LMIs in Control/KYP Lemmas/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]A link to CodeOcean or other online implementation of the LMI (in progress)
Related LMIs
[edit | edit source]KYP Lemma (Bounded Real Lemma)
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.