01 Fuzzy Model Reference Learning Control for Cargo Ship Steering

8/2/2019 01 Fuzzy Model Reference Learning Control for Cargo Ship Steering

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Fuzzy Mode ControlforJeffery R. Layne and Kevin M. Passino

h i l e f u z z y c o n t r o l h a semerged as an altemative tos o m e c o n v e n t i o n a l c o n t r o lschemes since it has shown successin some application areas (e.g., intrain control and camera auto-fo-cusing) , there are several draw-backs to this approach: a) the designof fuzzy controllers is usually per-formed in an ad hoc manner whereit is hard to justify the choice ofsome controller parameters (e.g.,the membership functions), and b)the fuzzy controller constructed forthe nominal plant may later performinadequately if significant and un-predictable plant parameter vari-a t ions occur . Whi l e t he fuzzymodel reference learning control-ler (FMRL C) introduced in [11,[ 2 ]and other adaptive fuzzy controlapproaches seek to address theseissues, they primarily focus on im-proving existing learning controlapproaches or i n t roducing newones. In this article we provide ac o m p a r a t i v e a n a l y s i s o f t h eFMRLC and conventional modelr e f e r e n c e a d a p t v e c o n r o 1(MRAC) for a cargo ship steeringapplication. Our main ob jective isto make an initial assessment ofwhat advantages (if any) a fuzzylearning control approach has overconventional adaptive control ap-proaches. For the cargo ship steer-

ing application, our simulation re-sul ts show that the FMRLC hasseveral potential advantages overMRAC including a) improved con-vergence rates, b) use of less con-t r o l e n e r g y, c ) e n h a n c e ddisturbance rejection properties,and d) lack of dependence on amathematical m odel . Using ourcomparative analysis we discusshow the well developed concepts inconventional adaptive control canbe used to evaluate fuzzy learningcontrol techniques.

Learning Control SystemsIn recent years, to improve fuelefficiency and reduce we ar on shipcomponent s , au topi lo t sys t emshave been developed and imple-mented for controlling the direc-tional heading of ships. Generally,the autop ilots utilize simple controls ch em es su ch a s PID cont ro l .Often, however, the capability formanual adjustments of the parame-ters of the controller is added tocompensate for disturbances actingupon the ship such as wind andcurrents. Once suitable controllerparameters are found m anually, thecontroller will generally work wellfor small variations in the operatingconditions; for large variations theparameters of the autopilot must becontinually modified. Such contin-ual adjustments are necessary because the dynamics of a shipvary it h speed, and loading. Also, it is useful to changelarge disturbances which result from changes in the wind, waves,

~ ~~J . Layne was with theDepartmentofElcc~tr.ic~ulr7girre~v.irrg.Th eH e is now wi th Wright Laboratories, CVLIAAAS-3. Wright Pur-Oh io StUte University,2015 Neil Avenue. Co//m1b/1.c.O H 43210 . the autopilot co n t r o l la w parameters when the ship is exposed to

current. and water depth. M anual adjustment of the controllerparameters is often a burden on the crew. Moreover, poor adjust-ment may result from human error. As a result, it is of greatinterest to hav e a method for automatically adjusting or modify-ing the underlying controller.In this article, we investigate the use of a learning controlsystem to maintain adequate performance of a cargo ship autopi-lot when there are process disturbances or variations as men-

terson AFB, OH 45433-6543.K . Passirlo i5 M i t h th c D e / w t n i e i l tof Electrical Engineerin


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tioned above. In general, a learning system possesses thecapability to improve its performance over time by interactionwith its environment. A learning control system is designed sothat its learning controller has the ability to improve theperformance of the closed loop system by generating commandinputs to the plant and utilizing feedback information from theplant. The leaming control algorithm con sidered here is based ona direct fuzzy controller. In genera l, a fuzzy controller utilizesa fuzzy system to capture a human experts knowledge abo ut howto control a process for use in a com puter algorithm. Often, thehuman experts knowledge must be known a priori for fuzzycontroller design. However, the learning control algorithm pre-sented here automatically generates the fuzzy controllers knowl-edge base on-line as new information on how to control the shipis gathered. For instance, the FMRLC can automatically synthe-size a fuzzy c ontroller for the cargo ship and later tune it if thereare significant disturbances/process variations.The FM RLC a lgorithm employs a reference model (a modelof how you would like the system to behave) to provide closedloop performance feedback for generating and modifying a fuzzycontrollers knowledge base. Consequently, this algorithm isreferred to as a fuzzy model reference leaming controller(FMRLC). Th e FM RLC algorithm, which was first introducedin [ 11-[3], grew from research performed on the linguistic self-organizing controller (SOC ) presented in [4] by Procyk an dMamdani and ideas in conventional model reference adaptivecontrol (MR AC) [5 ], [6]. Since the basic architecture and func-tionality of the FMRLC is consistent with the prevailing defini-tion of a leaming controller, the term leaming is used ratherthan adaptive (for a more detailed discussion on this issue see[3], [7]). A significant amount of research has been done on theSOC . For instance, the linguistic S OC has been used in roboticsapplications [8], [9], motor and temperature control [ IO ] , bloodpressure control [ll], and in satellite control 1121-[14]. Otherrelevant literature that focuses on adaptation of a direct fuzzycontroller includes the work in [15] where an adaptive fuzzysystem is developed for a continuous casting plant, and theapproach in [I61 where a fuzzy system ada pts itself to drivercharacteristics for an automotive speed control device. The useof fuzzy systems for estimation/identification [ 171-[23] is rele-vant, especially if indirect adaptive [SI, [6] fuzzy control tech-niques such as those in [24]-1261 are used. Sinc e the introductionof the FMR LC in [3] some other relevant new adaptivebeamingtechniques have been developed [27]-[30]. While we describehere the application of the FMRL C to cargo ship steering, i t hasalso been used for a robotics problem, a rocket velocity controlproblem, and a cart-pendulum system 111. [2] and it has beenbeen used to improve the performance of anti-skid brakes inadverse road conditions [3 ], [32].We provide a comparative analysis of FMRLC and severaltypes of MRA C for a ship steering application. In particular, forthe ship steering application described in [SI, [6] we develop aFMRLC and a gradient-based and Lyapunov-based MRAC,provide some simulation studies, and compare their perfom-ance. We find that for our simulations, the FMRLC leams tocontrol the ship faster, does so with less control energy, andaccomodates for a wind disturbance better than the two types ofMRA C considered. Moreover, the development of the FMRLC,although somewhat ad hoc, does not depend on the form of the

mathematical model of the cargo ship as the MRAC approachesdo .Next, we interpret the results of our comparative analysis ofFMRLC and MRAC. First, we emphasize that one must becareful not to over-generalize the results of our simulation-basedanalysis for this application. While the results look somewhatpromising, we have not performed a) a mathematical analysis ofthe stability and convergence properties of the FM RLC, b) ananalysis of persistency of excitation (which relates to how wellthe knowledge base of the fuzzy controller is filled in), c) ananalysis of the robustness properties of the FMRLC, or d) ananalysis of the computational properties of the FMRLC. Suchconventional approaches to the analysis of adap tive systems arerarely used for fuzzy leaming control (one exception lies in [29])but they do provide useful direction s for future work in the studyof fuzzy learning control systems.In the next section we provide a detailed overview of theFMR LC algorithm. Following this, we describe the cargo shipsteering problem, design an appropriate FMRLC, and developseveral conventional MRAC designs. Next, we provide the re-sults from several simulations to co mpare the performance of theFMR LC to the MRAC designs. Then we discuss the advantagesand disadvantages of FM RLC from a control-theoretic perspec-tive in order to lay a foundation for future work in modeling andanalysis of fuzzy leaming con trol systems. Finally, some con-cluding remarks are given.

Fuzzy Model Reference Learning ControlThe FMRLC, which is shown in Fig. 1, utilizes a learningmechanism that a) observes data from a fuzzy control system(Le., p,(kT) an d y (k T) ) ,b) characterizes its current performance,and c) automatically synthesizes and/or adjusts the fuzzy con-troller so that some prespecified performance objectives are met.Thes e performance objectives are characterized via the referencemodel shown in Fig. 1.In an analogous manner to conv entionalMRA C where conventional controllers are adjusted, the leamingmechanism seeks to adjust the fuzzy controller so that the closed-loop system (the map fromy,r(kT) o y ( k T ) )acts like a prespecifiedreference model (the map from y , ( k T ) to &( k T ) ) . Next wedescribe each component of the FMRL C in more detail.I Learning Mechanismi eference I v.(kTl I ,I I 1 i

Fig. 1 . Airhitecture fo r the FMRLC.

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Fuzzy ControllerThe process in Fig. 1is assumed to have r inputs denoted bythe r - dimensional vector g ( k T ) = [gl(kT) ... ur(kT)] (T is thesample period) and s outputs denoted by the s - dimensionalvector y (k T) = [ r l (k T) ... ys(kT)]. Most often the inputs to thefuzzy controller are generated via some function of the plantoutput y(kT) and reference input y,{kT). Fig. 1 shows a specialcase of such a map that was found useful in many applications.The inpu ts to the fuzzy controller are the error e(k7) [e i (kT)...

es(kr)]and change in error c(k7) = [c l (kT) ..c,(kT)]defined as

e(kT)- (k T - T )T(kT)=

respectively, where

denotes the desired process output.In fuzzy control theory, the range of values for a givencontroller input or output is often called the universe of dis-course [33]. Often, for greater flexibility in fuzzy controllerimplementation, the universes of discourse for each process inputare normalized to the interval [ - I , + I ] by means of constantscaling factors. For our fuzzy controller design, the gains g,, g(,an d pu were employed to normalize the universe of discourse forthe errore(kT), change in errorc(kT), and controller output u(kT),respectively (e.g., g , = [g,,, ..., geJf o that g, ,el(kT) is an inputto the fuzzy controller). The gains ge are chosen so that the rangeof values of g,?e,(kT) lie on [-1, 11 and g, is chosen by using theallowed range of inputs to the plant in a similar way. The gainsare determined by experimenting with various inputs to thesystem to determine the normal range of values that c(k7) willtake on; then gc s chosen so that this range of values is scaled to1-1, 11.We utilize r multiple-input single-ouput (MISO) fuzzy con-trollers, one for each process input un (equivalent to using oneMIMO controller). The knowledge base for the fuzzy controllerassociated with the nth process input is generated from IF-THE Ncontrol rules of the form:

where ZU an d C, enote the l inguistic variables associated withcontroller inputs ea an d ca ,respectively, U,, denotes the linguisticvariable associated with the controller output un , E$ an d ?,hdenote the hth linguistic value associate associated with Za an d C,,respectively, and fi$. k,..m denotes the consequent l inguisticvalue associated with U,. Hence, as an example, one fuzzy controlrule could be

If error is positive-large an d change-in-error is negative-small Then plant-input i s posi t ive-big

(in this case el = error,E? = positive-large,etc.). A set ofsuch rules forms the rule-base which characterizes how tocontrol a dynamical system.The above control rule may be quantified by utilizing fuzzyset theory to obtain a fuzzy implication of the form:If EI and ...and E t and E1 and ...and C yThen ,y{; .. J , ...m

where E t , C$ an d U~.3k*9.,menote the fuzzy sets that quantifyt h e l i ngu i s t i c s t a t ement s F a is Et, C, sc,n d

, respectively. For the example above, wemay use fuzzy sets on the e;( t )normalized universes of discourseas shown in Fig. 2. Assume that we use the same fuzzy sets onthe c;( t ) normalized universes of discourse. The membershipfunctions on the output universe of discourse are assumed to beunknown; they are what the FMRLC will automatically synthe-size. In fact, we will initialize the fuzzy controller knowledgebase with 121 rules (for our examples we utilize the fuzzy setsshown in Fig. 2and use all possible combinations of rules) whereall the right-hand-side membership fu nctions are triangular withbase widths of 0.4 and centers at zero (to model that the fuzzycontroller initially k nows nothing ab out how to control the plant;of course one can often make a reasonable best guess at how tospecify a fuzzy controller that is more knowledgable). Forexample, if s = 1 then all rules in our controller will take on theform

G,, is 6 ..., ,/ ...,m 73

if E{ an d Cl Then Uhwhere the membership functions for E4 an d C{ are shown in Fig.2 an d U+ is a fuzzy set with triangular membership function withbase width 0.4 centered at zero. In conventional direct fuzzycontroller development the de signer specifies a set of such con-trol rules where U$ are also specified a priori ; for the FMRLC,the system will automatically specify and/or modify the fuzzysets Ui to improve/main tain performance. Finally, we note thatwe use Zadehs comp ositional rule of inference and the standardcenter-of-gravity (COG) defuzzification technique [33].

Fig. 2. Fuzzy sets on a universe of discourse.Reference Model

The reference m odel provides a capability for quantifying thedesired performance. In general, the reference model may be anytype of dynam ical system (linear or nonlinear, time-invariant ortime-varying, discrete or continuous time, etc.). The performan ce

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of the overall system is computed with respect to the referencemodel by generating an error signal

where

Given that the reference mo del characterize s design criteria suchas rise time and overshoot and the input to the reference modelis the reference input y , {k r ) , the desired performance of thecontrolled process is met if the leaming m echanism forces X e ( k T )to remain very small for all time; hence, the error ye(k7)providesa characteriza tion of the extent to which the desired performanc eis met at time t = kT . If the performance is met b e ( k r ) = 0) thenthe leaming mechanism will not make significant modificationsto the fuzzy controller. On the other hand if ye(kT) is big, thedesired performance is not achieved and the leaming mechanismmust ad just the fuzz y controller.

Learning MechanismAs previously mentioned, the leaming mechanism performsthe function of modifying the knowledge base of a direct fuzzy

controller so that the closed loop system behaves like the refer-ence model. These knowledge base modifications are made byobserving data from the controlled process, the reference model,and the fuzzy controller. T he leaming mechanism con sists of twoparts: a fuzzy inverse mo del and a knowledge base modifier. Thefuzzy inverse model performs the function of mapping ye(kT)(representing the deviation from the desired behavior), tochanges in the process inputs p = [p i . p,-] hat are necessa ry toforce ge(kT) to zero. The know ledge base m odifier performs thefunction of modifying the fuzzy controllers knowledge b ase toaffect the needed changes in the process inputs. More details ofthis process are discussed next.

Using the fact that most often a con trol engineer will knowhow to roughly characterize the inverse model of the plant, theauthors in [ l ] introduce the idea of using a fuzzy system to ma py&r) and possibly functions of y&r) (or process operatingconditions), to the necessary changes in the process inputs p( kT ) .This map is called the fuzzy inverse model since informationabout the plant inverse dyn amics is used in its specification. Notethat similar to the fuzzy controller, the fuzzy inverse modelshown in Fig. 1contains normalizing scaling factors, namelygv,, nd g,,or each universe of discourse. Given thatgypjei nd gx,,yc, are inputs to the fuzzy inverse model, theknowledge base for the fuzzy inverse model associated with thenth process input is generated from fuzzy implications of theform:

If Yk l an d ...an d Y $ $ nd Y i , an d ... an d Yy : Then P{; , .nlwhere da nd YFd denote the hth fuzzy set for the error ye , an dchange in error y c o , espectively, associated with the ath processoutput and pi...AI9.. m denotes the consequent fuzzy set for thisrule describing the nece ssary chan ge in the nth process input. A swith the fuzzy controller, we utilize membership functions for

the normalized input universes of discourse as shown in Fig. 2,triangular membership functions for the output universes ofdiscourse, Zadehs compositional rule of inference, and COGdefuzzification. Given the information about the necessarychanges in the input as expressed by the vector p(kT) , th eknowledge base m o d i f e r changes the knowledge base of thefuzzy c ontroller so that the previously applied con trol action willbe modified by the amount p(kT). Therefore, consider the pre-viously computed con trol action g ( k T - T ) , which contributed tothe present goodbad system performance. Note that e(kT - T )an d r(kT - 7J ould have been the process error and change inerror, respectively, at that time. By modifying the fuzzy control-lers knowledge base we may force the fuzzy controller toproduce a desired output u(kT - r ) + p(kr ) . Assume that onlysymmetric membership functions are defined for the fuzzy con -trollers output so that c J i . . k . . . . . m denotes the center value of themembership function associated with the fuzzy set U $ . . k 3 . .9 m(initially, c i ; . - . .1, ... J(0 ) = 0). Knowledge base modification isperformed by shifting centers of the membership functions of thefuzzy sets u$...A13... which are associated with the fuzzy im-plications that contributed to the previous control action g(kT -r ) (initially possibly shifting them away from having centers atzero). This modification involves shifting these membershipfunctions by an amount specified by p ( k T )= [ p i ( k r ) . p,-(kT)]so that

The degree of contribution for a particular fuzzy implicationwhose fuzzy relation is denoted Rh...k,,...3mS determined by itsactivation level, defined

where denotes the membership function of the fuzzy set A .Only those rules wh ose activation level 8;..9k,l,...3m(kTr ) > 0are modified; all others remain unchanged. It is important to notethat our rule-base modification procedure implements a form oflocal leaming and hence utilizes memory. In other words, differ-ent parts of the rule-base are filled in based on differentoperating conditions for the system, and when one area of therule-base is updated, other rules are not affected. Hence, thecontroller adapts to new situations and also remembers ho w ithas adapted to past situations. This justifies the use of the termleaming rather than adaptive (for more details on this pointse e [ l l , P I , [71).For example, assum e that all the normalizing gains for boththe direct fuzzy controller and the fuzzy inverse model are unityand that the fuzzy inverse model produces a n outputp,(kT) = 0. 5indicating that the value of the o utput to the plant at time kT -Tshould have been u(kT- ) + 0.5 to improve performance (Le.,to force y e , = 0). Next, suppose that el(kT - 7) = 0.75 and ci(kT- T ) = -0.2. Then rules

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If E? and Ci ' Then Ui3 - ' , an d ing" effects of the ship are sma ll for large vessels. The motion ofthe ship is generally described by a coo rdinate system which isfixed to the ship [5], [6]. See Fig. 3.

are the only rules with activation levels greater than zero(62'-'=0.25 an d 6:'-'=0.75) so these rules will be the only onesthat have their consequent fuzzy sets (U;,-',;l.-') modified(See Fig. 2 ) . To modify these fuzzy sets we simply shift theircenters according to (1).

Design ProcedureSelection of the normalizing gains can impact the overallperformance so we provide a gain selection procedure in thefollowing. Note that although it is often not highlighted, mostleam indad aptive control approaches assume that you are givenan initial controller structure and param eters (e.g., initial control-ler gains must be chosen in adaptive control approaches). In whatfollows we provide a procedure to pick such initial parametersfor the FMR LC.1. Select the controller gains gv, ssociated with the desiredoutput change ye(k7') such that each universe of discourse ismapped to the interval [-1, 11.2 . Choo se the controller gain g, o be the sam e as for the fuzzycontroller output gain gu.This will allow the elements of p(k7')to take on values as large as the largest possible inputs.3. Assign the numerical value 0 to the scaling factors associ-ated with the changes in the desired output changes (Le., allelements of gv,are set equal to 0).4. Apply a step input to the process which is of a magnitudethat may be typical for the process during normal operation.Observe the process response a nd the reference model response.5. Three cases:a) If there exist unacceptable oscillations in a given processoutput response about the reference model response, then in-crease the associated element of gvc.Go to step 4.

b) If a given process output response is unable to "keep up"with the reference model respon se, then decrease the associatedelement of gv,.Go to step 4.c) If the process response is acceptable with respect to thereference model response, then the co ntroller design is com pleted.For the application presented in this paper, the above gainselection procedure has proven very suc cessful. However, giventhat the procedure is a result of practical experience with theFMRLC rather than strict mathematical analysis, it is possiblethat it will not work for all processes. For some applications(although none of the on es studied in [11, [3] , [32]), the proceduremay result in an unstable process. In such situations, it may benecessary to modify other controller parameters such as thecontroller sampling period T or the numbe r of fuzzy controllerrules. Clearly, the stability analysis of the FMRL C is an importantresearch direction.

Learning and Adaptive Autopilots for a Cargo ShipProblem StatementGenerally, ship dynamics are obtained by ap plying Newton'slaws of motion to the ship. For very large ship s, the motion in thevertical plane may be neglected since the "bobbing" or "bounc-

Fig . 3. Cargo ship.A simple model which d escribes the dynamical behavior ofthe ship may be expressed by the following differential equation:

where w is the heading of the ship and 6 is the rudder angle.Assuming zero initial conditions, (3) can be written

where K , 71, 72, and 23 are parameters which are a function of theship's constant orward velocity u and its length 1as expressedbelow:K = KO(71

where we assume that for a cargo ship KO= -3.86,710= 5.66,720= 0.38, T ~ O = .89, an d 1= 161 (m) [6]. Also, we w ill assume thatthe ship is traveling in the x irection at a velocity of 5 m/s.In normal steering, a ship often makes only small deviationsfrom a straight line path. There fore, the model in (3) is obtainedby linearizing the eq uations of motion around the zero rudderangle (6 = 0). As a result, the rudder angle should not exceedapproximately 5 " , otherwise the model will be inaccurate. Forour purposes, we need a model which is suited for rudder angleswhich are larger than 5" ; hence, we use the model proposed byBech and Smitt in [34]. This extended model is given by

(7)

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where H ( $ ) is a nonlinear function of @(t). he function H ( $ )can be found from the relationship be tween 6 an d @ in steadystate such that y = @ = 6 = 0.An experiment known as the spiraltest has shown that H ( $ ) can be approximated by

where a an d h are real valued constants such that a is alwayspositive. For our simulations we choose the values of both a an db to be 1.

FMRLC DesignIn this section we presen t a FMRLC algorithm for controlling

the directional heading of a cargo ship. The inputs to the fuzzycontroller are the heading error and change in heading errorexpressed as

e ( k 9 = Vr(k T)- V ( k T )an d

e (k T)- ( k T - T )T

( k T ) =

respectively, where yr(kT)s the desired ship heading (T = 50ms). The controller output is the rudder angle, 6 ( k T ) ,of the ship.Here we assume that the dynamics of the actuator which is usedto position the rudder are much faster than the ship dynamics sothat they may be neglected. In this fuzzy controller design, 11fuzzy sets are defined for each controller input such that themembership functions are triangular shaped and evenly distrib-uted on the appropriate universes of discourse (as shown in Fig.2 ) . The normalizing controller gains for the error, change error,and the controller output are chosen to be ge = l / n , gc = 100,an d gu = 8n/18, respectively, according to the design procedureabove. The fuzzy sets for the fuzzy controller output are alsoassumed to be triangular shaped with a width of 0.4, and centeredat zero on the normalized universe of discourse. The referencemodel was chosen soas to represent somew hat realistic perform-ance requirements as

respectively. For these inputs, 11 fuzzy sets are defined withtriangular shaped m embership functions which are evenly dis-tributed on the appropriate universes of discourse (as shown inFig. 2 ) .The normalizing controller gains associated with ve(kiJ,v,(kT),ndp(k T) are chosen to be

g+,?= -, gwc= 5 , an d gP =-TTn 18

respectively, according to the design procedure above. For acargo ship, an increase in the rudder angle 6(kT), will generallyresult in a decrease in the ship heading angle. The knowledgebase array shown in Table I was employed for the fuzzy inversemodel for the cargo ship (for more details on how to choose thefuzzy inverse model for other plants see [11, [2], [32]). In TableI, Y{ denotes thejth fuzzy set associated w ith the error signal yean d W f denotes the kth fuzzy set associated with the change inerror signalyc. he entries of the table represent the center valuesof triangular membership functions with base widths 0.4 fo rfuzzy sets P$kon the normalized universe of discourse.

It is important to note that in designin g the FMRL C, the onlyways in which the design procedure relied on the plant modelwas in the choic e of the normalizing gains an d in the specificationof the fuzzy inverse model. Although the design procedure forthe FMRLC is somewhat ad hoc it does have the adva ntage thatit neither relies on the explicit mathematical m odel of the processnor on the form of such a model.

c(r) 0.1 $,(t) + 0.0025 Vm(r) 0.0025 vr(t) (9) where

Model Reference Adaptive ControlIn this section we present two MRAC designs which utilizean underlying proportional derivative (PD) control law for thedirect controller. The PD control law is used to obtain a faircomparison with FMR LC algorithm where error and change inerror are employed as controller inputs. We will consider boththe gradient method and the Lyapunov stability method forMRAC design.Gradient Approach. The controller parameter adjustment

mechanism for the gradient approach to MRAC can be imple-mented via the MZT rule. For this, the cost function

wherevm(t)pecifies the desired system performance for the shipheading v(t).The input to the fuzzy inverse model includes the error andchange in error between the reference model and the ship headingexpressed as

~ l e ( k T ) V m (k T )- N k T )an d

is used and

28

so that

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-5-4-3-2-10

+1+2+3+4+.5

~

lable IKnowledge Base Array Table Employedin the Fuzzy Inverse Model for a Cargo ShipY

-5 -4 -3 -2 -1 +O +1 +2 +3 +4 +5+1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 0.0+1.0 +1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 0.0 -0.2+1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 0.0 -0.2 -0.4+1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 0.0 -0.2 -0.4 -0.6+1.0 +1.0 +0.8 +0.6 +0.4 +0.2 0.0 -0.2 -0.4 -0.6 -0.8+1.0 +0.8 +0.6 +0.4 +0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0+0.8 +0.6 +0.4 +0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0+0.6 +0.4 +0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0 -1.0+0.4 +0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0 -1.0 -1.0+0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0 -1.0 -1.0 -1.00.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0

which is comm only referred to as the M.I.T. rule.For developing the M.I.T. rule for the ship we assum e that theship may be modeled by a second order linear differential equa-where p is the differential operator. The reference model for thisprocess is chosen to be

tion. This model is obtained by eliminating the process poleresulting from 22 in (3) since its associated dynamics are signifi-

O nV m ( t )= 2 wl.(r).p 2 + j O n p + a n (14)where to be consistent with the FMRLC design we choose 5 = 1and On = o,o.5.Combining (14) and (13) an d finding the partialderivatives with respect to the proportional gain kp and the

cantly faster than those resulting from 21. Also, for small headingvariations the rudder a ngle derivative 6 is likely to be sm all andmay be neglected. Therefore we obtain the following reducedorder model for the ship

derivative gain kd we find thatK-10)

The PD-typ e control law which will be emplo yed for this processmay be expressed by--ve -

6(r )= kp (Vr(r )- V(r)) - d (11)kp and kd are the proportional and derivative gains, respectively,and V I { ? )s the desired process output. Substituting (10) into ( 1 1 )

and

we obtain a w e TIr-- - 1 2 [ ;kdjp+[K$ (16)P + -G( t )+ [y+t) +[y)V(f)( y ) V r ( r ) . (12) In general, (15) and (16) cannot be used because the controllerparameters kp and kdare not known. Observe that for the optimal

values of kp an d kd we haveIt follows from (1 2) that

Kk,21V(t) = Vr(r)

(17)(13) Furthermore, the term K / T I may be absorbed into the adaptationgain y. However, this requires that the sign of K / z l be known

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since, in general, y should be po sitive to ensure that the controllerupdates are ma de in the direction of the negative gradient. For aforward moving cargo ship the sign of K/zl happens to benegative which implies that the term y with K / z l absorbed intoit must be n egative to ac hieve the appropriate negative gradient.After making the above app roximations we obtain the followingdifferential equations for upda ting the PD co ntroller gains:

L .where yi an d y2 are negative real numbers.After many simula-tions, the best values that we cou ld find for the yi are yi = -0.005a n d m = - 0 . 1 .

Lyapunov Approach. Examples of Lyapunov based MRACdesigns are illustrated by Narendra and Annaswamy in [5] andby Ame rongen an d Cate in [35]; here we utilize the approach in[5] to design our Lyapunov-based MRAC. Recall from theProblem Statement section that the ship dynamics may be ap-proximated by a second order linear time-invariant differentialequation given by (10).Once again, we use the PD control lawdefined in (11). The dynamical equation which describes thecompensated system is

where y = [v IT an d

The reference model is given by

and where to be consistent with the FMR LC design we choose5 = 1 an d wn 0.05.The dyn amical equation which describes the error (ve(t)vm(t)~ ( t ) )ay be expressed by

The equilibrium point vc= 0 in (25) is asymptotically stable ifwe choose the adaptation laws to be

where P E 3Snm is a symm etric positive definite matrix which isa solution of the Lyapunov equation AZ P + PA = -Q < 0. As -suming that Q is a 2 x 2 identity matrix and solving for P we findthat

pi1 pi2 25.0015 200.000= [p21 p22] = [ 00.000 ?005.00] (28)Solving for kp an d kd in (26) and (27),respectively, the adaptationlaw in (26) and (27) may be implemented as

Of course, (29) and (30)assum e that the plant parameters anddisturbance are varying slowly. In obtaining (29) and (30) theterm K / T ~was absorbed into the adaptation gains yl an d p.Recall that for the cargo ship K / T ~ appens to be a negativequantity. Therefore, both yl an d y2 must be negative to com pen-sate for this fact. Again we found that yl = -0.005 an d p = -0.1were suitable. See Narendra an d Annaswa my [5] for more detailsabout obtaining (29) and (30).

Comparative Analysis of FMRLC and MRAC:Simulation ResultsFor the simulations for all three adaptive control methodspresented above (FMRLC, gradient MRAC, and LyapunovMRAC) we use the nonlinear process model given in (7 ) toemulate u sing the true ship dynamics. Fig. 4 shows the resultsfor the FMRLC controller. Recall that initially the right-hand-sides of the control rules have mem bership function s with centersall at zero (i.e., initially, the controller knows little about ho w tocontrol the plant). Note in Fig. 4 that the FMRL C algorithm wasquite successful in gene rating the appropriate control rules for agood process response since the reference model and the shipheading track almost perfectly. In fact the maximum deviationbetween the two signals was observed to be less than 1 . As aresult, the system exhibits a fast transient response with noovershoot. Also no te that the rate of convergence for the FMR LC

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60

-6 0 t

' Ref input- ef Model Response____..argo Ship Response.

i-80 I I0 1000 2000 3000 4000 5000 6000Time (sec)g 100u-p 5 0 -m5 0a -50

-100U 0 1000 2000 3000 4000 5000 6000Time (sec)

0 1000 2000 3000 4000 5000 6000Fig. 4 . Simulation results for the FM RLC algorithm when employedfo r a cargo ship.algorithm was very fast. In the last plot of Fig. 4 two large spikesof a magnitude of about 1" occur after the first two step inputchanges. Howev er, as time progresses the spikes resulting fromsubsequent step input cha nges are reduced to less than 0.5" as aresult of leam ing.Compare the results for the FMRLC with those obtained forthe gradient and the Lyapunov approach to MRAC shown inFigs. 5 an d 6, respectively. The con troller gains kp an d kd for bothMRA C algorithms where initially chosen to be 5 . This choice ofinitial controller gains happens to be an unstable case for thelinear second order process model (in the case where the adap-tation m echanism is disconnected). However, we felt this to be afair comparison since the fuzzy controller is initially chose n sothat it would put in zero degrees of rudder no matter what thecontroller input values were. We would have chosen both con-troller gains to be 0; but, this choice resulted in a very slowconvergence rate for the MRAC. It is easy to show that thecompensated second order process model (described by A,. an dBc) s equal to the reference model in (22) if the controller gainsare chosen to be k,, = -3.8 an d kd = -143.7. In Table 11 wesummarize the final values of the process gains shown in Figs. 5an d 6. Although the process gains for both algorithms convergedto values relatively close to the optim al values, they do not m atchexactly. This reason for this may be explained by a few simplefacts:

8o Cargo Ship Response - Gradient Approach to MRA C I-.....

-80 10 1000 2000 3000 4000 5000 60(Time (sec)

10 0 I I

0 1000 2000 3000 4000 5000 60(Time (sec)' ProoodibnalGain K01' 1

u -200 I I0 1000 2000 3000 4000 5000 60(

Fig. .Simulation resultsfo r the gradient appmach to MRAC whenemployed for a cargo ship.Both M RAC algorithms are designed to minimize the error

signal qr.This does not necessarily mean that the process pa-rameters will also converge.The reduced second order linear process model, for whichthe controller designs and the "optimal" process gains are based,is not a completely accurate characterization of the third order

nonlinear process model used in simulation.For both the gradient approach and the Lyapunov approach,the system response converged to track the reference model.However, the convergence rate of both algorithms was signifi-cantly slower than the FMRLC method.Another significant advantage of the FMRLC a lgorithm maybe seen in the amount of input energy which was spent at thesystem input to obtain accurate tracking with the referencemodel. D ue to the fact that for our simulations the magnitude ofthe rudder angle is generally larger for both MRAC approachesthan for the FMRLC algorithm, we may suspect that the inputenergy for the FMRLC is significantly less. The rudder angleplots shown in Figs. 4,5, an d 6 for each of the adaptive controlalgorithms represent 1200 sampled data points. We may obtaina measure of the input energy if we think of these data points as

a vector 6 = [(0)6(T) 6(2T) . 6(1199r)]', where the energy isthe square of the 2-norm for this vector (Le., energy = sT6). po nperforming this for the data shown in Figs. 4,5, an d 6 in radiansrather than degrees, we obtain the result shown in Table 111. AsDecember 1993 31


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8o60

Cargo Ship Response - Lyapunov Approach to MRA C' Ref Input- ef Model Response.__.-.argo Ship Response

FMRLC Gradient MRAC17.3368 458.6324

t I

Lyapunov MRAC425.7801

-80 I I0 1000 2000 3000 4000 5000 6000Time (sec)

Controller Parameter

- 1 0 0 , 1

MRAC ApproachGradient Lyapunov

z -100 r J0 1000 2000 3000 4000 5000 6000Time (sec)

kp(6000)

-150 L I0 1000 2000 3000 4000 5000 6000Time (sec)

Fig. 7. Simulation results compare disturbance rejection for theFMRL C, the gradient appi-ouch to MRAC , and the Lyapunov ap--4.7752 -3.0974

comparison with the FMRLC algorithm, we initially loaded thePD controllers in both MRAC algorithms with the controllergains shown in Table 11, which were previously found by eachmethod. However, the centers of the right-hand-sides of themembership functions for the knowledge base of the fuzzycontroller in the FMRL C algorithm w ere initialized with all zerosas before (hence, we are giving the MR AC an advantage). Noticethat the FM RLC algorithm was nearly able to completely cancelthe effects of the disturbance input. However, the gradient andthe Lyapunov approaches to MRA C where not nearly so success-ful.Control Engineering Perspective

We now summarize and more carefully analyze the conclu-sions from our simulation studies. Our goal is to provide anobjective contro l-theoretic assessment that will identify researchdirections focusing on a careful engineering evaluation of theFMRLC . T he results in the previous section indicate the follow-ing advantages of the FMRLC:

Fast convergence compared with MRAC.Minimal amount of control energy needed as compared toGood disturbance rejection properties com pared to MR AC.

MRAC.Fig. 6 . Simulation results or the Lyapunov approach to MRAC whenemployed or a cargo ship. The FMRLC design is independent of the particular formof the mathematical model of the underlying process (in theMRAC designs we need an explicit mathematical model of aexpected, the process input energy for the FM RLC was signifi-cantly less than that obtained for both M RAC approaches.

The final set of experiments performed for this process weredesigned to illustrate the ability of the leaming and adaptivecontrollers to compensate for disturbances at the process input.Fig. 7 illustrates the results obtained for this simulation. Thedisturbance added at the rudder was chosen to be be a sinusoidwith a frequency of one cycle per minute and a magnitude of 2"with a bias of 1".Th e effect of this disturbance is similar to thatof a gusting wind acting upon the ship since wind effects aregenerally modeled as a rudder disturbance. To provide a fair

particular form).

1 Lya pA iiA ppr dac h to'MRAC Gradient Approach o MRAC FMRLC ' 10= oCU

Im- -0.51 , , , . . . , , . I0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (sec)m Wind Disturbance As Seen At The Rudder9 5 , . . . . . , ' ' ' I

$ - S I , c 2 ' ' ' 1 '0 0 200 400 600 800 1000 1200 1400 1600 1800 20001 -


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Overall, the FMRLC provides a method to synthesize (i.e.,automatically design) and tune the knowledge base for a directfuzzy controller. As the direct fuzzy controller is a nonlinearcontroller, some of the above advan tages may be attributed to thefact that the underlying controller that is tuned inherently hasmore significant functional capabilities as compared to the PDcontrollers used in the MRAC designs.While our application may indicate that FMR LC is a promis-ing alternative to conventional MRA C, we must emphasize that:We have compared the FMRLC to only two types ofMRA Cs, for only on e application, for a limited class of referenceinputs, and only in simulation. There are a wide variety of otheradaptive control approaches that deserve consideration (e.g.,those of the self-tuning regulator type [6], [ 5 ] ) .

There are no g uarantees of stability or convergence; hence,we can simply pick a different reference input and the systemmay be unstable (w e have not found this in other simulations, butit is possible).There seem to be no investigations into persistency ofexcitation issues for the FMRLC or any other fuzzy leamingcontrol technique. Persistency of excitation is related to theleaming controllers ability to always generate an appropriateplant input and its ability to generalize the results of what it haslearned earlier and apply this to new situations. In this context,for the ship we ask the following questions:-What if we need to tum the ship in a different direction-will the rule base be filled in f or this direction?- Or will it have to leam for each new direction?- f it leams for the new directions, will it forget how tocontrol for the old ones?In terms of control energy we may have just gotten luckyfor this application and for the chosen reference input. Thereseem to be no analy tical results that guarantee that the FMRL Cor any other fuzzy leam ing control technique minimizes the useof control energy fo r a wide class of plants.This is a very limited investigation of the disturbancerejection properties (Le., only one type of wind disturbance isconsidered). As of yet there seem to be no results on mathem aticalrobustness analysis for a wide class of plants for the FMRL C orany other fuzzy leaming control technique.The design approach for the FMRL C, although it did notdepend on a mathematical model, it is somewhat ad hoc. Willthere be fundamental limitations on the FMRLC imposed bynonminim um phase systems? Certainly there will be limitationsfor classes of nonlinear sy stems. What will these limitations be?It is important to note that the use of a m athematical model helpsto show what these limitations will be (hence it cannot a lwa ys beconsidered an advantage that many fuzzy control techniques donot depend on the specification of the mathematical model). Alsonote that due to our avoidance of using a mathem atical model ofthe plant, we have also ignored the important model matchingproblem in adaptive control.There may be g ains in performance, but are these gains beingmade by paying a high price in computational comp lexity for theFMRLC? The FM RLC is somewhat computationally intensive

as are many neural and fuzzy leaming control approach es but wehave neither performed a careful study of the computationalproperties of the MRAC versus the FMRLC nor investigatedtechniques to simplify the FMRLC computations.

All things considered, the major conclusions one can drawfrom this work are: a) that the FMRLC looks promising anddeserves further attention - specially analysis in a control-theoretic framework, and b) the conventional control-theoreticviewpoint is quite useful fo r the study of this class of intelligentcontrollers.Laying Fou ndations for Com parative AnalysisWe have provided a detailed description of the FMRLCalgorithm, and developed a FM RLC and two MRACs for a shipsteering application. Then we cond ucted some sim ulation studiesto evaluate the performance of the FMRLC as compared to agradient-based and Lyapunov-based MRA C design. M oreover,we discuss the results from a control-theoretic perspective toprovide an objective assessment of the FMRLC and to indicatefuture research directions.We want to emphasize the importance of laying foundationsfor comparative analyses of conventional and intelligent con-trol techniques. Many concepts and results from conventionalcontrol (e.g., stability and stability analysis) can provide for amore careful engineering evalua tion of intelligent control tech-niques and provide for productive research directions. At thesame time, the intelligent control techniques have much to offerconventional control by infusing new concepts, approaches tocontrol, and new design methodologies. In this article we have

made a small move in the direction of bridging the gap betweenfuzzy learning control and conventional adaptive control; it ishoped that this will be beneficial to both fields.References

[ I ] J. Layne and K . Passino, Fuzzy model reference learning control, inProc. 1st IEEE Conf. Control Applications, Da yton, OH, Sept. 1992, pp.686-691.[2] J. Layne and K. Passino, Fuzzy model reference learning control,submitted for publication.[ 3 ] . Layne, Fuzzy model reference leamingcontro1,Masters thesis, Dept.of Elec. Eng., The Ohio State Univ., Mar. 3 1992.[4] T. Procyk and E. Mamdani, A linguistic self-organizing process control-ler,Automatica, vol. 15, no. 1, pp. 15-30, 1979.[ 5 ]K. Narendra and A. Annaswamy, Stable Adaptive Systems. EnglewoodCliffs, NJ: Prentice Hall, 1989.[6 ] K. h r o m and B. W ittenmark,Adaptive Control. Reading, MA: Addison-Wesley Pub. Co., 1989.[7 ] J. Farrell and W. Baker, Leaming control systems, in An Introductionto Intelligent and Autonomous Control Systems, P. Antsaklis and K . Passino,Eds. Norwell, MA: Kluwer, 1993.[8] E. Scharf and N . Mandic, The application of a fuzzy controller to thecontrol of a multi-degree-of-freedom robot arm, in Industrial Applicationsof Fuzzy Control. Amsterdam, The Netherlands: M . Sugeno , Ed., 1985, pp.41-62.[9] R. Tanscheit and E. Scharf, Experiments with the use of a rule-basedself-organising controller for robotics applications, Fuzzy Sets Syst., vol. 26,pp. 195-214, 1988.[ l o ] S. Shao, Fuzzy self-organizing controller and its application for dy -namic processes, F uzzy Sets Syst., vol. 26, pp. 151-16 4, 1988.[111S. Isaka, A . Sebald, A. Karimi, N. Sm ith, and M. Quinn, On the designand performance evaluation of adaptive fuzzy controllers, in Proc. 1988IEEE Conf Decision and Control, Austin, TX , Dec. 1 988, pp. 1068-1069.

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[I21 S. Daley and K.F. Gill, Comparison of a fuzzy logic controller with aP+D control law,J . Dynamical Syst., Meas. . Control,vol. 11 1, pp. 128-1 37,June 1989.[I31 S. Daley and K.F. Gill, Altitude control of a spacecraft using anextended self-organizing fuzzy logic controller, Proc. I . M r c h . E., vol. 201,no. 2, pp. 97-106, 1987.[ 141 S. Daley a nd K.F. Gill, A design study of a self-organizing fuzzy logiccontroller, Proc. 1.Mech. E.. vol. 200, pp. 59-69, 1986.[IS] G. Bartolini, G. Casalino, F. Davoli, R.M.M. Mastretta, and E. Morten,Development of performance adaptive fuzzy controllers with applicationsto continuous casting plants, Industrial Application ofF u z z y Control. pp.73-86, 1985.[161 H. Takahashi, Automatic speed control device using self-tuning fuzzylogic, in Proc. 1988 IEEE Workshop on Automotive Applications o Elec-tronics, Dearbom, M I, Oct. 1988, pp. 65-71.1171 T. Takagi and M. Sugeno, Fuzzy identification of systems and itsapplications to modeling and control,/EEE Trans. Syst..M a n . Cyhern., vol.15, no. 1, pp. 1161 32, 1985.[I81 L. Wang and J. Mendel, Generating fuzzy rules by leaming fromexamples,inProc. 1991IEEEInt . Symp . Intell. Control,Arlington,VA,Aug.1991, pp. 263-268.[191 R.M. Tong, Some properties of fuzzy feedback systems,/EEE Trans.Syst.,Man, Cybern. ,vol. 10, pp. 327-330, June 1980.[20] A. Cumani, On a possibilistic approach to the analysis of fuzzyfeedback systems,IEEE Trans.Syst.,M a n , Cyhrrn. ,pp. 4174 22 , May/June1982.(211 E. Czogala and W. Pedrycz, O n identification in fuzzy systems and itsapplications in control problems, Fuzzy Sers Syst., vol. 6, pp. 73-83, 1981.[22] E. Czogala and W. Pedrycz, Control problems in fuzzy systems,Fuz z ySets Syst., vol. 7, pp. 257-273, 1982.[23] C. Batur, A. Srinivasan, and C. Chan, Automatic rule based modelgeneration for uncertain complex dynamical system s, in Proc. 1991 IEEEInt . Symp. Intelligent Control, pp. 275-279, 1991.[24] F.V.D. Rhee, H.V.N. Lemke, and J. Dijkman, Knowledge based fuzzycontrol of systems, IEEE Trans. uto. Control, vol. 35, pp. 148-155, Feh.1990.1251 P. Gra ham and R. Newell, Fuzzy ada ptive control of a first-ordeiprocess, Fuzzy Sets Syst.,vol. 31, pp . 47 45 , 1989 .[26] P. Graham and R. Newell, Fuzzy identification and control of a liquidlevel rig, Fuzzy Sets Sysr., vol. 8, pp. 255-273, 1988.[27]Pror. IEEE Int. Conf. F uzzy Systems, San Diego, CA, 1992[28] R. Langari and H. Berenji, Fu zzy logic in control engineering , inHandhook of Intelligent Control, D. White and D. Sofge, Eds. New York,N Y Van Nostrand Reinhold, 1992, pp. 93-140.

[29] L. Wang, Stable adaptive fuzzy control of nonlinear syste ms, in Proc.IEEE Conf. Decision and Control,Dec. 1992, Tuscon, AZ, pp. 25 11-2516.[30] H. Berenji, Fuzzy and neural co ntrol, in An Introduction to Intelligentand Autonomous ControlSystems, P. Antsaklis and K. Passino, Eds. Norwell,MA: Kluwer, 1993.(311 J. Layne, K. Passino, and S. Yurkovich, Fuzzy learning control foranti-skid braking systems, in Proc. IEEE Conf. Decision an d Control.Tucson, AZ, Dec. 1992, pp. 2523-2528.[32] J. Layne, K. Passino, and S. Yurkovich, Fuzzy leaming control foranti-skid braking systems, IEEE Trans. Control Svst. Techno/. , vol. I , pp.122-129, June 1993.[33] C. Lee, Fuzzy logic in control systems: Fuzzy logic controller-part I, IEEE Trans. Syst., M a n . , Cyhern., vol. 20, pp . 4 0 4 4 1 8, Mar./Apr. 1990.[34] M. Bech and L. Smitt, Analogue simulation of ship m aneuvers, Tech.Rep., Hydro-og Aerodynmisk Laboratorium, Lyngby, Denmark, 1969.[35] J. Amerongen and A. Cate, Model reference adaptive autopilots forships, Automatica, vol. 11 , pp. 441449, 1975.

Jeffrey R. Layne was born in Ashland, KY, in 1966.He received the B.S. degree from Wnght StateUniversity, Dayton, OH , in 1 990, and the M S.degree from the Ohio State University in 1992, bothin electrical engineering. He has worked at ArmcoSteel Corporation in Middletown, OH, and is cur-rently with Wright Laboratory in Dayton, workingon research in the area of integrated avionic sys-tems His research interests include fuzzy systems,intelligent sys tem , sensor integration, ddta fusion,\tochastic systems, and Kalman filters.

Kevin M. Passino received the Ph D degree inelectrical engineering from the University of NotreDame in 1989 He has worked in the Control Sy?-tems Gro up at Magnavox Electronic Systems Co . inFt. Wayne, IN , and at McDonnell Aircratt Co., St.Louis, MO, on research in intelligent flight control.He spent a ye ar at Notre Dam e as a Visiting AssistantProfessor and is currently an Assistant Professor inthe Department of Electrical Engineenng at TheOhio State University. He serves as the Chair of

Student Activities for the IEEE Control Systems Society and is on theEditorial Board ot the Internutional Joui nul oi Engineerinp Applications ofAI ificial Intelligence He is an Associate Editor for the IEEE TIinsactionson Automatic Contiol. He is co-editor (with PJ Antsakli?) of the book ( A nIntroduction to Intelligent and Autonomous Control (Kluwer) His researchinterests include intelligent and autonomous control, fuzzy and expert sys-tems and control, discrete event and hybrid syrtems, stability theory, andfailure detection and identification system?

34 IEEE Control Systems

Documents

01 Fuzzy Model Reference Learning Control for Cargo Ship Steering