LMIs in Control/pages/TDSIC

The System

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

$\begin{array}{c}\{\begin{array}{cc}\dot{x}(t)& =Ax(t)+A_{d}x(t-d)\\ x(t)& =\varphi (t),t\in [-d,0],0<d\le \overline{d},\end{array}\end{array}$

where

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

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

The Data

The matrices $A,A_d$ 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 $P,S\in {𝕊}^n$ such that

$P>0$

$\left[\begin{array}{cc}{A}^{T}P+PA+S& P{A}_d\\ A_d^{T}P& -S\end{array}\right]$$\begin{array}{c}<0\end{array}$

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
$A^{T}P+PA+P{A}_d{S}^{-1}{A}_d^{T}P+S<0$

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

• [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.