== The System ==
The problem is to check the stability of the following linear time-delay system
{
x
˙
(
t
)
=
A
x
(
t
)
+
A
d
x
(
t
−
d
)
x
(
t
)
=
ϕ
(
t
)
,
t
∈
[
−
d
,
0
]
,
0
<
d
≤
d
¯
,
{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0
0
{\displaystyle P>0}
[
A
T
P
+
P
A
+
S
P
A
d
A
d
T
P
−
S
]
{\displaystyle {\begin{bmatrix}A^{T}P+PA+S&PA_{d}\\A_{d}^{T}P&-S\end{bmatrix}}}
<
0
{\displaystyle {\begin{aligned}<0\end{aligned}}}
This LMI has been derived from the Lyapunov Function for the system.
By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati Inequality
A
T
P
+
P
A
+
P
A
d
S
−
1
A
d
T
P
+
S
<
0
{\displaystyle A^{T}P+PA+PA_{d}S^{-1}A_{d}^{T}P+S<0}
== Conclusion: ==
We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition
== Implementation ==
The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks
== Related LMIs ==
Time Delay systems (Delay Dependent Condition)
== External Links ==
[1] - LMI in Control Systems Analysis, Design and Applications
LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
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