# LMIs in Control/pages/TDSDC

**The System**Edit

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

where

is the initial condition

represents the time-delay

is a known upper-bound of

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

**The Data**Edit

The matrices are known

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

*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

and a scalar such that

Here

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

This is obtained by replacing with

**Conclusion:**Edit

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

**Implementation**Edit

The implementation of the above LMI can be seen here

**Related LMIs**Edit

Time Delay systems (Delay Independent 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.