Robust decentralized parameter identification for two-input two-output process from closed-loop step responses

doi:10.1016/j.conengprac.2004.04.017

Control Engineering Practice

Volume 13, Issue 4, April 2005, Pages 519-531

https://doi.org/10.1016/j.conengprac.2004.04.017 Get rights and content

Abstract

In this paper, a novel parameter identification method for closed-loop two-input two-output (TITO) processes from step-test is proposed. Through sequential step change of set points, the coupled closed-loop TITO system is decoupled equivalently into four independent single open-loop processes with same input signal acting on the four transfer functions. Consequently, existing identification methods for single-loop process can be extended to TITO systems and the parameters of first- or second-order plus dead-time models for each transfer function can be directly obtained by using the linear regression equations derived for the decoupled identification system. The proposed method is simple for engineering application and robust in the presence of large amounts of measurement noise. Simulation examples are given to show both effectiveness and practicality of the identification method for a wide range of multivariable processes.

Introduction

In multivariable process control, most schemes such as inverse Nyquist array or characteristic locus methods Astrom & Hagglund (1984), Astrom & Hagglund (1995) require a full model of the process in the form of a transfer-function matrix or a frequency-response matrix over the entire working frequency range. In many cases, such a model is not available and physical modeling may require a prohibitive engineering effort. Therefore, practical and effective estimation of the full process models becomes appealing and has been an active research area of control engineering for a few decades. A considerable number of identification methods and their application to other engineering fields including advanced control strategy, optimization and signal processing have been reported in the literature (Loh & Vasnani, 1994; Poulin, Pomerleau, Desbiens, & Hodouin, 1996; Wang & Cai, 2003; Zhu & Butoyi, 2002).

There are two ways to identify a multivariable process for control application, i.e. open-loop and closed-loop ones. In any case, an excitation of the process is needed to extract useful information on process dynamics. For open-loop transient response experiments, step or pulse excitation signals are commonly injected at the process inputs, and the response is measured (Choi, Lee, Jung, & Lee, et al., 2000; Young, 1970). The main advantage of the step test is that the testing procedure is simple and requires little prior knowledge. However, it is quite sensitive to non-linearity in the system (Luyben, 1991). For closed-loop identification, a majority of existing techniques are in the frequency domain while the frequency range of interest for such applications is usually from zero up to the process critical frequency (Loh, Hang, Quek, & Vasnani, 1993; Shen, Wu & Yu, 1996; Wang & Shao, 1999; Wang & Cai, 2003). Since the closed-loop testing causes less perturbation to the process, it is preferred to open-loop one in process control practice.

As the step test is the simplest and dominant in process control applications, there has been strong research interest in using such a test to determine the dynamics of unknown processes. Bi, Cai, Lee, and Wang, et al., 1999 proposed a simple yet robust identification method to obtain a first-order plus dead-time model for a linear monotonic process from a step test of open-loop control systems. The identification method was later applied to design auto-tuning PID controllers for heating, ventilation and air-conditioning (HVAC) systems (Bi, Cai, Wang, & Hang, et al., 2000), experimental results have demonstrated the effectiveness of the technique. Wang and Cluett 1994 presented an identification algorithm for processes operating in closed-loop, the algorithm involves fitting two Laguerre models directly to the control signal and the process output signal generated by a step change in the set-point. Wang, Guo and Zhang (2001); Wang and Zhang (2001) proposed some robust identification methods for linear time-delay processes from step responses in both time domain and frequency domain. These results was also applied to PID controller auto-tuning for multivariable processes Wang, Huang and Guo (2000).

In this paper, an engineering oriented identification technique for multivariable process is proposed, which extend SISO identification method in Bi, Cai, Lee, and Wang, et al., (1999) to the multivariable systems. The proposed method only requires step response data of closed-loop process, and no prior knowledge of the process dynamics and of the controller dynamics is needed. Through sequential step change of set points, the coupled closed-loop TITO system is decoupled equivalently into four individual single open-loop processes with same input signal acting on the four transfer functions. Consequently, existing identification methods for single-loop process can be extended to TITO systems and the parameters of first- or second-order plus dead-time models for each transfer function can be directly obtained by using the linear regression equations derived for the decoupled identification system. The proposed method is simple for engineering application and robust in the presence of large amounts of measurement noise. Various typical multivariable processes have been employed to illustrate the effectiveness of the method. It offers a good engineering tool for control engineers in retuning an existing multivariable control system and designing advanced controllers for multivariable processes.

Section snippets

Formulation of decentralized TITO Identification systems

Consider a TITO process under decentralized control as shown in Fig. 1, where r_i,e_i,u_i and y_i, i,j=1,2 are set points, errors, controllers and process outputs, K_i,ζ_i and G_ij controllers, noises and process transfer functions, respectively. In general, K₁ and K₂ could be any type of controllers that make the closed loop system stable. To simplify our derivation, the notation $r_{i},e_{i},y_{i} i,j=1,2$ , are used in both s and t domain. The fundamental relationship between error signals and transfer function

Least squares method

For the decentralized identification systems, the forward transfer function can be represented by $G_{ij} (s)= b_{ij1} s^{n−1} +b_{ij2} s^{n−2} +⋯+b_{ijn} s^{n} +a_{ij1} s^{n−1} +a_{ij2} s^{n−2} +⋯a_{ijn} e^{−L_{ij}s}, i,j=1,2.$ Following Remark 2, initial conditions for the equivalent decentralized identification systems are zeros. Therefore, Eq. (11) can be written equivalently in differential equation form as $y_{ij}^{(n)} (t)+a_{ij1} y_{ij}^{(n−1)} (t)+⋯+a_{ijn} y_{ij} (t)=b_{ij1} u^{(n−1)} (t−L_{ij})+b_{ij2} u^{(n−2)} (t−L_{ij})+⋯+b_{ijn} u(t−L_{ij}).$ Define $∫ [0,t] (m) f(t)≜ ∫ 0 t ∫ 0 τ_{m} ⋯ ∫ 0 τ_{2} m mf(τ_{1}) d τ_{1} ⋯ d τ_{m}, t⩾ max (t_{1}$

Simulation example

The Wood and Berry binary distillation column plant (Wang, Guo & Zhang 2001c) is a typical TITO process with strong interaction and significant time delays, which transfer function matrix is given as $G(s)= 12.8 e^{−s} 1+16.7s −18.9 e^{−3s} 1+21s 6.6 e^{−7s} 1+10.9s −19.4 e^{−3s} 1+14.4s .$

For the decentralized closed-loop system with K_P1=0.5271, K_I1=0.0763, K_D1=0.45, and K_P2=−0.1064, K_I2=−0.018 K_D2=0.02, the step-test error signals without noise and the differentials of the equivalent signals u and y_ij (i,j=1,2) are

Application to multistage gas–liquid absorption column system

Consider a multistage gas–liquid absorption column shown in schematic form in Fig. 6 Giovani, Serkan and Oscar, 2003. The column features n trays where the transfer of a component of interest from the gas stream into the liquid stream is accomplished in each tray as the rising gas stream bubbles through a liquid layer that flows horizontally across the tray. The variable x and y, respectively, denote the composition of the species of interest in the liquid and gas phases. The manipulated

Conclusions

In this paper, a novel identification method based on step tests for multivariable process has been presented. By decoupling the closed-loop multivariable system into independent single open-loop systems with same input signal acting on the multiple transfer functions, most restrictions of existing multivariable process identification methods has been relaxed. The proposed method requires no prior knowledge of the process and the first-order or second-order plus delay model elements of the

Acknowledgements

The authors would like to acknowledge the financial support of the High Technology Research and Development Program of China (Grant No.2002AA412130) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.: 20020248028). The authors are grateful to the anonymous reviewers for their valuable recommendations.

References (20)

K.J. Astrom et al.
Automatic tuning of simple regulators with specifications on phase and amplitude margins
Automatica
(1984)
Q. Bi et al.
Robust identification for first-order plus dead-time model from step response
Control Engineering Practice
(1999)
Q. Bi et al.
Advanced controller auto-tuning and its application in HVAC systems
Control Engineering Practice
(2000)
E. Poulin et al.
Development and evaluation of an auto-tuning and adaptive PID controller
Automatica
(1996)
Q.G. Wang et al.
Auto-tuning of TITO decoupling controllers from step tests
ISA Transactions
(2000)
Q.G. Wang et al.
Robust identification of continuous systems with dead-time from step responses
Automatica
(2001)
Q.G. Wang et al.
Direct identification of continuous time delay systems from step responses
Journal of Process Control
(2001)
P.C. Young
An instrumental variable method for real time identification of a noisy process
Automatica
(1970)
Y.C. Zhu et al.
Case studies on closed-loop identification for MPC
Control Engineering Practice
(2002)
K.J. Astrom et al.
Pid controllersTheory, design, and tuning
(1995)

There are more references available in the full text version of this article.

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Robust decentralized parameter identification for two-input two-output process from closed-loop step responses

Abstract

Introduction

Section snippets

Formulation of decentralized TITO Identification systems

Least squares method

Simulation example

Application to multistage gas–liquid absorption column system

Conclusions

Acknowledgements

Automatica

Control Engineering Practice

Control Engineering Practice

Automatica

ISA Transactions

Automatica

Journal of Process Control

Automatica

Control Engineering Practice

Pid controllersTheory, design, and tuning