LMIs in Control/pages/TDSDC

The SystemEdit

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 DataEdit

The matrices   are known

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

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

ImplementationEdit

The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIsEdit

Time Delay systems (Delay Independent Condition)

External LinksEdit

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