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 $H_{\infty }$ norm of a system and is also useful for proving many LMI results.

The System

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

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

The Data

The matrices $A,B,C,D$ are known.

The Optimization Problem

The following optimization problem must be solved.

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

The LMI: The KYP or Bounded Real Lemma

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

$1)\quad \left\|G\right\|_{H_{\infty }}\leq \gamma$

$2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0$

$3)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB&C^{T}\\B^{T}X&-\gamma I&D^{T}\\C&D&-\gamma I\end{bmatrix}}<0$

Conclusion:

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

Implementation

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

Related LMIs

Positive Real Lemma

External Links

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.