Control Engineering Practice 11 (2003) 471–480

Designing a Fuzzy-like PD controller for an underwater robot

I.S. Akkizidisa,*, G.N. Robertsa, P. Ridaob, J. Batlleb

aMechatronics Research Centre, University of Wales College, Newport, Allt-yr-yn Campus, P.O. Box 180, Newport, NP9 5XR, Wales, UKb Computer Vision and Robotic’s Group, Institute of Informatics and Applications, Edifici Polit"ecnica II, Campus Montilivi, 7071-Girona, Spain

Received 15 June 2001; accepted 7 December 2001

Abstract

The design of a steering and depth control in terms of course-changing and course-keeping tracking mission and motion of an

underwater vehicle is described in this paper. Fuzzy-like proportional derivative (PD) controller is used where the Fuzzy-like part of

the controller is optimised based on its structure and parameter design aspects, whereas the scaling factors of the PD part is

optimised based on the minimum number of experiments in a real environment. The experiments were planned using Taguchi design

of experiments method. The experimental trials and their results are presented and analysed extensively. r 2003 Published by

Elsevier Science Ltd.

Keywords: Underwater vehicle control; Fuzzy like-PD controller; Yaw and depth control; Taguchi design of experiments

1. Introduction

Manoeuvring and depth control of an underwatervehicle (UV) is discussed in this paper. UVs are classifiedas systems possessing highly non-linear dynamics. Inaddition, the environment in which they operate has alot of disturbances. These give rise to special problemsthat may be solved using intelligent control techniques.

This paper presents the development of fuzzycontroller to control steering and depth of a low-costremote-operated vehicle (ROV) named GARBI devel-oped at the Polytechnic of Barcelona and the Universityof Gerona in Spain. The vehicle, as illustrated in Fig. 1,is used for underwater mission operations such asobservations and inspections. An umbilical cable carry-ing power and providing communication links to asurface ship or other operating platform (Amat, Batlle,Casals, & Forest, 1996).

The objective of this paper is to describe how todesign and apply Fuzzy-like proportional derivative(PD) controller in an UV to control the yaw and thedepth of the vehicle by keeping the path of the

navigation to a desired one, and/or changing the pathaccording to a set point. This makes the navigationsmoother and safer, the propulsion more economicaland more accurate path-keeping.

The structure of the Fuzzy-like PD controller is basedon the combination of fuzzy logic and conventional PDcontrol techniques.

The main advantage of the fuzzy logic controller(FLC) is that it can be applied to systems that are non-linear where their mathematical models are difficult toobtain. Another advantage is that the controller can bedesigned to apply heuristic rules that reflect experiencesof the human experts. The membership functions (MFs)of the associated input and output linguistic variablesare generally predefined according to non-linearities ofthe system. Conventional PD controllers provide highsensitivity and tend to increase the stability of theoverall feedback control system. Additionally, PDcontrollers can reduce overshoot and permit the use oflarger gain by adding damping to the system. Thederivative action is employed because it performs well inreducing disturbances and keeping the set point to thedesired one.

During the building of the ‘‘FLC’’ part of Fuzzy-likePD controllers the important tasks are the structure andparameter designs. Structure design means to determinethe architecture of a controller, the input/outputvariables of a controller, the format of the fuzzy control

*Corresponding author. National Technical University of Athens,

Dept. of Electrical and Computer Engineering Division of Signals,

Control and Robotics GR-157 73 Athens, Greece. Tel.: +301-

9318016; fax: +301-9318018.

E-mail address: iakkiz@softlab.ece.ntua.gr (I.S. Akkizidis).

0967-0661/03/$ - see front matter r 2003 Published by Elsevier Science Ltd.

PII: S 0 9 6 7 - 0 6 6 1 ( 0 2 ) 0 0 0 5 5 - 2

rules, and the number of rules. Parameter design meansdetermining the optimal parameters for a fuzzycontroller.

For the successful design of ‘‘PD’’ part of the Fuzzy-like PD controllers, proper selection of the optimalinput and output scaling factors (SFs) is required whichscales up or down the entire universe of discourse. Dueto their global effect on the control performance androbustness, input and output SFs play critical role in theFuzzy-like PD controller and they have the highestpriority in terms of tuning and optimisation (Mudi &Pal, 1999). Analysis of how to investigate their optimalvalues is presented in this paper. Experimental results ofthe Fuzzy-like PD controller are presented and dis-cussed extensively in the following sections.

2. The hydrodynamic forces and moments of GARBI

The motion study of marine vehicle involves sixdegrees of freedom as shown in Fig. 2, since sixindependent coordinates are necessary to determinethe position and orientation of a rigid body. The firstthree co-ordinates surge, sway and heave and their timederivatives correspond to the position and translationalmotion along the x-, y-, and z-axis. The last three co-ordinates roll, pitch and yaw and their time derivativesare used to describe orientation and rotational motion.In GARBI the motions in the x and z direction (Surgeand Heave) are controlled from the horizontal propel-lers (T1; T2) and vertical propellers (T3; T4), respectively(Fig. 2). However, no correction in y direction (sway) isapplied.

The structure of GARBI is designed in such a waythat pitch and roll cross-coupling is virtually non-existent. However, minor coupling appears between yawand surge only when the vehicle has initial speed.Nevertheless, this coupling is expected and acceptable.Similarly, coupling between yaw and pitch and yaw anddepth is also minor and can be neglected.

3. Control tasks of GARBI underwater vehicle

When designing GARBI’s controller it is necessary tocompensate for its non-linear dynamics and kinematics,non-linearities due to thrusters and pressure hysteresis,barometer dead-zones, and the noise in yaw and depthmeasurements. GARBI’s Fuzzy-like PD controller isdesigned to make the vehicle follow the commands fromthe pilot in terms of course-changing and course-keepingof both yaw angle and depth of the robot.

Controllers for course-keeping and/or course-chan-ging are normally based on feedback from a gyro-compass measuring the heading for the yaw and airpress-sensors measuring the difference of the pressureinside and outside of the robot and therefore the depth.

The control objective for a course-keeping controllercan be expressed as c; z ¼ constant: For course-chan-ging, the objective is to follow the changes of the pilotcommands with the best control performance in terms ofsmall overshoot, settling time and steady-state error.

Fig. 3 shows a simplified scheme of course-keepingand course-changing controller. The structure uses twoindependent FLCs for each controlled variable (yaw anddepth), greatly simplifying the design at the cost of somedecrease in performance. As shown in Figs. 4 and 5 thecorresponding inputs of these controllers are the errorecðnTÞ ¼ csp � cðnTÞ between the real and the desiredyaw angle and the error ezðnTÞ ¼ zsp � zðnTÞ betweenthe real and the desired heave position as well as thechange of the above errors DecðnTÞ ¼ ecðnTÞ �ecðnT � 1Þ; DezðnTÞ ¼ ezðnTÞ � ezðnT � TÞ whereeðnTÞ; DeðnTÞ and c; zðnTÞ designate crisp error, rateand process output at sampling time nT ; respectively.The computed rate (De) may not be the actual one dueto delays and noise of the measurements. To overcomethe above problem a rate giro should be used.Unfortunately, during the experiments the above device

Fig. 1. Photo of GARBI underwater robot.

Fig. 2. GARBI body-fixed reference frames showing the six degrees of

freedom.

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was not available. Limiters are used to avoid saturationof inputs in the universe of discourse. The correspondingoutputs of these controllers are; for the first controllersthe moment N around the z-axis, and for the secondcontroller the force Z of the two propellers in the z-direction. The rotation N is related to the difference ofpower between the propellers T1 and T2 in the x-direction. The motion Z in the z-direction relates to thepower of the propellers T3 and T4; which is always equaland of the same polarity.

4. Design of the Fuzzy-like PD controller for GARBI

In studying the dynamic properties of the fuzzycontroller, the model of the process is needed so that the

impact of the successive control actions may bemonitored. Since a model of GARBI is not available(due to its very complicated shape), the dynamicproperties of the closed-loop structure have to bederived intuitively and experimentally. This is simply acornerstone feature of the idea of fuzzy controllers.However, the tuning of Fuzzy-like PD controllersystems is a fundamental problem, specially for opti-mum performance. There are two main different aspectsin the design of Fuzzy-like PD controllers. The firstaspect includes the structure (as described in Section 3),the rule base, the antecedent and consequent member-ship functions together with their distribution, theinference mechanism and the defuzzification strategy.The second aspect is how to optimise the input/outputscaling factors of the Fuzzy-like PD controller. Bothaspects are described as follows.

Fig. 3. Control loop for GARBI.

Fig. 4. Yaw Fuzzy-like PD controller.

Fig. 5. Depth Fuzzy-like PD controller.

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4.1. Design aspects of the ‘‘FLC’’ part of Fuzzy-like PD

controller

4.1.1. Input/output universe of discourse

Both yaw and depth controllers use the whole rangeof its universe. Therefore, the maximal values of theerror and its change should be equal to the limit of theuniverse. Thus, for the yaw controller

eyawmaxSyawe

¼ DeyawmaxSyawDe

¼ YawUniversemax; ð1Þ

where the universe is in a range of �1801 to +1801 (thepositive sign is for port and the negative sign is forstarboard turns). The depth controller is

edepthmaxSdepthe

¼ DedepthmaxSdepthDe

¼ DepthUniversemax; ð2Þ

where the universe is in a range of 10 m. In both casesthe scale is normalised in the range [�1,1]. It should benoted that the depth controller starts using the range ofits universe when the robot is within 10 m of the setpoint. In any other case the control output has themaximum value.

4.1.2. Input/output linguistic variable—MFs

The choice of the shape of the antecedent MFs istriangular ðm1ðxÞ;m2ðxÞ;y;mnðxÞÞ with a specific overlapof 50% to ensure that each value of the universe is amember of at least two sets, except possibly for elementsat the extreme ends. This means that the height of theintersection of the two successive fuzzy sets ishgtðmi-miþ1Þ ¼ 1=2:

The selection of MF has two important character-istics: one is its optimal interface design and the other isits semantic integrity (Pedrycz, 1993).

The first characteristic refers to error-free reconstruc-

tion, where in the fuzzy system the numerical values areconverted into linguistic values by means of fuzzificationand in the defuzzification method the linguistic valuescan be reconstructed in the same numerical value, i.e.8xA½a; b� : f �1½f ðxÞ� ¼ x:

The second characteristic refers to three main designaspects: justifying the number and the labels of MFs,

distinguishability and completeness as described in thefollowing paragraphs.

In the first aspect, the number of the input/outputfuzzy sets is seven. This number comes from therecommendation that the number of the sets should becompatible with the number of ‘‘quantifiers’’ thathuman beings can handle which actually is within thelimit of 772 distinct terms (Espinosa & Vandewalle,1997). Thus, each of the FLC blocks contains 49 rules.The input/output variables of GARBI’s FLC arequantified into sets of classes defined by linguistic labelssuch as ‘‘Positive Big’’ (PB), ‘‘Positive Small’’ (PS),‘‘Zero’’ (ZO), etc. These variable in the premise parts arefuzzy, while in the consequence part are singleton withvalues between 0 and 1. By using MFs in the input and

singletons in the output of GARBI’s control system, theactual Takagi–Sugeno fuzzy system approach (Takagi &Sugeno, 1985) is utilised. Mamdani and Assilian (1975)control approach is not used due to its computationalcomplexity during the defuzzification procedure which istime consuming. It is well known, however, that fuzzyrules with singletons can be used without loosing theperformance of the control (Sugeno & Yasukawa, 1993).It is therefore recommended for real-time fuzzy controlapplications to use singletons in the output resulting insimpler and faster control action (Jantzen, 1998; Nguyen& Prasad, 1999).

In the second aspect, distinguishability, each of thelinguistic labels should have semantic meaning and thefuzzy sets should clearly define a range in the universe ofdiscourse. So, the MFs should be clearly different. Theassumption of the overlap equal to 0.5 assures that thesupport of each fuzzy set will be different. The distancebetween the model values of the MFs defined as the a—

cut with a ¼ 1miða¼1ÞðxÞ; i ¼ 1; 2;y;N; assures that theMFs can be distinguished.

The third aspect, completeness, express the ability ofthe fuzzy control algorithm to infer a control action withconfidence not less than a minimal level e; for which thethreshold e—cuts of all terms covering the intervaluniverse. Decreasing this parameter decreases thefuzziness of the partitioning of the input space of theFLC. Based on heuristic considerations and the 50%overlapping of the MFs, the level of completeness isdefined as e ¼ 0:5: This actually is a fixed structure forinitial setting of most FLCs.

4.1.3. Construction of the rule (knowledge) base

The construction of the rule base is based on thetemplate rule-base method that is regarded as a basictool uniting the common engineering sense and experi-ence in fuzzy logic control. MacVicar-Whelan (1977)developed this type of rule-base template which wasintroduced in the first FLC, (Mamdani & Assilian, 1975;King & Mamdani, 1977). The MacVicar-Whelan rule-base summarises the rules used in the rule bases of theseFLCs, and in addition includes situations (combinationsof linguistic labels of input and output variables of theFLC) that were not defined. Thus, the rule-basetemplate used in the FLC part of GARBI’s Fuzzylike-PD controller is as presented in Table 1. The celldefined by the intersection of the first row and the firstcolumn represents a rule such as

if eðnTÞ is NB and DeðnTÞ is NB then uðnTÞ is NB.Analytical explanation of how the rule-base table is

designed can be found in Reznik (1997).

4.1.4. Operators

Using the min operation for the aggregation AND

(outer product) of the fuzzy rules, the output fuzzy set isgiven by mu ¼ Minðme;mDeÞ: Thus, the Fuzzy-like PD

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controller is a controller where the output is a non-linearfunction of the error e and its derivative de=dtðu ¼F ðe;de=dtÞÞ; where F is a non-linear function of twovariables.

4.1.5. Defuzzification

The control signal results from the defuzzificationmethod that uses the degree of membership functions ofthe antecedent and the singleton of the conseqeuences ofthe MFs obtained by

u ¼XR

i¼1ximi=

XR

i¼1mi; ð3Þ

where mi is the degree of MF, xi is the singleton’s valueand R is the number of rules.

4.2. Design aspects of the SFs of the Fuzzy-like PD

controller

One way of improving the dynamic properties of theFuzzy-like PD control systems is to optimise its SFs. SFsplay an important role in the formation of the dynamicsof the closed-loop structure leading to the desiredresponse of the controlled system (Zheng, 1992). Theimportance of an optimal choice of input SFs isevidently shown by the fact that inappropriate scalingset values is either shifting the operating area to theboundaries or utilising only a small area of the normal-ised universe of discourse. Additionally, the adjustmentsof the output SF affects the closed-loop gain, which hasdirect influence on stability and oscillation tendency.

There is no general method to optimise the SFs in theFuzzy-like PD control systems. Most successful resultsreported are based on the combination of expertunderstanding about the controlled object and the useof the analogies between the FL and PID controllers(Zheng, 1992). There are some general directions tooptimise the SFs (Reznik, 1997; Procyk & Mamdani,1979), however, their effectiveness is bounded by thecontradictory requirements resulting from differentperformance measures.

For GARBI’s Fuzzy-like PD controller, the corre-sponding SFs are for the inputs Se; SDe and for theoutput Su; respectively. A systematic approach based on

design of experiments is being proposed and applied tooptimise GARBI’s controllers SFs as described in thenext sections.

5. Designing the experiments to obtain the SFs of

GARBI’s Fuzzy-like PD Controller

A large amount of engineering effort is consumed inconducting experiments to produce the informationrequired to make decisions about how different factorsaffect performance under different usage conditions. Asystematic methodology of how to identify the optimumvalues of Fuzzy-like PD controllers in terms of controlperformance is proposed and developed herein. Theproposal is based on the combination of the widely usedmethod called the Taguchi design of experiment

(Fowlkes, 1995) and a proposed Fuzzy Combined

Scheduling System approach.Various types of matrices are used for planning

experiments to study one factor at a time, where eachindividual factor is varied while all the other factors arefixed. This is known as the full factorial method thatinvestigate all possible combinations of all factors andtheir levels, where the possible combinations can rise tothe order of yx; x being the number of factors and y thedifferent levels. This approach investigates all thepossible combinations, maximising the possibility offinding the optimum result, but large numbers ofexperiments are required and thus time consuming andcostly. Alternatively, orthogonal arrays, extensivelyused in the Taguchi method, studies several factors atdifferent levels simultaneously, but only require afraction of the full factorial combinations. The ortho-gonal array imposes an order on the way the experimentis carried out. The combinations are chosen to provideenough information to determine the factor effects usingthe analysis of mean values. In order to use a standardorthogonal array provided by the Taguchi method, thedegrees of freedom (number of independent measure-ments available to estimate sources of information) ofthe factors and levels must be matched with the degreesof freedom for that orthogonal array (Taguchi, 1987).

The SFs are the parameters/factors of both yaw anddepth Fuzzy-like PD controllers of GARBI have beendefined. When a SF is changed, it is assumed that thedefinition of each membership function will be changedby the same ratio. Hence, changing of any SF canchange the meaning of one part, the IF-part or THEN-part, in any rule. Therefore, it can be said that thechange of SFs may affect all of the control rules Table 1.

Three factorial levels are chosen initially for all SFs ofboth yaw and depth controllers. The choice of theirinitial values is based on min, max and intermediate

value that excite the response of the system (i.e., Se ¼f0; 5; 0; 75; 1g; SDe ¼ f0; 5; 1; 2g; Su ¼ 3; 7; 10).

Table 1

The rule base of a Fuzzy-like PD in tabular form

e=De NB NM NS ZO PS PM PB

NB NB NB NB NB NM NS ZO

NM NB NB NB NM NS ZO PS

NS NB NB NM NS ZO PS PM

ZO NB NM NS ZO PS PM PB

PS NM NS ZO PS PM PB PB

PM NS ZO PS PM PB PB PB

PB ZO PS PM PB PB PB PB

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Note that, increasing the number of the levelsincreases the number of experiments. This is veryimportant issue as increasing the number of experimentsis both time consuming and costly. Moreover, thedefinition of the above levels indicates the initial valuesused to define the subspace where the optimal values lie,as described in the next sections.

Using a full factorial, three levels would result in 33

(27) different experiments. However, for a three factorlevel experiment, six degrees of freedom exists, so anorthogonal array with nine experimental runs can beemployed instead reducing the number of experiments(Phadke, 1989). Table 2 shows the orthogonal array forboth yaw and depth experiments that is sufficient for thisstudy.

6. Conducting the experiments (in a real environment)

and optimising the SFs

Experimental trials were undertaken in Lake Ba-nyoles, Spain to test both depth and yaw Fuzzy-like PDcontrollers. The power of the propellers is controlledwith power cards for voltage tuning. The voltage usedwas in the range of 3–10 V. For heading control, theopposite voltage between the horizontal propellers (T1;T2) is used i.e. +V1, �V2. So, if the heading angle isturned to a1 clockwise, for instance, the voltage in theright propeller T1 is reduced and the voltage in the leftpropeller T2 is increased by this amount.

6.1. Initial experiments and results

Using Table 2, the experiments to investigate bothyaw and depth control performances were undertaken.The navigation plan for the yaw and depth experimentswas

* initial voltage of the horizontal and vertical propel-lers is set to 3V for a period of 60 s to ensure that thevehicle goes straight ahead. Equal power to the

horizontal propellers are employed and the vehiclemoves away from the platform in the water,

* manoeuvring with set point of 1801. After themanoeuvring, the task is to keep the vehicle movingin the same direction, and

* at the same time changing the depth course from 0 to10 m and then to 5m and then keeping it at thisdepth.

6.2. Analysis of the results to optimise the SFs

After the nine trials (Table 2), the results of the yaw

and depth response (shown in Table 3) were analysed interms of the integral absolute-error (IAE) pIAEyaw

;pIAEdepth

and integral-of-time-multiplied by absolute-error (ITAE) pITAEyaw

; pITAEdepthperformance criteria.

The analysis of means (Phadke, 1989) is the methodthat is used to investigate the possible optimal levels ofthe SFs (optimal in this case is the levels that minimisesthe above integrals). Analytically, for each factor level,the mean responses are obtained as in Table 3. Theseresponses together with the levels are the co-ordinatesthat construct the plots (Figs. 6 and 7) used in theanalysis of the graphical representation of factor levels.

As the aim of the controller is to minimise both IAEand ITAE, the objective characteristic of these targetvalues is ‘‘smaller-the-better’’. Therefore, for the yaw

response shown in the IAE and ITAE mean responseplot (Fig. 6) the smallest yaw mean (average) responses

are the 9.79, 11.25, and 9.31 for Sec ; SDecSuc and 354.79,453.03, and 387.37 for Sec ; SDecSuc ; respectively. Thus,the possible optimal levels of the SFs that minimise theIAE and ITAE are {0.5,0.5,7} and (1,0.5,7}, respec-tively.

For the depth response shown in the IAE meanresponse plot (Fig. 7) the smallest ‘‘depth mean re-

sponses’’ for Sez; SDez

and Suzare 15.93, 21.89 and 15.52,

respectively. Thus, the possible optimal levels of the SFsthat minimise the IAE are {0.5,0.5,10}. Note here thatthese factor levels are not defined as a combination in

Table 2

Orthogonal array with nine experiments at three levels for each SF

No. exp. Se SDe Su

1 0.5 0.5 3

2 0.5 1 7

3 0.5 2 10

4 1 0.5 7

5 1 1 10

6 1 2 3

7 0.75 0.5 10

8 0.75 1 3

9 0.75 2 7

Table 3

Results of yaw and depth performance in terms of IAE and ITAE from

the real experiments

Depth performance Yaw performance

No. exp. pIAEdepthpITAEdepth

pIAEyawpITAEyaw

1 29.92 633.25 11.38 547.33

2 8.515 137.58 9.06 556.41

3 9.36 64.64 8.914 290.28

4 20.86 294.22 7.9 141.2

5 22.32 1265.25 9.87 277.8

6 80.62 4190.8 14.32 645.37

7 14.89 1118.72 14.4 670.55

8 84.17 5111.58 16.95 729

9 21.68 314.2 10.92 464.5

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the experiment shown in Table 2, however, because ofthe orthogonality of the Taguchi method, combinationsof factor levels not appearing in the standard orthogonalarray table can be predicted by the method (Taguchi,1987). This is one of the most important features of theabove approach, that is, to be able to identifyexperimental combinations that were not originallyspecified in the orthogonal array. For the ITAE meansplot shown in Fig. 7 the smallest ‘‘depth mean responses’’for Se; SDe and Su are 278.49, 682.07 and 248.67,respectively. Thus, the possible optimal levels of the SFsthat minimise the ITAE are {0.5,0.5,7}.

Note that, with the above analysis, the possible

optimal factor levels may not be defined in terms oftheir actual optimal values but their direction to thesubspace that they belong.

Using the Fuzzy combined scheduling system as brieflyexplained in Appendix A the SFs is tuned to the final

optimal level. Thus, for the yaw controller the finaloptimal levels are Sec ¼ 0:5; SDec ¼ 0:5; Suv

¼ 7 and for

the depth controller the final optimal levels are Sez ¼0:51; SDez

¼ 0:48; Suz ¼ 8:05:

6.3. Final experiments and results

Applying these SFs, new experiment was heldfollowing the same navigation scenario as in Section6.1 to verify the applicability of the final optimal SFsvalues. The new performances of the controllers incourse-changing and course-keeping are shown inFigs. 8–11.

Fig. 8(a) illustrates the yaw, Fig. 8(b) the controlleroutput of the power horizontal propellers (T1; T2) (notethat as mentioned before they have opposite sign),Fig. 8(c and d) the error and the change of error betweenthe desired and the actual yaw. Additionally, Fig. 9(a)shows the depth, Fig. 9(b) the controller output of thepower of the vertical propellers (T3; T4), Fig. 9(c and d)the error and the change of error between the desired

Fig. 6. Plot used in analysis of means to investigate the optimal levels that minimise IAE and ITAE (yaw response).

Fig. 7. Plot used in analysis of means to investigate the optimal levels that minimise IAE and ITAE (depth response).

I.S. Akkizidis et al. / Control Engineering Practice 11 (2003) 471–480 477

and depth. Finally, Figs. 10 and 11 shows the corre-sponding pitch and roll motions.

The performance of the yaw controller in changingthe course (heading/depth) is satisfactory, as both

overshoot and rise time are small. Due to buoyancyeffects the depth control dynamics vary, i.e. the vehiclerises faster than it descends. From Fig. 9 it can be seenthat the controller has accommodated this variation

Fig. 8. The plan of the ‘‘Yaw’’ experiment (briefly) was as follows: Change the course from 2701 to 1351 and then to 2251 and then keep heading in

this direction.

Fig. 9. The plan of the ‘‘Depth’’ experiment (briefly) was as follows: Change the course from 0 to 10m and then to 5 m and then keep depth at this

level.

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producing very acceptable rise times with a very smallovershoot when ascending. In course-keeping control,both controllers perform quite well (Figs. 8 and 9).Finally, in Figs. 10 and 11, the pitch and roll appearsmostly due to the heading changes. These are very smalland can be considered as fractional and thus negligible.

7. Conclusions

The proposed approaches presented in this paper wereapplied to define the Fuzzy-like PD controllers forGARBI underwater robot. The design of GARBI’scontroller involves course-changing and course-keepingfor both steering and depth-control performances. Thearchitecture of the controller was based on twoindependent Fuzzy-like PD controllers for each con-trolled variable, yaw and depth. The design aspects ofthe FLC as well as for the ‘‘PD’’ part for these types ofcontrollers were presented.

It was shown that the flexible structure of the FLCleaves the designer to decide about the input/outputuniverse of discourse and linguistic variables in terms oftheir shape, number and meaning, the construction ofthe rule base and the meaning of their operators and thedefuzzification method. The above decisions wereguided from particular design aspects coming from the

fuzzy system theory as well as from the expert’sknowledge in terms of system’s dynamics and behaviouridentification. Moreover, the gains of the input/outputSFs of the PD part of the controller were definedaccording to their global effect on the dynamics of theclosed-loop control system. Although the SFs weredefined based on some general instructions comingmostly from the classical PID design theory, a moresystematic procedure to investigate and optimise itscontroller’s parameters was introduced. This optimisa-tion was based on experiments in a real environmentthat were planned using the Taguchi design of experi-ments method.

It was shown that the Taguchi design of experimentmethod could help to minimise the number of experi-ments, using only nine experiments in the orthogonalarray, without the risk of losing vital information. Thiswas very important as the experiments were held in areal environment, where time and money is an issue.

The results of the experiments were presented andanalysed extensively. By analysing the mean response ofthe performance criteria measurements, it was shownthat the possible optimal factor levels could be defined.Finally, applying the fuzzy combined scheduling system

approach, the possible optimal factor levels of both yaw

and depth controllers’ SFs were tuned to obtain the final

optimal values of the SFs.Using, therefore the proposed systematic method,

GARBI acquired a Fuzzy-like PD controller thatperforms satisfactorily since both course-changing andcourse-keeping performance is within the desiredresponse with minimal error. It has been shown there-fore, that Fuzzy-like PD controllers designed, optimisedand tuned by the proposed approaches possessesfeatures that are attractive in navigation controlproblems posed by underwater vehicles.

It has also been shown in this paper, how a fuzzycontroller combined with conventional PD controltechniques can help to design Fuzzy-like PD controller,dealing with the uncertainties and non-linearities of anunderwater vehicle.

Acknowledgements

The authors wish to acknowledge the support for thiswork provided by the British Council under the British/Spanish Acciones Integradas programme.

Appendix A. Fuzzy combined scheduling system (FCSS)

approach

As the possible optimal factor levels define thesubspace that the actual optimal values belong, theFuzzy combined scheduling system approach drives

Fig. 10. The roll due to the experiment is not more than 51.

Fig. 11. The pitch due to the experiment is not more than 2.51.

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the possible optimal factor levels to the final optimal

ones. The discriminant function w ¼ f ðLowMF ;HighMF Þcombines the current factor level with the possible

optimal ones. This function defines the weight vectorof both current and possible optimal factor levels w ¼fwc1

;wpo1;wc2

;wpo2;y;wcj

;wpojg constructed from the

weight values of the fuzzy sets of the performance states.Therefore, the construction of the fuzzy rules thatdefines the relationship between the performance state inthe antecedent and the current lfc

or possible optimal lfpo

factor levels in the consequences are as

If Performance State is Low Then factor level is lfc

If Performance State is High Then factor level is lfpo

Thus, for each performance state output, the degreeof the current and the possible optimal factor levels arerelated to the degree of the low and high membershipfunctions (MFs). The shapes of the MFs, as well as theirlinguistic labels may vary and depend mostly on the setcriteria. The overlapping between them is 50% to ensurethat both rules are excited for each performance statevalues. Fig. 12 shows the setting of these two MFs basedon the criterion to minimise the value of a performancestate. Therefore, when the performance state increases,the weight of the current level should reduce, whereasthe weight of the possible optimal factor level shouldincrease accordingly and vice-versa.

If the discriminant function is such that wA½0; 1� andmore than one w is different from 0, then the transitionfrom one factor level to another will be smooth.Therefore, each of the overall factor levels will beoptimal even for a non-linear control system, since theweights themselves are functions of the performancestates.

The overall final optimal factor level vector resultingfrom the composite current and possible optimal factorlevels defined by equation LoF ¼ fLoF1

;LoF2;y;LoFj

gwhere LoFj

is the final optimal level for each factor j;calculated as the weighted mean described in thefollowing equation:

LoFj¼

Pmk¼1ðwpok

lfpoj þ wcklfcjÞPj

k¼1ðwpokþ wck

Þ; ðA:1Þ

where m is the number of the performance states.In this equation multi-performance-criteria is used

where more than one objective is maximised and/orminimised.

More analytical explanation of this approach can befound in Akkizidis (2000).

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Fig. 12. The low and high performance state membership functions

where the shapes are Z and S, respectively, with 50% overlap.

I.S. Akkizidis et al. / Control Engineering Practice 11 (2003) 471–480480