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 $\left |T(j\omega) \right |_{\infty }<\gamma$, where $ \left |T(j\omega) \right |_{\infty }$ is the H-infinity of the transfer function $T(j\omega)$.

I know that for the bounded real lemma, if the eigenvalues of $A$ have a 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 a 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 thought that if the A matrix is negative definite, it has all eigenvalues with a 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 completely on the wrong path.

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


Finding the $H_\infty$ norm of a linear system is not trivial. There are many numerical methods. A classic paper is: http://stanford.edu/~boyd/papers/bisection_hinfty.html

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