A general methodology for online TS fuzzy modeling by the extended Kalman filter
Introduction
Obtaining an accurate model of a system is a fundamental part of its study, however, not always have enough information to obtain an acceptable mathematical model, it is necessary to use modeling techniques based on input–output data [1]. This process is even more critical in control systems, since both, system analysis [2], [3], [4], [5], [6], [7], [8] and controller design [9], [10], [11], require to obtain a model as accurate as possible. One exception to this may be the Mamdani [12] type control systems, although its great heuristic load difficult formal stability analysis.
Takagi–Sugeno (TS) fuzzy modeling are universal approximators [13], [14], [15], [16], and allows to obtain, in most cases, highly accurate models from a small number of rules [17], and therefore is an ideal tool for this purpose. Up to date, many fuzzy modeling algorithms based on input–output data have been proposed [18], [19], [20], [21]. However, in many cases it is required that modeling algorithm works online with the system, and do it properly in the presence of noise.
In order to design fuzzy modeling algorithm based on input–output data, which can work online with the system, properly in the presence of noise and can be very efficient computationally, this paper presents a general methodology that, by the extended Kalman filter [22], it allows for online accurate models, in the presence of noise, and without renounce the computational efficiency that characterizes the Kalman filter. The presented algorithms allow to adjust both antecedents and consequents of TS fuzzy models, imposing restrictions, if desired, in the distribution of the antecedents of the model.
The Kalman filter is an efficient recursive filter that estimates the internal states of a linear dynamic system from a series of noisy measurement. It is used in a wide range of engineering and econometric applications, from radar and computer vision to estimation of structural macroeconomic models, and it is an important topic in control theory and control systems engineering.
The Kalman filter is the minimum-variance state estimator for linear dynamic systems with white noise with zero-mean value. Various modifications of the Kalman filter can be used to estimate the state in nonlinear systems, like the extended Kalman filter (EKF) [23], which linearizes the system around the current parameters. This algorithm updates the parameters been consistent with previous data, and usually converges in a few iterations.
The Kalman filter has been previously used with fuzzy logic in various applications [24], [25], [26], [27]. In 2002, Simon introduced the use of Kalman filter for adjusting the parameters of a fuzzy model [28], [29], assuming that antecedents were membership functions of triangular type, and using its center of gravity to perform the adaptation process. Later, other proposals have been made [30], [31], however, the complexity of the calculation for others types of membership functions has meant that this proposal has not been widespread so far.
Motivated by the successful use of Kalman filter in the works presented above, a theoretically general methodology for use of the extended Kalman filter (EKF) to estimate the adaptive parameters of a TS fuzzy model is presented in this paper. This methodology is theoretically general because it adjusts both, the antecedents such as the consequents, it does not limit the size of input/output vectors, neither the type or distribution of the membership functions used in the definition of fuzzy sets of the model. So, the authors try to use the excellent features of Kalman filter to obtain fuzzy models of unknown systems from input–output data, and also allowing its application in real-time [32].
In recent years, evolving fuzzy systems (EFS) have had a major advance [33], [34], [35], especially evolving Takagi–Sugeno (eTS) systems [36], [37], [38]. eTS systems evolve online with the process, adapting the consequents and updating the rule base using an online clustering mechanism. The algorithms presented in this paper should not be understood as an alternative to eTS, but as a method of adjusting the parameters of the model that does not change its structure. Therefore, these algorithms may be complementary. In any case, in order to evaluate the fitness of the presented algorithms, a comparative example with the eTS has been included.
This paper is organized as follows: Section 2 presents the problem and the formulation used throughout the entire paper. Section 3 presents the Kalman filter and its extended version, which will be used in Section 4 to present three algorithms to fuzzy modeling. The proposed algorithms are illustrated by several examples in Section 5. Finally, Section 6 provides some conclusions.
Section snippets
Problem formulation
As it is known, TS models are universal approximators, and they can achieve high accuracy with a small number of rules [14], [13], [15], [16]. On the other hand, is relatively easy to convert them into nonlinear state models [11], [39], which support formal analysis to use in control engineering. However, it is also known that the number of rules in TS models is increased as a lower approximation error is desired [40]. This implies that the modeling process is very important both for analysis
Extended Kalman filter
Kalman filter was developed by Rudolph E. Kalman [22], [46] and allows to construct an optimal observer for linear systems in the presence of white noise both in model and in measures. The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. The basic
Application of the extended Kalman filter to fuzzy modeling
A so interesting application of extended Kalman filter is the adaptive identification of parameters in nonlinear systems, which allows the online obtaining of the adaptive parameters set of a discrete nonlinear model with noise presence and in a pseudo-optimal way (is optimal in linear systems). The identification of a TS fuzzy model can be seen as the obtaining of parameters of a nonlinear model, so the Kalman filter can be applied using the extended algorithm for estimating these parameters.
Examples
To demonstrate the practical application of the extended Kalman filter to fuzzy modeling, several examples will be shown in this section. For each case, the three algorithms presented in the previous section are run to evaluate their performance: EKF(c), EKF(ac) and EKF(c+a). These algorithms has been programmed using the Fuzzy Logic Tools (FLT) library [51].
Examples 1, 2 and 3a carried out 10 times with a different noise signal (tenfold cross validation [52]), while the example 4 does not
Conclusions
This paper has addressed the problem to apply the extended Kalman filter in estimation of adaptive parameters of a completely general TS fuzzy system based on input–output data. The generality lies with at there are no restrictions on the size of the input or the output vectors, or the type or distribution of membership functions used in the rules of the model. The presented algorithms allows to adjust both antecedents and consequents, and in addition, it allows to imposing restrictions, if
Acknowledgments
This work is a contribution of the DPI2010-17123 Project supported by the Spanish Ministry of Education and Science and the TEP-6124 Project supported by the Regional Government of Andalusia (Spain). Both Projects are also supported by the European Union Regional Development Fund. This work is also funded by Spanish Ministry of Innovation and Science (ARABOT project DPI2010-21247-C02-01).
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