[<< wikibooks] LMIs in Control/pages/TDSIC
== 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)

[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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