Elsevier

Applied Soft Computing

Volume 77, April 2019, Pages 76-87
Applied Soft Computing

Online distributed fuzzy modeling of nonlinear PDE systems: Computation based on adaptive algorithms

https://doi.org/10.1016/j.asoc.2018.12.035Get rights and content

Abstract

With the emergence of novel model-based controllers for partial differential equation (PDE) systems, identifying the mathematical model of PDE systems has become a promising and complicated research topic. This paper suggests a new method to identify an adaptive Takagi–Sugeno (TS) fuzzy PDE model for nonlinear multi-input multi-output (MIMO) first-order PDE systems. The proposed approach is performed online based on the measured input and output data of the nonlinear PDE systems. Furthermore, the identification process will be obtained for the cases that the noise is either white or colored. For the case of white noise, a nonlinear recursive least square (NRLS) approach is applied to identify the nonlinear system. On the other hand, when the colored noise is exerted to the nonlinear PDE system, the fuzzy PDE model of the nonlinear PDE system and also nonlinear colored noise are identified based on the nonlinear extended matrix methods (NEMM). Moreover, the problem of identification for both colored and white noise cases is investigated when premise variables of membership functions are known or unknown. Finally, in order to illustrate the effectiveness and merits of the proposed methods, the identification method is applied to a practical nonisothermal Plug-Flow reactor (PFR) and a hyperbolic PDE system with Lotka–Volterra type applications. As it is expected, the evolutions of the error between the state variables for the obtained TS fuzzy PDE model and the output data converge to the zero in the steady-state conditions. Thus one concludes, the proposed identification algorithm can accurately adjust both consequents and antecedents parameters of TS fuzzy PDE model.

Introduction

A significant number of physical phenomena in the real-world such as industrial process and biological systems inherently depend on spatially position as well as time (i.e. their behaviors are distributed in space) [1], [2]. Whereas their dynamics depend on more than one independent variable thus they are well-known as a partial differential equation (PDE) systems [3], [4]. Based on spatially distributed points, PDE systems are classified into three categories: (1) hyperbolic [5] (2) parabolic [4] (3) elliptic [6]. Consequently, due to infinite dimensional and spatially distributed behaviors of PDE systems, more effort is needed to design the controller, analyze the stability and also identify the PDE systems. Moreover, it is generally more difficult to directly apply the existing lumped parameter systems techniques to the distributed ones [7].

Recently, a significant number of research has been devoted to the problem of stability and stabilization of nonlinear PDE systems based on TS fuzzy PDE model [5], [8], [9]. TS fuzzy PDE modeling of parabolic PDE systems is presented in [10], [11], [12], [13] and the hyperbolic ones is investigated in [2], [7], [14]. In the literature of PDE systems, it is assumed that the nonlinear system equations exist and subsequently, the exact TS fuzzy model has been obtained based on sector nonlinearity approach [7], [15].

According to the control and system engineering points of view, the fundamental part of a study is achieving an accurate model for the existing linear or nonlinear physical system [16]. Since enough information for obtaining a suitable mathematical model does not exist, the exact mathematical dynamic representation of real-world systems is seldom available [17]. On the other hand, in real applications, we encounter with colored noise instead of white noise. Subsequently, the effect of the colored noise is as critical as the un-modeled dynamic in the system identification and modeling [18]. Hence, an important problem is to model and identify the linear or nonlinear system based on input–output data. The identified model must describe the physical behavior of the original plant with an adequate level of accuracy [19], [20].

Most of the real-world systems are inherently nonlinear [21], [22]. Takagi–Sugeno (TS) fuzzy models provide a powerful and systematic framework to analyze the stability and synthesize the controller for nonlinear systems [19], [23]. Moreover, it can describe the complicated smooth nonlinear systems in the convex structure [24], [25], [26]. Thus, lots of attention has been focused on the TS fuzzy systems during the last two decades [23], [27], [28]. A TS fuzzy model represents the nonlinear system via some local linear subsystems that will be introduced in fuzzy IF–THEN rules structure. Then, by fuzzy blending of the local linear subsystems, the overall fuzzy model will be obtained. Such models have the capability to approximate a wide range of nonlinear systems [29]. There exist two approaches to obtain a TS fuzzy model. The first and more attractive one is based upon the identification using validation input–output data when the system is unknown and the second one is derived from the given nonlinear system equations [29]. This paper focuses on the first approach which involves a technique to find optimal values of (1) premise and (2) consequent parameters sets [30]. The premise parameters set constructs the characteristics of fuzzy membership functions and the consequent one contains the coefficients of the local linear subsystems [19]. It is generally difficult to quantify these parameters based on an expert man’s validation knowledge. Hence, the parameters will be usually approximated based on the least squares estimate (LSE) [31], recursive least square (RLS), Kalman filter, extended Kalman filter (EKF) [25] and data-driven approach [32].

Numerous researches have argued about the identification of linear and nonlinear ODE systems. Ref. [18] has suggested a method to estimate the states and parameters of the linear dynamic system which is affected by the colored noise. Furthermore, it considers the minimum discrepancy measure criterion to model the colored noise. However, the problem of identifying nonlinear system has not been addressed in Ref. [18]. In addition, if the sampling times get lower or sampling frequencies get higher, then due to the computational time, the approach [18] cannot be applicable. Thus, constructing a fast computational algorithm is necessary. In [33], the extended recursive least square (ERLS) algorithm has been presented to estimate the parameters of discrete-time nonlinear stochastic systems. Based on the ERLS algorithm in [33], the consistency of the parameters has been guaranteed without any restrictive conditions such as (1) the persistent excitation condition (2) the noise condition and (3) the variance functions condition. However, the identification algorithm [33] is only valid for a class of nonlinear systems called polynomial systems. Due to the linearization processes, the presented algorithms [18] and [33] identify the nonlinear system in a small vicinity of operating point or equilibrium point. Whereas TS fuzzy models create a powerful algorithm to represent the nonlinear systems, respectable amount of studies have been focused on TS fuzzy modeling of nonlinear ODE systems based on input–output data [19]. Recently, several approaches are presented to identify the TS fuzzy model of a nonlinear system such as: genetic algorithm [34], artificial neural networks (ANN) [32], gravitational search-based hyper-plane clustering algorithm [26], self-organizing migration algorithm [35] [36], least square (LS) algorithm [30] and EKF [25]. The propose of Ref. [34] is to present a new encoding scheme for identifying the TS fuzzy model by the nondominated strong genetic algorithm. In addition, identification of multiple input multiple output (MIMO) systems based on MIMO TS fuzzy model is presented in [35], [37]. In [38], the Kalman filter is utilized to design a state estimator for each local model of the fuzzy system. Then, the states of the overall time-varying discrete-time system are estimated by aggregating the local models. A small number of researches have been focused on the problem of identifying TS fuzzy model-based on the Kalman filters. Ref. [39] uses the Kalman filter and Gustafsone Kessel clustering algorithm (GKCA) to update the information of the consequent and antecedent parts, respectively. In other words, the parameters and structure of the fuzzy model are identified in two separate steps. Thus, the accuracy of the obtained TS fuzzy model is reduced significantly. Refs. [38], [40] use EKF to adjust the parameters of the TS fuzzy model. In Refs. [38], [40], the structure of the membership functions is assumed to be triangular. However, because of the complexity of the learning algorithm, the efficiency and the applicability of the approaches [38], [40] for other types of membership functions are reduced. Ref. [25] also identifies the TS fuzzy model by utilizing the EKF algorithm. The method presented in Ref. [25] is simpler than the ones [38], [40]. In addition, in the situation that the membership functions are overlapped by pairs, the approach [25] is not applicable. Ref. [41] employs the EKF to approximate the parameters of the antecedent and consequent parts of TS Type-2 fuzzy systems. In addition, the high-speed convergence and desirable accuracy of the learning algorithm in comparison with the PSO algorithm are improved significantly [41]. However, according to the best knowledge of the authors, the references [25] and [41] have some main drawbacks. First, the problem of identifying the system in the presence of colored-noise has not been addressed in the literature of identifying TS fuzzy model-based on KF. Second, as we mentioned previously, large numbers of phenomena are described by PDE systems. The presented approaches are not capable to identify TS fuzzy model of such systems. Thus, it is essential to construct a symmetric approach to identify the TS fuzzy model of PDE systems. To the best knowledge of the authors, the identification of TS PDE fuzzy model of nonlinear PDE systems based on input–output data has not been addressed yet, which is the main contribution of this paper.

This paper presents a novel approach for online adaptive TS fuzzy PDE modeling of nonlinear MIMO first-order PDE systems. The proposed identification algorithm investigates the cases that the system is affected by the white noise or the colored one. The important feature of the proposed approach is adjusting the antecedent and consequence parts of the TS fuzzy PDE model of nonlinear PDE system without limiting the size of the input–output data. To cope with these difficulties, the authors create a suitable structure to identify the nonlinear PDE system with NRLS and NEMM approaches. Generally, the main contributions of the proposed approach can be classified as follows:

  • Identifying the nonlinear PDE systems based on input–output data

  • For the cases that the colored noise affects the nonlinear PDE system, not only the TS PDE fuzzy model of the nonlinear PDE system is identified but also the TS fuzzy model of the colored noise is identified.

  • The TS fuzzy PDE model is defined in a suitable structure such that deploying the NRLS and NEMM approaches will be possible.

To illustrate the efficiency of these key ideas, a practical PFR system and a hyperbolic PDE system with Lotka–Volterra type are considered. The identification is obtained for two cases: the premise variables in membership functions are known or unknown. The results will be indicated that the nonlinear first-order PDE system can be suitably approximated by the obtained TS fuzzy PDE first-order model. Moreover, in the case that the measurement colored noise is presented, the measurement colored noise dynamic will be correctly approximated by the TS fuzzy PDE model.

The remainder of the paper is organized as follows. In Section 2, the problem formulation regarding MIMO TS fuzzy PDE models is reviewed. Section 3 focuses on two methods. The first one investigates the nonlinear least square (NLS) method and the second one studies the extended matrix method (EMM). In Section 4, TS fuzzy PDE modeling of nonlinear PDE systems in the presence of white and colored measurement noise are discussed. Then in Section 5, the simulation results are presented to identify the nonisothermal PFR based on the identification methods. Finally, the conclusions will close the paper in Section 6.

Section snippets

Problem formulation

TS fuzzy models are known as universal approximators. Thus, any smooth nonlinear system can be approximated via a TS fuzzy model with any desired degree of accuracy [29], [30]. The TS fuzzy model has been widely used to analyze and synthesize the nonlinear ODE or PDE systems. Furthermore, it is suitable for designing fuzzy ODE or PDE controllers [42], [43], [44]. Therefore, fuzzy modeling and identification of nonlinear PDE processes are very essential. In this Section, a TS fuzzy MIMO

Nonlinear least square and extended matrix method

Rudolf E. Kalman developed the Kalman filter that is defined as a linear combination of measurements [45]. It is well-known as an optimal linear filter and also, it is the best recursive state estimator for linear systems in the presence of zero-mean white noise in measurements and model [45]. In general, the real systems are inherently nonlinear and complex. For the nonlinear systems, several kind of nonlinear Kalman filters are formulated to approximate the solutions, such as: linearized

Application of the proposed methods to fuzzy pde modeling of nonlinear pde systems

Recently, one of the most interesting and efficient applications of LS and EKF is TS fuzzy modeling of nonlinear ODE systems. The LS algorithm presents an offline approximation [30], while the EKF algorithm presents an online one [25]. This paper obtains an online fuzzy model for nonlinear first-order PDE systems, which the identified parameters are modified during the adaptive process. The mentioned identification is achieved based on NLS in a pseudo-optimal way (i.e. optimal for linear

Examples

In this section, the proposed online distributed fuzzy modeling approach is applied on two examples: PFR [7] and a nonlinear hyperbolic PDE system with Lotka–Volterra type [2].

Conclusions

From this paper, one can conclude that a general structure for identifying the TS fuzzy PDE model of nonlinear MIMO first-order PDE systems in the presence of white and colored noises was proposed. Against the existing approaches on TS fuzzy PDE modeling of nonlinear PDE systems, the identification method in this paper was based on input–output data. For PDE systems with white noise, we can conclude that the NRLS method was able to identify the fuzzy PDE model. When the colored noise affects

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