value of $\gamma$ in the H-infinity norm

Question

Suppose I have the system:

$\dot{x} = Ax+Bu$

$y=Cx+Du$

and the following Hamiltonian matrix:

$H=\begin{pmatrix} A & \frac{1}{2}B^TB\\ -CC^T&-A \end{pmatrix}$

I want to find the value of $\gamma$ which is the bound of the $H_{\infty}$ norm, so it is the value such that $\left |T(j\omega) \right |_{\infty }<\gamma$.

I know that for the bounded real lemma, if the eigenvalues of $A$ have negative real part, and $I\gamma^2-DD^T>0$, then the Hamiltonian have no eigenvalues on the imaginary axis. I also know that if the eigenvalues of $A$ have negative real part, then $\left |T(j\omega) \right |_{\infty }<\gamma$ holds.

But my question is : how do I find the value of $\gamma$?

I have been told the result is $\gamma=0.5$ but I really can't get to this result. I have tried using the Shur's complement to see if this matrix is negative definite (so before doing that i switched sign to the Hamiltonian). In this way i thiught that if the A matrix is negative definite, it has all eigenvalues with negative real part, so the resulting value of $\gamma$ form the computation would have been the searched value. But I don't find the desired result. Maybe I am missing a point and doing something wrong, or maybe I am complitely on the wrong path.

Can somebody please help me? Thank's in advance.

Answer 1

I believe you should take a look of the real bounded lemma. Because the hamiltonian matrix has another shape. I quote the book from Zhou - Essentials of Robust Control p.238. $$ H=\begin{bmatrix} A+BR^{-1}D^{\top}C & BR^{-1}B^{\top} \\ -C^{\top}(I+DR^{-1}D^{\top})C & -(A+BR^{-1}D^{\top}C)^{\top} \end{bmatrix} \ $$ where $R=\gamma^2I-D^{\top}D$.

I would compare both hamiltionian matrixes and try to find $\gamma$.