LMIs in Control/pages/KYP Lemma (Bounded Real Lemma)
KYP Lemma (Bounded Real Lemma)
The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the norm of a system and is also useful for proving many LMI results.
The System[edit | edit source]
where , , , at any .
The Data[edit | edit source]
The matrices are known.
The Optimization Problem[edit | edit source]
The following optimization problem must be solved.
The LMI: The KYP or Bounded Real Lemma[edit | edit source]
Suppose is the system. Then the following are equivalent.
Conclusion:[edit | edit source]
The KYP Lemma can be used to find the bound on the norm of a system. Note from the (1,1) block of the LMI we know that is Hurwitz.
Implementation[edit | edit source]
Since the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:
This dual KYP LMI is now linear in both and .
This implementation requires the use of Yalmip and Sedumi.
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.