demonstrates the simulation results for speed

control of 1400 rpm with external load disturbance 0.1 N-m

added at t= 1.5 sec and without parameter uncertainties,

TABLEI

BLDCMOTOR PARAMETERS

Symbol Description ValueVDC Voltage source 24VDC

Rs phase resistance 101

Ls phase inductance

5104HJ moment of inertia 6.5105Kg-m2Ke flux linkage constant 3102V/B damping constant 5106N-M/

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Fig. 6. Speed tracking error for speed control of 1400 rpm with flux linkage

constant (KE) increased by 100%.

according to a ramp speed trajectory. It can be observed that

the PIC approaches present a large overshoot response, but

the proposed SMC controller provides a smooth and stable

speed response without the overshoot phenomenon. The

speed tracking error according to SMC approach is smaller

than 0.5 rpm, except the lunching portion. Additionally,

when external load disturbance of 0.1 N-m is suddenly

added at t=0.15 sec, it is also found that the proposed SMC

controller still maintains the desired speed tracking

performance by means of quickly compensating for systemdisturbance; by contrast, the PIC approaches will

unexpectedly increase the speed tracking error at the point of

load disturbance applied (see Fig. 4(b)). Furthermore, from

Fig 4(c) and (d), the current waveform for each phase circuit

is just like the demanded current command in Fig. 2, except

very small current noise due to PWM chopping operation.

Figure 5 shows the speed tracking error response for a low

speed control of 10 rpm when the moment of inertia for

BLDCM is considered to reduce by 75%. It is noted that the

conventional PIC approach brings a large speed tracking

error and obviously induces a speed tracking error of 2 rpm

even during the steady state operation. By contrast, the

proposed SMC approach consistently provides an excellent

speed response under the influence of moment of inertia.

Since flux linkage constant plays an important role for speed

control, Fig. 6 considers the simulation case with the flux

linkage constant (KE) increased by 100%. Compared with

speed tracking errors, it is seen that the proposed SMC

approach can continuously give a superior speed response,

compared with speed tracking performance of PIC

approach.

Fig. 4. Simulation results for speed control of 1400 rpm with external load

disturbance 0.1 N-m added at t =0.15 sec, (a) Speed response, (b) Speed

tracking error, (c) phase current response for SMC, (d) phase current

response for PIC.

V. CONCLUSIONS

This paper proposes an integrated control scheme, which

is combination of a PWM current regulator and a sliding

mode controller, to achieve high-performance speed control

of a BLDC drive. Simulation studies demonstrated that theproposed SMC controller, compared with the PIC

approaches, can produce a better speed response for

different speed commands, load disturbances, and parameter

uncertainties. In our future work, the proposed sliding mode

controller will be implemented in a real control system with

digital signal processing (DSP) chips.Fig. 5. Speed tracking error for speed control of 10 rpm under the conditionof J reduced by 75%.

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