ning

to anel. Themancedesignthe twoutational

© 2004

ISATRANSACTIONS®

ISA Transactions 43~2004! 611–621

Comparison of two approaches to automated PI controller tufor an industrial weigh belt feeder

Yanan Zhao,* Emmanuel G. Collins, Jr.†

Department of Mechanical Engineering, Florida A&M University–Florida State University, Tallahassee, Florida 32310, USA

~Received 17 September 2003; accepted 27 March 2004!

Abstract

In this paper, two advanced PI controller tuning methods, unfalsified control and fuzzy control, are appliedindustrial weigh belt feeder that has significant nonlinearities. Both methods do not require an explicit plant modadvantage of the unfalsified PI control design method is that it is able to directly incorporate multiple perforcriteria, while the advantage of fuzzy logic is that it is able to directly incorporate human reasoning in theprocess. Experimental results exhibit the effectiveness of both control methods. A detailed comparison ofapproaches is given in the areas of allowed design specifications, process knowledge requirements, comprequirements, controller development effort, transient performance, and the ability to handle motor saturation.ISA—The Instrumentation, Systems, and Automation Society.

Keywords: Fuzzy logic control; Unfalsified control; PI control; Weigh belt feeder

-in-rgyeonsoreceis

lyors inod.

ofter-

yssary

nshat

ch-lt

r-ad,Toon,inI

gi-rePI

1. Introduction

Automated controller tuning is of significant interest to control engineers since it can lead tocreased performance while saving time and ene@1,2#. The original work on tuning focused on thdevelopment of procedures and recommendatito be used manually by operators. However, msophisticated methods of autotuning have sinbeen proposed. For example, automatic tuningnow used extensively in PID control@1,3,4#.

PID controllers have been the most commonused controllers in industrial control practice fthe past 60 years, even though great progrescontrol theory has been made over this periThis is because of their simple structure, easedesign and implementation, and transparent in

*E-mail address: yzhao@eng.fsu.edu†E-mail address: ecollins@eng.fsu.edu

0019-0578/2004/$ - see front matter © 2004 ISA—The Instru

pretation. A poorly tuned control system mawaste energy and cause excessive and unnecewear of actuators@2#. High performance is alwaysthe design target in industrial control applicatioand recently many modern control techniques timprove PID tuning have been reported@1,5,6#. Inthis research, automated PI controller tuning teniques are studied for an industrial weigh befeeder.

The weigh belt feeder used in this research~seeFig. 1! was designed and manufactured by Merick Industries, Inc. of Lynn Haven, Florida. It isprocess feeder that is typically used in a foochemical, or plastics manufacturing process.ensure a constant feed rate in industrial operatia PI control law is designed and implementedthe Merrick controller. In current practice the Ptuning process is performed manually by an enneering technician. However, for better and moconsistent quality, it is desired to use automatedtuning @7#.

mentation, Systems, and Automation Society.

i-n-

i-onn-

sly.r-r,as

orgeat-

oisves

thethes oyerser,serur-

-l-

hisn-ntign

tlyedalsto

ern-ibleses-dn

ersof

theeds

-e-ut

eIthasur-the

er-alnd-

in

thee

re

Ithes.onts,p-ityri-ost

n.e-

tic

612 Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

The dynamics of the weigh belt feeder are domnated by the motor. To protect the motor, the cotrol signal is restricted to lie in the interval@0,10#V. The motor also has significant friction. In addtion, the sensors exhibit significant quantizatinoise. Hence the weigh belt feeder exhibits nolinear behavior @7#. The system nonlinearitiemake standard tuning methods difficult to appFor example, in attempting to apply the ZiegleNichols tuning method to the weigh belt feedethe system saturated before the ultimate gain wobtained. Relay feedback autotuning@1,8#, anotherwidely used method, does not perform well fhighly nonlinear systems or systems with lardisturbances. When relay feedback tuning wastempted, because of the sensor quantization nand the motor friction, the desired square wawith symmetric positive and negative half cyclecould not be achieved, even after consideringrelay hysteresis and compensating for part ofload disturbance. There are many other methodPID controller tuning in the literature, usuallbased on knowledge of the process parametDue to the nonlinearities of the weigh belt feedits process parameters change with time andpoint. For example, friction is highly nonlineaand depends on multiple parameters that vary ding the process@9#. In an effort to design PI controllers for this type of process, two non-modebased PI tuning methods were studied in tresearch: unfalsified PI control and fuzzy PI cotrol. Both methods do not require an explicit plamodel, hence they are suitable for control desfor nonlinear plants that are difficult to model.

The unfalsified design concept@10,11# pointsout that the control law can be obtained direcfrom a set of candidate controllers by using storsensor output signals and actuator input signConsequently, it is not necessary for a controller

Fig. 1. The Merrick weigh belt feeder.

e

f

.

t

.

actually be inserted in the feedback loop in ordto be falsified. Thus the adaptive unfalsified cotrol processes may be significantly less susceptto poor transient response than other procesthat require inserting controllers in the loop. Another particularly attractive feature of unfalsifiecontrol is the ability to find control laws that cameet multiple objectives. For the weigh beltfeeder, the objectives are to design PI controllto minimize the transient and steady-state errorsthe feed rate and to avoid actuator saturation atsame time. To reduce the computational time usfor the unfalsification, a genetic algorithm waadopted to perform the unfalsification.

Fuzzy logic control provides a formal methodology for representing, manipulating, and implmenting a human’s heuristic knowledge abohow to control a system@12#. It has been foundparticularly useful for controller design when thplant model is unknown or difficult to develop.does not need an exact process model andbeen shown to be robust with respect to distbances, large uncertainty, and variations inprocess behavior@13#. Two types of fuzzy logiccontrollers~FLC’s!, PI-like FLC’s ~including gainscheduled and self-tuning PI-like FLC’s! and PIFLC’s, were designed for the weigh belt feed@14#. PI-like FLC’s do not have explicit proportional and integral gains; instead the control signis directly deduced from the knowledge base athe fuzzy inference. In contrast, PI FLC’s are composed of the conventional PI control systemconjunction with a set of fuzzy rules~knowledgebase! and a fuzzy reasoning mechanism to tunePI gains online. Since the PI FLC’s outperform thPI-like FLC’s for the weigh belt feeder@14# andthey have explicit P and I gains, only PI FLC’s acompared with unfalsified PI controllers.

Both unfalsified PI control and fuzzy logic Pcontrol are effective solutions to systems witime-varying parameters and other uncertaintiIn this paper, a detailed comparison of the twmethods from the point of view of allowed desigspecifications, process knowledge requiremencomputational requirements, controller develoment effort, transient performance and the abilto handle motor saturation is given. This compason can help a control engineer choose the msuitable type of controller for a given applicatio

The paper is organized as follows. Section 2 dscribes unfalsified PI control design with gene

ltrolde-h-ns

s aest

e

2.

f

hepernd

p

at

ist

e

noalstheg-

ol-

-he

s-d

n

-etm-oe

h

o

613Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

algorithm implementation for the weigh befeeder. Section 3 describes fuzzy logic PI contfor the weigh belt feeder. Section 4 presents atailed comparison of the two control design metods. Finally, Section 5 presents some conclusio

2. Unfalsified PI control

The unfalsified control concept is used here ameans of using either open- or closed-loop tdata to identify a subset of controllers~from aninitial set! that is not proved to violate the multiplobjectives specified by the control engineer@11#.Consider the feedback control system of Fig.Given K5$K1 ,K2 ,...,KN%, a finite, initial set ofcausally left-invertible control laws, the goal ounfalsified control is to determine aKiPK ( i51,2,...,N) such that for a set of plantsP, repre-senting the variations of the real system, tclosed-loop system response satisfies a set offormance specifications involving the commasignal r (t), the command inputu(t), and themeasured outputy(t).

Let ( y,u) denote a set of open- or closed-lootest data, taken with sample periodh in the timeinterval @ t0 ,t01ph# wherep is some positive in-teger, andr Ki

is the set of command signals thwould have yielded the signalsu andy if the con-troller Ki were in the loop. Assume that there exscalar cost functionsJj :R

(p11)3R(p11)3R(p11)

→R, j 51,...,q, and associated performancspecifications@7#

Jj~ r Ki,y,u!<s j , j 51,...,q. ~1!

Then, for each control lawKi ( i 51,2,...,N) onecan compute the set of costs$Jj( r Ki

,y,u)% j 51q . A

controller is falsified if there existsj P$1,2,...,q%such that

Jj ~ r Ki,y,u!.s j . ~2!

Fig. 2. Closed-loop system with unfalsified controller.

.

-

2.1. Performance specifications for the weighbelt feeder

The cost functions used in the unfalsificatioprocess are based on the actual engineering gand the observed closed-loop performance offeeder. As previously mentioned, the control sinal u(t) is restricted to lie in the interval@0, 10# V.Also, in practice,r (t)<5 V ~corresponding to abelt speed of2.5431022 m/sec). It was experi-mentally observed that saturation occurs if the flowing constraint is violated:

J1~ r Ki,u!5r max

tPTuu~ t !u/min

tPTur ~ t !u<2, ~3!

where T5(t0 ,t01h,...,t01ph) and r takes ondifferent values for different set points. In particular, r increases as the set point increases. Tnumber 2 in Eq.~3! is the ratio of the absolutevalue of the saturation control signal~10 V! andthe maximum reference signal~5 V!. Since actua-tor saturation is a hard constraint~i.e., it must besatisfied for proper operation of the controlled sytem!, the constraint~3! is also treated as a harconstraint.

To achieve a good step response~i.e., low over-shoot and fast settling time! another cost functionis needed for unfalsification. The cost functioconstructed using experimental observations is

J2~ r Ki,y!5

i r Ki2 yi2

i r Kii2

1r1Kp1r2Ki , ~4!

where Kp and Ki are, respectively, the proportional and integral gains of the PI controller. Thtermsr1Kp andr2Ki are used to take into accounthe transient overshoot of the system. The paraetersr1 andr2 are selected to make the latter twterms in Eq.~4! comparable in magnitude to thfirst term.

In the unfalsified tuning process for the weigbelt feeder a controller is falsifed if Eq.~3! is vio-lated. AmongKu , the current set of unfalsifiedcontrollers, the ‘‘best’’ controller is considered tbe the controller that satisfies

minKPKu

J2~ r K ,y!. ~5!

614 Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Fig. 3. Step responses for set point51 ~left! and 5~right! with unfalsified PI control.

m

usen-theoled

tion

al-in

etu-llrsAsle

ersnd

o-en

ac-rolheef.e-nal

ys-

n-I

he-as

a-s:sl-es.alarra-s

or-alndzylo-Inm-in

2.2. Experimental results with a genetic algorithimplementation

A genetic algorithm~GA! @15,16#, which maybe viewed as a random search process, can beto avoid computing the costs for each of the ufalsified controllers and hence can increasecomputational efficiency of the unfalsified contrprocess. The unfalsified control problem describabove can be viewed as a constrained optimizaproblem, i.e., solve the optimization problem~5!subject to the constraint~3!. A GA was adapted tosuch a constrained optimization problem for unfsified PI control implementation as describedRef. @7#.

Due to the nonlinearities in the dynamics of thfeeder, specifically the friction and actuator saration, one fixed controller does not perform wefor each set point. Hence different PI controlleneed to be designed for different set points.mentioned above, 5 V is the maximum possibvalue of the reference command and PI controllwere designed for set points equal to 1, 2, 3, 4, a5 V, respectively. The initial set of candidate prportional and integral gains were both choswithin the range of@0.1 3.2# with the grid corre-sponding to 0.1.

A closed-loop data set, generated from thetual hardware, was used for the unfalsified contdesign process for each of the five set points. Tdetailed experimental setup is described in R@17#. Fig. 3 shows the corresponding step rsponses of the plant output and the control sig

d

at set points of 1 and 5 V~the curves are similarunder the other set points!. Clearly the control sig-nals are within the range of saturation and the stem outputs have desirable performance~i.e., noovershoot and fast rise times!. The results in thisresearch clearly demonstrated the ability of the ufalsification procedure to yield high performing Pcontrollers. Also, the results demonstrated tability of the GA to reduce the computational requirements of an exhaustive search, especiallythe size of the initial candidate set increased.

3. Fuzzy PI control

Fuzzy PI control can be classified into two mjor categories according to their constructionfuzzy PI-like controllers and fuzzy PI controller@14#. In the first category, a fuzzy PI-like controler is constructed as a set of heuristic control rulThat is, the control signal or the incrementchange of the control signal is built as a nonlinefunction of the error, change of error and acceletion error, where the nonlinear function includefuzzy reasoning. Thus there are no explicit proptional and integral gains; instead the control signis directly deduced from the knowledge base athe fuzzy inference. They are referred to as fuzPI-like controllers because their structure is anagous to that of the conventional PI controller.the second category, a fuzzy PI controller is coposed of the conventional PI control systemconjunction with a set of fuzzy rules~knowledge

615Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Fig. 4. Closed-loop system with fuzzy PI controller.

thea-

der

al

ol-n-nd

t,at

.in-in-

e-tpec

the

-t,

ofzy

dts

-ed

onses.c-tolerrgetheidld

ral

base! and a fuzzy reasoning mechanism to tuneproportional and integral gains online. In this pper, only the second category of controllers~i.e.,conventional PI controllers with fuzzy tuned P anI gains! is introduced as they outperformed PI-likfuzzy logic controllers for the weigh belt feede@14# and have explicit proportional and integrgains.

3.1. System diagram

Fig. 4 shows the system diagram of the contrler. For fuzzy PI control, the control signal is geerated according to the online tuning of the P aI gains based on the transfer function,

H~z!5Kp1KiTs

z

z215KpS 11

Ts

Ti

z

z21D ,

~6!

whereKp is the proportional gain,Ki is the inte-gral gain,Ti5Kp /Ki is the integral time constanand Ts is the sampling period. It is assumed thKp is in the prescribed range@Kp,min,Kp,max#; theappropriate range is determined experimentally

There are two fuzzy logic reasoning systemscluded in the diagram. The first one has twoputs, the errore(k) and change of errorDe(k),which are defined bye(k)5r (k)2y(k) andDe(k)5e(k)2e(k21), where r and y denotethe applied set point input and plant output, rspectively. Indicesk andk21 indicate the presenstate and the previous state of the system, restively. The output is the proportional gainKp . Thesecond fuzzy system has the same inputs, butoutput is the integral time constantTi . The inte-gral gainKi was then obtained byKp /Ti .

3.2. Membership functions and rule bases

To make implementation of a fuzzy logic controller possible with limited processor throughpu

-

this research focused on reducing the numberfuzzy sets and thereafter the number of fuzrules. Here, the membership functions~MF’s! fore andDe are defined on the common normalizedomain@21,1#, where each has three fuzzy seN ~negative!, ZE ~zero!, and P~positive! as shownin Fig. 5. The two MF’s ofKp , corresponding tothe fuzzy sets S~small! and B~big!, are shown inFig. 6. The MF’s of Ti , corresponding to thesingleton fuzzy sets S~small!, M ~medium!, and B~big!, are also shown in Fig. 6. Below, it is explained how the singleton MF’s can be adjustfor different set points.

The fuzzy rules in Tables 1 and 2 are basedthe desired characteristics of the step responFor example, at the beginning of the control ation, a big control signal is needed in orderachieve a fast rise time. Thus the PI controlshould have a large proportional gain and a laintegral gain. When the step response reachesset point, a small control signal is needed to avoa large overshoot. Thus the PI controller shouhave a small proportional gain and a small integgain.

Fig. 5. Membership functions fore andDe.

616 Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Fig. 6. Membership functions forKp ~left! andTi ~right!.

larn

n-r-r-e

rstheer-thelts

orn-atce

in.

.of

al

t,

hethe

intV

for

of

taheantheo

der-

3.3. Tuning algorithm for the fuzzy PI controller

The scaling factors which describe the particuinput normalization and output denormalizatioplay a role similar to that of the gains of a convetional controller. Hence they are of utmost impotance with respect to controller stability and peformance @18#. The relationship between thscaling factorsGe andGDe and the input variablesof the FLC areeN5Gee and DeN5GDeDe. Se-lection of suitable values of these scaling factoare made based on expert knowledge aboutprocess to be controlled and through trial andror. Adjustment rules have been developed forscaling factors by evaluating control resu@19,20#. Here, the scaling factorsGe50.1 andGDe51 were chosen.

There are two other tuning algorithms used fthe fuzzy PI controller design. Due to the nonliearity of the feeder, to avoid high overshoothigher set points, it is necessary to suitably reduboth the proportional gain and the integral gaHence we chose a gain scheduling coefficientr351/(110.23sp), wheresp stands for set pointThis coefficient was used for the online tuningboth the range ofKp and the MF’s ofTi . For

Table 1Fuzzy rules for computation ofKp .

e(k)\De(k) N ZE P

N B B BZE S B SP B B B

different set points the range of the proportiongain was chosen as@0,Kp,max#, whereKp,max5r33Kp,max 0, andKp,max 053.2was chosen accordingto experimental experience.

The singleton membership function ofTi

was adjusted online along with the set poinm f5m f0 /r3 . Thus these MF’s shift right asthe set point increases, while they shift left as tset point decreases as shown in Fig. 6, wheresolid lines represent the MF’s at the 5 V set poand the dotted lines represent the MF’s at the 1set point. Here, the singleton values chosensp50 is S50.75, M51, and B51.5. Sugeno-type inference was used for the fuzzy reasoningTi .

3.4. Experimental results

The maximum possible sampling rate of the daacquisition hardware is 0.001 sec; however, tprocessor overload point may be much higher th0.001 sec depending on the size of the model,type of simulation integration algorithms and son. Thus the sampling time is chosen asTs

50.01 sec in the experimentation. It is observethat this sampling rate can avoid processor ov

Table 2Fuzzy rules for computation ofTi .

e(k)\De(k) N ZE P

N S S SZE B M BP S S S

617Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Fig. 7. Step responses of the fuzzy PI controller at set point51 ~left! and 5~right!.

be

heofr-

ts.

-r,

ndde-

-PI

fieslsi-tht at

-he

load while allowing acceptable performance toachieved.

Fig. 7 shows the experimental results for tfuzzy PI controllers implemented for set points1 and 5 V. It is clearly that the desired perfomance~fast rise time, no overshoot! is achieved.The performance is similar at the other set poin

4. Comparison of unfalsified PI control andfuzzy PI control

The unfalsified control and fuzzy control provide two different approaches for PI controlletuning for the weigh belt feeder. In the following

the two approaches are compared in detail apossible future research for each method isscribed.

4.1. Proportional and integral gains

For unfalsified PI control, proportional and integral gains are obtained by selecting the bestcontroller among the PI candidate set that satisthe performance specifications. Since the unfafied controllers were implemented off-line, bothe proportional and integral gains are constana fixed set point.

In contrast, fuzzy PI controllers obtain their proportional and integral gains online based on t

Fig. 8. Proportional and integral gains of the fuzzy PI controller at set point51 ~left! and 5~right!.

ing8ra

of

ds,the

dinsetho

ursetPwhere

rbi-ofional

dapeor-chheaseor

t’’

pe-d.

dl-

esPI-PI-

ntitheyc-r-tallforri-or-

ethe

heisy

u-r-to

es.hegicn

-ostr-

zy

618 Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

error and change of error signal at each samplperiod, thus both gains are time varying. Fig.shows the changes of the proportional and integgains of the fuzzy PI controllers as a functiontime at set points of 1 and 5 V~the curves aresimilar at the other set points!. It is seen that bothgains converge very fast in the first few seconand are subsequently only finely tuned aroundmean steady-state values.

Table 3 lists the P and I gains of the unfalsifiecontrollers and the mean value of the P and I gaof the fuzzy controllers from 8 to 40 sec at the fivset points. It is seen that for fuzzy controllers boP and I gains decrease along with the increasethe set points. This trend matches well with oexperimental observation that the higher thepoint, the lower the control effort. The values ofand I gains of unfalsified controller do not shosuch an obvious trend. This is partially due to tfact that both the candidate P and I gains awithin the set of@0.1, 0.2, 0.3,..., 3.2# and hencethe unfalsified control process cannot choose atrary values for P and I. Also, the constructionthe performance specifications and the terminatcriterion of the implementation influence the finvalues of the gains.

4.2. Classification as adaptive control

Both approaches can be categorized as an ative control method. Fuzzy logic control can bclassified as adaptive control, because its proptional and integral gains are tuned online at easample period to improve the performance of tsystem. Although, unfalsified control design wimplemented off-line in this research, it can bimplemented online as an adaptive method. Fthis online implementation, a switch to the ‘‘bes

Table 3Comparison of PI gains of unfalsified control and fuzcontrol.

Set point~V!

Unfalsified PI Fuzzy PI

P I P I

1 1.5 1.2 1.5698 1.30852 2.6 0.7 1.3742 0.98173 1.8 0.6 1.2262 0.77684 1.4 0.5 1.1097 0.61715 1.0 0.5 1.0153 0.5080

l

f

-

unfalsified controller can occur at each sampleriod or at an integer multiple of the sample perio

4.3. Allowed design specifications

Multiple criteria can be proposed for unfalsifiePI controller design. Then the optimal PI controler is selected to meet the multiple objectivspecified by the designer. In contrast, fuzzycontrol cannot explicitly incorporate multiple criterion. The rule bases constructed for fuzzycontrol must implicitly represent the multiple control objectives.

4.4. Process knowledge

Both methods do not require an explicit plamodel. However, experimental experience wthe plant is required in the design process. A kto the unfalsified control design was the constrution of the cost functions reflecting the perfomance specifications, which required experimenexperience with the plant. In fuzzy logic controexpert experience, i.e., the ‘‘rules of thumb’’ ohow to achieve a good control, is the basis fconstructing the rule bases. Experimental expeence is also needed to select the range of proptional gain.

4.5. Computational requirements

For unfalsified control, the computational timis dependent on the size of the candidate set,computational cost of the cost function and tefficiency of the search method. Currently, itmore computationally intensive than the fuzzcontrol.

Fuzzy logic control requires less online comptational effort. At each sample period the propotional and integral gains are updated accordingthe reasoning of the proposed fuzzy rule basSince much effort was devoted to reducing tsize of the rule bases in this research, fuzzy locontrol was more computationally efficient thaunfalsified control.

4.6. Controller development effort

In unfalsified control design significant development efforts are needed in the construction of cfunctions that correctly reflect the underlying pe

619Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Fig. 9. Performance comparison of unfalsified PI control and fuzzy PI control at set point51 ~left! and 5~right!.

aren-

p-thnt

erisa

ro-of

n

ts.bye-re-fiedof

ll.onper-

thectn

troloronol-

sesn-hPI

nle.celere.

er-roln.odngad-

edAofe

e-ngllyisutToothp-es

formance specifications. These cost functionskey to guaranteeing good performance of the ufalsified controller.

Fuzzy controller design required less develoment effort. First, fuzzy reasoning rules for bothe proportional gain and integral time constawere constructed based on the desired characttics of the step responses. Meanwhile, onlysimple tuning mechanism for the range of the pportional gain and the membership functionsthe integral time constant was built.

4.7. Transient performance and motor saturatio

Unfalsified control design benefited from iability to explicitly handle multiple objectivesHence it was possible to avoid motor saturationfalsifying the candidate controllers that were prdicted to cause saturation. Also, good transientsponse was achieved since the best unfalsicontroller was chosen based on the optimizationa carefully chosen cost function.

The fuzzy logic controllers also performed weThe control signal was generated online basedthe error and change of error at each sampleriod. The fuzzy rules yielded good transient peformance. As the online reasoning based onerror and change of error has the ability to correthe control effort promptly, the fuzzy system caadjust the control signal around the desired coneffort, which can prevent the system from motsaturation. In our experiments, motor saturatinever occurred in the process of fuzzy PI contrler design.

-

-

Fig. 9 shows the comparison of step responof an unfalsified PI controller and a fuzzy PI cotroller under set points of 1 and 5 V. Even thougthe step response curves are very similar, fuzzycontrol has a faster rise time.

The best feature of unfalsified PI control desigis that it can explicitly incorporate multiple controobjectives including motor saturation avoidancHowever, fuzzy PI control has better performanin terms of computational requirements, controldevelopment effort and transient performancBoth methods have the potential for improved pformance. In this research the unfalsified contconcept was used for off-line controller desigHowever, as previously mentioned, the methhas the potential to be implemented on-line usia computationally efficient GA. For example,GA with a real-valued representation can be stuied as a means of implementing the unfalsificontrol online, since it has been reported that a Gusing real-valued representation is an ordermagnitude more efficient in terms of CPU timthan a GA based on a binary representation@16#.

In this research, fuzzy PI controllers were dsigned with the membership functions, scalifactors, and fuzzy reasoning rules tuned manuaby trial and error. The controller developed in thmanner can yield satisfactory performance bmay not yield the best achievable performance.save the time and cost of the tuning process, bgenetic fuzzy or neural fuzzy systems can be aplied for the tuning of parameters and rule basof fuzzy logic controllers@21–24#.

ern,firsheheslyheateh

d to

a-S-

l,

ers.

r:ci-

gsn

g

l-

icr ao-

u-C,

elheE

nh-

rol

l.

l-r-

e-s.

o-n,

tic

e/

-te

nr-

er-r.ce9–

y

-

czy

n-

o-nt

l

-

.

620 Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

5. Conclusions

Two approaches for tuning PI controllers undhigh nonlinearity due to motor saturation, frictioand sensor quantization were described. Themethod uses the unfalsified concept in which tcontrol objectives can be clearly specified. Tsecond method uses fuzzy logic to continuoutune the proportional and integral gains. Tpower of these approaches has been demonstrthrough their applications to an industrial weigbelt feeder. The two approaches were compareshow their strengths and limitations.

Acknowledgment

This research was supported in part by the Ntional Science Foundation under Grant CM9802197.

References

@1# Astrom, K. J. and Wittenmark, B., Adaptive Contro2nd ed. Addison-Wesley, Reading, MA, 1995.

@2# Bi, Q., Cai, W., and Wang, Q., Advanced controllauto-tuning and its application in HVAC systemControl Eng. Pract.8, 633–644~2000!.

@3# Astrom, K. J. and Hagglund, T., PID ControlleTheory, Design and Tuning, 2nd ed. Instrument Soety of America, Research Triangle Park, NC, 1995.

@4# Astrom, K. J., Tuning and adaptation. In Proceedinof the 13th World Congress of IFAC, Vol. K, SaFrancisco, CA, 1996, pp. 1–18.

@5# He, J., Wang, Q., and Lee, T., PI/PID controller tuninvia LQR approach. Chem. Eng. Sci.55 ~13!, 2429–2439 ~2000!.

@6# Wang, Y. and Shao, H., Optimal tuning for PI controler. Automatica36 ~1!, 147–152~2000!.

@7# Collins, E. G., Jr., Zhao, Y., and Millett, R., A genetsearch approach to unfalsified PI control design foweigh belt feeder. Int. J. Adapt. Control Signal Prcess.15 ~5!, 519–534~2001!.

@8# Hang, C. C. and Lee, T. H., Adaptive Control. Instrment Society of America, Research Triangle Park, N1993.

@9# Teeter, J. T., Chow, M., and Brickley, J. J., Novfuzzy friction compensation approach to improve tperformance of a DC motor control system. IEETrans. Ind. Electron.43 ~1!, 113–120~1996!.

@10# Brozenec, T. F., Controller Validation, Identificatioand Learning. Ph.D. Dissertation, University of Soutern California, Los Angeles, CA, 1996.

@11# Safonov, M. G. and Tsao, T. C., The unfalsified contconcept and learning. IEEE Trans. Autom. Control42~6!, 843–847~1997!.

t

d

@12# Passino, K. M. and Yurkovich, S., Fuzzy ControAddison-Wesley, Menlo Park, CA, 1998.

@13# Li, W. and Chang, X., Application of hybrid fuzzylogic proportional plus conventional integraderivative controller to combustion control of stokefired boilers. Fuzzy Sets Syst.111, 267–284~2000!.

@14# Zhao, Y. and Collins, E. G., Jr., Fuzzy PI control dsign for an industrial weigh belt feeder. IEEE TranFuzzy Syst.11 ~3!, 311–319~2003!.

@15# Chambers, L., Practical Handbook of Genetic Algrithms: Applications, Vol. 1. CRC Press, Boca RatoFL, 1995.

@16# Houck, C. R., Joines, J. A., and Kay, M. G., A genealgorithm for the function optimization: A Matlabimplementation, see www.ie.ncsu.edu/miragGAToolBox/gaot/

@17# Zhao, Y., Automated Controller Tuning for an Industrial Weigh Belt Feeder. Ph.D. Thesis, Florida StaUniversity, 2001.

@18# Driankov, D., Hellendoorn, H., and Reinfrank, M., AIntroduction to Fuzzy Control. Springer-Verlag, Belin, 1993.

@19# Daugherity, W. C., Rathakrishnan, B., and Yen, J., Pformance evaluation of a self-tuning fuzzy controlleIn Proceedings of the IEEE International Conferenon Fuzzy Systems, San Diego, CA, 1992, pp. 38397.

@20# Maedo, M. and Murakami, S., A self-tuning fuzzcontroller. Fuzzy Sets Syst.51, 29–40~1992!.

@21# Magdalena, L., Adapting the gain of an FLC with genetic algorithms. Int. J. Approx. Reason.17, 327–349~1997!.

@22# Arslan, A. and Kaya, M., Determination of fuzzy logimembership functions using genetic algorithms. FuzSets Syst.118, 297–306~2001!.

@23# Jang, J. R., ANFIS: Adaptive-network-based fuzzy iference system. IEEE Trans. Syst. Man Cybern.23~3!, 665–685~1993!.

@24# Nurnberger, A., Nauck, D., and Kruse, R., Neurfuzzy control based on the NEFCON-model: Recedevelopment. Soft Comput.2 ~4!, 168–182~1999!.

Yanan Zhao received thePh.D. degree in MechanicaEngineering from the FloridaState University in 2001. Sheis currently a Research Associate in the Department of Me-chanical Engineering at theFlorida A&M University—Florida State University Col-lege of Engineering. Her re-search interests includeautomated controller tuning,intelligent control systems forautonomous vehicles, fluidic

thrust vector control, and evolutionary methods for control design

-

-

ms,

621Y. Zhao, E. G. Collins, Jr. / ISA Transactions 43 (2004) 611–621

Emmanuel G. Collins re-ceived the Ph.D. degree inAeronautics and Astronauticsfrom Purdue University in1987. He worked for sevenyears in the Controls Technology Group at Harris Corpora-tion, Melbourne, FL beforejoining the Department of Me-chanical Engineering at theFlorida A&M University—Florida State University Col-lege of Engineering, Tallahassee, FL, where he currently

serves as Professor and Director of the Center for Intelligent Syste

Control, and Robotics~CISCOR!. Dr. Collins is the author of over 140technical publications in controls and robotics.