adaptive position tracking control of a bldc motor using a ...international journal of computational...

International Journal of Computational Intelligence in Control, 1(2), 2009, pp. 95-103

Adaptive Position Tracking Control of a BLDC Motorusing a Recurrent Wavelet Neural Network

Chun-Fei Hsu1 Chih-Min Lin2* Chao-Ming Chung2

1Department of Electrical Engineering, Chung Hua University, Hsinchu, 300, Taiwan, Republic of China2Department of Electrical Engineering, Yuan Ze University, Chung-Li, Tao-Yuan, 320

Taiwan, Republic of China

Abstract: The recurrent wavelet neural network (RWNN) has the advantages such as fast learning property, goodgeneralization capability and information storing ability etc. Based on these advantages of RWNN, an adaptive positiontracking control (APTC) system, which is composed of a neural controller and a robust controller, is proposed in this paper.The neural controller uses the RWNN structure to online mimic an ideal controller, and the robust controller is designed toachieve tracking performance with desired attenuation level. The adaptive laws of APTC system are derived based on theLyapunov stability theorem and gradient decent method. Finally, the proposed APTC method is applied to a brushless DC(BLDC) motor. The hardware implementation of APTC scheme is developed on a field programmable gate array (FPGA)chip. Experimental results verify that a favorable tracking response can be achieved by the proposed APTC method evenunder the change of position command frequency after training of RWNN.

Keywords: Adaptive control, recurrent neural network, wavelet neural network, BLDC,

1. INTRODUCTION

With the learning ability of neural network, neural networkshave widely been recognized as a powerful tool in industrialcontrol, commercial prediction, image processingapplications and etc. (Mendel, 1994). The characteristics offault tolerance, parallelism and learning suggest that theymay be good candidates for implementing real-time adaptivecontrol for unknown nonlinear dynamic systems (Omidvarand Elliott, 1997). The successful key element is theapproximation ability of neural network, where theparameterized neural network can approximate an unknownsystem dynamics through some learning algorithms. Manyauthors have hinted the neural networks as powerful buildingblocks for a wide class of complex nonlinear system controlstrategies when there exists no complete model informationor a controlled plant is considered as a “black box” (Hsu etal., 2005; Leu et al., 2005; Polycarpou, 1996; Tang et al.,2006).

Recently, some researchers have developed the structureof neural network based on the wavelet functions to constructthe wavelet neural network (WNN) (Billings and Wei, 2005;Zhang, 1997). Unlike the sigmoidal functions used inconventional neural networks, wavelet functions are spatiallylocalized, so that the learning capability of WNN is moreefficient than the conventional sigmoidal function neuralnetwork for system identification. The training algorithmsfor WNN typically converge in a smaller number of iterationsthan for the conventional neural networks (Zhang, 1997).Up to now, there has been considerable interest in exploring

the applications of WNN to deal with nonlinearity anduncertainties of real-time adaptive control system (Hsu etal., 2006; Lin, 2002; Sousa et al., 2002). These WNN-basedadaptive neural controllers combine the capability of neuralnetworks for learning ability and the capability of waveletdecomposition for identification ability.

Although the control performances are acceptable in(Hsu et al., 2005; Hsu et al., 2006; Leu et al., 2005; Lin,2002; Sousa et al., 2002; Tang et al., 2006), these neuralnetworks are feedforward neural networks belonging to staticmapping networks. Without aid of tapped delay, afeedforward neural network is unable to represent a dynamicmapping. In addition, the neural network must be selectedwith a sufficiently large number of neurons in the hiddenlayer, which consumes a large amount of processing timefor real-time applications (Juang et al., 2007; Lin and Lee,1996). According to the structure, the recurrent neuralnetwork has superior capabilities as compared to feedforwardneural networks, such as their dynamic response and theirinformation storing ability. Since the recurrent neuralnetwork captures the dynamic response of a system, thenetwork model can be simplified (Lin and Lee, 1996).

This paper proposes a recurrent wavelet neural network(RWNN), which has superior capability to the conventionalWNN in an efficient learning mechanism and dynamicresponse. Temporal relations are embedded in RWNN byadding feedback connections, so that the RWNN provides adynamical mapping. Then, an adaptive position trackingcontrol (APTC) system using RWNN approach is proposed

International Journal of Computational Intelligence in ControlVol. 12 No. 2 (July-December, 2020)

96 International Journal of Computational Intelligence in Control

to tackle the control problem of the brushless DC (BLDC)motor, which is ideal for use in expensive environments suchas aeronautics, robotics, electric vehicles and dynamicactuation. All the parameters of the proposed APTC systemare online tuned in the Lyapunov sense and gradient decent,thus the stability of the closed-loop system can beguaranteed. Finally, the proposed APTC system isimplemented on a field programmable gate array (FPGA)chip for possible low-cost and high-performance industrialapplications. The experimental results demonstrate that theproposed APTC scheme can achieve favorable positiontracking control for the BLDC motor even under the changeof position command frequency. Moreover, the bettertracking performance can be achieved as a specifiedattenuation level is chosen smaller.

2. PROBLEM STATEMENT OF BLDC MOTOR

During two decade years, brushless DC (BLDC) motors havegained widespread use in electric drivers. BLDC motors areideal for use in expensive environments such as aeronautics,robotics, electric vehicles and dynamic actuation (Dote andKinoshita, 1990). Unfortunately, the BLDC motor is anonlinear system whose internal parameter values willchange slightly with different input command andenvironments. Using these BLDC motors in high-performance drivers require advance and robust controlmethods. The system equations of BLDC motor driver in ad-q model can be expressed as (Rubaai et al., 2002; Rubaaiet al., 2007).

1s d mqs qs r ds qs r

q q q q

R Li i i v

L L L L

�� (1)

1qsds ds r qs ds

d d d

LRi i i v

L L L� � � � � (2)

Lq = L

is + L

mq(3)

Ld = L

is + L

md(4)

where ids

and iqs

represent the d and q axes stator currents,respectively, V

ds and V

qs are the d and q axes stator voltage,

respectively, Ld and L

q are the d and q axes stator inductances,

respectively, Rs is the stator resistance, L

is is the stator leakage

inductance, Lmd

and Lmq

are the d and q axes magnetizinginductances, respectively, �

r is the electrical rotor angular

velocity, and �m is the flux linkage of permanent magnet.

The torque equation is expressed as (Rubaai et al., 2007).

2 2r r e LJ B T T

N N� � � � �� (5)

where N is the number of poles, J is the inertia of the rotor,B is the damping coefficient, T

e is the electromagnetic torque

and TL is the load disturbance. By using the field-oriented

control, the electromagnetic torque of BLDC motor drivercan be expressed as (Slotine and Li, 1991).

3

2 2e m qs t qs

NT i k i� � � (6)

where 3

2 2t m

Nk � � is the constant gain. From (5) and (6),

the system dynamic equation can obtain

f gu h� � �� (7)

where � is a position of rotor, , ,2

tkB Nf g

J J� � �

,2 L

Nh T

J� � and u = i

qs is the control effort. The control

objective of BLDC motor is to find a control law so that therotor position � can track the position command �

c closely.

Define the tracking error as

e = �c – � (8)

Assume that all the parameters in (7) are well known,there exits an ideal controller (Slotine and Li, 1991).

* 11 2( )cu g f h k e k e�� (9)

where k1 and k

2 are non-zero positive constants. Applying

the ideal controller (9) into (7), it is obtained that

1 2 0e k e k e� � �� (10)

If k1 and k

2 are chosen to correspond the coefficients of

a Hurwitz polynomial, it implies that lim 0t

e��

� for any

starting initial conditions. Since the system dynamics andthe load disturbance may be unknown or perturbed inpractical applications, the ideal controller u* in (9) can notbe precisely obtained.

Much research has been done to apply variousapproaches in the control field of BLDC motor (Liu et al.,2005; Rubaai et al., 2002; Rubaai et al., 2007). A PIcontroller is proposed based on the completely understandingof the model and through some time-consuming designprocedures; however, their performances generally dependon the working point, thus the control parameters which wantto ensure proper behavior in all operating conditions aredifficult to design (Liu et al., 2005). Rubaai et al. (2002)proposed an adaptive fuzzy controller. In order to ensurethe system stability, a compensation controller will bedesigned to dispel the approximation error. However, themost frequently used of compensation controller is like asliding-mode control, which requires the bound of theapproximation error. To solve this chattering problem, arobust adaptive fuzzy-neural-network controller had beendeveloped (Rubaai et al., 2007). Though the robust trackingperformance can be achieved, the used neural network is afeedforward neural network. It may be selected with asufficiently large number of hidden neurons, in which thecomputation loading is heavy.

Adaptive Position Tracking Control of a BLDC Motor using a Recurrent Wavelet Neural Network 97

3. APTC DESIGN

3.1. Description of RWNN

As shown in Fig. 1, the RWNN, is comprised of an inputlayer, a mother wavelet layer, a product layer and an outputlayer, is adopted to implement the neural controller. TheRWNN output can perform the mapping according to

11 1

( )M M L

j j j ij ijij j

y z��

� � � � � �� (11)

where the subscript ij indicates the ith input term of the jthwavelet, �

j is the connection weight between the product

nodes and output node, and ( ).pij ij i ij ij ijz x r m� � � � � The

�ij, m

ij and r

ij are the dilation factor, translation factor and

recurrent factor, respectively, and the memory term pij� is

the self-recurrent term of a wavelet. In addition, the motherwavelets are chosen as

2 2 2( ) (1 )exp( ).ij ij ij ijz z z� � �� (12)

For ease of notation, (11) can be expressed in a compactvector form as

y(x, , m, r) = T �(x, , m, r) (13)

where x = [x1 x

2 ... x

L]T is the input vector, � = [�

1 �

2 ... �

M]T,

�= [�1 �

2 ... �

M]T, = [�

11 ... �

L1 �

12 ... �

L2 ...... �

1M ... �

LM]T,

m = [m11

... mL1

m12

... mL2

...... m1M

... mLM

]T and r = [r11

... rL1

r12

... rL2

...... r1M

... rLM

]T. The architecture of the RWNN

used in this paper is designed to keep the advantage of simplestructure and to consider the dynamic characteristics. Themeaning of the recurrent network is to consider the pastoutput of translation layer in the input space since the inputof translation layer is related to its output. Thus, the RWNNhas dynamic characteristics.

3.2. Design of APTC System

This study proposes an adaptive position tracking control(APTC) system as shown in Fig. 2, where the control law isdesigned as

Figure 1: The Architecture of RWNN

Figure 2: The Block Diagram of APTC System for BLDC Motor

uac

= unc

+ urb

(14)

in which a sliding surface is defined as

1 2 0( ) .

ts e k e k e d� � � � �� (15)

The neural controller unc

uses a RWNN to onlineapproximate the ideal controller in (9), and uses a robustcontroller u

rb is designed to achieve L

2 tracking performance

with desired attenuation level. Substituting (14) into (7) andusing (15), the error dynamic equation can be obtained as

*1 2 ( ) .nc rbe k e k e g u u u s� � � � � �� (16)

According to the approximation theory, there exists anoptimal RWNN approximator to approximate the idealcontroller, which is denoted as (Billings and Wei, 2005).

u* = *T � + � (17)

where � denotes the approximation error and * is theoptimal parameter vector of . In fact, the optimal parametervector for neural network to approximate the ideal controlleru* is difficult to determine. To solve this difficulty, anestimative parameter vector is required to estimate theoptimal value, and the neural network with estimativeparameter vector is

ˆ Tncu � (18)

where ˆ is the estimative parameter vector of �. Then theestimative error between ideal controller and neuralcontroller is defined as

* * ˆT T Tncu u u� � � � � � � � �� (19)


where * ˆ .� �� Substitute equation (19) into equation

(16) can obtain

( ).Trbs g u� � � �� (20)

In case of the existence of �, consider a specified L2

tracking performance (Lin and Lin, 2002; Wang et al., 2002;Yang and Wang, 2007)

22 2 2

0 01

(0) (0) (0)TT Tss dt dt

g

� ��

��

(21)

where �1 is a positive constant. If the system starts with initial

conditions s(0) = 0 and (0) 0,�� the L2 tracking

performance in (21) can be rewritten as

2 [0, ]supL T

s

��

� (22)

where 2 2

0

Ts s dt� � and

2 2

0.

Tdt� � �� The attenuation

constant � can be specified by the designer to achieve desiredattenuation ratio between and || s | and | � ||. If � = � , this isthe case of minimum error tracking control without disturbanceattenuation. To guarantee the stability of the APTC system,the Lyapunov function candidate is defined as

2

1

1.

2 2Tg

V s� �� (23)

Taking the derivative of Lyapunov function in (23) andusing (20), yields

V� =1

Tgss �

��

=1

ˆ( )� � � ��

�� T Trb

gsg u

=1

ˆ( ).

� ��

�� T

rbg s sg u (24)

If the parameter tuning law is chosen as

1ˆ s� �� (25)

and the robust controller is chosen as

urb

=2

2

1

2s

� �� (26)

then (24) can be rewritten as

V� =

22

2

1

2g s s� ��

=

222 21 1

2 2 2

s sg� ��

�2

2 21

2 2

sg� ��

. (27)

Assume � � L2[0, T], �T � [0, �], integrating (27)

from t = 0 to t = T, yields

2 2 2

0 0

1 1( ) (0) .

2 2

T T

V T V g s dt g dt� � � � � �� (28)

Since V(T) � 0, the above inequality implies thefollowing inequality

2 2 2

0 0

1 1(0) .

2 2

T T

g s dt V g dt� � � �� (29)

Using (23), (29) is equivalent to (21). Since V(0) isfinite, if the approximation error � � L

2, that is

2

0( ) ,

tt dt� � �� using the Barbalat’s lemma (Slotine and Li,

1991), the APTC system is stable with L2 tracking

performance in Lyapunov sense.

3.3. Full-tuned Online Learning Laws

Although the parameter tuning law in previous sub-sectioncan modify the weights of output layer to the optimal values,the performance and converge speed are still affected bythe parameter vectors in translation layer. If the variances,means and recurrent weights in translation layer are selectedin appropriate value, the network will converge at high speedas well as high performance. However, the optimal valuesof parameter vectors in translation layer are not easy to find.In order to get the optimal parameter values of translationlayer, the online tuning laws which are derived by Lyapunovfunction and gradient decent method are proposed to tunethese parameters. To obtain the online tuning laws, first, theadaptive law in equation (25) can be rewritten as

1ˆ

j js� � � �� (30)

According to gradient decent method, this adaptive lawcan also be presented as

1 1ˆ

ˆj jj

V u V

u u

� � ��

� ��

(31)

Comparing equation (31) with equation (30), theJacobian term of the controlled system can be obtained as

.V

su

��

� Therefore, the online tuning laws of variances,

means and recurrent weights can be presented as

ij�� = 2 2ˆ ˆij j ij

V u V u

u u

� � � � ��

� ��

=

2 22

2 2 2 2

2 ( )ˆ 2 ( )

1 ( )

pij i ij ij ij p

j j ij i ij ij ijpij i ij ij ij

w x r ms x r m

w x r m

� ��

� � � � ��

(32)

3 3 ˆj

ijij j ij

V u V um

u m u m

��

� � � ��


2 22

3 2 2 2

2 ( )ˆ 2 ( )

1 ( )

pij i ij ij ij p

j j ij i ij ij ijpij i ij ij ij

w x r ms x r m

w x r m

� ��

� � � � �� (33)

4 4ˆ ˆijij j ij

V u V ur

u r u r

� � � � ��

� � � ��

2 22

4 2 2 2

2 ( )ˆ 2 ( )

1 ( )

p pij ij i ij ij ij p p

j j ij ij i ij ij ijpij i ij ij ij

w x r ms x r m

w x r m

� ��

� � � � ��

(34)

Apply these online tuning laws into RWNN, theparameters in translation layer can be tuned to appropriatevalues. Therefore, the convergence can still be at high speedeven if inappropriate initial parameters are given.

4. EXPERIMENTAL RESULTS

This study used the Altera Stratix II series FPGA chip, theAltera Quartus II software, the Nios II processor, the NiosII Integrated Development Environment (IDE) and theverilog hardware description language to implement thehardware control system. Field programmable gate array(FPGA) is a fast prototyping IC component. This kind of ICincorporates the architecture of a gate array andprogrammability of a programmable logic device. Theadvantage of controller implement by FPGA includes shorterdevelopment cycles, lower cost, small size, fast systemexecute speed, and high flexibility. The Quartus II softwareis the development tool for programmable logic devices. TheNios II processor is a configurable, versatile, RISCembedded processor. It can be embedded into Altera FPGA,and allow designers to integrate peripheral circuits andprocessors in the same chip. Additionally, the PC-developedalgorithm and C language program can be rapidly migratedto the Nios II processor to shorten the system developmentcycle. The Nios II IDE can be accelerates softwaredevelopment (http://www.altera.com).

The external peripheral interfaces are used to transmitand receive the motor driver signals including motorrotational direction control signal circuit, encoder signalcircuit, and 12-bits D/A converter circuit. The motorrotational direction control signal circuit uses the operationalamplifier IC to raise the motor rotational direction controlvoltage up from the FPGA. The encoder signal circuit raisesthe encoder signal voltage up from the motor driver. The12-bits D/A converter IC with dual channel voltage outputis used to control the BLDC motor. Additionally, every ICthat connects with the FPGA chip uses asynchronous bustransceiver IC to protect the current reflow to FPGA chip.The experimental setup is shown in Fig. 3. The proposedcontrol algorithm is realized in the Nios II programminginterface. The software flowchart of the control algorithmis shown in Fig. 4. In the main program, the initialization ofcontroller parameters is preceded. Next, the interrupt interval

for the interrupt service routine (ISR) with a 1msec samplingrate is set. Then, the controller sample times can be governedby the built-in timer, which generates periodic interrupts.

The BLDC motor system offers high performance andsimple operation from a compact driver and motor. Thespecifications of the adopted BLDC motor systemmanufactured by the Orientalmotor Company are outlinedin Table I (http://www.orientalmotor.com). Modernmechanical systems often require high-speed high-accuracylinear motions. These linear motions are usually realizedusing the rotary motors with a mechanical transmission. Thecommand using alternating sinusoidal and alternatingstepped can supply the different linear motion speed. Asecond-order transfer function with 0.3 sec rise time of thefollowing form is chosen as the reference model for theperiodic step command

2

2 2 2

400

2 40 400n

n n

��

� � �� (35)

where � is the Laplace operator, � is damping ratio (set atone for critical damping) and �

n is undamped natural

frequency. The periodic step command can be specified inthe reference model to smooth the reference trajectory.Moreover, in the proposed control system, without thesecond-order reference model the control input at thebeginning will be very large due to the tracking error in thecontrol algorithm. In addition, when the command is asinusoidal reference trajectory, the trajectory of sinusoidalcommand doesn’t need to change; therefore, the referencemodel is set as one.

First, an H� adaptive fuzzy tracking control systemproposed in Ref. (Rubaai et al., 2007) is applied to BLDCmotor. The experimental results of H� adaptive fuzzytracking control are shown in Fig. 5. The tracking responsesare shown in Figs. 5(a) and 5(d), associated control effortsare shown in Figs. 5(b) and 5(e); and tracking errors areshown in Figs. 5(c) and 5(f), due to a sinusoidal commandand a periodic step command, respectively. From theexperimental results, though the H� adaptive fuzzy trackingcontrol can achieve tracking performances and there are nochattering phenomena in the control efforts; however, theconvergence of the tracking error and controller parametersare very slow at beginning.

Then, the proposed APTC is applied to BLDC motoragain. It should be emphasized that the development ofAPTC system does not need to know the system dynamicsof the controlled system. For practical implementation, theparameters of the APTC system can be online tuned by theproposed adaptive laws without the need of systemparameters. The control parameters for adaptive laws arechosen as k

1 = k

2 = 4, �

1 = 0.02 and �

2 = �

3 = �

4 = 0.0002.

All the gains in the proposed control systems are chosen toachieve the best transient control performance in theconsidering the requirement of stability and possible


operating conditions. The experimental results of APTC with��= 0.8 are shown in Fig. 6. The tracking responses are shownin Figs. 6(a) and 6(d); associated control efforts are shownin Figs. 6(b) and 6(e); and tracking errors are shown in Figs.6(c) and 6(f) due to a sinusoidal command and a periodicstep command, respectively. If a specified attenuation level� is chosen smaller, the experimental results of APTC with��= 0.5 are shown in Fig. 7. The tracking responses are shownin Figs. 7(a) and 7(d); associated control efforts are shownin Figs. 7(b) and 7(e); and tracking errors are shown in Figs.7(c) and 7(f) due to a sinusoidal command and a periodicstep command, respectively. It is shown that the proposed

Figure 3: The Experimental Setup

Figure 4: The Control Design Flow Chart

Figure 5: Experimental Results of H� Adaptive Fuzzy TrackingControl


Figure 6: Experimental Results of APTC with � = 0.8 Figure 7: Experimental Results of APTC with � = 0.5


step command, respectively. From the experimental results,it is seen that the tracking performance of the trained APTCis further improved when the initial values are trained, andthey can achieve favorable robust characteristics for thecommand frequency variation.

5. CONCLUSIONS

This paper has successfully implemented an adaptiveposition tracking control (APTC) scheme for a brushless DC(BLDC) motor position tracking control on a fieldprogrammable gate array (FPGA) chip. Using the FPGA toimplement, the APTC system can achieve the characteristicsof small size, fast execution speed, less memory. Then, theeffectiveness of the proposed APTC system is verified bysome experimental results. The major contributions of thispaper are: (1) the successful development of an APTC, inwhich the Lyapunov stability theorem and gradient decentis used to derive the online tuning algorithms. (2) the L

2

tracking performance can be achieved with a desiredattenuation level using the proposed learning mechanism.(3) the successful applications of APTC to control a BLDCmotor. And, the proposed APTC methodology can be easilyextended to other motors. (4) the FPGA implementationconsumes less power, in terms of core IC power consumptionand especially in terms of the board-level powerconsumption, than the PC and DSP implementation.

Acknowledgments

The authors appreciate the partial financial support from theNational Science Council of Republic of China under grantNSC 96-2218-E-216-001.

References

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APTC can achieve favorable tracking performance;moreover, the better tracking performance can be achievedas a specified attenuation level � is chosen smaller.

Further, the trained APTC is applied to the BLDCsystem again. The experimental results of trained APTC with��= 0.5 are shown in Fig. 8. The tracking responses are shownin Figs. 8(a) and 8(d); associated control efforts are shownin Figs. 8(b) and 8(e); and tracking errors are shown in Figs.8(c) and 8(f) due to a sinusoidal command and a periodic

Figure 8: Experimental Results of Trained APTC with � = 0.5

Table IThe Specifications of BLDC Motor System

Output power HP (W) 1/25HP (30W)

Power supply Single-phase 100~115VAC

Rated current 1.4 A

Gear/ shaft type Round Shaft

Variable speed range 30 ~ 3000 r/min

Rated torque 0.1 N.m

Moment of inertia 1.5�10–4 kg�m2

Load of inertia 0.088 � 10–4 kg � m2

Components BXD30A-A (Driver)BXM230-A2 (Motor)

Control detection system Optimal encoder (500P/R)


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