**5. Illustrative training examples of fuzzy control**

### **5.1 Fuzzy control**

This section presents the tests with fuzzy controls that have had their relevance adjusted using meta-heuristic methods. These tests demonstrate the efficiency of such mechanisms, allowing an objective assessment of results found. The original relevance functions are shown in Figure 11. This control has 148 rules, 15 functions relevant for the *x* and *y* input variables and car angle, and 7 output functions of angle of the wheel.

Table 2 shows the training results for the fuzzy functions shown in Fig. 11. Three initial positions have been used in this test. This table has the number of iterations that are generated by the vehicle to park using the original relevance functions.


Table 2. Initial positions for training and number of iterations

Figure 12 shows the vehicle in each of the initial positions. These positions were chosen according to the points where the vehicle doesn't develop a good trajectory until park and therefore generating an excessive number of iterations. The main idea is setting several

The evaluation function has the role to assess the level of fitness (adaptation) of each chromosome generated by algorithms. The problem goal is to minimize the trajectory of the

> 1 <sup>1</sup> *<sup>f</sup> <sup>I</sup>*

where *I* is the total number of iterations until the final position into the park lot. According to the fitness function, the fitness of each chromosome is inversely proportional to the

The integration of meta-heuristic training algorithms with fuzzy model has made as follow:

3. To check the performance of the fuzzy system it is rolled up from an initial set of

4. This information is used for set up each individual adjustment (adaptability) and the

5. The cycle repetition is made up to complete the defined meta-heuristic method iteration number made by the user. To each meta-heuristic method iteration the best values set

This section presents the tests with fuzzy controls that have had their relevance adjusted using meta-heuristic methods. These tests demonstrate the efficiency of such mechanisms, allowing an objective assessment of results found. The original relevance functions are shown in Figure 11. This control has 148 rules, 15 functions relevant for the *x* and *y* input

Table 2 shows the training results for the fuzzy functions shown in Fig. 11. Three initial positions have been used in this test. This table has the number of iterations that are

Position X Y Angle of the car Iterations without training

1 2.5 12.0 180 330 2 16.0 13.0 -90 888 3 27.5 16.0 -40 655

Figure 12 shows the vehicle in each of the initial positions. These positions were chosen according to the points where the vehicle doesn't develop a good trajectory until park and therefore generating an excessive number of iterations. The main idea is setting several

1. The individual is defined as a link of the membership functions adjustment values. 2. The parameters are the centers and widths of each fuzzy set. These parameters compose

(13)

vehicle to be parked. In case the evaluation function is given by:

number of iterations.

the individual.

**5.1 Fuzzy control** 

possible parameters.

making of the evolution of the particle.

for the membership functions parameters is found.

variables and car angle, and 7 output functions of angle of the wheel.

generated by the vehicle to park using the original relevance functions.

Table 2. Initial positions for training and number of iterations

**5. Illustrative training examples of fuzzy control** 

initial positions will not only minimize the trajectories for these points, but as well as for other points, thus achieving a global minimization of space covered. Figures 13 show the trajectories for each initial position.

Fig. 11. Original relevance functions

Fig. 12. Initial positions training

An Evolutionary Fuzzy Hybrid System for Educational Purposes 417

sets of fuzzy membership are computed one for each meta-heuristic method of training (GA, PSO and HPSO). For example, the resultant GA fuzzy membership functions after

Fig. 14. Membership functions after the genetic algorithm adjustment

positions could create other fuzzy membership functions.

Position Iterations without

training

Table 5. Iterations after the meta-heuristic training

The generated results by meta-heuristic methods are shown in Table 5. The reduction of 941 iterations (50,2%) for GA training, 1116 iterations (59,6%) for PSO training, and 930 iterations (49,6%) were made for parking the vehicle starting from the three initial positions. These results are not optimal. Other control setups could be chosen in oder to get better results from these three initila start positions. The idea of theses simulations is presented possible adjustements of the fuzzy memberships. Also, ohter silmulations with other initial

Other kind of possible simulation is to verify the quality of the resultant fuzzy membership for other initial position different from the initial position used in the training. Table 6 presents results of simulations results made starting from initial positions not used in the training for 4 types of adjustments of the fuzzy functions: human setting, and GA, PSO and HPSO trainning methods (from the functions setting by the human). The average of results

of meta-heuristic methods are able to improve the better chose of the human being.

Iterations with GA training

1 330 280 285 278 2 888 384 592 402 3 655 277 239 250 Total 1873 941 1116 930 Average 624,33 331.67 372 310

Iterations with PSO training

Iterations with HPSO training

adjustment are shown in Figure 14.

Fig. 13. Simulation results with fuzzy control without training for the following initial position: (a) position 1, (b) position 2, (c) position3

#### **5.2 Meta-heuristic methods training fuzzy control memberships**

The definition of several initial positions will not only minimize the routes referred to these points but also for other points, resulting a global minimization of traveled space. The defined GA and PSO parameters for the training are shown in Tables 3 and 4.


Table 4. PSO parameters

After the training if the algorithm described in Section 4.2 with the fuzzy membership functions presented in Figure 11 and for three initial positions presented in Table 2, three

(a) (b)

(c)

**5.2 Meta-heuristic methods training fuzzy control memberships** 

defined GA and PSO parameters for the training are shown in Tables 3 and 4.

position: (a) position 1, (b) position 2, (c) position3

Table 3. GA parameters

Table 4. PSO parameters

Fig. 13. Simulation results with fuzzy control without training for the following initial

The definition of several initial positions will not only minimize the routes referred to these points but also for other points, resulting a global minimization of traveled space. The

> Population Size 14 Generations Number 30 Crossover Probability 90% Mutation Probability 1%

Size of Population 14 Number of Iterations 30

After the training if the algorithm described in Section 4.2 with the fuzzy membership functions presented in Figure 11 and for three initial positions presented in Table 2, three

Vmax 10

sets of fuzzy membership are computed one for each meta-heuristic method of training (GA, PSO and HPSO). For example, the resultant GA fuzzy membership functions after adjustment are shown in Figure 14.

Fig. 14. Membership functions after the genetic algorithm adjustment

The generated results by meta-heuristic methods are shown in Table 5. The reduction of 941 iterations (50,2%) for GA training, 1116 iterations (59,6%) for PSO training, and 930 iterations (49,6%) were made for parking the vehicle starting from the three initial positions. These results are not optimal. Other control setups could be chosen in oder to get better results from these three initila start positions. The idea of theses simulations is presented possible adjustements of the fuzzy memberships. Also, ohter silmulations with other initial positions could create other fuzzy membership functions.

Other kind of possible simulation is to verify the quality of the resultant fuzzy membership for other initial position different from the initial position used in the training. Table 6 presents results of simulations results made starting from initial positions not used in the training for 4 types of adjustments of the fuzzy functions: human setting, and GA, PSO and HPSO trainning methods (from the functions setting by the human). The average of results of meta-heuristic methods are able to improve the better chose of the human being.


Table 5. Iterations after the meta-heuristic training

An Evolutionary Fuzzy Hybrid System for Educational Purposes 419

optimization or the hybrid particle swarm optimization, substituting for the "try-and-error"

The meta-heuristic methods provided distinctive advantages for the optimization of membership functions, resulting in a global survey, reducing the chances of ending into a local minimum, once it uses several sets of simultaneous solutions. The fuzzy logic supplied the evaluation function, a stage of the meta-heuristic methods where the

The authors would like to express their thanks to the financial support of this work given by

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method, as used before by students for this purpose, with no good results.

the Brazilian research agencies: CNPq, CAPES, and FAPEMIG.

adjustment is settled.

**7. Acknowledgment** 

Netherlands.

Boca Raton, USA.

Boston, USA.

Kaufmann, San Francisco, USA.

**8. References** 


Table 6. Results of simulations for different initial position from the used to training
