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Automatica 45 (2009) 2467–2470

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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 Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden



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
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1. Poles (x) and zeros (o) plot of an identified Laguerre model (10 coefficients, ˆ pole at 0.8) with a successful time-delay estimation (Td = 9.7114 sampling intervals). No zero guarding (see Section 2.3). The poles of the model should all be located at α = 0.8, but due to well-known numerical problems with multiple poles they are somewhat spread in the figure.

Fig. 2. Pole-zero plot of an identified Laguerre model with a failing time-delay ˆ estimation (Td = 1321.86 sampling intervals). The simulation setup is the same as in Fig. 1. Only the noise realization differs.

Pzmap whole model, 3 std, t125a1, Trial 3 0.8 0.6


ˆ The time-delay estimate Td (in number of sampling intervals) is given by an approximation of the derivative of the phase ϕ(ω) = arg Gap (eiωTs ) at the frequency zero and with an addition of 1 because of the extra time delay that is created by the sampling: ˆ Td = − arg Gap (eiω1 Ts )

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 0.5 1 1.5

ω 1 Ts

+ 1;




This is the DAP method. Of course, (1) cannot be evaluated for ω1 = 0. In Horch (2000); Isaksson et al. (2001) ω1 = 10−4 is suggested, since it ‘‘has been found to be sufficiently small when using typical industrial data’’. This value is also used in this article. Other approximations of the derivative should also be possible. In Horch (2000); Isaksson et al. (2001) a Laguerre model (Wahlberg, 1991) was used.

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