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

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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 System[edit | edit source]

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 Data[edit | edit source]

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

The Optimization Problem[edit | edit source]

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 Condition[edit | edit source]

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 | edit source]

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 .

Implementation[edit | edit source]

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:


External Links[edit | edit source]

A list of references documenting and validating the LMI.

Return to Main Page:[edit | edit source]