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[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]
External Links[edit | edit source]
A list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.