Elsevier

Automatica

Volume 37, Issue 3, March 2001, Pages 377-390
Automatica

Robust identification of continuous systems with dead-time from step responses

https://doi.org/10.1016/S0005-1098(00)00177-1Get rights and content

Abstract

In this paper, a simple yet robust method is proposed for identification of linear continuous time-delay processes from step responses. New linear regression equations are derived from the solution and its various-order integrals of the process differential equation. The regression parameters are then estimated without iterations, and explicit relationship between the regression parameters and those in the process are given. Due to use of the process output integrals in the regression equations, the resulting parameter estimation is very robust in the face of large measurement noise in the output. The proposed method is detailed for a second-order plus dead-time model with one zero, which can approximate most practical industrial processes, covering monotonic or oscillatory dynamics of minimum-phase or non-minimum-phase processes. Such a model can be obtained without any iteration. The effectiveness of the identification method has been demonstrated through simulation and real-time implementation.

Introduction

System identification has been an active area of automatic control for a few decades and it has strong links to other areas of engineering including signal processing, optimization and statistics. A considerable number of identification methods have been reported in the literature (Kurz & Goedecke, 1981; Sagara & Zhao, 1990; IEEEAC 1992 v.37(7); Automatica 1995 v.31(12)), and they are generally classified into parametric and non-parametric ones (Wellstead, 1981). Transfer functions might be the most welcome parametric model. Methods of filtering non-parametric time response to transfer functions are illustrated in Unbehauen and Rao (1990). Fitting parametric models to measured frequency data is another viable approach (Ninness, 1996).

However, most of the existing methods for transfer function identification do not consider the process delay (or dead-time) (Ljung, 1985; Sagara & Zhao, 1990) or just assume knowledge of the delay. It is well known that the delay is present in most industrial processes, and has a significant bearing on the achievable performance for control systems. Thus, there have been continuing interests in identification of delay processes. A frequently used method for dealing with unknown delays has been to use a shift operator model with an expanded numerator polynomial (Kurz & Goedecke, 1981). Another popular approach is based on the approximation of the dead-time by a rational transfer function such as the polynomial approximation, Padé approximation and Laguerre expansion. Such approaches require estimation of more parameters because the order of the approximated system model is increased, and an unacceptable approximation error may occur when the system has a large delay. The two-step procedure (Elnaggar, Dumont & Elshafei, 1989) first assumes that a known delay and estimates the other transfer parameters, then minimizes the least-squares error performance index with respect to the delay value. In a somewhat dual way, Ferretti, Maffezzoni and Scattolini (1991) suggests an algorithm to recursively update the value of a small delay by inspection of the phase contribution of the real negative zero arising in the corresponding sampled system. The main drawback of these methods is that iteration on delay is needed to estimate the parameters and this makes on-line implementation difficult. These methods are mostly developed for discrete systems while continuous systems are more familiar to practicing control engineers. Moreover, the identification robustness is yet a big concern with the methods. In Pintelon and Biesen (1990), a Gaussian frequency-domain maximum likelihood estimator of the transfer function of linear continuous time systems with time delay from the test of multisine input signal is presented.

The input signal can have significant influence on identification results. Its selection depends on several factors: available instrumentation, allowed disturbance, etc. The test signal should enable one to inject as much energy as possible (Schoukens & Pintelon, 1991). Popular test signals include pulse, pseudo-random binary sequence, step, ramp and sinusoidal functions (Unbehauen & Rao, 1987). Of all these tests, the step test is the simplest and dominant in process control applications. In the context of process control, continuous-time transfer function models are preferred and are essential to employ popular tuning techniques such as internal model control (IMC). It is noted that the existing identification methods using a step test result in a first-order plus dead-time (FOPDT) or rational dead-time free transfer function (Astrom & Hagglund, 1995; Rake, 1980), and the accuracy of the estimated model can be degraded significantly with noise since most methods only use a few points of the activated response which is usually contaminated with noise. Moreover, such methods are difficult to be extended to a second-order plus dead-time (SOPDT) or even high-order systems with delay. In Mamat and Fleming (1995), Rangaiah and Krishnaswamy 1996a, Rangaiah and Krishnaswamy 1996b, graphical methods are proposed to identify a SOPDT model depending on different types of response, i.e., underdamped, mildly underdamped, or overdamped. However, such a SOPDT model cannot well describe non-minimum-phase system. On the other hand, though model parameters can be easily calculated using several points from the system response, such graphical methods may not be robust to noise.

In this paper, a simple, general and robust method is proposed for identification of linear continuous time-delay processes from step responses. The solution and its various-order integrals of the process differential equation to a step input are derived and new linear regression equations are formed. The regression equations are solved using least-squares method and the parameters in the transfer function model are then recovered.

The paper is organized as follows. In Section 2, the proposed identification method is presented for SOPDT model. The method is further extended to general nth-order model in Section 3. Guidelines for implementation are discussed in Section 4. Simulation and real-time implementation are shown in Section 5. Conclusions are drawn in Section 6.

Section snippets

Second-order modelling

This section focuses on SOPDT modelling. It serves for motivation of the general method to be described in the next section and for recommended use in applications since such SOPDT models can essentially cover most practical industrial processes.

Assume that a stable process is represented byY(s)=G(s)U(s)=b1s+b2s2+a1s+a2e−LsU(s).Assume also that the process has distinct (possibly complex) poles and that the input is of step type, i.e., U(s)=h/s. Then the process output isY(s)=Kp1λ2βs+1s(s+λ1

nth-order modelling

Suppose that a time-invariant stable process with distinct poles (possibly complex) is represented by a NOPDT model:Y(s)=G(s)U(s)=b1sn−1+b2sn−2+⋯+bn−1s+bnsn+a1sn−1+⋯+an−1s+ane−LsU(s)=Kpi=1nλii=1n−1is+1)i=1n(s+λi)e−LsU(s),whereak=1≤i1<i2<⋯<ik≤nλi1λi2⋯λik,k=1,2,…,n,bk=Kpi=1nλi1≤i1<i2<⋯<in−k≤n−1βi1βi2⋯βin−k,k=1,2,…,n−1,bn=Kpi=1nλi.Assume that the input is of step type, i.e., U(s)=h/s, then the process output becomesY(s)=α0s+α1s+λ1+α2s+λ2+⋯+αns+λne−Ls,whereαj=−Kphi=1nλii=1n−1(1−βiλj)λj

Implementation issues

In this section, several practical issues in implementation of the proposed algorithm are discussed.

Choice oft1: It is noted from the above development that the first sample y(t1) should not be taken into the algorithm until t1L, when the output deviates from the previous steady state. In practice, the selection of the logged y(t) after tiL can be made as follows. Before the step test starts, the process output will be monitored for a period called the ‘listening period’, during which the

Simulation and real-time test

The proposed step identification method is now applied to several typical processes. Without loss of generality, a unit step is employed in all the simulation below. For a better assessment of its accuracy, identification errors in both the time domain and the frequency domain are considered. This is because some step identification methods are found to be able to fit the process response well in the time domain, but the frequency response of the model sometimes deviates too far away from the

Conclusions

In this paper, a new method has been developed for robust identification of linear continuous-time processes from step responses. The proposed method is based on the newly derived linear regression equations and instrumental variable least-squares technique. Guidelines for implementing the method is clearly set up. Simulation shows that the method gives a better identification result than existing ones using step test. The field test of the method achieved excellent performance.

Acknowledgements

The authors are indebted to the many constructive comments and suggestions made by the anonymous referees.

Qing-Guo Wang received, respectively, the B.Eng. in Chemical Engineering in 1982, the M. Eng. in 1984 and Ph.D. in 1987, both in Industrial Automation, all from Zhejiang University of the PRC. Since 1992 he has been with the Department of Electrical Engineering of National University of Singapore where he is currently an Associate Professor. He held a Alexander-von-Humboldt Research Fellowship of Germany with Duisburg University and Kassel University, from 1990 to 1992. His present research

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    Qing-Guo Wang received, respectively, the B.Eng. in Chemical Engineering in 1982, the M. Eng. in 1984 and Ph.D. in 1987, both in Industrial Automation, all from Zhejiang University of the PRC. Since 1992 he has been with the Department of Electrical Engineering of National University of Singapore where he is currently an Associate Professor. He held a Alexander-von-Humboldt Research Fellowship of Germany with Duisburg University and Kassel University, from 1990 to 1992. His present research interests are mainly in systems theory, robust, adaptive and multivariable control and optimization with emphasis on their applications in process, chemical and environmental industries.

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