LMIs in Control/Stability Analysis/Continuous Time/Transient Output Bound for Non-Autonomous LTI Systems

The System

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Consider a continuous-time LTI system with a state-space representation of:

 

 

where  ,  ,   and  ,

the transient bound on the output can be evaluated with the following LMI.

The Data

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 ,  ,   and  .

The LMI:

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The Euclidean norm of the output satisfies

 

if there exists   and  , where  , such that

  •  
  •  
  •  

Conclusion

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By using these LMIs, the transient output bound can be found for a given non-autonomous LTI system such that if   and   is a unit-energy input (i.e.,  ), then the preceding conditions ensure that  .

Implementation

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The LMI given above can be implemented and solved using a parser such as YALMIP, along with an LMI solver such as SeDuMi, or MOSEK. (For a detailed list of solvers, please visit this page on Relevant Solvers).

References

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Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.