Computational Intelligence and Neuroscience

Volume 2017, Article ID 8942394, 9 pages

https://doi.org/10.1155/2017/8942394

## Extracting T–S Fuzzy Models Using the Cuckoo Search Algorithm

Research Unit of Industrial Systems Study and Renewable Energy (ESIER), National Engineering School of Monastir (ENIM), 5019 Monastir, Tunisia

Correspondence should be addressed to Mourad Turki; rf.oohay@ikrutssedaruom

Received 18 March 2017; Revised 15 May 2017; Accepted 23 May 2017; Published 6 July 2017

Academic Editor: Paolo Gastaldo

Copyright © 2017 Mourad Turki and Anis Sakly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new method called cuckoo search (CS) is used to extract and learn the Takagi–Sugeno (T–S) fuzzy model. In the proposed method, the particle or cuckoo of CS is formed by the structure of rules in terms of number and selected rules, the antecedent, and consequent parameters of the T–S fuzzy model. These parameters are learned simultaneously. The optimized T–S fuzzy model is validated by using three examples: the first a nonlinear plant modelling problem, the second a Box–Jenkins nonlinear system identification problem, and the third identification of nonlinear system, comparing the obtained results with other existing results of other methods. The proposed CS method gives an optimal T–S fuzzy model with fewer numbers of rules.

#### 1. Introduction

To control any system, it is necessary to obtain an exact model of it but in many cases, it has not enough information to get an acceptable mathematical model, and it is required to use modelling techniques based on input–output data.

Fuzzy models are used due to their excellent performance in the modelling of nonlinear systems and being easy to implement. A fuzzy model is constructed from a basis of rules formed by inputs and outputs of a system [1].

The Takagi–Sugeno (T–S) fuzzy model is a type of fuzzy models which is able to give a local linear representation of the nonlinear system. Such a model is able to approximate a wide class of nonlinear systems because they are considered powerful in modelling and control of complex dynamic systems.

A T–S fuzzy model is powerful if it allows obtaining highly accurate models from a small number of rules but the majority of works in literature provide a large number of rules.

Many works concerning the T–S fuzzy models especially discrete-time type are done in literature such as in [2, 3]. The optimization of T–S fuzzy models is to determine the structure and parameters of model. The methods used to tune the antecedent and consequent parameters are the clustering algorithms [4] and the linear least squares [5–7]. The design of a fuzzy model is a search problem where each point represents a possible fuzzy model with different structures and parameters [1, 8]. In the aim to obtain optimal fuzzy models, many evolutionary algorithms (EAs) such as genetic algorithms (GAs) [9], genetic programming [10], evolutionary programming [11], evolution strategy [12], and differential evolution (DE) [13] have been used [14]. These methods tune the parameters and the structure is predefined.

Though, all parameters of model such as structure and parameters are linked and should be optimized simultaneously. Thereby, in [1] the authors have presented the optimization of rule structure where all the information is encoded into a chromosome.

The particle swarm optimization (PSO) is a novel metaheuristic algorithm used recently in many domains [15]. The PSO algorithm is used to elicit fuzzy models such as in [16] in which PSO optimize the structure, the number of membership functions, and the singleton consequent parameters. In [17], the results of PSO and GA are compared for the same method given in [16] with fixed number of rules and membership function for the same example of simulation.

The structure of the fuzzy model is identified using an online fuzzy clustering method and the parameters are optimized by PSO [20]. In [21], the fuzzy model is extracted using PSO with the recursive least-squares method. The ant colony and PSO were used to obtain T–S fuzzy model in [22]. Lin [23] used immune-based symbiotic PSO to obtain T–S models for the prediction of the skin color detection. In [24], Niu et al. used multiswarm PSO to tune fuzzy systems parameters. In [25], the subtractive clustering is utilized to extract a set of fuzzy rules and a variant of PSO called CPSO algorithm in the aim to find the optimal membership functions and the consequent parameters. In [19], the CRPSO is employed to tune all parameters of the fuzzy models. In [26], the GA is used for learning the T–S fuzzy model from data with a new encoding scheme.

DE and quantum inquired differential evolution is utilized to learn the T–S fuzzy model in [27]. T–S fuzzy models are also developed for modelling industrial systems such as the moving grate biomass furnace in [28].

Other metaheuristic named hunting search (HUS) is used in [18] to determine the parameters of the T–S fuzzy model.

A recent metaheuristic algorithm named the cuckoo search (CS) is proposed in 2009 by Yang. The CS algorithm has given best results compared to other metaheuristics such as PSO and GA. The CS is used to learn neural networks [29] and in the reliability optimization [30]. In [31], the CS is used to tune parameters of Sugeno-Type Fuzzy Logic Controller for liquid level control. Optimizing fuzzy controller using CS in the case study of computer numerical control of a steam condenser is done in [32]. In [33], a prediction of academic performance of student based on the CS is proposed. In [34, 35], the cuckoo search is used in the reduction of high order. In [36], Hammerstein model trained by the cuckoo search algorithm is proposed to identify nonlinear system. In [37], CS is used in the structural damage identification.

The goal and the motivation of this contribution is to obtain a T–S model with minimum number of rules by using a CS method. The objective is to have a model with less complexity able to be easily implemented in embedded systems and with a minimum of errors which proves its efficiency and precision.

This paper describes the use of CS to obtain the optimal structure in terms of a number of rules and also the parameters premises and consequents of T–S model simultaneously in the aim to explore the advantages of CS in optimization which are better compared to other metaheuristics in many examples in literature. The optimal T–S fuzzy models extracted are compared on the same examples with other methods in terms of MSE and number of rules.

This paper is organized as follows. Section 2 describes the structure of T–S fuzzy system. The CS algorithm is introduced in Section 3. Section 4 explains the encoding scheme of T–S model method used. Section 5 presents the different metaheuristic algorithms. Section 6 presents the application examples with results and discussions. Finally, conclusions are given in Section 7.

#### 2. T–S Fuzzy Model

T–S fuzzy model presented in [38] is given by the following basis of rules.

*Rule *. If is and and is , thenwhere , is the number of rules, represents the input, is the size of input, are the consequent parameters, is the fuzzy rule output, and is a fuzzy subset.

The output of the model* y* is obtained as follows:where of the rule is computed aswith being the membership function’s grade of and is characterized by a Gaussian function as and are, respectively, the mean and the deviation of the MF. The premise parameters and are adjusted [20].

#### 3. Cuckoo Search

CS is developed by Yang and Deb in 2009 [39–41]. CS imitates the parasitism of cuckoo and used Lévy Flights which are better than random walks [32].

The CS has the specificity that the cuckoos lay their eggs in the nests of host birds. Some cuckoos can mimic the properties of the host eggs.

As a result, the number of the eggs abandoned is reduced and their reproductivity increases [42].

The CS models are used in many optimization problems. In [40, 43], it is demonstrated that Lévy Flights are better than random walk in CS algorithm.

In CS algorithm, the solution is given by an egg in a nest, and a new solution is represented by a cuckoo egg. The goal is to use the newer cuckoos or solutions to substitute worst solutions. In the classical algorithm, each nest has one egg; however the algorithm can be extended to complex problem [40, 43].

In the cuckoo search, the rules are as follows:(i)Every cuckoo lays a single egg and throws it in a random nest.(ii)The best nests or solutions will be transferred to the next generations.(iii)The number of host nests is predefined, and a host can detect a stranger egg with probability . In that event, the host bird skips the egg or abandons the nest and builds a new nest in another place [43].The CS algorithm is described in Pseudocode 1 [43].