Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 198371, 18 pages

http://dx.doi.org/10.1155/2015/198371

## Synchronization of Discrete-Time Chaotic Fuzzy Systems by means of Fuzzy Output Regulation Using Genetic Algorithm

^{1}Instituto Politécnico Nacional, SEPI-ESIME Zacatenco, Avenida IPN S/N, 07738 México, DF, Mexico^{2}Instituto Politécnico Nacional, SEPI-ESIME Azcapotzalco, Avenida de las Granjas No. 682, 02250 Azcapotzalco, México, DF, Mexico

Received 7 August 2015; Accepted 4 November 2015

Academic Editor: Rongwei Guo

Copyright © 2015 Tonatiuh Hernández Cortés et al. 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

The synchronization of chaotic systems, described by discrete-time T-S fuzzy models, is treated by means of fuzzy output regulation theory. The conditions for designing a discrete-time output regulator are given in this paper. Besides, when the system does not fulfill the conditions for exact tracking, a new regulator based on genetic algorithms is considered. The genetic algorithms are used to approximate the adequate membership functions, which allow the adequate combination of local regulators. As a result, the tracking error is significantly reduced. Both the Complete Synchronization and the Generalized Synchronization problem are studied. Some numerical examples are used to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

A special nonlinear dynamical phenomenon, known as chaos, emerged in mid-1960s and reached applicable technology in the late 1990s and was considered as one of the three monumental discoveries of the twentieth century. On the other hand, fuzzy logic, a set theory and then an infinite-valued logic, gets a wide applicability in many industrial, commercial, and technical fields, ranging from control, automation, and artificial intelligence, just to name a few. Fuzzy logic and chaos had been considered by many researches and engineers as fundamental concepts and theories and their broad applicability in technology as well. The interaction between fuzzy logic and chaos has been developed for the last 20 years leading to research topics as fuzzy modeling of chaotic systems using Takagi-Sugeno models, linguistic descriptions of chaotic systems, fuzzy control of chaos, synchronization, and a combination of fuzzy chaos for engineering applications [1, 2].

In the 1960s, Rechenberg [3] introduced “evolution strategies,” a method to optimize real-valued parameters for devices such as airfoils. This idea was further developed by Schwefel in [4]. Genetic algorithms (GAs) were initially developed by Bremermann [5] in 1958 but popularized and developed by Holland in the 1960s. In contrast with evolution strategies and evolutionary programming, Holland's idea was not to design algorithms to solve specific problems but rather to formally study the phenomenon of adaptation, as it occurs in nature, and develop ways in which the mechanisms of natural adaptation might be transferred into computer systems [6]. The genetic algorithm is presented as abstraction of biological evolution and theoretical framework for adaptation for moving from one population of “chromosomes” (e.g., strings of ones and zeros, or “bits”) to a new population by using a kind of “natural selection” together with the genetics-inspired operators of crossover, mutation, and inversion. Each chromosome consists of “genes” (e.g., bits); each gene is being an instance of a particular “allele” (e.g., 0 or 1). The operator selection chooses those chromosomes in the population that will be allowed to reproduce and those adjusted chromosomes produce more offspring than the less ones [7].

According to Fogel and Aderson [8], Bremermann was the first to implement real-coded genetic algorithms as well as providing a mathematic model of GA known as the one-max function. In contrast to genetic algorithms, Evolutionary Strategies were initially developed for the purpose of parameter optimization. The idea was to imitate the principles of organic evolution in experimental parameter optimization for applications such as pipe bending or PID control for a nonlinear system [9].

Synchronization of chaotic systems is one of the more exiting problems in control science and can be referred at least to Huygen’s observations [10]; it is understood as one of the trajectories of two autonomous chaotic systems, starting from nearly initial conditions and converging to the other, and remains as ; in [11] it was reported that some kind of chaotic systems possesses a self-synchronization property. However, not all chaotic systems can be decomposed in two separate responses subsystems and be able to synchronize the drive system. The ideas of these works have led to improvement in many fields, such as communications [12], encrypted systems, the complex information processing within the human brain, coupled biochemical reactors, and earthquake engineering [13].

Synchronization can be classified as follows: Complete Synchronization: it is when two identical chaos oscillators are mutually coupled and one drives to the other; Generalized Synchronization: it differs from the previous case by the fact that there are different chaos oscillators and the states of one are completely defined by the other; Phase Synchronization: it occurs when the coupled oscillators are not identical and have different amplitude that is still unsynchronized, while the phases of oscillators evolve to be synchronized [14]. It is worth mentioning that studies in synchronization of nonlinear systems have been reformulated based on the previous results from classical control theory such as [15–18].

In this paper, the fuzzy output regulation theory and Takagi-Sugeno (T-S) fuzzy models are used to solve the Complete and Generalized Synchronization by using linear local regulators. Isidori and Byrnes [19] showed that the output regulation established by Francis could be extended for a nonlinear sector as a general case, resulting in a set of nonlinear partial differential equations called Francis-Isidori-Byrnes (FIB). Unfortunately these equations in many cases are too difficult to solve in a practical manner. For this reason in [20] the approach based on the weighted summation of local linear regulators is presented and in [21] the new membership functions in the regulator are approximated by soft computing techniques.

So, the main contribution of the present work is to find a control law for synchronizing of chaotic systems described by discrete-time Takagi-Sugeno fuzzy models, first when the system fulfills the following: () the input matrix for all subsystems is the same and () the local regulators share the same zero error manifold . In this way, the results given in [20] are extended to the discrete-time domain. On the other hand, when the system* master-slave* does not fulfill the aforementioned conditions, new membership functions are computed in order to enhance the performance of the fuzzy regulator. Such proposed membership functions are different from those given in the plant or exosystem and are tuned by using the GA. The tuning of the new membership functions, which is as generalized bell-shaped function, is given by optimization of the form parameter.

The rest of the paper is organized as follows. In Section 2 the discrete-time output regulation problem formulation is given with a brief review of the Takagi-Sugeno models and the discrete-time fuzzy regulation problem. In Section 3 the tuning of membership functions by means of GAs is thoroughly discussed. In Section 4 Complete and Generalized Synchronization with some examples are presented and finally, in Section 5, some conclusions are drawn.

#### 2. The Discrete-Time Fuzzy Output Regulation Problem

Consider a nonlinear discrete-time system defined bywhere is the state vector of the plant, is the state vector of the exosystem, which generates the reference and/or the perturbation signals, and is the input signal. Equation (5) refers to difference between output system of the plant () and the reference signal (); that is, and take into account that . Besides, it is assumed that , , and are analytical functions and also that , , and [22].

Clearly, by linearizing (1)–(5) around , one getsThus, the Nonlinear Regulator Problem [19, 23] consists of finding a controller , such that the closed-loop system has an asymptotically stable equilibrium point, and the solution of system (6) satisfies .

So, by defining as the steady-state zero error manifold and as the steady-state input, the following theorem gives the conditions for the solution of nonlinear regulation problem.

Theorem 1. *Suppose that is Poisson stable and there exists a gain such that the matrix is stable and there exist mappings and with and satisfyingThen the control signal for the nonlinear regulation is given by*

The equation set (7) is known as Discrete-Time Francis-Isidori-Byrnes (DTFIB) equations and linear counterpart is obtained when the mappings and transform into and , respectively. Thus, the problem is reduced to solve linear matrix equations [25] given by

##### 2.1. The Discrete-Time Output Fuzzy Regulation Problem

Takagi and Sugeno proposed a fuzzy model composed of a set of linear subsystems with IF-THEN rules capable of relating physical knowledge, linguistic characteristics, and properties of the system. Such a model successfully represents a nonlinear system at least in a predefined region of phase space [15]. The T-S model for the plant and exosystem can be described as follows [26]:

*Plant Model* Rule : IF is and and is ,where is the number of rules in the model of the plant and the sets are the fuzzy sets defined based on the previous dynamic knowledge of the system.

*Exosystem Model* Rule : **IF ** is and and is ,where is the number of rules in the model of the exosystem and are the fuzzy sets.

Then, the regulation problem defined by (1)–(5) can be represented through the T-S discrete-time fuzzy model; that is, [20]where is the state vector of the plant, is the state vector of the exosystem, is the input signal, , and is the normalized weight of each rule, for the plant and for the exosystem, which depends on the membership function for the premise variable in ; that is,with as a function of and/or , and .

The* discrete-time fuzzy output regulation problem* consists of finding a controller , such that the closed-loop system with no external signal has an asymptotically stable equilibrium point.

The solution of system (12) satisfies

In order to achieve the synchronization of chaotic systems described by a T-S discrete-time fuzzy model it is necessary to fulfill (7) [27, 28]. Thenwhere is the zero error steady-state manifold which becomes invariant by the effect of the steady-state input .

Assuming the mappings and asrespectively, with and as a solution of lineal local problems,for all and , the following control law can be obtained [20, 22, 23]:However, by substitution of and in (16) and (17) and considering(1)the steady-state zero error manifold , that is, ,(2)the input matrices and/or ,for all and , the following control signal emerges:

On the other hand, the existence of a fuzzy stabilizer of the form , ensuring that the tracking error converges asymptotically to zero, can be obtained from the Parallel Distributed Compensator (PDC) [29, 30] or another stability analysis for T-S fuzzy models such as [31].

*Remark 2. *The control signal in (22) is given by the substitution of (18) and (19) in (16) and (17); the proposed controller provides the following advantages:(1)All parameters included in the controller are known; this includes the membership functions of the plant and exosystem, which are well defined in the T-S fuzzy model. On the other hand, and come directly from solving of local linear problems equivalent to solving the Francis equations; such problems can be easily solved by using programs like Matlab or Mathematica.(2)In the case when the local linear problems lead to , then at least a bounded error is ensured.(3)It is clear that when , the term changes to leading to controller defined in (23).(4)The following condition: the input matrices and/or avoids the crossed terms in the solutions of (16) and (17), allowing the exact fuzzy output regulation.(5)The proposed controller can be seen as a simple substitution of the aforementioned elements.On the other hand, the following disadvantages can appear: (1)If the condition is that the steady-state zero error manifold , that is, , then, it will be necessary to adjust the local regulator by means of new membership functions. Please refer to Section 3.(2)As expected, the complexity of the controller increases according to the number of local subsystems.

The following theorem provides the conditions for the existence of the exact fuzzy output regulator for a discrete-time T-S fuzzy models.

Theorem 3. *The exact fuzzy output regulation with full information of systems defined as (12) is solvable if (a) there exists the same zero error steady-state manifold ; (b) there exist for the fuzzy system; (c) the exosystem is Poisson stable, and the input matrices for all subsystems are equal. Moreover, the Exact Output Fuzzy Regulation Problem is solvable by the controller*

*Proof. *From the previous analysis, the existence of mappings and is guaranteed when the input matrices for all subsystems are equal, and the solution of lineal local problems isleading to for all and .

On the other hand, the inclusion of condition (b) has been thoroughly discussed in [22, 23, 27, 28, 31–33], and it implies the existence of a fuzzy stabilizer.

Condition (c) ensures the nonexistence of crossed terms in the local Francis equations. Finally, condition (d) is introduced to avoid the fact that the reference signal converges to zero, which would turn the regulation problem into a simple stability problem. The rest of the proof follows directly from the previous analysis.

#### 3. The Output Regulator by means of Local Regulators and Tuning of New Membership Functions

In this section, a discrete-time T-S fuzzy model is considered to solve the exact output regulation on the basis of linear local controllers. So, the main goal is to find a complete regulator based on the fuzzy summation of local regulators using adequate membership functions, such that the result given in Theorem 3 can be relaxed [34]. These membership functions are not necessarily the same included in the fuzzy plant, as is described in (19). Thus, the steady-state input can be defined aswhere and are new membership functions, such that the fuzzy output regulator obtained from local regulators provides the exact fuzzy output regulation. This approach requires the computation of the linear local controllers and the computation of the new membership functions. In this work, such functions are represented by the following expression: and are well known as* generalized bell-shaped membership functions* and the parameters , , , , , and determine the form, center, and amplitude, respectively. Therefore, from (22), the input can be defined bybecause is a function of and in steady-state .

Then, for tuning the membership functions (27) and (28), the parameters and will be optimized by means of genetic algorithms, ensuring the correct interpolation between the local linear regulators. The foregoing can be summarized in the control scheme depicted in Figure 1.