The System

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

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

where

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

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

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

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

The Data

The matrices ${\displaystyle A,A_{d}}$ 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
${\displaystyle X}$ and a scalar ${\displaystyle 0<\beta <1}$ such that

${\displaystyle {\begin{bmatrix}\Phi (X)&{\bar {d}}XA^{T}&{\bar {d}}XA_{d}^{T}\\{\bar {d}}AX&-d\beta I&0\\{\bar {d}}A_{d}X&0&-{\bar {d}}(1-\beta )I\end{bmatrix}}}$${\displaystyle {\begin{aligned}<0\end{aligned}}}$

Here ${\displaystyle \Phi (X)=X)A+A_{d})^{T}+(A+A_{d})X+{\bar {d}}A_{d}A_{d}^{T}<0}$

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

${\displaystyle P(A+A_{d})+(A+A_{d})^{T}P+{\bar {d}}PA_{d}A_{d}^{T}P+{\frac {\bar {d}}{\beta }}A^{T}A+{\frac {\bar {d}}{1-\beta }}A_{d}^{T}A_{d}<0}$
This is obtained by replacing ${\displaystyle X}$ with ${\displaystyle P^{-1}}$

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)

External Links

• - 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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