Direct identification of continuous time delay systems from step responses

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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 directly derived from the process differential equation. The regression parameters are then estimated without iterations, and an 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.

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 [1]. A considerable number of identification methods have been reported in the literature [Automatica 1981 v.17(1), Automatica 1990 v.26(1), IEEEAC 1992 v.37(7), Automatica 1995 v.31(12)], and they are generally classified into parametric and non-parametric ones [2]. Transfer functions might be the most welcome parametric model. Methods of filtering non-parametric time responses to transfer functions are illustrated in Unbehaue and Rao [3]. Fitting parametric models to measured frequency data is another viable approach [4].

However, most of the existing methods for transfer function identification do not consider the process delay (or dead-time) [5], [6], [7], [8] 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 has been continuing interest in identification of delay processes. A frequently used method for dealing with unknown delays is to use a shift operator model with an expanded numerator polynomial [9]. Another popular approach is based on the approximation of the dead-time by a rational transfer function such as the polynomial approximation [10], Padé approximation [11] and Laguerre expansion [12]. 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 [13] first assumes 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 et al. [14] suggest 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. Furthermore these methods are mostly developed for discrete systems while continuous systems are more familiar to practising control engineers. Identification robustness is a big concern with the methods. In recent years, the genetic algorithm (GA) has received considerable attention in various fields, because it has a high potential for global optimization. Studies on control and identification problems using GA are made, for example, by [15], [16]. However, GA is always computationally demanding.

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) [17] and integral of the time-weighted absolute value of the error (ITAE) [18]. In general, processes are of high order and have certain nonlinearity. But control engineers usually use a first-order plus dead-time (FOPDT) model as an approximation to such processes for control practice:Gs=KpTs+1e−Ls.

For such an FOPDT model, area-based methods are more robust than other methods such as the graphical method or two-point method [19]. This FOPDT model is able to represent the dynamics of many processes over the frequency range of interest for feedback controller design [20]. Yet, there are certainly many other processes for which the model (1) is not adequate to describe the dynamics, or for which higher-order modelling could improve accuracy significantly.

The input signal can have significant influence on identification results. Popular test signals include pulse, pseudo-random binary sequence, step, ramp and sinusoidal functions [21]. Of all these tests, the step test is probably the simplest. The step test needs little equipment, and can even be performed manually. A step test can be easily implemented on programmable logic controllers (PLC) or distributed control systems (DCS), since the step function is usually available as a standard module in most PLCs and DCSs. Therefore, step tests are dominant in process control applications. It is noted that the existing identification methods using a step test result in a FOPDT or rational dead-time free transfer function [19], [22], 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 extend to a second-order plus dead-time (SOPDT) or even higher order systems with delay.

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 directly derived from the process differential equation. The regression parameters are then estimated, without iteration, and explicit relationships between the regression parameters and those in the process are given. Due to the 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 SOPDT 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.

The paper is organized as follows. In Section 2, the proposed identification method is presented for an SOPDT model. The method is extended to general n-th order models in Section 3. Guidelines for implementation are discussed in Section 4. Simulations are shown in Section 5, and conclusions are drawn in Section 6.

Section snippets

Second-order modelling

This section focuses on SOPDT modelling. It motivates the general method to be described in the next section, and is itself very useful as SOPDT models can essentially cover most practical industrial processes [20].

Assume that a stable process is represented in the Laplace domain byYs=GsUs=b1s+b2s2+a1s+a2e−LsUs,or in time domain byÿt+a1ẏt+a2yt=b1u̇t−L+b2ut−L,under zero initial conditions with the time delay, L⩾0. Integrating both sides of (3) givesy+a10tyτdτ+a20t0τyτ1dτ1dτ=b10tuτ−Ldτ+b20t

n-th order modelling

Suppose that a time-invariant stable process is represented byYs=GsUs=b1sn−1+b2sn−2+…+bn−1s+bnsn+a1sn−1+…+an−1s+ane−LsUs,or equivalently byynt+a1yn−1t+…+an−1y1t+anyt=b1un−1t−L+b2un−2t−L+…+bn−1u1t−L+bnut−L,where L⩾0. For an integer m⩾1, define0,tmf=0t0τm0τ2fτ1dτ1dτm.

Under zero initial conditions,0,tmut−L=1m!t−Lmh.

Integrating (15) n times givesyt=−a10,t1y−a20,t2y…−an−10,tn−1y−an0,tny+hb1t−L+12hb2t−L2+…+1n−1!hbn−1t−Ln−1+1n!hbnt−Ln,=−a10,t1y−…−an0,tny+ht0j=1nbj−Ljj!+ht1j=1nbj−Lj−1j−1!

Implementation issues

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

Simulation

The proposed step identification method was applied to several typical processes. Without loss of generality, a unit step was 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 fit the time domain well, but the frequency response of the model sometimes deviates too far away from the real process frequency response. To

Conclusions

In this paper, a new method has been developed for direct and robust identification of linear continuous time processes from step responses. The proposed method is based on linear regression equations and the instrumental variable least squares technique. Guidelines for implementing the method are given. Simulation shows that the method gives a better identification result than existing methods using step tests.

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