# LMIs in Control/KYP Lemmas/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 ${\displaystyle H_{\infty }}$ norm of a system and is also useful for proving many LMI results.

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}}

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$, ${\displaystyle y(t)\in \mathbb {R} ^{m}}$, ${\displaystyle u(t)\in \mathbb {R} ^{q}}$, at any ${\displaystyle t\in \mathbb {R} }$.

## The Data

The matrices ${\displaystyle A,B,C,D}$ are known.

## The Optimization Problem

The following optimization problem must be solved.

{\displaystyle {\begin{aligned}&{\underset {\gamma ,\;X}{\operatorname {minimize} }}\quad \gamma \\&\operatorname {subject\;to} &X>0\\&&{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}}

## The LMI: The KYP or Bounded Real Lemma

Suppose ${\displaystyle {\hat {G}}(s)(A,B,C,D)}$ is the system. Then the following are equivalent.

${\displaystyle 1)\quad \left\|G\right\|_{H_{\infty }}\leq \gamma }$
${\displaystyle 2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}}$
${\displaystyle {\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0}$
${\displaystyle 3)\quad {\text{There exists a}}\;X>0\;{\text{such that}}}$
${\displaystyle {\begin{bmatrix}A^{T}X+XA&XB&C^{T}\\B^{T}X&-\gamma I&D^{T}\\C&D&-\gamma I\end{bmatrix}}<0}$

## Conclusion:

The KYP Lemma can be used to find the bound ${\displaystyle \gamma }$ on the ${\displaystyle H_{\infty }}$ norm of a system. Note from the (1,1) block of the LMI we know that ${\displaystyle A}$ is Hurwitz.

## Implementation

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