## 5 MIMO Extension
### Construction of Upper Bound for MIMO System  
In this section, it is shown that the function \(f\) in (3) exists for a MIMO model if each element of the transfer matrix is interpolated piecewise as a SISO system. Thus, it is shown that a uniform bound can be given on the deviation of MIMO models \(G(\rho)\) from \(G_{1}\) in the following form:  
\[\left\|G(\rho)-G_{1}\right\|_{\infty}\leq g_{1}(G_{1},G_{2}). \tag{35}\]  
The deviation of \(G(\rho)\) from \(G_{2}\) can be calculated analogously. Let the transfer matrix of the system \(G(\rho)-G_{1}\) be \(M_{\rho}(z)\). The problem is to find an upper bound estimate on the \(H_{\infty}\) norm of \(M_{\rho}(z)\) that depends only on \(G_{1}\) and \(G_{2}\). The \(H_{\infty}\) norm of \(M_{\rho}(z)\) at a given parameter \(\rho\) is given by [36]:  
\[\left\|G(\rho)-G_{1}\right\|_{\infty}=\left\|M_{\rho}(z)\right\|_{\infty}= \operatorname*{ess\,sup}_{\omega\in\mathbb{R}}\sigma\left(M_{\rho}(e^{i\omega })\right) \tag{36}\]  
where \(\tilde{\sigma}\big{(}M_{\rho}(e^{i\omega})\big{)}\) is the maximal singular value of the matrix \(M_{\rho}(e^{i\omega})\) at a given \(\omega\). Every member of the studied class of systems (see Section 2) reaches its maximum on the unit circle. Therefore, there is no need for the essential supremum; a maximum can simply be used. \[\left\|G(\rho)-G_{1}\right\|_{\infty}=\left\|M_{\rho}(z)\right\|_{\infty}= \max_{\omega\in\mathbb{R}}\tilde{\sigma}\Big{(}M_{\rho}(e^{i\omega})\Big{)} \tag{37}\]  
The upper bound of the \(H_{\infty}\) norm of \(M_{\rho}(z)\) is  
\[\left\|M_{\rho}(z)\right\|_{\infty}\leq\max_{\rho}\left\|M_{\rho}(z)\right\|_ {\infty}=\max_{\rho}\max_{\omega\in\mathbb{R}}\tilde{\sigma}\Big{(}M_{\rho}(e ^{i\omega})\Big{)} \tag{38}\]  
Let \(\omega_{0}\) and \(\rho_{0}\) be the discrete frequency and parameter value where the maximal singular value of the matrix \(M_{\rho}(e^{i\omega})\) reaches its maximum. Let \(m_{ij}\in\mathbb{C}\) be an element of the matrix \(M_{\rho_{0}}(e^{i\omega_{0}})\). Compute Gerschgorin estimate \(\sigma^{\prime}\) of the maximal singular value \(\tilde{\sigma}\) of \(M_{\rho_{0}}(e^{i\omega_{0}})\). As the first step, the elements of \(M_{\rho_{0}}^{*}(e^{i\omega_{0}})M_{\rho_{0}}(e^{i\omega_{0}})\) need to be expressed. Let \(m_{ij}^{\prime}\) be an element of \(M_{\rho_{0}}^{*}(e^{i\omega_{0}})M_{\rho_{0}}(e^{i\omega_{0}})\in\mathbb{C}^{ n\times n}\) then:  
\[m_{ij}^{\prime}=\sum_{k=1}^{n}\overline{m_{ki}}m_{kj} \tag{39}\]  
where bar means complex conjugate. So the Gerschgorin estimate \(\tilde{\sigma}^{\prime}\) according to (34) is the following:  
\[\sigma^{\prime}=\sqrt{\max\left\{\sum_{j=1}^{n}\left|m_{ij}^{\prime}\right|,i= 1\ldots n\right\}}=\sqrt{\max\left\{\sum_{j=1}^{n}\left|\sum_{k=1}^{n} \overline{m_{ki}}m_{kj}\right|,i=1\ldots n\right\}} \tag{40}\]  
we can move the absolute value operation inside the sum and apply it to each factor  
\[\tilde{\sigma}^{\prime}=\sqrt{\max\left\{\sum_{j=1}^{n}\sum_{k=1}^{n}\left| \overline{m_{ki}}\right|\left|m_{kj}\right|,i=1\ldots n\right\}}. \tag{41}\]The value of \(\sigma^{\prime}\) is the Gersgorin estimate for \(\max_{\rho}\max_{\omega\in\mathbb{R}}\sigma\big{(}M_{\rho}(e^{i\omega})\big{)}\). According to Equation (34), this estimate is greater than the upper bound of the \(H_{\infty}\) norm of \(M_{\rho}(z)\). That is,  
\[\max_{\rho}\max_{\omega\in\mathbb{R}}\sigma\big{(}M_{\rho}(e^{i\omega})\big{)} \leq\sigma^{\prime}. \tag{42}\]  
Notice that \(\left|m_{ij}\right|\) is a value on the magnitude curve of a SISO transfer function at \(\omega_{0}\) and \(\rho_{0}\) on the unit circle. The transfer function is \(G(\rho_{0})_{ij}-G_{1 ij}\), following the definition of \(M_{\rho}(z)\). The maximum of the magnitude curve of a SISO transfer function is equal to its \(H_{\infty}\) norm.