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 b

a Mechatronics Research Centre, University of Wales College, Newport, Allt-yr-yn Campus, P.O. Box 180, Newport, NP9 5XR, Wales, UK 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 underwater vehicle (UV) is discussed in this paper. UVs are classified as systems possessing highly non-linear dynamics. In addition, the environment in which they operate has a lot of disturbances. These give rise to special problems that may be solved using intelligent control techniques. This paper presents the development of fuzzy controller to control steering and depth of a low-cost remote-operated vehicle (ROV) named GARBI developed at the Polytechnic of Barcelona and the University of Gerona in Spain. The vehicle, as illustrated in Fig. 1, is used for underwater mission operations such as observations and inspections. An umbilical cable carrying power and providing communication links to a surface ship or other operating platform (Amat, Batlle, Casals, & Forest, 1996). The objective of this paper is to describe how to design and apply Fuzzy-like proportional derivative (PD) controller in an UV to control the yaw and the depth of the vehicle by keeping the path of the *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.: +3019318016; fax: +301-9318018. E-mail address: [email protected] (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

navigation to a desired one, and/or changing the path according to a set point. This makes the navigation smoother and safer, the propulsion more economical and more accurate path-keeping. The structure of the Fuzzy-like PD controller is based on the combination of fuzzy logic and conventional PD control techniques. The main advantage of the fuzzy logic controller (FLC) is that it can be applied to systems that are nonlinear where their mathematical models are difficult to obtain. Another advantage is that the controller can be designed to apply heuristic rules that reflect experiences of the human experts. The membership functions (MFs) of the associated input and output linguistic variables are generally predefined according to non-linearities of the system. Conventional PD controllers provide high sensitivity and tend to increase the stability of the overall feedback control system. Additionally, PD controllers can reduce overshoot and permit the use of larger gain by adding damping to the system. The derivative action is employed because it performs well in reducing disturbances and keeping the set point to the desired one. During the building of the ‘‘FLC’’ part of Fuzzy-like PD controllers the important tasks are the structure and parameter designs. Structure design means to determine the architecture of a controller, the input/output variables of a controller, the format of the fuzzy control

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Fig. 1. Photo of GARBI underwater robot.

rules, and the number of rules. Parameter design means determining the optimal parameters for a fuzzy controller. For the successful design of ‘‘PD’’ part of the Fuzzylike PD controllers, proper selection of the optimal input and output scaling factors (SFs) is required which scales up or down the entire universe of discourse. Due to their global effect on the control performance and robustness, input and output SFs play critical role in the Fuzzy-like PD controller and they have the highest priority in terms of tuning and optimisation (Mudi & Pal, 1999). Analysis of how to investigate their optimal values is presented in this paper. Experimental results of the Fuzzy-like PD controller are presented and discussed extensively in the following sections.

2. The hydrodynamic forces and moments of GARBI The motion study of marine vehicle involves six degrees of freedom as shown in Fig. 2, since six independent coordinates are necessary to determine the position and orientation of a rigid body. The first three co-ordinates surge, sway and heave and their time derivatives correspond to the position and translational motion along the x-, y-, and z-axis. The last three coordinates roll, pitch and yaw and their time derivatives are used to describe orientation and rotational motion. In GARBI the motions in the x and z direction (Surge and Heave) are controlled from the horizontal propellers (T1 ; T2 ) and vertical propellers (T3 ; T4 ), respectively (Fig. 2). However, no correction in y direction (sway) is applied. The structure of GARBI is designed in such a way that pitch and roll cross-coupling is virtually nonexistent. However, minor coupling appears between yaw and surge only when the vehicle has initial speed. Nevertheless, this coupling is expected and acceptable. Similarly, coupling between yaw and pitch and yaw and depth is also minor and can be neglected.

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

3. Control tasks of GARBI underwater vehicle When designing GARBI’s controller it is necessary to compensate for its non-linear dynamics and kinematics, non-linearities due to thrusters and pressure hysteresis, barometer dead-zones, and the noise in yaw and depth measurements. GARBI’s Fuzzy-like PD controller is designed to make the vehicle follow the commands from the pilot in terms of course-changing and course-keeping of both yaw angle and depth of the robot. Controllers for course-keeping and/or course-changing are normally based on feedback from a gyrocompass measuring the heading for the yaw and air press-sensors measuring the difference of the pressure inside and outside of the robot and therefore the depth. The control objective for a course-keeping controller can be expressed as c; z ¼ constant: For course-changing, the objective is to follow the changes of the pilot commands with the best control performance in terms of small overshoot, settling time and steady-state error. Fig. 3 shows a simplified scheme of course-keeping and course-changing controller. The structure uses two independent FLCs for each controlled variable (yaw and depth), greatly simplifying the design at the cost of some decrease in performance. As shown in Figs. 4 and 5 the corresponding inputs of these controllers are the error ec ðnTÞ ¼ csp  cðnTÞ between the real and the desired yaw angle and the error ez ðnTÞ ¼ zsp  zðnTÞ between the real and the desired heave position as well as the change of the above errors Dec ðnTÞ ¼ ec ðnTÞ  ec ðnT  1Þ; Dez ðnTÞ ¼ ez ðnTÞ  ez ðnT  TÞ where eðnTÞ; DeðnTÞ and c; zðnTÞ designate crisp error, rate and process output at sampling time nT; respectively. The computed rate (De) may not be the actual one due to delays and noise of the measurements. To overcome the above problem a rate giro should be used. Unfortunately, during the experiments the above device

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Fig. 3. Control loop for GARBI.

Fig. 4. Yaw Fuzzy-like PD controller.

Fig. 5. Depth Fuzzy-like PD controller.

was not available. Limiters are used to avoid saturation of inputs in the universe of discourse. The corresponding outputs of these controllers are; for the first controllers the moment N around the z-axis, and for the second controller the force Z of the two propellers in the zdirection. The rotation N is related to the difference of power between the propellers T1 and T2 in the xdirection. The motion Z in the z-direction relates to the power of the propellers T3 and T4 ; which is always equal and of the same polarity.

4. Design of the Fuzzy-like PD controller for GARBI In studying the dynamic properties of the fuzzy controller, the model of the process is needed so that the

impact of the successive control actions may be monitored. Since a model of GARBI is not available (due to its very complicated shape), the dynamic properties of the closed-loop structure have to be derived intuitively and experimentally. This is simply a cornerstone feature of the idea of fuzzy controllers. However, the tuning of Fuzzy-like PD controller systems is a fundamental problem, specially for optimum performance. There are two main different aspects in the design of Fuzzy-like PD controllers. The first aspect includes the structure (as described in Section 3), the rule base, the antecedent and consequent membership functions together with their distribution, the inference mechanism and the defuzzification strategy. The second aspect is how to optimise the input/output scaling factors of the Fuzzy-like PD controller. Both aspects are described as follows.

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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 range of its universe. Therefore, the maximal values of the error and its change should be equal to the limit of the universe. Thus, for the yaw controller eyawmax Syawe ¼ Deyawmax SyawDe ¼ YawUniversemax ;

ð1Þ

where the universe is in a range of 1801 to +1801 (the positive sign is for port and the negative sign is for starboard turns). The depth controller is edepthmax Sdepthe ¼ Dedepthmax SdepthDe ¼ DepthUniversemax ; ð2Þ where the universe is in a range of 10 m. In both cases the scale is normalised in the range [1,1]. It should be noted that the depth controller starts using the range of its universe when the robot is within 10 m of the set point. In any other case the control output has the maximum value. 4.1.2. Input/output linguistic variable—MFs The choice of the shape of the antecedent MFs is triangular ðm1 ðxÞ; m2 ðxÞ; y; mn ðxÞÞ with a specific overlap of 50% to ensure that each value of the universe is a member of at least two sets, except possibly for elements at the extreme ends. This means that the height of the intersection of the two successive fuzzy sets is hgtðmi -miþ1 Þ ¼ 1=2: The selection of MF has two important characteristics: one is its optimal interface design and the other is its semantic integrity (Pedrycz, 1993). The first characteristic refers to error-free reconstruction, where in the fuzzy system the numerical values are converted into linguistic values by means of fuzzification and in the defuzzification method the linguistic values can be reconstructed in the same numerical value, i.e. 8xA½a; b : f 1 ½f ðxÞ ¼ x: The second characteristic refers to three main design aspects: justifying the number and the labels of MFs, distinguishability and completeness as described in the following paragraphs. In the first aspect, the number of the input/output fuzzy sets is seven. This number comes from the recommendation that the number of the sets should be compatible with the number of ‘‘quantifiers’’ that human beings can handle which actually is within the limit of 772 distinct terms (Espinosa & Vandewalle, 1997). Thus, each of the FLC blocks contains 49 rules. The input/output variables of GARBI’s FLC are quantified into sets of classes defined by linguistic labels such as ‘‘Positive Big’’ (PB), ‘‘Positive Small’’ (PS), ‘‘Zero’’ (ZO), etc. These variable in the premise parts are fuzzy, while in the consequence part are singleton with values between 0 and 1. By using MFs in the input and

singletons in the output of GARBI’s control system, the actual Takagi–Sugeno fuzzy system approach (Takagi & Sugeno, 1985) is utilised. Mamdani and Assilian (1975) control approach is not used due to its computational complexity during the defuzzification procedure which is time consuming. It is well known, however, that fuzzy rules with singletons can be used without loosing the performance of the control (Sugeno & Yasukawa, 1993). It is therefore recommended for real-time fuzzy control applications to use singletons in the output resulting in simpler and faster control action (Jantzen, 1998; Nguyen & Prasad, 1999). In the second aspect, distinguishability, each of the linguistic labels should have semantic meaning and the fuzzy sets should clearly define a range in the universe of discourse. So, the MFs should be clearly different. The assumption of the overlap equal to 0.5 assures that the support of each fuzzy set will be different. The distance between the model values of the MFs defined as the a— cut with a ¼ 1miða¼1Þ ðxÞ; i ¼ 1; 2; y; N; assures that the MFs can be distinguished. The third aspect, completeness, express the ability of the fuzzy control algorithm to infer a control action with confidence not less than a minimal level e; for which the threshold e—cuts of all terms covering the interval universe. Decreasing this parameter decreases the fuzziness of the partitioning of the input space of the FLC. Based on heuristic considerations and the 50% overlapping of the MFs, the level of completeness is defined as e ¼ 0:5: This actually is a fixed structure for initial setting of most FLCs. 4.1.3. Construction of the rule (knowledge) base The construction of the rule base is based on the template rule-base method that is regarded as a basic tool uniting the common engineering sense and experience in fuzzy logic control. MacVicar-Whelan (1977) developed this type of rule-base template which was introduced in the first FLC, (Mamdani & Assilian, 1975; King & Mamdani, 1977). The MacVicar-Whelan rulebase summarises the rules used in the rule bases of these FLCs, and in addition includes situations (combinations of linguistic labels of input and output variables of the FLC) that were not defined. Thus, the rule-base template used in the FLC part of GARBI’s Fuzzy like-PD controller is as presented in Table 1. The cell defined by the intersection of the first row and the first column 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 is given by mu ¼ Minðme ; mDe Þ: Thus, the Fuzzy-like PD

I.S. Akkizidis et al. / Control Engineering Practice 11 (2003) 471–480 Table 1 The rule base of a Fuzzy-like PD in tabular form e=De

NB

NM

NS

ZO

PS

PM

PB

NB NM NS ZO PS PM PB

NB NB NB NB NM NS ZO

NB NB NB NM NS ZO PS

NB NB NM NS ZO PS PM

NB NM NS ZO PS PM PB

NM NS ZO PS PM PB PB

NS ZO PS PM PB PB PB

ZO PS PM PB PB PB PB

controller is a controller where the output is a non-linear function of the error e and its derivative de=dtðu ¼ F ðe; de=dtÞÞ; where F is a non-linear function of two variables. 4.1.5. Defuzzification The control signal results from the defuzzification method that uses the degree of membership functions of the antecedent and the singleton of the conseqeuences of the MFs obtained by XR XR u¼ x m = m; ð3Þ i i i¼1 i¼1 i where mi is the degree of MF, xi is the singleton’s value and 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 the Fuzzy-like PD control systems is to optimise its SFs. SFs play an important role in the formation of the dynamics of the closed-loop structure leading to the desired response of the controlled system (Zheng, 1992). The importance of an optimal choice of input SFs is evidently shown by the fact that inappropriate scaling set values is either shifting the operating area to the boundaries or utilising only a small area of the normalised universe of discourse. Additionally, the adjustments of the output SF affects the closed-loop gain, which has direct influence on stability and oscillation tendency. There is no general method to optimise the SFs in the Fuzzy-like PD control systems. Most successful results reported are based on the combination of expert understanding about the controlled object and the use of the analogies between the FL and PID controllers (Zheng, 1992). There are some general directions to optimise the SFs (Reznik, 1997; Procyk & Mamdani, 1979), however, their effectiveness is bounded by the contradictory requirements resulting from different performance measures. For GARBI’s Fuzzy-like PD controller, the corresponding SFs are for the inputs Se ; SDe and for the output Su ; respectively. A systematic approach based on

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design of experiments is being proposed and applied to optimise GARBI’s controllers SFs as described in the next sections.

5. Designing the experiments to obtain the SFs of GARBI’s Fuzzy-like PD Controller A large amount of engineering effort is consumed in conducting experiments to produce the information required to make decisions about how different factors affect performance under different usage conditions. A systematic methodology of how to identify the optimum values of Fuzzy-like PD controllers in terms of control performance is proposed and developed herein. The proposal is based on the combination of the widely used method 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 each individual factor is varied while all the other factors are fixed. This is known as the full factorial method that investigate all possible combinations of all factors and their levels, where the possible combinations can rise to the order of yx ; x being the number of factors and y the different levels. This approach investigates all the possible combinations, maximising the possibility of finding the optimum result, but large numbers of experiments are required and thus time consuming and costly. Alternatively, orthogonal arrays, extensively used in the Taguchi method, studies several factors at different levels simultaneously, but only require a fraction of the full factorial combinations. The orthogonal array imposes an order on the way the experiment is carried out. The combinations are chosen to provide enough information to determine the factor effects using the analysis of mean values. In order to use a standard orthogonal array provided by the Taguchi method, the degrees of freedom (number of independent measurements available to estimate sources of information) of the factors and levels must be matched with the degrees of freedom for that orthogonal array (Taguchi, 1987). The SFs are the parameters/factors of both yaw and depth Fuzzy-like PD controllers of GARBI have been defined. When a SF is changed, it is assumed that the definition of each membership function will be changed by the same ratio. Hence, changing of any SF can change the meaning of one part, the IF-part or THENpart, in any rule. Therefore, it can be said that the change of SFs may affect all of the control rules Table 1. Three factorial levels are chosen initially for all SFs of both yaw and depth controllers. The choice of their initial values is based on min, max and intermediate value that excite the response of the system (i.e., Se ¼ f0; 5; 0; 75; 1g; SDe ¼ f0; 5; 1; 2g; Su ¼ 3; 7; 10).

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Note that, increasing the number of the levels increases the number of experiments. This is very important issue as increasing the number of experiments is both time consuming and costly. Moreover, the definition of the above levels indicates the initial values used 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 factor level experiment, six degrees of freedom exists, so an orthogonal array with nine experimental runs can be employed instead reducing the number of experiments (Phadke, 1989). Table 2 shows the orthogonal array for both yaw and depth experiments that is sufficient for this study.

6. Conducting the experiments (in a real environment) and optimising the SFs Experimental trials were undertaken in Lake Banyoles, Spain to test both depth and yaw Fuzzy-like PD controllers. The power of the propellers is controlled with power cards for voltage tuning. The voltage used was in the range of 3–10 V. For heading control, the opposite voltage between the horizontal propellers (T1 ; T2 ) is used i.e. +V1, V2. So, if the heading angle is turned to a1 clockwise, for instance, the voltage in the right propeller T1 is reduced and the voltage in the left propeller T2 is increased by this amount. 6.1. Initial experiments and results Using Table 2, the experiments to investigate both yaw and depth control performances were undertaken. The navigation plan for the yaw and depth experiments was *

initial voltage of the horizontal and vertical propellers is set to 3 V for a period of 60 s to ensure that the vehicle goes straight ahead. Equal power to the

Table 2 Orthogonal array with nine experiments at three levels for each SF

*

*

horizontal propellers are employed and the vehicle moves away from the platform in the water, manoeuvring with set point of 1801. After the manoeuvring, the task is to keep the vehicle moving in the same direction, and at the same time changing the depth course from 0 to 10 m and then to 5 m and then keeping it at this depth.

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 in terms of the integral absolute-error (IAE) pIAEyaw ; pIAEdepth and integral-of-time-multiplied by absoluteerror (ITAE) pITAEyaw ; pITAEdepth performance criteria. The analysis of means (Phadke, 1989) is the method that is used to investigate the possible optimal levels of the SFs (optimal in this case is the levels that minimises the above integrals). Analytically, for each factor level, the mean responses are obtained as in Table 3. These responses together with the levels are the co-ordinates that construct the plots (Figs. 6 and 7) used in the analysis of the graphical representation of factor levels. As the aim of the controller is to minimise both IAE and ITAE, the objective characteristic of these target values is ‘‘smaller-the-better’’. Therefore, for the yaw response shown in the IAE and ITAE mean response plot (Fig. 6) the smallest yaw mean (average) responses are the 9.79, 11.25, and 9.31 for Sec ; SDec Suc and 354.79, 453.03, and 387.37 for Sec ; SDec Suc ; respectively. Thus, the possible optimal levels of the SFs that minimise the IAE and ITAE are {0.5,0.5,7} and (1,0.5,7}, respectively. For the depth response shown in the IAE mean response plot (Fig. 7) the smallest ‘‘depth mean responses’’ for Sez ; SDez and Suz are 15.93, 21.89 and 15.52, respectively. Thus, the possible optimal levels of the SFs that minimise the IAE are {0.5,0.5,10}. Note here that these factor levels are not defined as a combination in

Table 3 Results of yaw and depth performance in terms of IAE and ITAE from the real experiments Depth performance

Yaw performance

No. exp.

Se

SDe

Su

No. exp.

pIAEdepth

pITAEdepth

pIAEyaw

pITAEyaw

1 2 3 4 5 6 7 8 9

0.5 0.5 0.5 1 1 1 0.75 0.75 0.75

0.5 1 2 0.5 1 2 0.5 1 2

3 7 10 7 10 3 10 3 7

1 2 3 4 5 6 7 8 9

29.92 8.515 9.36 20.86 22.32 80.62 14.89 84.17 21.68

633.25 137.58 64.64 294.22 1265.25 4190.8 1118.72 5111.58 314.2

11.38 9.06 8.914 7.9 9.87 14.32 14.4 16.95 10.92

547.33 556.41 290.28 141.2 277.8 645.37 670.55 729 464.5

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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).

the experiment shown in Table 2, however, because of the orthogonality of the Taguchi method, combinations of factor levels not appearing in the standard orthogonal array table can be predicted by the method (Taguchi, 1987). This is one of the most important features of the above approach, that is, to be able to identify experimental combinations that were not originally specified in the orthogonal array. For the ITAE means plot 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 SFs that 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 of their actual optimal values but their direction to the subspace that they belong. Using the Fuzzy combined scheduling system as briefly explained in Appendix A the SFs is tuned to the final optimal level. Thus, for the yaw controller the final optimal levels are Sec ¼ 0:5; SDec ¼ 0:5; Suv ¼ 7 and for

the depth controller the final optimal levels are Sez ¼ 0:51; SDez ¼ 0:48; Suz ¼ 8:05:

6.3. Final experiments and results Applying these SFs, new experiment was held following the same navigation scenario as in Section 6.1 to verify the applicability of the final optimal SFs values. The new performances of the controllers in course-changing and course-keeping are shown in Figs. 8–11. Fig. 8(a) illustrates the yaw, Fig. 8(b) the controller output of the power horizontal propellers (T1 ; T2 ) (note that as mentioned before they have opposite sign), Fig. 8(c and d) the error and the change of error between the desired and the actual yaw. Additionally, Fig. 9(a) shows the depth, Fig. 9(b) the controller output of the power of the vertical propellers (T3 ; T4 ), Fig. 9(c and d) the error and the change of error between the desired

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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 10 m and then to 5 m and then keep depth at this level.

and depth. Finally, Figs. 10 and 11 shows the corresponding pitch and roll motions. The performance of the yaw controller in changing the course (heading/depth) is satisfactory, as both

overshoot and rise time are small. Due to buoyancy effects the depth control dynamics vary, i.e. the vehicle rises faster than it descends. From Fig. 9 it can be seen that the controller has accommodated this variation

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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.

producing very acceptable rise times with a very small overshoot 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 appears mostly due to the heading changes. These are very small and can be considered as fractional and thus negligible.

7. Conclusions The proposed approaches presented in this paper were applied to define the Fuzzy-like PD controllers for GARBI underwater robot. The design of GARBI’s controller involves course-changing and course-keeping for both steering and depth-control performances. The architecture of the controller was based on two independent Fuzzy-like PD controllers for each controlled variable, yaw and depth. The design aspects of the FLC as well as for the ‘‘PD’’ part for these types of controllers were presented. It was shown that the flexible structure of the FLC leaves the designer to decide about the input/output universe of discourse and linguistic variables in terms of their shape, number and meaning, the construction of the rule base and the meaning of their operators and the defuzzification method. The above decisions were guided from particular design aspects coming from the

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fuzzy system theory as well as from the expert’s knowledge in terms of system’s dynamics and behaviour identification. Moreover, the gains of the input/output SFs of the PD part of the controller were defined according to their global effect on the dynamics of the closed-loop control system. Although the SFs were defined based on some general instructions coming mostly from the classical PID design theory, a more systematic procedure to investigate and optimise its controller’s parameters was introduced. This optimisation was based on experiments in a real environment that were planned using the Taguchi design of experiments method. It was shown that the Taguchi design of experiment method could help to minimise the number of experiments, using only nine experiments in the orthogonal array, without the risk of losing vital information. This was very important as the experiments were held in a real environment, where time and money is an issue. The results of the experiments were presented and analysed extensively. By analysing the mean response of the performance criteria measurements, it was shown that 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 that performs satisfactorily since both course-changing and course-keeping performance is within the desired response with minimal error. It has been shown therefore, that Fuzzy-like PD controllers designed, optimised and tuned by the proposed approaches possesses features that are attractive in navigation control problems posed by underwater vehicles. It has also been shown in this paper, how a fuzzy controller combined with conventional PD control techniques can help to design Fuzzy-like PD controller, dealing with the uncertainties and non-linearities of an underwater vehicle.

Acknowledgements The authors wish to acknowledge the support for this work 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 the subspace that the actual optimal values belong, the Fuzzy combined scheduling system approach drives

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The overall final optimal factor level vector resulting from the composite current and possible optimal factor levels defined by equation LoF ¼ fLoF1 ; LoF2 ; y; LoFj g where LoFj is the final optimal level for each factor j; calculated as the weighted mean described in the following equation: Pm ðwpok lfpo j þ wck lfc j Þ LoFj ¼ k¼1 ; ðA:1Þ Pj k¼1 ðwpok þ wck Þ 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/or minimised. More analytical explanation of this approach can be found in Akkizidis (2000). Fig. 12. The low and high performance state membership functions where the shapes are Z and S, respectively, with 50% overlap.

the possible optimal factor levels to the final optimal ones. The discriminant function w ¼ f ðLowMF ; HighMF Þ combines the current factor level with the possible optimal ones. This function defines the weight vector of both current and possible optimal factor levels w ¼ fwc1 ; wpo1 ; wc2 ; wpo2 ; y; wcj ; wpoj g constructed from the weight values of the fuzzy sets of the performance states. Therefore, the construction of the fuzzy rules that defines the relationship between the performance state in the 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 degree of the current and the possible optimal factor levels are related to the degree of the low and high membership functions (MFs). The shapes of the MFs, as well as their linguistic labels may vary and depend mostly on the set criteria. The overlapping between them is 50% to ensure that both rules are excited for each performance state values. Fig. 12 shows the setting of these two MFs based on the criterion to minimise the value of a performance state. Therefore, when the performance state increases, the weight of the current level should reduce, whereas the weight of the possible optimal factor level should increase accordingly and vice-versa. If the discriminant function is such that wA½0; 1 and more than one w is different from 0, then the transition from one factor level to another will be smooth. Therefore, each of the overall factor levels will be optimal even for a non-linear control system, since the weights themselves are functions of the performance states.

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