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

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

The LMI: The KYP or Bounded Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad \left\| G \right\|_{H_\infty} \leq \gamma $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0$$
 * $$ 3)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB & C^T\\B^TX & -\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