Pulse response method for the Wiener-type nonlinear process identification

https://doi.org/10.1016/j.compchemeng.2023.108178Get rights and content

Highlights

  • A new Wiener-type process identification method is proposed.

  • The proposed method is developed based on a narrow pulse response.

  • It requires a relatively short process excitation than previous methods but estimates the process almost exactly.

  • The performance of the proposed method is verified by the simulation and experiment.

Abstract

The Wiener-type nonlinear process where a nonlinear static block follows a linear dynamic subsystem is widely used to describe the dynamics of chemical processes. The present study proposes a new Wiener-type process identification method based on a narrow pulse response to cope with the relatively long process excitation requirement in the previous methods. The proposed identification method is developed based on the property of the Wiener-type nonlinear process that outputs of a linear dynamic subsystem is the same at two points when process outputs are equivalent at those points. The proposed method requires a relatively short process excitation (a single pulse response) but almost exactly estimates the Wiener-type nonlinear process. The simulation and experimental studies show the outstanding performance of the proposed method.

Introduction

The industrial processes are widely approximated into a linear process within a restricted operating region in order to implement systematic approaches such as control and optimization. However, the nonlinearity of processes cannot be neglected when the operating regions are extended due to significant disturbance or set point change. To describe nonlinear processes precisely, a nonlinear block-oriented model, the Wiener-type nonlinear model consisted of a nonlinear static function and a linear dynamic subsystem, is widely used to express the nonlinear behavior of chemical processes. Acid-base titration (Lee and Choi, 2000; Wright and Kravaris, 1991), preferential oxidation (Heo et al., 2020), fuel cell (Xia et al., 2018), polymerization reaction (Jeong et al., 2001), and high purity distillation column (Bloemen et al., 2001) are representative examples of a Wiener-type nonlinear chemical process. In the case of acid-base titration, the mass balance of the system matches with a dynamic linear subsystem, and the dynamics of hydrogen ion concentration is a nonlinear static function. However, its complex model identification procedure is remained as an obstacle to implementing the Wiener-type nonlinear model in practice.

Diverse nonlinear process identification methods including the SINDy, Koopman operator, and dynamic mode decomposition have been proposed to cover the effect of nonlinearity in a process (Bevanda et al., 2021; Bhadriraju et al., 2020; Brunton et al., 2016; Narasingam and Kwon, 2019; Proctor et al., 2016; Williams et al., 2015). These methods theoretically can guarantee fine performance regardless of a process's type. However, they require a relatively complex calculation as well as a large amount of process data (requires long process excitation). A classical pulse response method can be easily identified with simple excitation, although it can only be used in restricted processes (e.g., Wiener-type nonlinear process). Various linear model identification methods have been available for identifying the Wiener-type nonlinear process (Hagenblad, 1999), however, they cannot be easily available in practice due to requirement of iterative calculation to estimate model parameters. Furthermore, the correlation between the parameters in the block-oriented nonlinear model makes it challenging to identify physically meaningful parameters. Even if the model parameters are well defined, sufficient-long process excitation time is required to obtain an accurate model. Many research works have focused on identifying linear/nonlinear blocks of the Wiener-type nonlinear model independently using a designed process excitation signal. A model identification using two step response with different sizes (Park and Lee, 2006) and a sinusoidal response (Sung et al., 2008) are representative examples of improved identification methods that can reduce iterative computation as well as experimental time.

Here, a simple method is proposed to estimate the Wiener-type nonlinear model from a single pulse response. The Wiener-type nonlinear process has the property that if process outputs are the same at certain points, then a linear dynamic subsystem's outputs are equivalent at those points. In the proposed method, the linear dynamic subsystem and the nonlinear static function are estimated sequentially using a single pulse response in accordance with this property. Various works have been carried out to identify a linear dynamic subsystem through process data that have the same values at particular times. For instance, in the relay feedback method (Åström and Hägglund, 1984), which is the widely used process identification method, the process's relative magnitude of the time delay or the shape factor that determines the order of a model is estimated using this information at which outputs become the equivalent (Luyben, 2001). In the present research, two methods are proposed to identify the Wiener-type nonlinear process; (1) estimating parameters of a low-order parametric model by solving the nonlinear least square method with iterative calculation, and (2) approximating the linear dynamic subsystem by using a sum of Laguerre functions. The summation of Laguerre functions can describe the transfer function of general processes (Park et al., 1997; Zervos et al., 1988), and its parameters can be obtained by solving the linear least square method. The proposed method sequentially identifies a linear dynamic subsystem and a nonlinear static function by using a single pulse response. The performance of the proposed method is verified with both the simulation study of the pH process and the experimental study of the liquid level process.

Section snippets

Previous identification methods for the Wiener-type nonlinear process

This study takes below Wiener-type nonlinear process into account:dz(t)dt=Az(t)+bu(t)x(t)=cz(t)y(t)=g(x(t))where u(t) and y(t) are process input and output. The n-vector state variable of a linear dynamic subsystem is given as z(t), and x(t) is the intermediate scalar variable. A, b, and c denote the matrixes of the linear dynamic subsystem. g(x(t)) is the nonlinear static function. The linear dynamic subsystem can be represented in Eq. (2) and Fig. 1 by using the Laplace transform.X(s)=c(sIA)

Simple and fast identification method for the Wiener-type nonlinear process

The mTSR method uses pulse-type input to obtain the process response. The pulse width of the mTSR method should have a sufficient size to obtain a perfect step response. Here, we propose the identification method using the pulse response having a brief pulse width. The pulse response with the width of pw and magnitude of M is demonstrated in Fig. 3a. The nonlinear static function g(x) is assumed to be an invertible, strictly monotone function. According to the assumption, x=g1(y), if the

Simulation study of pH process

The pH process is a representative example of the Wiener-type nonlinear process. In this simulation study, the pH process where the acid of acetic acid (CH3COOH) and hydrochloric acid (HCl) mixture is titrated by the strong based of sodium hydroxide (NaOH) in a continuous-stirred tank reactor (CSTR) (Lee and Choi, 2000; Wright and Kravaris, 1991). This study takes below material balance and pH equations.Vz˙1(t)=(F+u(t))z1(t)+c1u(t)Vz˙2(t)=(F+u(t))z2(t)+c2FVz˙3(t)=(F+u(t))z3(t)+c3Fz1(t)z2(t)

Conclusion

The Wiener-type nonlinear process, which connects a linear dynamic subsystem followed by a nonlinear static function, has to require a well-designed process response and accurate calculation for the process identification. Otherwise, it is hard to obtain physically meaningful parameters due to the high correlation of process parameters. This study proposes the identification method using a single pulse response to estimate a linear dynamic subsystem and a nonlinear static function sequentially.

CRediT authorship contribution statement

Sanghun Lim: Conceptualization, Methodology, Software, Writing – original draft. Jea Pil Heo: Formal analysis. Kyung Hwan Ryu: Supervision, Writing – original draft, Writing – review & editing. Su Whan Sung: Conceptualization, Supervision, Writing – review & editing. Jietae Lee: Formal analysis, Methodology, Conceptualization, Software. Friedrich Y. Lee: Formal analysis.

Declaration of Competing Interest

The authors declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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