User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Time-Delay system on delay Dependent Condition

The System
The problem is to check the stability of the following linear time-delay system on a delay dependent condition



\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_dx(t-d)\\ x(t)&=\phi(t), t\in[-d,0], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} {A, A_d}\in\mathbb{R}^{n\times n}, A\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ \end{align}$$

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

For the purpose of the delay dependent system we rewrite the system as

\begin{cases} \dot x(t) &= Ax(t)+A_dx(t-d)\\ \dot x(t) &= (A+A_d)x(t)-A_d(x(t)-x(t-d) \end{cases} $$

The Data
The matrices $$A, A_d$$ are known

The LMI: The Time-Delay systems (Delay Dependent Condition)
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix $$X$$ and a scalar $$0< \beta <1$$ such that



\begin{bmatrix} \Phi(X) & \bar dXA^T & \bar dXA_d^T\\ \bar dAX & -d \beta I & 0\\ \bar dA_dX & 0 & -\bar d(1- \beta )I \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

Here $$ \Phi(X) =X)A+A_d)^T + (A+A_d)X+ \bar d A_dA_d^T<0 $$

This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if

$$ P(A+A_d) + (A+A_d)^TP + \bar dPA_dA_d^TP + \frac {\bar d}{\beta}A^TA +\frac {\bar d}{1- \beta} A_d^TA_d<0$$ This is obtained by replacing $$X$$ with $$P^{-1}$$

Conclusion:
We can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Independent Condition)