Predictor De Smith

Automatica 45 (2009) 2467–2470

Contents lists available at ScienceDirect

Automatica
journal homepage: www.elsevier.com/locate/automatica

Technical communique

An improved phase method for time-delay estimation$
Svante Björklund a,∗ , Lennart Ljung b
a b

Swedish Defence Research Agency, Box 1165, SE - 581 11, Linköping, Sweden Division of Automatic Control, Department of ElectricalEngineering, Linköping University, SE-581 83 Linköping, Sweden

article

info

abstract
A promising method for estimation of the time-delay in continuous-time linear dynamical systems uses the phase of the all-pass part of a discrete-time model of the system. We have discovered that this method can sometimes fail totally and we suggest a method for avoiding such failures. © 2009 Elsevier Ltd. All rightsreserved.

Article history: Received 23 November 2007 Received in revised form 7 February 2009 Accepted 4 July 2009 Available online 4 August 2009 Keywords: Time-delay Dead-time Laguerre All-pass Phase Zero System identification Estimation

1. Introduction Estimation of time-delays (or dead-times) in linear dynamical systems with additive noise is a common engineering problem, e.g. in controlperformance monitoring of industrial processes, in design and tuning of controllers, in range estimation in radar and in direction finding in signal intelligence. It is also a necessary part in all system identification. In Horch (2000); Isaksson, Horch and Dumont (2001) a timedelay estimation method is described: A discrete-time model of a continuous-time system is identified using a Laguerre basis. Thediscrete-time non-minimum phase zeros of the model constitute the all-pass part of the model and directly represent the timedelay. The time-delay is estimated by studying the slope at low frequencies of the phase of the all-pass part. The method shows very good results in both open loop and closed loop for input and reference signals in the form of steps in Horch (2000); Isaksson et al. (2001).This method is in this paper called Laguerre DAP. The part of the method from the discrete-time model to the time-delay

estimate is here called the DAP (Discrete-time All-pass part Phase) method. In this paper we show that the DAP method is non-robust and we improve the method to also handle some cases where it otherwise totally fails. A more thorough description of the work underlying this paperis given in Björklund (2002). 2. The DAP method 2.1. The principles of the DAP method

$ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Guoxiang Gu under the direction of Editor André L. Tits. ∗ Corresponding author. Tel.: +46 13 378000; fax: +46 13 378100. E-mail addresses: svabj@foi.se (S. Björklund),ljung@isy.liu.se (L. Ljung).

¯ ¯ Assume the true continuous-time system is G(s) = G1 (s) · ¯ ¯ ¯ e−sTd = G1 (s) · Gap (s). The system G1 (s) is a SISO (single-input single-output) time-invariant linear rational transfer function. A ¯ discrete-time rational linear model G(z ) of G(s) is estimated and factorized into a minimum-phase system G1 (z ) and an all-pass system Gap (z ) as G(z ) = G1 (z )Gap (z ). Then,Gap (eiωTs ) is considered ¯ to be an approximation of the time-delay system Gap (iω) = e−iωTd (Horch, 2000; Isaksson et al., 2001), which should agree well for low frequencies since the frequency function of a sampled system agrees well with its continuous counterpart for low frequencies (Ljung & Glad, 1994). Ts is the sampling interval. The all-pass part Gap (z ) of G(z ) is formed by thenon-minimum phase (positioned outside the unit circle) zeros of G(z ) and with poles added to Gap (z ), which are these zeros mirrored in the unit circle.

0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.07.001

2468

S. Björklund, L. Ljung / Automatica 45 (2009) 2467–2470

Pole–zero map Pzmap whole model, t122b1, Trial 3 0.8 0.6 0.4 Imag Axis...