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 System

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where  ,  ,  , at any  .

The Data

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The matrices   are known.

The LMI: The Positive Real Lemma

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Suppose   is the system. Then the following are equivalent.

 
 
 

Conclusion:

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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.

Implementation

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

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KYP Lemma (Bounded Real Lemma)

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A list of references documenting and validating the LMI.


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