Background and Motivation

The development in the control area has been fueled by three major needs: the need to deal with increasingly complex systems, the need to accomplish increasingly demanding design requirements, and the need to attain these requirements with less precise advanced knowledge of the plant and its environment [1]. Hence, many researches are interested in some intelligent control design or intelligent systems to attain these needs.

In the past two decades, fuzzy systems have replaced conventional technologies in many scientific applications and engineering systems, especially in control systems. Fuzzy sets, introduced by Zadeh in 1965 [2] as a mathematical way to represent vagueness in linguistics, can be considered a generalization of classical set theory. Fuzzy sets are a generalization of conventional set theory and contain objects that belong imprecisely to the set. The degree of belonging is defined by the value of a membership function, which usually has values between 0 and 1. One of the biggest differences between crisp and fuzzy sets is that the former always have unique membership functions, whereas every fuzzy set has an infinite number of membership functions that may represent it. Fuzzy logic control (FLC) system, which induces human experience and human decision-making behavior, has been developed over 20 year. In the design of a FLC system, the sensory variables are converted into the fuzzy numbers by membership functions and they are matched with the preconditions of linguistic IF-THEN rules (fuzzy logic rules) and then the response of each rule is obtained through fuzzy computation. As a result, it will generally lead to fuzzy outputs. Finally, the fuzzy outputs are inverted into a crisp result to obtain the appropriate control signal. One major feature of fuzzy logic is its ability to express the amount of ambiguity in human thinking and subjectivity. In summary, the advantages of fuzzification include greater generality, higher expressive power, an enhanced ability to model real-world problems, and a methodology for exploiting the tolerance for imprecision. Hence, this algorithm provides a way of representing the uncertainties in a complex model. However, system designers must spend more time to ascertain how many rules are best [3] and fuzzy systems do not have

much learning capability [4].

The concept of neural network (NN) was first proposed by McCulloch and Pitts in 1943 [5]. NNs are a new generation of information processing systems that are deliberately constructed to make use of some of the organizational principles. They have a large number of highly interconnected processing elements (nodes) that usually operate in parallel and are configured in regular architectures. A NN has a massively parallel and distributed structure that is composed of many simple processing elements i.e., artificial neurons with nonlinear mapping functions. The neurons in a NN can communicate with each other through the links i.e., weights between the neurons [6]. The collective behavior of an NN is like a human brain to demonstrate the ability to learn, recall, and generalize from training patterns or data. NNs offer the salient characteristics and properties, such as nonlinear input-output mapping, generalization, adaptation, fault tolerance, and evidential response etc. Therefore, the NN has been applied to various areas [7-9]. However, because the internal layers of neural networks are always opaque to the user, the mapping rules in the network are not visible and are difficult to understand. The convergence of learning is usually very slow and not guaranteed [4].

Recently, the fuzzy neural network (FNN), which incorporates the advantages of fuzzy inference and neuro-learning, has been an interesting topic. Fuzzy logic and NNs are complementary technologies in the design of intelligent systems. The FNN possesses the merits of the low-level learning and computational power of NN, and the high-level human knowledge representation and thinking of fuzzy theory [4, 10]. Due to their learning ability, FNNs are increasingly receiving attention in solving the control problems [11-14]. Hence, the FNN will be a focus of our researches. Although the neuro-learning structure can tune membership functions and fuzzy rules automatically, the structure of the FNN should be determined in advance by trial-and-error. It is difficult to consider the balance between the rule number and the desired performance. As a result, if the number of fuzzy rules is chosen too large, the computation loading is heavy so that it is not suitable for practical applications.

If the number of fuzzy rules is chosen too small, the control performance may be not good enough to achieve the desired performance.

To solve the problem of determining the structure in FNN approaches, much interest has been focused on the self-structuring fuzzy neural network (SFNN) approach [15-19]. The self-structuring approach demonstrates the properties of automatic generating rules for FNN without needing preliminary knowledge. In general, the mathematical description of the existing rules can be expressed as a set of clusters. As usually seen in other self-structuring

approaches, the new membership function is generated when a new input signal is too far from the current clusters, and an existing rule is deleted when the fuzzy rule is insignificant.

SFNNs also have been adopted widely for the control of complex dynamic systems due to their good generalization capability, structural adaptation, and simple computation [20-25].

Some of them use the gradient descent method to derive the parameter learning algorithms;

however, they can’t guarantee the system stability [22, 23]. Some of them derive the parameter learning algorithms based on the Lyapunov function to guarantee system stability;

however, the structure learning algorithm is too complex [20, 24, 25]. Some of them proposed a simple growing-and-pruning algorithm to online self-structure the FNN with symmetric membership functions; however, the bounds of parameters are not stated [21].

In addition, system identification also plays an important role in control field. It is an important task for control engineer to acquire system information so as to design a proper control law based on a good understanding of the plant under consideration and its environment. It has been clear that a mathematical description of a plant is often a prerequisite for system analysis and controller design in control system theory. System identification, whether online or offline, is an essential part of any control system design. The processes of system identification mainly consists of two steps: the first is to choose an appropriate identification model and the second is to adjust parameters of the selected model according to some derived adaptive laws so that the output of the selected model can approach the response of the real system under the same input [26]. Hence, the nonlinear system identification process has turned out to be one of central parts in various control researches.

Recent research results show that NN techniques seem to be very effective to identify a wide class of complex nonlinear systems when the complete model information can not be available [27-29]. NNs have been an interested focus because they have good learning, noise-tolerance, and generalization abilities to solve the nonlinear problem. According to the used types of NNs, they can be qualified as static (feed-forward) or as dynamic (recurrent) nets. The first one deals with the class of global optimization problems. The universal approximation property of static NNs makes them be a useful tool for modeling nonlinear systems. The designers try to adjust weights of such NNs to achieve favorable performance.

The second approach, which converts the partial learning (training) focuses to an adequate feedback design, permits to avoid many problems related to global extremum search [30].

When outputs are directed back as inputs to the same or the preceding layer node, the network is a feedback network. Feedback networks that have closed loop are called recurrent networks.

From a system theoretical point of view, multilayer networks represent static nonlinear maps

while recurrent networks are represented by nonlinear dynamic feedback systems [27].

However, an important viewpoint is that static NNs are unable to represent dynamic system mapping without the aid of tapped delay, which results in long computation time, high sensitivity to external noise, and a large number of neurons when high dimensional systems are considered [31, 32]. This drawback severely affects the applicability of static NNs to system identification, which is the central part in some control techniques for nonlinear systems. Dynamic neural networks (DNNs) can deal with this disadvantage since they have dynamic memory, which makes them more suitable for representing dynamic systems than static NNs. Hence, if the mathematical model of a considered process is incomplete or partially known, the DNN approach provides an effective instrument to research a wide spectrum of problems such as identification, state estimation, trajectories tracking, etc. [33].

Recurrently connected NNs, sometimes called Hopfield neural networks (HNN), which is a special kind of DNN, have been extensively studied in recent years. The HNN is first proposed by Hopfield J.J. in 1982 and 1984 [34, 35]. Because of the easy implementation of the HNN circuit, the characteristic of decreasing in energy by finite number of node-updating steps, and the dynamical behavior of the networks, the HNN has found many applications in different areas, such as optimization [36, 37], system identification [38, 39], and image processing [40, 41]. However, in [38, 39], the system identification via HNN involved a learning process which has no guarantee for convergence.