A Simplified Adaptive Neuro-Fuzzy Inference System -ANFIS- Controller Trained by Genetic Algorithm to Control Nonlinear Multi-Input Multi-Output Systems

Scientific Research and Essays Vol. 6(31), pp. 6475-6486, 16 December, 2011 Available online at http://www.academicjournals.org/SRE DOI: 10.5897/SRE11.1059 ISSN 1992-2248 2011 Academic Journals

Full Length Research Paper

A simplified adaptive neuro-fuzzy inference system (ANFIS) controller trained by genetic algorithm to control nonlinear multi-input multi-output systems

Omar F. Lutfy*, Samsul B. Mohd Noor and Mohammad H. Marhaban

Department of Electrical and Electronic Engineering, Faculty of Engineering, University Putra Malaysia, Serdang 43400, Selangor, Malaysia.

Accepted 24 August, 2011

This paper presents a simplified adaptive neuro-fuzzy inference system (ANFIS) controller to control nonlinear multi-input multi-output (MIMO) systems. This controller uses only few rules to provide the control actions, instead of the full combination of all possible rules. Consequently, the proposed controller possesses several advantages over the conventional ANFIS controller especially the reduction in execution time, and hence, it is more appropriate for real time control. A real-coded genetic algorithm (GA) was utilized to optimize the premise and the consequent parameters of the ANFIS controller, instead of the hybrid learning methods that are widely used in the literature. Accordingly, the necessity for the teaching signal required by other optimization techniques has been eliminated. Furthermore, the GA was employed to determine the input and output scaling factors for this controller, instead of the widely used trial and error method. Two nonlinear MIMO systems were chosen to be controlled by this controller. In addition, the controller robustness to output disturbances was also investigated and the results clearly showed the notable accuracy and the generalization ability of this controller. Moreover, the result of a comparative study with a conventional MIMO ANFIS controller has indicated the superiority of the simplified MIMO ANFIS controller. Key words: Neuro-fuzzy systems, ANFIS, genetic algorithms, nonlinear MIMO systems.

INTRODUCTION The nonlinear universal function approximation property of fuzzy logic (FL) systems and artificial neural networks (ANNs) qualifies them to be powerful candidates for identification and control of nonlinear dynamical systems.

As a result, these techniques have been successfully applied to solve various kinds of problems in control *Corresponding author. E-mail: [email protected]. Abbreviations: ANFIS, Adaptive neuro-fuzzy inference system; MIMO, multi-input multi-output; GA, genetic algorithm; FL, fuzzy logic; ANNs, artificial neural networks; LSE, least square estimator; SISO, single-input single-output; PID, proportional integral derivative; PI, proportional integral; FIS, fuzzy inference system; MF, membership function; UOD, universe of discourse; ISE, integral square of errors.

system design resulting in the emergence of intelligent control systems. However, each of these two intelligent techniques has its own drawbacks which limit its usefulness for certain situations and not for others. For instance, fuzzy logic controllers suffer from some problems like the selection of appropriate membership functions, the selection of fuzzy if-then rules, and furthermore how to tune both of them to achieve the desired performance. On the other hand, ANNs have some problems such as their black-box nature, the lack of knowledge representation power, and the selection of the proper structure and size to solve a specific problem.

In order to overcome these drawbacks, the integration of both FL systems and ANNs in a unified system has received great attention in the literature which resulted in the appearance of a rapidly emerging field of neuro-fuzzy systems. One of the most widely used neuro-fuzzy

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6476 Sci. Res. Essays systems is the ANFIS network, which was proposed by Jang (Jang et al., 1997; Jang, 1993). The hybrid learning rule, which combines the gradient descent technique and the least square estimator (LSE), was proposed in (Jang, 1993) to optimize the premise and the consequent parameters of the ANFIS network, respectively. However, as a supervised learning method, this hybrid learning rule requires a teaching signal. This teaching signal can not be easily provided in some situations. For example, in control system design, when the objective is to utilize the ANFIS network as a feedback controller, the teaching signal represents the desired control actions which are simply unknown.

In order to alleviate this difficulty, some ANFIS learning approaches were proposed in the literature. One of these learning approaches to train the ANFIS as a controller is represented by the inverse learning method in which the ANFIS network is trained to acquire the inverse dynamics of the plant to be controlled. This approach has been successfully employed to control nonlinear MIMO systems (Toha and Tokhi, 2009; Yao and Chai, 2007; Zhou and Jagannathan, 1996). However, the efficiency of this ANFIS learning approach depends on three crucial issues. Firstly, the availability of an accurate model for the process to be controlled, which can not be easily developed for complex systems; Secondly, the existence of the inverse dynamics of the system, which is not always available; and thirdly, the way the training data are distributed. More precisely, it is desired to make the training data uniformly distributed across the controller input space. However, this data distribution may not be achieved for all the systems due to either the small number of data or the constraints imposed by the system dynamics.

Another learning method to train the ANFIS as a MIMO controller was proposed in (Djukanovi et al., 1997). The basic idea of this method, which was called temporal back propagation, is to consider both the controller and the plant as a single unit at each time step. Several ANFIS controllers were adopted to handle different operating points and a neural network classifier was used to settle on the best controller for a particular operating point. However, this ANFIS training method is characterized by its heavy computational load and the complexity in design and implementation.

In addition to the above mentioned methods, a different training technique in training the ANFIS as a controller for nonlinear MIMO systems was utilized in (Mahmoud et al., 2010; Cao et al., 2007; Touati et al., 2002). This technique is based on training the ANFIS network by using data which are collected from another working controller. The disadvantage of this method is that, the effectiveness of the resulting ANFIS controller depends heavily on the performance of the original controller, which is mostly a linear one.

One of the most powerful approaches to train a neuro-fuzzy system to act as a feedback controller is

represented by using derivative-free methods as the optimization technique to train the controller. These derivative-free methods do not need the teaching signal information in training the neuro-fuzzy system as a controller. The genetic algorithm is one of the most widely used derivative-free techniques and many researchers have successfully utilized it in various control applications to solve the teaching signal problem.

For example, some researchers used binary-coded GA to tune a neuro-fuzzy controller utilizing off-line learning algorithms (Chou, 2006; Seng et al., 1999). In addition, they used the grid partitioning method to define the structure of the neuro-fuzzy controller. However, the grid partitioning method leads to the exponential increase in the number of fuzzy rules as the number of input variables increases. Moreover, the binary-coded GA requires encoding and decoding operations to construct the chromosomes of the GA which result in some resolution problems, the increment in chromosome lengths, and the need to select the values of some user-defined upper and lower limits of the genes.

Zhou and Lai (2000) used real-coded GA to train a fuzzy controller to control a linear single-input single-output (SISO) system. They also used the grid partitioning method to define the structure of the fuzzy controller. Their control system consisted of a combination of a classical proportional-integral-derivative (PID) controller as the master controller and the fuzzy controller as the slave controller to enhance the performance of the PID controller.

Serra and Bottura (2006) proposed a gain scheduling adaptive control scheme based on a genetically-tuned fuzzy proportional-Integral (PI) controller.

It is worth highlighting that, in (Chou, 2006; Serra and Bottura, 2006; Zhou and Lai, 2000; Seng et al., 1999) the controller was used to control SISO systems only. Hence, the applicability of these control algorithms to control MIMO systems is questionable.

In order to handle the teaching signal problem in this work, a real-coded GA, which does not use a teaching signal, is utilized to optimize the premise and the consequent parameters of a simplified ANFIS structure, which was proposed in (Lutfy et al., 2011; Lutfy et al., 2010) to control (SISO) systems. In the present work, this controller is trained to act as a PID-like feedback controller for nonlinear MIMO systems. This simplified ANFIS controller requires less training time and memory resources compared to the conventional ANFIS controller. Moreover, the input and output scaling factors for this simplified ANFIS controller are also determined by the real-coded GA. Structure of the simplified pid-like adaptive neuro-fuzzy inference system (ANFIS) controller In a conventional fuzzy reasoning system, if there are n

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Figure 1. Structure of the simplified PID-like ANFIS controller.

input variables each of which is partitioned into p fuzzy sets, then p

n fuzzy rules are required to construct the

corresponding fuzzy inference system (FIS) for a particular application. As a fuzzy reasoning system, the ANFIS network utilizes the Sugeno fuzzy models, which can be either zero-order or first-order models depending on the consequent part of the fuzzy rules. For both of these models, the total number of fitting parameters in the ANFIS structure is given by:

f(n,p,m)=n.p.m+p

n.(n+1) for the first-order Sugeno fuzzy

model (1)

f(n,p,m)=n.p.m+p

n for the zero-order Sugeno fuzzy

model (2)

Where m is the number of fitting parameters per membership function (MF) (Jovanovi et al., 2004). In this work, the zero-order Sugeno fuzzy model is used since it requires less number of parameters compared to the first-order Sugeno fuzzy model. Therefore, according to Equation 2, if three input variables, seven fuzzy sets per input, and two fitting parameters per MF are used, then the resulting number of parameters will be 385. Obviously, this number of parameters will complicate the task of the optimization technique in finding the optimal values for these parameters in terms of accuracy and computational complexity, not to mention the case of MIMO systems which requires more ANFIS parameters. The so called curse of dimensionality can be evidently seen from the p

n term included in Equations 1 and 2. In

order to overcome this dimensionality problem, the ANFIS structure (Lutfy et al., 2011; Lutfy et al., 2010) shown in Figure 1 is used in this work.

In this structure, the input variables 1x , 2x , and 3x are

the error of the plant being controlled, the error rate of change, and the summation of errors respectively, while the output variable, y, is the control action of this ANFIS controller. Seven MFs are used for each of the three input variables to the ANFIS controller. This number of MFs was found to be adequate to achieve satisfactory control results, where increasing this number into eleven terms did not improve the performance significantly. On the other hand, when five terms for each input variable were used the control accuracy has deteriorated. In the following, to describe the function of each layer in the ANFIS structure, the output of the ith node in layer k is expressed as Ok,i. Layer 1: Each node in this layer is used to assign the degree of membership for each input variable to the ANFIS network. In particular, the function of each node is defined by the following:

21...,,16,15),(

14...,,9,8),(

7...,,2,1),(

3,1

2,1

1,1

14

7

ixO

ixO

ixO

i

i

i

Ai

Ai

Ai

(3)

The bell-shaped activation functions are used in this work to be the MFs for each input variable. These bell-shaped

6478 Sci. Res. Essays activation functions are defined by the following expression:

2)(2/1exp)(

k

a

k

a

a x

A

x

Ak

kA

Cxx

(4)

where xk, k = 1, 2, 3, are the scaled input variables after multiplying them by the input scaling factors (c1, c11, and

c111) as illustrated in Figure 1. k

a

x

AC and k

a

x

A are the

centers and the widths of the MFs, respectively, and they are known as the premise parameters. Layer 2: Each node in this layer generates the firing strength of the corresponding fuzzy rule utilizing the multiplication operation. As can be seen from Figure 1, this simplified ANFIS structure uses only seven fuzzy rules in this layer, instead of the 343 rules used in the conventional ANFIS structure (that is, p

n , where p = 7

and n = 3), see Equation 2. Consequently, the output of Layer 2 is expressed by:

7,...,2,114,17,1,1,2 iOOOO iiii (5)

Layer 3: This layer has seven nodes as well. The ith node in this layer determines the ratio of the ith rules firing strength to the sum of the firing strengths of all the rules, as expressed by the following:

7

1

,3

j

j

iii

w

wwO where i=1 to 7. (6)

Layer 4: Similar to Layers 2 and 3, this layer has also seven nodes. Each node in this layer multiplies the corresponding output from Layer 3 by the consequent parameter of the ANFIS structure. The output of each node is given by:

iii kwO 0,4 (7)

where i=1 to 7, iw is the ith output from Layer 3, and k0i is

the ith consequent parameter of the ANFIS structure. Layer 5: This layer includes only one node which calculates the overall output as follows:

7

1

,45

i

iOO (8)

Finally, O5 is multiplied by the output scaling factor (c2) to determine the final output of this ANFIS controller, as

expressed by the following:

52 Ocuy (9)

For the SISO case, 53 parameters are required to represent this simplified ANFIS structure in the GA (49 modifiable parameters plus 4 scaling factors). In this work, two ANFIS controllers are used to control the MIMO plants. Each of these two controllers has exactly the same structure described above. As a result, 106 parameters are used to represent the simplified MIMO ANFIS controller in the GA. The real-coded genetic algorithm (GA) implementation to train the simplified multi-input multi-output (MIMO) adaptive neuro-fuzzy inference system (ANFIS) controller The MATLAB software was used in this work to implement the real-coded GA for training the simplified MIMO ANFIS controller using a specific program which has been written in an m-file. As mentioned before, each chromosome in the GA consists of 53 genes which represent all the ANFIS parameters. These genes include 3 genes for the input scaling factors, 1 gene for the output scaling factor, 42 genes for the premise parameters, and 7 genes for the consequent parameters of the ANFIS structure. The universe of discourse (UOD) for each input variable was selected to be from -6 to 6, keeping in mind that other range can also be used since there are input and output scaling factors. Four real-coded GA operators were used in this work. These operators include the hybrid selection, the elitism, the crossover, and the mutation operators. The hybrid selection operator, which was proposed in (Al-Said, 2000), is a combination of the roulette wheel and the deterministic selection. In this operator, only the chromosomes that have better fitness values than the worst individual in the old population are accepted in the new population. In the elitism operator, the best two chromosomes in a given generation are copied directly into the next generation as they are in order to prevent the best fitness value in a given generation from becoming worse than that in the previous one (Sivanandam and Deepa, 2008). In the crossover operator, a pair of selected chromosomes exchanges their information by exchanging a subset of their components, where an integer position k is selected uniformly at random along the chromosome length. Then two new chromosomes are generated by swapping all the genes from positions k+1 to L, where L is the chromosome length (Sivanandam and Deepa, 2008). Finally, the real-coded mutation operator causes random changes in the components of the new population of chromosomes by replacing the mutated gene with another random number chosen in the same range

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assigned for that gene.

The procedure of the real-coded GA adopted in this work to train the simplified MIMO ANFIS controller is summarized in the following steps: Step 1: Initialize the crossover probability (Pc), the mutation probability (Pm), the population size, and the maximum number of generations. Step 2: Generate randomly the initial population of chromosomes within certain limits. Each of these chromosomes represents the entire premise and consequent parameters along with the input and output scaling factors for one MIMO ANFIS controller. Step 3: Evaluate the objective function for each chromosome in the population using the integral square of errors (ISE) criterion, which has the following form:

)()(5.05.0 220

2

1 kekeISET

k

(10)

where e1(k) and e2(k) are the error between the desired output and the plant output at sample k for each of the two outputs of the system (that is, e1(k) = r1(k) y1(k), e2(k) = r2(k) y2(k)), and T is the observation time. For the current application, and after a fair amount of simulation tests, the 0.5ISE was found to be the best performance index in training the simplified MIMO ANFIS controller to control nonlinear MIMO systems in this work. Then, for each chromosome in the current population, determine the fitness function using the Darwinian fitness equation which has the following form:

functionobjectivefitness

1 (11)

where is a small constant used to avoid division by zero. Step 4: Put in descending order all the chromosomes in the current population (that is, the first one is the fittest). Then apply the elitism strategy described before. Step 5: Select two individuals from the current population utilizing the hybrid selection method, and then apply the real-coded crossover and mutation operators described previously to generate two new chromosomes.

Step 6: Put the resulting two chromosomes in the new population.

Step 7: Repeat Step 5 until all the chromosomes in the new population are generated, that is, until the new population size is equivalent to the initial (old) population size.

Lutfy et al. 6479 Step 8: Replace the initial (old) population with the new population. Step 9: Stop if the maximum number of generations is reached, and the first chromosome in the last generation is the optimal controller found by the GA, otherwise increase the generation counter by one and go to step 3. RESULTS AND DISCUSSION In order to evaluate the effectiveness of the proposed simplified MIMO ANFIS controller, two nonlinear MIMO plants have been chosen to be controlled by this controller. Unlike the control of SISO systems in (Lutfy et al., 2011; Lutfy et al., 2010), the simplified MIMO ANFIS controller has to deal properly with the variable interactions in the MIMO systems considered in this work. In this regard, the objective of the following simulation tests is to investigate the controller ability in decoupling of loop-interactions while at the same time tracking the given testing signals. Therefore, the training and the testing signals were deliberately selected as shown in Figures 2 (a, b) and 3 (a, b), respectively. From these figures, it can be seen that there is a difference between the training and the testing signals. This difference is important in order to evaluate the generalization ability of the simplified MIMO ANFIS controller. The parameters of the real-coded GA were set to the following values; population size: 50, maximum number of generations: 300, Pc: 0.8, and Pm: 0.05. For the current application, and after several simulation tests, these parameter settings were found to be adequate to guarantee a satisfactory performance in training the simplified MIMO ANFIS controller by the GA to control the following nonlinear MIMO plants: Plant 1: A nonlinear MIMO plant is described by the following (Narendra and Parthasarathy, 1990):

)1(

)1(

)1(1

)1()1(

)1(1

)1(

)(

)(

2

1

2

2

21

2

2

1

2

1

ku

ku

ky

kyky

ky

ky

ky

ky (12)

Figure 3 shows the output responses, control actions and the best 0.5ISE against the generations for each output of this plant.

As can be seen from Figures 3a and b, the simplified MIMO ANFIS controller has done well in both tracking and de-coupling in controlling this nonlinear MIMO plant. More specifically, the tracking of the testing signals was achieved with zero steady-state error and with some overshoots at the beginning of each step change. Figure 3c shows a sharp control signal when the testing signal has changed its amplitude. This sharp control action is

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necessary to compensate for the interactions between the two channels of this nonlinear MIMO system, and consequently, to achieve the good tracking performance in Figure 3a. On the other hand, Figure 3d shows that the transition in the second control signal was not as large as that of the first control signal. Bearing in mind the difference between the training and the testing signals, it can be concluded that this controller has achieved good generalization ability. From Figure 3e, it can be seen that the 0.5ISE has reached its near optimal value from the first few generations. Therefore, the selection of the maximum number of generations to be 300 seems to be adequate. This fact indicates the fast convergence to the optimal solution that has been achieved by the GA in training the simplified MIMO ANFIS controller. Plant 2: This nonlinear MIMO plant is described by the following (Petlenkov, 2007):

)2(2.0)1()2(5.0)2()1(1

))2(sin()1(5.0)(

)2(2.0)1()2(3.0)2()1(1

)2()1(7.0)(

1222

1

2

2

222

2112

2

2

1

111

kukukukyky

kykyky

kukukukyky

kykyky (13)

Figure 4 shows the simulation results of controlling this nonlinear MIMO plant.

In spite of the complexity of this MIMO plant, the performance of the simplified MIMO ANFIS controller was good in de-coupling of loop interactions and in tracking the desired testing signals with zero steady-state error and some oscillations at the start of each change in the testing signals (Figures 4a and b). Even though the training signals are different from the testing signals, the simplified MIMO ANFIS controller has successfully generalized its learning to effectively handle the unexpected testing signals. This performance for the controller is due to its control signal given in Figures 4c and d. Similar to the first plant, the 0.5ISE has reached to its near optimal value from the first few generations, as can be seen from Figure 4e.

Robustness test

In order to assess the robustness of the simplified MIMO ANFIS controller, a disturbance rejection test was conducted on Plant 2 using the training and the testing signals of the previous tests. This robustness test was done by applying a bounded external disturbance of 20%

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Figure 3. Plant 1 (a) first testing input and plant output (b) second testing input and plant output (c) first control signal (d) second control

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of the first output and -20% of the second output during only the testing phase, which means that the controller was not trained to handle these disturbances during the training phase. The first disturbance was applied for 20

samples at the interval ( 4930 k ) while the second one was applied for 20 samples at the interval

( 149130 k ) of the two testing signals. Figure 5 shows the two plant outputs together with the two control

actions of the simplified MIMO ANFIS controller for this test.

Referring to Figures 5a and b, it can be seen that the simplified MIMO ANFIS controller has overcome the external disturbances, during their effect and after their disappearance as well, during the two disturbance periods. Figures 5c and d illustrate the adaptations in control actions achieved by the ANFIS controller to

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Figure 5. Plant 2 subjected to 20% external disturbances; a, first testing input and plant output;

b, second testing input and plant output; c, first control signal; d, second control signal.

Lutfy et al. 6485 Table 1. Comparison results of the conventional and the simplified MIMO ANFIS controllers

Controlled plant

Conventional MIMO ANFIS

controller

Simplified MIMO ANFIS

controller

Average Training

0.5ISE

Average testing 0.5ISE

Average time (sec.)

Average Training

0.5ISE

Average testing 0.5ISE

Average time (sec.)

Plant 1 0.422 0.345 652.526 0.200 0.108 89.858

Plant 2 0.184 0.102 649.471 0.168 0.083 90.125

handle these disturbances. This result indicates the ability of the simplified MIMO ANFIS controller to deal appropriately with external disturbances.

A comparative study with the conventional multi-input multi-output (MIMO) adaptive neuro-fuzzy inference system (ANFIS) controller

As previously mentioned, only few rules, in particular p rules, were utilized in the rule base of the simplified ANFIS controller to provide the control actions, instead of the full combination of all possible rules, that is, the p

n

rules, resulted from applying the grid partitioning method to partition the input space in the conventional ANFIS structure, where n is the number of input variables and p is the number of fuzzy sets for each input variable. As a result, the simplified ANFIS controller has several advantages over the conventional ANFIS controller, particularly the reduction in processing time without sacrificing the controller performance, as will be seen in this section. In this regard, it is important to compare the performances of the simplified and the conventional MIMO ANFIS controllers, and this section is devoted for this purpose. The structure of the simplified ANFIS controller was described at the beginning of this paper. This structure requires 53 genes in the GA to optimize the ANFIS parameters to control SISO systems, and 106 parameters to control MIMO systems in this work. The conventional MIMO ANFIS controller has exactly the same structure as that of the simplified MIMO ANFIS controller except that the former requires 343 fuzzy rules (that is, p

n rules, where p = 7 and n = 3). Therefore, each

of Layers 2, 3, and 4 of the conventional ANFIS controller consist of 343 nodes. According to Equation 2, 385 parameters are required to optimize the parameters of the conventional ANFIS controller to control SISO systems. On the other hand, for MIMO systems, two of the above conventional ANFIS controllers are required. In order to achieve a fair comparison, all the tests in this study were done under identical conditions for the two controllers. Specifically, the same training and testing signals considered before were used for all the comparison tests. In addition, the real-coded GA, with exactly the same operators and settings, was used to train the two controllers. Plants 1 and 2 described before were used to conduct this comparative study. Due to the

stochastic nature of the GA, the result obtained from one simulation run might be different from that of other runs. For this reason, and in order to achieve a more reliable comparison study, five runs were conducted for each controller to control each plant and the average objective function and average time for each of these five runs were considered as the basis in this comparison. Table 1 above summarizes the results of comparing the simplified MIMO ANFIS controller with the conventional MIMO ANFIS controller to control the two plants.

The main advantage of the simplified MIMO ANFIS controller, which is the reduction in training time, can be easily seen from Table 1 for the two plants. At the same time, by carefully examining Table 1, it can be concluded that the simplified MIMO ANFIS controller not only outperforms the conventional MIMO ANFIS controller in terms of training time but also in terms of control accuracy. More specifically, the average training and testing objective functions are less for the two plants in the case of the simplified MIMO ANFIS controller. In fact, the average testing 0.5ISE indicates the generalization ability of a given controller, and as can be seen from Table 1, this average testing 0.5ISE was 0.345 and 0.102 for the conventional ANFIS controller against only 0.108 and 0.083 for the simplified ANFIS controller in controlling Plants 1 and 2, respectively. This result confirms the fact that, as the controller complexity increases, its generalization ability deteriorates, as was highlighted in (Mascioli and Martinelli, 1998).

It is worth mentioning that although the control algorithm utilized in this work is based on an off-line design technique, the training time of the simplified MIMO ANFIS controller proposed in this work is dramatically less than that of the conventional MIMO ANFIS controller. This short processing time is a very desirable attribute in designing an on-line training method for the controller. Therefore, an on-line training algorithm, utilizing the same real-coded GA used in this work, is currently being developed to provide an on-line self-tuning method for the simplified MIMO ANFIS controller, and hence to promote the off-line design method presented in this work into an adaptive learning approach.

Conclusions

A genetically-trained simplified PID-like ANFIS controller

6486 Sci. Res. Essays was utilized in this work to control nonlinear MIMO systems. The GA does not require a teaching signal in its operations and hence, it is a more suitable technique to train the ANFIS network as a feedback controller in this work compared to the commonly used method to train the ANFIS namely, the hybrid learning method. Moreover, the real-coded GA was used to determine the optimal settings for the controller scaling factors, instead of the widely used trial and error method. The simulation results showed the effectiveness of the real-coded GA in training the simplified MIMO ANFIS controller in terms of control accuracy and generalization ability. Moreover, from the robustness test, this controller has shown a notable ability in eliminating the effects of external disturbances during their effect and after their disappearance as well. In addition, from a comparative study with a conventional MIMO ANFIS controller, the simplified MIMO ANFIS controller has shown its superiority in terms of the reduced training time and the control accuracy. REFERENCES Al-Said IAM (2000). Genetic algorithms based intelligent control. PhD

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