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

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t)\\ y(t)& =Cx(t)+Du(t)\\ x(0)& =x_0\end{array}$

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

The Data

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

The Optimization Problem

The following optimization problem must be solved.

$\begin{array}{cc}& \underset{\gamma ,X}{\mathrm{minimize}}\gamma \\ & \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& X>0\\ & & \left[\begin{array}{cc}{A}^{T}X+XA& XB\\ B^{T}X& -\gamma I\end{array}\right]+\frac{1}{\gamma }\left[\begin{array}{c}{C}^T\\ D^T\end{array}\right]\left[\begin{array}{cc}C& D\end{array}\right]<0\\ \end{array}$

The LMI: The KYP or Bounded Real Lemma

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

$1){\Vert G\Vert }_{H_{\mathrm{\infty }}}\le \gamma$

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

$\left[\begin{array}{cc}{A}^{T}X+XA& XB\\ B^{T}X& -\gamma I\end{array}\right]+\frac{1}{\gamma }\left[\begin{array}{c}{C}^T\\ D^T\end{array}\right]\left[\begin{array}{cc}C& D\end{array}\right]<0$

Conclusion:

The KYP Lemma can be used to find the bound $\gamma$ on the $H_{\mathrm{\infty }}$ 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

• 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.