,英3RCMAC的自适应控制设计无刷直流电机

ORIGINAL ARTICLE

RCMAC-based adaptive control design for brushless DC motors

Chih-Min Lin ?Chun-Fei Hsu ?Chao-Ming Chung

Received:27October 2007/Accepted:24November 2008/Published online:19December 2008óSpringer-Verlag London Limited 2008

Abstract This paper proposes a recurrent cerebellar model articulation controller (RCMAC)-based adaptive control for brushless DC motors.This control system is composed of a RCMAC and a compensation controller.RCMAC is used to mimic an ideal controller,and the compensation controller is designed to compensate for the approximation error between the ideal controller and RCMAC.The Lyapunov stability theory is utilized to derive the parameter tuning algorithm,so that the uni-formly ultimately bound stability of the closed-loop system can be achieved.For comparison,a fuzzy control,an adaptive fuzzy control and the developed RCMAC-based adaptive control are implemented on a ?eld programmable gate array chip for controlling a brushless DC motor.Experimental results reveal that the proposed RCMAC-based adaptive control system can achieve the best tracking performance.Moreover,since the developed RCMAC-based adaptive control scheme uses a hyperbolic tangent function to compensate for the approximation error,there is no chattering phenomenon in the control effort.Thus,the proposed control method is more suitable for real-time practical control applications.

Keywords Recurrent CMAC á

Uniformly ultimately bound stability áBrushless DC motor áFPGA

1Introduction

Recently,brushless DC (BLDC)motors have gradually been used in the motion control,because their advantages include simple to construct,high-torque capability,small inertia,low noise and long-life operation [1].Unfortu-nately,BLDC motor is a non-linear system whose internal parameter values will change slightly with different input commands and environments.Several investigations have been carried out in recent years to apply various control approaches for BLDC motor control [2–4].Liu et al.[2]proposed a PI controller to achieve instantaneous torque control with reduced torque ripple.However,the selection of PI gains is a trade-off between robustness and fast transient response.Rubaai et al.[3]proposed an adaptive fuzzy-neural-network controller to achieve satisfactory tracking performance.However,the computational loading is too complex.Rubaai et al.[4]proposed a robust adaptive fuzzy controller.The experimental results show that robust tracking performance can be achieved;however,the con-trol effort may lead to a large control signal as the speci?ed robustness is increased.

Based on the approximation ability of neural network (NN),the NN-based adaptive controllers have been developed to compensate for the effects of non-linearities and system uncertainties [5–8].To obtain the fast learn-ing property and good generalization capability,the cerebellar model articulation controller (CMAC)has been proposed [9].CMAC is classi?ed as a non-fully con-nected perceptron-like associative memory network with

C.-M.Lin (&)áC.-M.Chung

Department of Electrical Engineering,Yuan Ze University,Chung-Li,Tao-Yuan 320,Taiwan,ROC e-mail:cml@https://www.360docs.net/doc/4218599162.html,.tw C.-M.Chung

e-mail:s948507@https://www.360docs.net/doc/4218599162.html,.tw

C.-F.Hsu

Department of Electrical Engineering,

Chung Hua University,Hsinchu 300,Taiwan,ROC e-mail:fei@https://www.360docs.net/doc/4218599162.html,.tw

Neural Comput &Applic (2009)18:781–790DOI 10.1007/s00521-008-0230-2

overlapping receptive-?elds.CMAC has been already validated that it can approximate a non-linear function over a domain of interest to any desired accuracy.There has been considerable interest in exploring the applica-tions of CMAC to deal with the non-linearity and uncertainty of control systems[10–13].It has been shown that the CMAC-based adaptive control systems can achieve better control performance than NN-based adap-tive control systems by appropriately designing the CMAC controller[14,15].

According to the structure,the NNs can be mainly classi?ed as feedforward neural networks(FNNs)and recurrent neural networks(RNNs)[16].As known,FNN is a static mapping.Without the aid of tapped delays, FNNs are unable to represent a dynamic mapping.As far as RNNs are concerned,their ability to deal with time varying input or output through their own natural tem-poral operation is of particular interest.Thus,RNN is a

dynamic mapping and demonstrates good control perfor-mance in the presence of unmodeled dynamics[17,18]. In this study,a recurrent CMAC(RCMAC),which involves dynamic elements in the form of feedback con-nections that are used as internal memories,is developed to design the controller.RCMAC has advantages over CMAC in its dynamic response and its information stor-ing ability.

To tackle the non-linear problem of BLDC motor,this paper proposes a RCMAC-based adaptive control system with a new training algorithm.The RCMAC-based adaptive control system is composed of an RCMAC and a compensation controller.RCMAC is used to mimic an ideal controller and it presents the main controller.The compensation controller using a hyperbolic tangent func-tion is derived to compensate for the difference between the ideal controller and RCMAC.Since RCMAC has an internal feedback loop,it captures the dynamic response of system with external feedback through delays.More-over,all the parameters of the RCMAC-based adaptive control system are tuned in the Lyapunov sense,thus the uniformly ultimately bounded stability can be guaranteed. The stability analysis shows that the output of the system can exponentially converge to a small neighborhood of the trajectory command.Finally,the proposed RCMAC-based adaptive control system is implemented based on a ?eld programmable gate array(FPGA)chip for possible low-cost and high-performance industrial applications.A comparison between a fuzzy controller,an adaptive fuzzy controller and the proposed RCMAC-based adaptive controller is made.The experimental results demonstrate that the proposed RCMAC-based adaptive control scheme can achieve better control performance than the other control methods.2RCMAC approximator

The network structure of RCMAC is shown in Fig.1.The architecture of RCMAC includes input space,association memory space,receptive-?eld space,weight memory space,output space and recurrent weight.The signal propagation and the basic function in each space are introduced in the following.

1.Input space:For a given z??z1;z2;...;z n T2R n;

where z j represents the j th input to the node of layer1.

Each input state variable z j can be quantized into discrete regions(called an element)according to a given control space.

2.Association memory space:In this space,several

elements can be accumulated as a block.Each block performs a receptive-?eld basis function.The Gaussian function is adopted as the receptive-?eld basis function which can be represented as

u jkez jT?exp

àez rjkàm jkT2

r

jk

"#

;for k?1;2;...;n Be1T

where u jk(z j)presents the k th block of the j th input z j with the mean m jk and variance r jk and n B is the number of blocks.In addition,the input of this block can be represented as

z rjk?z jetTtr jk u jketàTTe2Twhere r jk is the recurrent weight of the recurrent unit, and u jk(t-T)denotes the value of u jk through delay time T.It is clear that the input of this block contains a memory term u jk(t-T),which stores the past infor-mation of the network and presents a dynamic

characteristic.

Fig.1The architecture of RCMAC

3.

Receptive-?eld space :In this receptive-?eld space,the multidimensional receptive-?eld function is de?ned as

b q ez T?Y n j ?1u jk ez rjk T?exp X n j ?1àez rjk àm jk T

2r jk "#

;for q ?1;2;...;n R

e3T

where b q is associated with the q th receptive-?eld.

De?ne the vectors m ,r and r collecting all parameters of the membership layer as

m ?m 11;...;m n 1;m 12;...;m n 2;...;m 1n B ;...;m nn B ? T e4Tr ?r 11;...;r n 1;r 12;...;r n 2;...;r 1n B ;...;r nn B ? T e5Tr ?r 11;...;r n 1;r 12;...;r n 2;...;r 1n B ;...;r nn B ? T :

e6T

The multidimensional receptive-?eld function can be expressed in a vector notation as

H ?b 1;...;b q ;...;b n R ??T

:e7T4.

Weight memory space :Each location of R to a particular adjustable value in the weight memory space with n R components can be expressed as

w ?w 1;...;w q ;...;w n R ??e8Twhere w q denotes the connecting weight value of the q th receptive-?eld.

5.

Output space :The output of RCMAC is the algebraic sum of the activated weights in the weight memory,and is expressed as y ?

X n R q ?1

w q b q :e9T

The output of RCMAC can be expressed in a vector

notation as y ?w T H ez ;m ;r ;r T:

e10T

This implies that there exists an RCMAC of (10)such that it can uniformly approximate a non-linear even time-varying function X .By the universal approximation theorem,there exists ideal weight vectors such that [19]X ?y ?

tD ?w ?T

H ez ;m ?

;r ?

;r ?

TtD ?w ?T

H ?

tD

e11T

where D denotes the approximation error,w *and H *are the optimal parameter vectors of w and H ,respectively,and m *,r *and r *are the optimal parameter vectors of m ,r and r ,respectively.In fact,the optimal parameter vectors that are needed to best approximate a given non-linear function X cannot be determined.Thus,an estimation function is de?ned as

^y ?^w

T H ez ;^m ;^r ;^r T?^w T ^H e12T

where ^w

and ^H are the estimated parameter vectors of w and H ,respectively,and ^m ;^r ;and ^r are the estimated

parameter vectors of m ,r and r ,respectively.De?ne the estimation error as

~y

?X à^y ?y ?à^y tD ?~w T ~H t^w T ~H t~w T ^H tD e13T

where ~w

?w ?à^w and ~H ?H ?à^H :In the following,some tuning laws will be derived to on-line tune the parameters of RCMAC to achieve favorable estimation of a non-linear function.To achieve this goal,the Taylor expansion linearization technique is employed to transform the non-linear function into a partially linear form,i.e.[19]~H ?A T ~m tB T ~r tC T ~r th e14Twhere ~m

?m ?à^m ;~r ?r ?à^r ;~r ?r ?à^r ;

h is a

vector of higher-order terms;

A ?o H 1o m o H 2o m áááo H n R o m ! m ?^m

B ?

o H 1o r o H 2o r áááo H n R o r ! r ?^r C ?

o H 1o r o H 2o r áááo H n R o r ! r ?^r

:Substituting (14)into (13),yields

~y

?~w T ~H t^w T eA T ~m tB T ~r tC T ~r th Tt~w T ^H tD ?~w

T ^H t~m T A ^w t~r T B ^w t~r T C ^w te e15T

where ^w T A T ~m ?~m T A ^w ;^w T B T ~r ?~r T B ^w and ^w T C T ~r ?~r T C ^w

are used since they are scales,and the uncertain term e ?^w

T h t~w T ~H tD denotes the approximation error,and is assumed to be bounded by 0B |e |B E *,in which E *is a positive constant.However,this uncertainty bound E *cannot be obtained in practice,so that it will be on-line estimated and will be discussed in the following section.3RCMAC-based adaptive controller design 3.1Modeling of a brushless DC motor

The system equations of BLDC motor driver in a d –q model can be expressed as [4]i qs ?àR s L q i qs àL d L q x r i ds t1L q v qs àk m

L q x r

e16Ti ds ?à

R s L d i ds tL q L d

x r i qs t1

L d v ds

e17T

L q?L istL mqe18TL d?L istL mde19Twhere i ds and i qs represent the d and q axes stator currents,respectively,R s is the stator resistance,L d and L q are the d and q axes stator inductances,respectively, L is is the stator leakage inductance,L md and L mq are the d and q axes magnetizing inductances,respectively,x r is the electrical rotor angular velocity,and k m is the?ux linkage of permanent magnet.The torque equation is expressed as

T e?3

2

N

2

?k m i qsteL dàL qTi qs i ds ?J

N

2

_x rtB

N

2

x rtT L

e20T

where N is the number of poles,J is the inertia of the rotor, B is the damping coef?cient,and T L is the load disturbance. By using the?eld-oriented control,the torque equation of BLDC motor driver can be expressed as

T e?3

2

N

2

k m i qs?k t i qse21T

where k t?3

2N

2

k m is the constant gain.From(20)and(21),

it can obtain

€h?f_htguthe22T

where h?R

x r d t is a position of rotor,f?àB

J

;g?

2 N k t

J

;h?à2

N

1

J

T L;and u=i qs is the control effort.

3.2Control system design

The control objective of the BLDC motor driver is to ?nd a control law so that the rotor position h can track the position command h c closely.De?ne the tracking error as

e?h càh:e23TAssume that the parameters of the controlled system in (22)are well known,there exits an ideal controller

u??gà1eàf_hàht€h ctk1_etk2eTe24Twhere k1and k2are positive constants.Applying the ideal controller(24)into(22)results in the following error dynamics

€etk1_etk2e?0:e25TIf k1and k2are chosen such that all roots of the polynomial hesTD s2tk1stk2lie strictly in the open left

half of the complex plane,then it implies that lim

t!1e?0for

any starting initial conditions.However,since the system dynamics f and g,and the disturbance h may be unknown or perturbed in practical applications,the ideal controller u*in(24)cannot be precisely obtained.

To tackle this problem,the RCMAC-based adaptive control system for BLDC motor driver is proposed and shown in Fig.2,where the control law is designed as

u?u RCMACtu rb:e26TThe RCMAC u RCMAC is used to mimic the ideal con-troller u*and the robust controller u rb is used to compensate for the difference between RCMAC and ideal controller.Substituting(26)into(22)and using(15),the error dynamic equation can be obtained as

_e?A m etbeu?àu RCMACàu rbT

?A m etbe~w T^Ht~m T A^wt~r T B^wt~r T C^wteàu rbT

e27Twhere A m?

01

àk2àk1

!

and b=[0,1]T.To guarantee the stability of the RCMAC-based adaptive control system, the Lyapunov function is de?ned as

V?

1

2

e T Pet

1

2g1

~w T~wt

1

2g2

~m T~mt

1

2g3

~r T~rt

1

2g4

~r T~r

t

1

2g5

~E2e28T

where the positive constants g1,g2,g3,g4and g5are the learning-rates,the estimation error of error bound is de?ned as~E?E?à^E where^E is the estimation of E*, and P is a symmetric positive de?nite matrix that satis?es the Lyapunov equation

A T

m

PtP T A m?àQe29Tand Q is a positive de?nite matrix.Taking the derivative of Lyapunov function in(28)and using(27),

yields

_V?1

2_e T Pet

1

2

e T_Pet

1

g1

~w T_~wt

1

g2

~m T_~wt

1

g3

~r T_~r

t1

g4

~r T_~wt

1

g5

~E_~E

?1

2

e T A T

m

PtPA m

àá

ete T Pb

à

~w T^Ht~m T A^wt~r T B^w

t~r T C^wteàu rb á

à

1

g1

~w T_^wà

1

g2

~m T_^mà

1

g3

~r T_^r

à1

g4

~r T_^rà

1

g5

~E_^E

?à1

2

e T Qet~w T e T Pb^Hà

_^w

g1

!

t~m T e T PbA^wà

_^m

g2

!

t~r T e T PbB^wà

_^r

g3

!

t~r T e T PbC^wà

_^r

g4

!

te T Pbeeàu rbTà

1

g5

~E_^E:

e30T

If the adaptation laws of RCMAC are chosen as

_^w?g

1

e T Pb^Hàl1e^wà^w0T

h i

e31T

_^m?g

2

e T PbA^wàl2e^mà^m0T

??

e32T

_^r?g

3

e T PbB^wàl3e^rà^r0T

??

e33T

_^r?g

4

e T PbC^wàl4e^rà^r0T

??

e34T

where l1,l2,l3and l4are positive small constants, ^w0;^m0;^r0and^r0are initial estimation vectors of w*,m*, r*and r*,respectively.The robust controller with bound estimation law is chosen as

u rb?^E tan h

e T Pb

d

e35T

_^E?g

5e T Pb tan h

e T Pb

àl5e^Eà^E0T

!

e36T

where tan h(á)is a hyperbolic tangent function,d and l5 are small positive constants,and^E0is initial estimation value of E*.The adaptation laws comprise a variant of the l-modi?cation,which allow the designer to incorporate any a priori parameter knowledge.In the absence of any a priori information,^w0;^r0and^E0can be simply set to zero,^m0can be assigned with equal distribution in the considered ranges and^r0can be set as a value to let the intersection of two neighbor receptive-?eld basis function will be equal to0.5.Then(30)can be rewritten as

_V?à1

2

e T Qet~w T l1e^wà^w0Tt~v T l2e^mà^m0T

t~r T l3e^rà^r0Tt~r T l4e^rà^r0T

te T Pb eà^E tan h

e T Pb

d

!

à~E e T Pb tan h

e T Pb

d

àl5e^Eà^E0T

!

à

1

2

e T Qet~w T l1e^wà^w0Tt~v T l2e^mà^m0T

t~r T l3e^rà^r0Tt~r T l4e^rà^r0T

tE?e T Pb

àe T Pb tan h

e T Pb

d

!

t~E l5e^Eà^E0T:

e37TIt can be found that the following inequality holds for any d[0[20]

0e T Pb

àe T Pb tan h

e T Pb

d

jde38T

where j is a constant satisfying j=exp(-(j?1)). Using the inequality(38),(37)can be rewritten as

_Và1

2

e T Qet~w T l1e^wà^w0Tt~v T l2e^mà^m0T

t~c T l3e^rà^r0Tt~r T l4e^rà^r0T

t~E l5e^Eà^E0TtE?k d

à

1

2

e T QetE?jdà

1

2

l1~w

k k2tl2~m

k k2tl3~r k k2

tl4~r k k2tl5~E

2

t

1

2

l1w?à^w0

k k

eT2

h

tl2m?à^m0

k k

eT2tl3r?à^r0

k k

eT2tl4r?à^r0

k k

eT2tl5E?à^E0

àá2i

e39TChoosing Q=I and considering the Lyapunov function (28),(39)can be obtained as

_Vàa Vtbe40Twhere a and b are positive constants given by

a?min

1

k maxePT

;l1g1;l2g2;l3g3;l4g4;l5g5

e41Tb?

1

2

l1w?à^w0

k k

eT2tl2m?à^m0

k k

eT2

h

tl3r?à^r0

k k

eT2tl4r?à^r0

k k

eT2tl5E?à^E0

àá2i tE?jd:e42TThus,the uniformly ultimately bound stability of the closed-loop system can be achieved[20].

4Experimental results

Field programmable gate array is a fast prototyping IC

component.This kind of IC incorporates the architecture of a gate array and programmability of a programmable logic device.The advantage of controller implemented by FPGA includes short development cycles,low cost,small size,fast system execution speed,and high ?exibility.In the FPGA design,the Altera Quartus II software and hardware description language Verilog are used in this study.The hardware is implemented by Altera Stratix II series FPGA chip with 50MHz clock frequency.The external peripheral interfaces are used to transmit and receive the motor driver signals through a 12-bits encoder counter circuit and a 12-bits D/A converter circuit.The encoder counter circuit is comprised of an encoder signal delay IC,motor rota-tional direction gauge IC,and three 4-bits up–down counter ICs for calculating the rotor position of BLDC motor.The 12-bits D/A converter IC with dual channel voltage output is used to control the BLDC motor.Additionally,every IC that connects with the FPGA chip uses asynchronous bus transceiver IC to protect the current re?ow to FPGA chip.The experimental setup is shown in Fig.3.To illustrate the effectiveness of the proposed design method,a comparison between a fuzzy controller,an adaptive fuzzy controller,and the proposed RCMAC-based adaptive controller is made.

4.1Comparison of different control methods

To compare the tracking ef?ciency,?rst a fuzzy controller is applied to control the BLDC motor.The fuzzy control rules are given in the following form:

Rule i :IF e is F i e and _e is F i

_e ;

THEN u is q i

e47T

where q i ,i =1,2,…,n are the singleton control actions,

and F e i and F i

_e are the labels of the fuzzy sets with Gaussian membership function.The fuzzy rules are summarized in Table 1,where the fuzzy labels used in this paper are negative big (NB),negative medium (NM),negative small (NS),zero (ZO),positive small (PS),positive medium (PM)and positive big (PB).The fuzzy rules in Table 1are

constructed in such a way that e and _e

will approach to zero with fast rise time and without large overshoot.The de-fuzzi?cation of the controller output is accomplished by the method of sum of weightings.The experimental results of the fuzzy controller are shown in Fig.4.The tracking response is depicted in Fig.4a,the associated control effort is depicted in Fig.4b,and the tracking error is depicted in Fig.4c,respectively.From the experimental results,the fuzzy controller can achieve tracking performance for a start;however,the degenerate tracking response occurs as the frequency of the input command is increased at the 5.5th second.Next,an adaptive fuzzy controller proposed in [3]is applied to control the BLDC motor again.The experimental results of the adaptive fuzzy controller are shown in Fig.5.The tracking response is depicted in Fig.5a,the associated control effort is depicted in Fig.5b,and the tracking error is depicted in Fig.5c.Though the tracking performance can be achieved after learning,there exists the undesirable control chattering in Fig.5b.4.2RCMAC-based adaptive control

It should be emphasized that the development of RCMAC-based adaptive control system does not need to know the system dynamics of the controlled system.For practical implementation,the parameters of the RCMAC-based adaptive control system can be on-line tuned by the pro-posed adaptive laws without the need of system parameters.Finally,the proposed RCMAC-based adaptive control system is applied to control the BLDC motor to illustrate its effectiveness.For the control system,choosing Q =I ,k 1=2,k 2=1and solving A m T P ?PA m =-Q ,it can be

obtained

Fig.3The experimental setup

Table 1Fuzzy control rules base _

e e NB

NS ZO PS PB NB -2.0-2.0-2.0-1.00.0NS -2.0-2.0-1.00.0 1.0ZO -2.0-1.00.0 1.0 2.0PS -1.00.0 1.0 2.0 2.0PB

0.0

1.0

2.0

2.0

2.0

P?1:76250:7812

0:78120:8088

!

:e48T

The parameters of adaptive laws are chosen as g1?0:02;g2?g3?g4?0:0002;g5?0:00002;d?0:1and l1?l2?l3?l4?l5?0:01:The initial estimation vectors of^w0;^r0;and^E0are all set to0,^m0is set as equal distribution in the range betweenà1:92;1:92

? and ^r0is set as0.58.All the gains are chosen to achieve satisfactory transient control performance in considering the requirement of stability and possible operating conditions.The experimental results of the RCMAC-based adaptive control system are depicted in Fig.6.The tracking response is depicted in Fig.6a,the associated control effort is depicted in Fig.6b,and tracking error is depicted in Fig.6c.It can be seen that there is no chattering phenomena in the control effort and perfect tracking response can be obtained after initial transient response. From the comparison of experimental results,it is shown that the RCMAC-based adaptive control system

can

achieve better tracking performance than other control methods.Additionally,the performance comparison between a fuzzy controller,an adaptive fuzzy controller, and the proposed controller is made as shown in Table2, which shows that the proposed RCMAC-based adaptive control pay the price of a little larger computational load for achieving better control performance.However, this computational load is acceptable for practical application.5Conclusions

In this paper,an RCMAC-based adaptive control system with a novel training algorithm for a brushless DC motor has been successfully developed and implemented based on the FPGA approach.The implementation of the control system using the FPGA can achieve the charac-teristics of small size,fast execution speed and less memory.All the parameters of the proposed

RCMAC-system

based adaptive control system can be tuned based on the Lyapunov stability theorem;thus,the uniformly ultimately bound stability of the control system is guaranteed.The effectiveness of the proposed RCMAC-based adaptive control has been veri?ed by experimental

results.

control system

Table 2Controller

characteristic comparison

Controller Controller parameters Stability proof Robustness Chattering phenomena Fuzzy control Trial and error No Middle No Adaptive fuzzy control On-line learning Yes Good Yes RCMAC-based adaptive control

On-line learning

Yes

Excellent

No

Acknowledgments The authors appreciate partial support from the National Science Council of Republic of China under grant NSC 95-2622-E-155-004-CC3.

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