I'm applying loop shaping on a MIMO system. We know that the sensitivity is $S = (I+PC)^{-1}$ and the control effort is given by $CS = C(I+PC)^{-1}$.
Through loop shaping theory for $H_\infty$ I'm trying to synthesize a controller such that
\begin{equation} \left\|\begin{array}{c} W_s S\\ W_u CS\end{array} \right\|_\infty \leq \gamma \end{equation}
so if $\gamma \leq 1$ we have that $\left\|S\right\|_\infty \leq \left\|W_s^{-1}\right\|_\infty$ and $\left\|CS\right\|_\infty \leq \left\|W_u^{-1}\right\|_\infty$.
My question is the following. Since $S$ is a MIMO transfer matrix, and I chose $W_s$ and $W_u$ to be diagonal weighting functions the proper channel-by-channel analysis should verify that
$\left\|S(i,:)\right\|_\infty \leq W_{s,i}^{-1}$
or I'm imposing
$\left\|S(i,i)\right\|_\infty \leq W_{s,i}^{-1}$?
where $i$ is the specific error channel I want to constrain frequency-wise. In other words if the plant is not diagonal and I choose diagonal weighting functions, are those weighting functions representing an upper bound for the specific output channels or for the I/O combinations of the sensitivity? Thanks!