Code covered by the BSD License

# The simulation of time-delay systems

10 Aug 2012 (Updated 08 Oct 2013)

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

yxiao_fun1(t,y,Z)
```% The simulation of time-delay systems
% Copyright (C2012-2015) Yang XIAO, Beijing Jiaotong University, Aug.10, 2012, E-Mail: yxiao@bjtu.edu.cn.
% In our book [1], we 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
% the time-delay system of case 1 is unstable, 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.

function v = yxiao_fun1(t,y,Z)
ylag1 = Z(:,1);
ylag2 = Z(:,2);
ylag3 = Z(:,3);
ylag4 = Z(:,4);
v = zeros(4,1);
A= [0 1 0 0;0 0 1 0;0 0 0 1;-29.17 -56 -36.7 -10.1];
B1=[-1.55 1 0 0;-1 -0.3 0 0;0 0 0.5 0;-.7 0 -.34 -2.6];
B2=[0 0 0 0;1 1.5 4 0;0 0 0 0;-.33 0 0 -1.1];
B3=[0 0 0 0;0 0 0 0; 0 0 0 0;-0.08 -0.7 0 -1];
B4=[0 0 0 0;0 0 0 0; 0 0 0 0; 0 0 -3 0];
v=A*y+B1*ylag1+B2*ylag2+B3*ylag3+B4*ylag4; ```