# LMIs in Control/pages/TDSIC

< LMIs in Control | pages

**The System**Edit

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

where

is the initial condition

represents the time-delay

is a known upper-bound of

**The Data**Edit

The matrices are known

**The LMI:*** The Time-Delay systems (Delay Independent Condition) *Edit

*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 such that

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

**Conclusion:**Edit

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

**Implementation**Edit

The implementation of the above LMI can be seen here

**Related LMIs**Edit

Time Delay systems (Delay Dependent Condition)

## External LinksEdit

- [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.