A new type of recurrent fuzzy neural network for modeling dynamic systems
Introduction
As an important part of intelligent control, fuzzy control is the quintessence that researchers explore for mimicking and coordinating human intelligence. We have witnessed rapidly growing interest in the theoretical research and engineering applications of fuzzy systems control in the last three decades [4], [7], [10].
The conventional methods of designing control systems start with mathematical models that describe the linear or nonlinear dynamic behaviors of systems. A variety of methods for designing fuzzy control system have been introduced. However, many issues regarding fuzzy control need to be further addressed. In fact, they have received a lot of attention. Mathematical modeling plays an important role in control system design. This role can be classified into two categories. One is the explicit model of physical plant, such as linear and nonlinear differential equations, as well as transfer functions or other unconventional methods. The other is the implicit one. Assuming a modeler has captured the key characteristics of a system, she/he will be able to establish a model for the system, although it may not be in the form of an explicit model. For example, if a controller is designed based on the knowledge of the experienced human operator, the operator's understanding of the physical plant is implicitly used. In fact, numerous successful fuzzy controllers are designed based on the knowledge of experts and experience of operators without use of the explicit model. It is obvious that modeling is significant in any methods for designing control systems.
It is well known that fuzzy control has always been a controversial topic, especially in the control community. Many researchers in the control community are skeptical about fuzzy control. Some of the criticism originates from the fact that the fuzzy controllers have not made use of an explicit model [3]. Not using the explicit model inevitably brings about the system design lack of stability assurance.
In comparison with traditional control theory, fuzzy control has integrated the implicit model into control system design. However, new problems including the stability of closed loop systems are arising [6]. It is difficult to avouch for the stability and robustness for closed loop systems. This indicates that the gap between the mode of fuzzy control and the expectation for modeling and analysis is getting wider.
In order to narrow down the gap and respond to the concern, more researchers in fuzzy control community attempted to introduce mathematical modeling to fuzzy control [3], [5]. Controllers have been designed based on modeling and those methods adapted from modern control theory that allow both qualitative and quantitative information integrate well. The fuzzy control not relying on mathematical modeling was called the ‘classical fuzzy control’, while the control based on the methods of modeling the ‘modern fuzzy control’. The new labeling clearly indicates the research direction of fuzzy control. The objective of this paper is to propose a fuzzy dynamical model (FDM) for studying complex systems.
Despite many successful applications of fuzzy control, the research efforts that have been made on useful mathematical tools and methods for analyzing fuzzy dynamical behavior of complex systems through FDM are still limited due to its complexity and nonparameterization. Fuzzy dynamical behavior analysis (FDBA) of complex systems can be conducted in conjunction with other methods and techniques. Dynamical neural networks including recurrent neural network (RNN) provides a new tool for FDBA [11], [12], [13]. More research efforts have recently been made on the topic [1], [2]. Two key problems need to be further addressed. The network architecture and formalization of FDM, i.e. the dynamical feedback in FDBA, should be integrated into the network design corresponding to FDM.
Order derivative is a powerful mathematical tool that can provide RNN with an important and useful method for building FDM and analyzing fuzzy dynamical behavior of complex system. In Section 2, FDM is represented by NN. Section 3 describes the training process of the network. Section 4 gives the simulation result.
Section snippets
FDM with RNN
To explore the intrinsic properties of FDM with RNN, it is necessary to design a highly effective training algorithm. The concept of ordered derivatives introduced in the following is central for deriving such algorithms.
Training algorithm of RFNN
The training algorithm given hereinafter is composed of premise parameter learning and consequent parameter learning. These are derived in detail in Section 5. In the FDS described by , , the fuzzy variables of the premise part are characterized by the premise parameters.
Letwhere denote the parameters defining those membership functions' shape, position, etc. The consequent parameters are referred to as
Simulation
Suppose the dynamical plant that has been studied is highly nonlinear and can be described by the difference equation,where u(k) and x(k) are the external input and output, respectively, and x(k−1) and x(k−2) are the two state feedback signals.
The RFNN shown as follows will be trained to capture the dynamic property of the plant:where the fuzzy variables in premise part are assumed to be the Gauss function, the
Conclusions
From the perspectives of artificial intelligence, the task of modeling the complex control aspects of a plant is actually a method of representing deep knowledge. Such deep knowledge includes the intrinsic knowledge about the control aspects of a plant. For a complex system involving uncertainties and fuzziness, it is difficult to employ conventional modeling methods as well as acquiring deep knowledge. In contrast with deep knowledge, shallow knowledge can be the simple knowledge about the
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