Elsevier

European Journal of Control

Volume 54, July 2020, Pages 119-128
European Journal of Control

A spatial multivariable SVR method for spatiotemporal fuzzy modeling with applications to rapid thermal processing

https://doi.org/10.1016/j.ejcon.2019.11.006Get rights and content

Highlights

  • Spatial multivariable SVR based 3-D fuzzy modeling methodology is proposed for distributed parameter systems.

  • Multivariable SVR with spatial kernel functions is proposed to cope with a set of spatiotemporal data.

  • Spatial multivariable SVR builds up a complete 3-D fuzzy rule-base.

  • 3-D fuzzy model is constructed in the form of data-driven design.

Abstract

Many industrial processes have significant spatiotemporal dynamics and they are usually called distributed parameter systems (DPSs). Modeling such system is challenging due to its nonlinearity, time-varying dynamics, and spatiotemporal coupling. Using model reduction techniques, traditional DPS modeling methods usually reduce an infinite-dimensional system to a finite-dimensional system, which leads to unknown nonlinearity and unmodeled dynamics. The modeling method and the established model are hard to understand. Here, we propose a spatial multivariable support vector regression (SVR) based three-domain (3-D) fuzzy modeling method for complex nonlinear DPSs. The proposed 3-D modeling method integrates the time-space separation and time-space synthesis into a 3-D fuzzy model. Therefore, it does not require model reduction and owns the capability of linguistic interpretability. A spatial multivariable SVR with spatial kernel functions is proposed to deal with spatiotemporal data. The spatial fuzzy basis functions from a 3-D fuzzy model are spatial kernel functions for a spatial multivariable SVR, which satisfy Mercy theorem. Hence, the spatial multivariable SVR can be directly employed to build up a complete 3-D fuzzy rule-base of the 3-D fuzzy model. The proposed modeling method integrates the merits of learning ability from a spatial multivariable SVR and fuzzy space processing and fuzzy linguistic expression from a 3-D fuzzy model. The proposed 3-D fuzzy modeling method is successful applied to a simulated rapid thermal processing system. In comparison with several newly developed modeling methods for DPSs, the simulation results validate the superiority of the proposed modeling method.

Introduction

Many industrial processes have significant spatiotemporal dynamics and they are usually called distributed parameter systems (DPSs) [5]. For example, flexible beam [6], transport-reaction processes [9], and thermal process [3] are typical spatiotemporal systems. These systems are often represented by partial differential equations (PDEs). The PDE descried system can be transformed into an infinite-dimensional system of ordinary differential equations (ODEs). The spatially distributed feature requires an infinite-dimensional modeling [5], [10], [18], which is more difficult and complicated than modeling of lumped parameter systems (LPS). In spite of the difficulty, it is indispensable to model DPSs for control design, optimization, and dynamic prediction.

DPS modeling has been broadly investigated since 1960s. Much research is based on the first-principle model, which is represented by PDEs. The DPS system is an inherent infinite-dimensional system. However, because of a finite number of actuators and sensors for practical sensing and control and limited computing power for implementation, such infinite-dimensional system needs to be approximated by a finite-dimensional system [10], [18]. In traditional DPS modeling methods, therefore, the model reduction is critical to derive a low-order model for practical application. The finite-element method [8] and the finite-difference method [25] are commonly used to transformed a first-principle PDE into high-order ODEs, which is also called early lumping [18]. Additionally, spectral method [5] and other methods can be employed to reduce a PDE model, which is also called late lumping [18]. Furthermore, fuzzy PDE [22], [23] has been reported in recent years to model a DPS. All these methods are only applicable for situations where the PDEs of a DPS are known.

Whereas, the PDEs of many practical DPSs are often unknown. Therefore, data-driven based modeling methods [7] are usually used. In recent decades, many researchers have done related research. Much research is developed on the basis of a time-space separation framework [10] where a spatiotemporal variable is decomposed to a series of spatial functions and temporal coefficients. In some methods, the spatial functions are chosen beforehand. For instance, Jacobi polynomials, Fourier series, and Spline functions can be used as spatial functions [10]. In addition, the number of the spatial functions are also determined by prior knowledge. Traditional modeling methods for lumped parameter systems (LPSs) can be employed to estimate the time model. Different to the prior selection of the spatial functions, the spatial functions can be estimated from a set of spatiotemporal data using Karhunen-Love (KL) decomposition [17]. Wiener models [15], Hammerstein [16], and Spatiotemporal Volterra [11] have been used to model DPSs based on KL decomposition. For purpose of acquiring low-order ODEs, model reduction is required. However, it will bring about unmodeled dynamics and unknown nonlinearity [30]. The standard KL decomposition has the feature of linearity, which means that it ignores the nonlinear variations among data. To deal with the nonlinearity, the authors in Ref. [13] introduce adaptive KL decomposition to a fuzzy TS modeling method, real-time update spatial functions using adaptive KL decomposition, and update temporal model using T-S fuzzy model.

3-D fuzzy modeling [27], [30] has been developed as a new modeling method for DPSs in recent years. The functions of time-space separation and time-space synthesis are naturally implemented in each 3-D fuzzy model. On the one hand, the time-space separation is naturally realized in a 3-D fuzzy rule, i.e., the computation of the antecedent part represents the temporal coefficient and the consequent part represents spatial function. On the other hand, the time-space synthesis is realized by the Union of all fired 3-D fuzzy rules. In comparison with the traditional time-space based modeling method, the 3-D fuzzy modeling has two obvious features: no reliance on model reduction and linguistic interpretability. Currently, the developed 3-D fuzzy models [27], [30] have successfully applied to RTP systems. In [27], an embryonic 3-D fuzzy model is constructed via KL decomposition for spatial functions and particle swarm optimization (PSO) for membership functions in antecedent part of 3-D fuzzy rules. Since KL decomposition is employed, this method still relies on model reduction. In [30], a 3-D fuzzy model is achieved using nearest neighborhood clustering (NNC) algorithm and similarity measure for antecedent part and multiple support vector regressions (SVR) for spatial functions. It is the first 3-D fuzzy model without model reduction.

Here, we develop a novel 3-D fuzzy modeling method using spatial multivariable support vector regression (SVR). Traditional SVR [21], [26] has not inherent capability of handling with spatiotemporal data; therefore, when traditional SVR is used for modeling a DPS, it is often required to combine with KL decomposition method, i.e. spatial functions are acquired by KL decomposition and time coefficients are estimated by SVR [17]. The conventional KL-SVR modeling method relies on model reduction and has not the ability to linguistically explanation. In this study, a multivariable SVR with spatial kernel functions is proposed to deal with spatiotemporal data. We call it spatial multivariable SVR (abbreviated as spatial MSVR). Different to the multiple traditional SVRs used to learn spatial functions in [30], here the spatial MSVR is directly employed to build up a complete 3-D fuzzy rule-base. The spatial fuzzy basis functions from a 3-D fuzzy model are spatial kernel functions for a spatial MSVR that satisfy Mercy theorem. The proposed 3-D fuzzy modeling method naturally integrates the advantage of learning ability from a spatial multivariable SVR and fuzzy space processing and fuzzy linguistic expression from a 3-D fuzzy system.

The main contributions of this paper are given as follows.

  • 1)

    Spatial multivariable SVR is proposed to cope with a set of spatiotemporal data.

  • 2)

    Spatial multivariable SVR is used to build up a complete 3-D fuzzy rule-base.

  • 3)

    A data-driven 3D fuzzy modeling method is proposed via using a spatial multivariable SVR.

The paper is organized as follows. Section 2 presents the problem description. In Section 3, the spatial multivariable SVR based 3-D fuzzy modeling approach is described in detailed. In Section 4, an RTP system is taken as an application to validate the effectiveness of the proposed 3-D fuzzy modeling method. Finally, the conclusion is given in Section 5.

Section snippets

Problem description

Here a nonlinear DPS is considered. Let u(θ, t) ∈ R be the spatiotemporal output, c(t) ∈ Rm be the temporal input, θΘ¯ be the spatial variable, Θ¯ be the spatial domain, and t be the time variable. The considered DPS is of infinite-dimension. However, limited number of sensors and actuators are applied for the purpose of actual needs. It is assumed that P sensors are located at spatial points θ1,θ2,,θP for measuring the output of the system and m actuators are spatially distributed on space

Methodology

Via utilizing KL decomposition technique, the traditional SVR can be applied to model a DPS. However, the traditional SVR only models the time coefficients. In this study, since 3-D fuzzy model has the characteristics of space-time separation, we investigate a multivariable SVR with spatial kernel function (called spatial MSVR), combine this spatial MSVR with 3-D fuzzy model, and construct a novel spatial MSVR based 3-D fuzzy modeling method. This method integrates two distinct merits: the

RTCVD system

In this work, an important RTP process, rapid thermal chemical vapor deposition (RTCVD) in semiconductor manufacturing process [20], is investigated. As shown in Fig. 7, the RTCVD system is divided into three zones for heating, i.e., Lamp banks 1, 2, and 3. A 6-in silicon wafer is positioned on a rotatable support to guarantee azimuthal temperature uniformity. Silane (SiH4) injected from the top of the reactor is resolved into silicon (Si) and hydrogen (H2). At temperature close to 1000 K

Conclusions

A spatiotemporal 3-D fuzzy modeling approach using spatial multivariable SVR was put forward for modeling unknown nonlinear distributed parameter systems. The proposed modeling method naturally fuses the time-space separation and the time-space synthesis into a 3-D fuzzy model. SFBFs from the 3-D fuzzy model is spatial kernel functions for the spatial multivariable SVR. Therefore, the proposed spatial MSVR based 3-D fuzzy model fully integrates the advantages both from 3-D fuzzy model and

Acknowledgments

This work was supported by the project from the National Science Foundation of China under Grant no. 61273182.

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