LMIs in Control/pages/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The SystemEdit

 

where  ,  ,  , at any  .

The DataEdit

The matrices   are known.

The LMI: The Positive Real LemmaEdit

Suppose   is the system. Then the following are equivalent.

 
 
 

Conclusion:Edit

The Positive Real Lemma can be used to determine if the system   is passive. Note from the (1,1) block of the LMI we know that   is Hurwitz.

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

Related LMIsEdit

KYP Lemma (Bounded Real Lemma)

External LinksEdit

A list of references documenting and validating the LMI.


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