Discussion: “Analysis of a System of Linear Delay Differential Equations” (Asl, F. M., and Ulsoy, A. G., 2003, ASME J. Dyn. Syst., Meas., Control, 125, pp. 215–223)

The exact solution of Eq. 1 cannot, in general, be obtained. Special cases of Eq. 1 were considered by Chen et al. (1) and exact analytical solutions were obtained. In (2), Asl and Ulsoy offer a closed form solution to the general case in matrix form and compute the stability lobes numerically. Both studies in (1,2) are based on a solution of a transcendental equation expressed in terms of the Lambert function, which was first derived by Briggs in (3). Because the solution involves the Lambert function and it is given as a series, reaching an exact analytical stability bound for Eq. 1 is still a problem.

Let us reconsider the case study in (2) with

In Fig. 1, the data points $N=1∕T,Kc∕Km$⁠, represented by a diamond sign, are obtained from 8 numerically for the principal branch of the Lambert function. An important observation is that these data points do not match with Fig. 9 of (2). Figure 1 also displays, by the solid curve, the actual stability lobe diagram predicted analytically (5).

Using certain data points from Fig. 1, one can easily verify that 8 is satisfied but 7 is not. It is easy to check that $(19.9,0.193993)$ at $s=180.520834i$ is such a point. On the other hand, it can also be shown that 7 holds but 8 does not for some data points from Fig. 1. For instance, this can be verified by using $(14.245976,0.105)$ at $s=157.321327i$⁠.

As a result, the method given by Asl and Ulsoy (2) results in $s$ values which do not satisfy 7. Thus, the closed form solution of Eq. 1 given in (2) is not correct.

I would like to thank the anonymous referee for the valuable suggestions.