Shown below is a question from a model reduction exam. I'm not sure how to answer the questions and I'm wondering if my approach is correct.
A continuous time system relates the inputs $u_1$ and $u_2$ to the output $y$ according to the differential equation.
$$\dot{y}+ \rho y= u_1 + 2u_2$$
Where $\rho$ is a real parameter.
a) $\quad$ Determine for arbitrary $\rho > 0$ the Hankel norm of this system.
b) $\quad$ Determine for arbitrary $\rho > 0$ the $H_{\infty}$ norm of this system.
For the hankel norm we first must determine the state space representation. We assume $\dot{y} = \dot{x}$. Which leads to: $$\dot{x}=-\rho x+u_1+2u_2, \quad y=1$$ So the state space form becomes: $$\dot{x} = \begin{bmatrix} -\rho \end{bmatrix}x + \begin{bmatrix} 1&2 \end{bmatrix} \begin{bmatrix} u_1\\u_2 \end{bmatrix}, \quad y = \begin{bmatrix} 1 \end{bmatrix} x $$ So $A = \begin{bmatrix} -\rho \end{bmatrix}, \quad B = \begin{bmatrix} 1&2 \end{bmatrix}, \quad C = \begin{bmatrix} 1 \end{bmatrix}$ and $D = 0$
Next, we need to determine the continuous time $\infty$ horizon reachability and observability gramians using. $$0 = AP+PA^{\top}+BB^{\top}$$ $$0 = A^{\top}Q + QA +C^{\top}C$$ This leads to $P = \frac{5}{2 \rho}$ and $Q = \frac{1}{2 \rho}$
The Hankel norm can then be determined using: $||\Sigma||_H=\sqrt{\lambda_{max}(PQ)}= \sqrt{\lambda_{max}(\frac{5}{4 \rho^2})}$
The $H_{\infty}$ norm can be determined using $||\Sigma||_{H_{\infty}}=sup \ \sigma_{max}(G(i \omega))$
In which $G(i \omega)=C(SI-A)^{-1}B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don't have an idea on how to calculate the $||\Sigma||_{H_{\infty}}$ norm.
The dimensions of the matrices are consistent, $C$ is a $1\times 1$ matrix, $sI-A$ is a $1\times 1$ matrix and $B$ and $D$ are $1\times 2$ matrices.
You have found $\|\Sigma\|_H = \frac{\sqrt{5}}{2\rho}$ and you should find $\|\Sigma\|_{H_\infty} = \frac{\sqrt{5}}{\rho}$. You can check that $\|\Sigma\|_H\leq \|\Sigma\|_{H_\infty}$.