# LMIs in Control/pages/TDSIC

## The System

The problem is to check the stability of the following linear time-delay system

{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0

where

{\displaystyle {\begin{aligned}{A,A_{d}}\in \mathbb {R} ^{n\times n}{\text{, }}{A}\in \mathbb {R} ^{n\times r}{\text{ are the system coefficient matrices,}}\\\end{aligned}}}

${\displaystyle \phi (t)}$  is the initial condition
${\displaystyle d}$  represents the time-delay
${\displaystyle {\bar {d}}}$  is a known upper-bound of ${\displaystyle d}$

## The Data

The matrices ${\displaystyle A,A_{d}}$  are known

## The LMI: The Time-Delay systems (Delay Independent Condition)

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists two symmetric matrices ${\displaystyle P,S\in \mathbb {S} ^{n}}$  such that

${\displaystyle P>0}$

${\displaystyle {\begin{bmatrix}A^{T}P+PA+S&PA_{d}\\A_{d}^{T}P&-S\end{bmatrix}}}$ {\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
${\displaystyle A^{T}P+PA+PA_{d}S^{-1}A_{d}^{T}P+S<0}$

## 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

## Related LMIs

Time Delay systems (Delay Dependent Condition)