LMIs in Control/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

 

where  ,  ,  , at any  .

The Data edit

The matrices   are known.

The Optimization Problem edit

The following optimization problem must be solved.

 

The LMI: The KYP or Bounded Real Lemma edit

Suppose   is the system. Then the following are equivalent.

 
 
 

Conclusion: edit

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

A link to CodeOcean or other online implementation of the LMI (in progress)

Related LMIs edit

Positive Real Lemma

External Links edit


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