LMIs in Control/pages/Lin Sys Time Delay Stability LMI

LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

For systems experiencing time-varying delays where the delays are bounded, the feasibility LMI in this section can be used to determine if the system is -exponentially stable.


The SystemEdit

 

where   is the state,   are the matrices of delay dynamics, and   is the initial function with norm   and it is continuously differentiable function on  . The tyime-varying delay function   satisfies:

 

The DataEdit

The matrices   are known, as well as the bounds   of the time-varying delay.

The Optimization ProblemEdit

For a given  , the zero solution of the system described above is  -exponentially stable if there exists a positive number   such that every solution   satisfies the following condition:

 

The LMI: -Stability ConditionEdit

The following feasibility LMI can be used to check if the system is  -exponentially stable or not for a given  :

 

The above LMI can be combined with the bisection method to find  .

Conclusion:Edit

For systems with time-varying delays with intervals, the LMI in this section can be used to check if the system is exponentially stable with a certain  . The bisection algorithm can be additionally used to compute  .

ImplementationEdit

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Intervaled_Delay_Sys_Stability_example.m

External LinksEdit

A list of references documenting and validating the LMI.

Return to Main Page:Edit