LMIs in Control/pages/Positive Real Lemma

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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[edit | edit source]

where , , , at any .

The Data[edit | edit source]

The matrices are known.

The LMI: The Positive Real Lemma[edit | edit source]

Suppose is the system. Then the following are equivalent.

Conclusion:[edit | edit source]

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[edit | edit source]

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

Related LMIs[edit | edit source]

KYP Lemma (Bounded Real Lemma)

External Links[edit | edit source]

A list of references documenting and validating the LMI.


Return to Main Page:[edit | edit source]