Elsevier

Applied Soft Computing

Volume 8, Issue 1, January 2008, Pages 676-686
Applied Soft Computing

Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification

https://doi.org/10.1016/j.asoc.2007.05.006Get rights and content

Abstract

In this paper, a new encoding scheme is presented for learning the Takagi–Sugeno (T–S) fuzzy model from data by genetic algorithms (GAs). In the proposed encoding scheme, the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the antecedent membership function (MF) parameters of the T–S fuzzy model are all represented in one chromosome and evolved together such that the optimisation of rule structure, input structure, and MF parameters can be achieved simultaneously. The performance of the developed evolving T–S fuzzy model is first validated by studying the benchmark Box–Jenkins nonlinear system identification problem and nonlinear plant modelling problem, and comparing the obtained results with other existing results. Then, it is applied to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper of which identification problem is significantly difficult due to its inherently hysteretic and highly nonlinear dynamics. It is shown by the validation applications that the developed evolving T–S fuzzy model can identify the nonlinear system satisfactorily with acceptable number of rules and appropriate inputs.

Introduction

The Takagi–Sugeno (T–S) fuzzy model is a system described by fuzzy IF–THEN rules which can give local linear representation of the nonlinear system by decomposing the whole input space into several partial fuzzy spaces and representing each output space with a linear equation. Such a model is capable of approximating a wide class of nonlinear systems. For the reason that it employs linear model in the consequent part, conventional linear system theory can be applied for the system analysis and synthesis accordingly. And hence, the T–S fuzzy models are becoming powerful engineering tools for modelling and control of complex dynamic systems.

The tasks for learning T–S fuzzy models from data are based on the idea of consecutive structure and parameter identification [1], and main methods such as clustering algorithms, linear least squares, and/or nonlinear optimisation for tuning both antecedent and consequent parameters have been applied. To accommodate new input data, adaptive online learning of T–S fuzzy model has been developed [4] as well. From another point of view, design of a fuzzy model can be formulated as a search problem in multidimensional space where each point represents a possible fuzzy model with different rule structure, membership functions (MFs), and related parameters [11]. Due to the capability of search irregular multidimensional solution, evolutionary algorithms (EAs), such as genetic algorithms (GAs) and evolution strategies (ESs), have been utilised greatly in evolutionary fuzzy modelling [7].

At the beginning of EA-based fuzzy modelling, only parameters of fuzzy models are optimised using EAs while the structure itself is fixed. However, since parameters and rule structure of fuzzy models are co-dependent, they should be designed or evolved simultaneously. Thus, methodologies that try to change the rule structure by encoding all the information into the chromosome have been developed [5]. On the other hand, for a real-world modelling problem, it is very common to have tens of potential inputs to the model. However, the excessive number of inputs not only affects the concise and transparency of the underlying model, but also increases the complexity of computation necessary for building the model. Therefore, it is necessary to do input selection that finds the priority of each candidate input and uses them accordingly [3]. Although some efforts [3], [6], [8], [13] have been done in finding the possible input candidates to the model, the EA-based methods have not been fully explored in this area for doing automatic input selection.

With above discussions, in this paper, an evolving T–S fuzzy model based on GAs is developed. Rather than considering all three factors, i.e. encoding scheme, evaluation method, and evolutionary operation, in the GA-based T–S fuzzy model to improve the fuzzy model performance, the proposed algorithm mainly consists of a new encoding scheme that consists of three parts in one chromosome, which allows simultaneous optimisation of the rule structure (including selection of rules and number of rules), the input structure (including number of inputs and selection of inputs), and the antecedent membership function (MF) parameters. In practice, it will be very difficult to know a priori exactly how many rules are required to be included in the rule set and how many inputs to the model need to be used, only the maximal number of rules or inputs can be guessed or estimated [11]. Hence, in the proposed encoding scheme, we only define the maximum numbers of rules and inputs. The actual numbers of rules and inputs will be evolved with the MF parameters. The evaluation method used in this paper only considers one evaluation criterion, i.e. accuracy, in terms of the mean square error (MSE) between the predicted output and the target output. The other aspect, such as the compactness of the fuzzy model with respect to the numbers of rules and inputs, is constrained by the maximum limits but will be automatically determined at the end of the evolution process to get the best accuracy.

To validate the performance of the proposed algorithm and to apply it to the real engineering problem, the developed evolving T–S fuzzy models are applied to three nonlinear system identification examples. The first two applications are to the Box–Jenkins benchmark problem and a nonlinear plant-modelling problem, which are well-known benchmark examples and are used here for the comparison with other existing models. The third application is to the identification of the magneto-rheological (MR) dampers. Recently, the use of MR dampers to solve vibration problems has received a great deal of attention in engineering. However, the practical use of MR dampers for control is significantly hindered by their inherently hysteretic and highly nonlinear dynamics. This makes the modelling of MR dampers very important for their applications. In order to characterise the performance of MR dampers, several models have been proposed to describe their dynamic behaviours. These include the phenomenological model, neural network model, fuzzy model, nonlinear blackbox model, NARX model, viscoelastic-plastic model, and polynomial model, etc. Among these models, phenomenological model [12] can accurately describe the dynamic behaviours of the MR dampers, but the corresponding models for the dynamics of the MR dampers are often nonlinear to make their control problem being difficult. A multi-layer perception (MLP) neural network [2] and an adaptive neuro-fuzzy inference system (ANFIS) [10] models can be used to emulate the dynamics of the MR dampers, but the selection of network structure is essential to obtain the accurate results. This paper applies the presented approach to approximate the forward and inverse dynamic behaviours of an MR damper in the form of the T–S fuzzy model. The numbers of rules and inputs, values of centres and widths of the T–S model are all evolved by GA simultaneously. It is validated by numerical values that the use of the developed evolving T–S model to emulate the dynamic behaviours of the MR damper can have less number of rules and inputs with acceptable accuracy. And, most of the important improvement is that the selection of rules and inputs are achieved automatically without the use of trial and error method.

The rest of this paper is organised as follows: Section 2 introduces the basic structure of the T–S fuzzy model and the developed evolving T–S model, where the new encoding scheme is given in details. The uses of the evolving T–S fuzzy models in modelling Box–Jenkins gas furnace data, nonlinear plant, and the forward and inverse dynamic behaviours of an MR damper are presented in Section 3. Conclusions are given in Section 4.

Section snippets

T–S fuzzy model

Using the rule base structure as shown in Fig. 1, the Takagi–Sugeno (T–S) fuzzy model considered is described by the following fuzzy IF–THEN rules:Rulei:IFx1isAi1andandxNIisAiNI,THENyi=ai0+ai1x1++aiNIxNI,where i = 1, …, NR, NR is the number of IF–THEN rules, x=[x1xNI] the premise input variable, NI the number of input variable, aij,j=1,,NI the consequent parameters, yi an output from the ith IF–THEN rule, and Aij is a fuzzy variable.

Given the input x=[x1xNI], the final

Application examples

In this section, the developed evolving T–S models are applied to three nonlinear system identification problems. The first two applications are to the Box–Jenkins gas furnace benchmark study and a nonlinear plant modelling problem. The third application is to the identification problem for the highly nonlinear MR damper model. The commonly used parameters associated with the genetic algorithm presented in Section 2.3 are given as follows: population size = 100, crossover probability = 0.8,

Conclusions

In this paper, a new encoding scheme is presented for learning T–S fuzzy model using GAs. The advantage of the developed evolving T–S fuzzy model is that the rule structure, the input structure, and the MF parameters can be optimised simultaneously such that the selection of rules and inputs can be achieved automatically. With the objective to reduce the MSE between the predicted output and the target output, the obtained T–S fuzzy model will have compact number of rules and inputs with the

Acknowledgments

The financial support of this work by the University of Technology Sydney, Early Career Research Grant, and the Australian Research Council's Discovery Projects funding scheme (project number DP0560077) is gratefully acknowledged.

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