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 SystemEdit


where  ,  ,  , at any  .

The DataEdit

The matrices   are known.

The Optimization ProblemEdit

The following optimization problem must be solved.


The LMI: The KYP or Bounded Real LemmaEdit

Suppose   is the system. Then the following are equivalent.



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.


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 LMIsEdit

Positive Real Lemma

External LinksEdit

A list of references documenting and validating the LMI.

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