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 ${\displaystyle H_{\infty }}$ 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.

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

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

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:

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

This dual KYP LMI is now linear in both ${\displaystyle X}$ and ${\displaystyle \gamma }$.

This implementation requires the use of Yalmip and Sedumi.