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# The simulation of time-delay systems

### Yang Xiao (view profile)

10 Aug 2012 (Updated )

This program of time-delay systems can verify stability test theorems.

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In our book [1], we proposed some stability test theorems for linear time-delay system.
To verify our stability test theorems and others' stability test algorithms [2], we provide this simulation program of time-delay systems.
You can use the program, to find that the results of [2] are incorrecct.
The following example is taken from the case 1 of [2], where [2] claimed to found a stable region for the time-delay system of case 1,however, the simulation shows that the time-delay system of case 1 is unstable,
dx(t)/dt=Ax(t)+B1x(t-t1)+B2x(t-t2)+B3x(t-t1)+B4x(t-t4)
with delay parameters: t1=.4;t2=.5; t3=.169;t4=0.26.
Thus, the main results of [2] are incorrect.

[1] Yang Xiao, Yingkang Zhang, Multidimensional Signal Processing and
Multidimensional Systems, Publishing House of Electronics Industry,
Beijing, 2011.
[2] Rifat Sipahi and Ismail Ilker Delice, Advanced Clustering With Frequency Sweeping Methodology for the Stability Analysis of Multiple Time-Delay Systems,IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 2, 2011, pp.467-472.

MATLAB release MATLAB 7 (R14)
07 Oct 2013 Yang Xiao

### Yang Xiao (view profile)

Most of stability test algorithms are sufficient condition for time-delay systems. To verify the conservatism of the algorithms we need some examples of time-delay systems. We are interested in the recent new stability test algorithms including your algorithm in [2].
However, we disappointed that [2] claimed to found a stable region for the time-delay system of case 1, our simulation shows that the time-delay system of case 1 is unstable in fact. It is possible for the example in [2] to mislead readers of IEEE Trans AC.

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05 Oct 2013 Rifat Sipahi

### Rifat Sipahi (view profile)

The numerical example in [2] was accidentally labeled stable as this can be trivially confirmed with TRACE DDE software. On the link

http://www1.coe.neu.edu/~rifat/publist.htm

the correction is clearly mentioned.

It is questionable why [1] is needed to simulate time-delay systems; as multiple and world-wide established resources already exist:

Simulink toolbox can simulation LTI delay systems easily, and rightmost roots can be computed easily by the work of Dimitri Breda (and colleagues), Dirk Roose (and colleagues), Tomas Vyhlidal (and colleagues), and Galip Ulsoy (and colleagues), using respectively TRACE DDE, BIF TOOL, QPMR, and Lambert W function approaches.

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