The System

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

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

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

https://github.com/yashgvd/LMI_wikibooks

Related LMIs

Time Delay systems (Delay Dependent 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.

Return to Main Page: