LMIs in Control/KYP Lemmas/KYP Lemma (Bounded Real Lemma)

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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]

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

Related LMIs[edit | edit source]

Positive Real Lemma

External Links[edit | edit source]

A list of references documenting and validating the LMI.

Return to Main Page:[edit | edit source]