## Appendix B _Calculations for solving Problem 1_  
Assume that the matrix \((I_{m}-K_{c}D_{c})\) is non-singular with \(\|K_{c}D_{c}\|_{2}<1\) and \(A=A_{c}=A_{d}\). Then, one gets from Equations (10) and (13) that:  
\[ \begin{array}{l}A_{d}+B_{d}(I_{m}-K_{d}D_{d})^{-1}K_{d}C_{d}\\ =A+\left(B_{c}-\overline{B}\right)\left(I_{m}-K_{c}D_{c}-\overline{\Delta}_{0 }\right)^{-1}\left(K_{c}D_{c}-\overline{\Delta}_{1}\right)\\ =A+\left(B_{c}-\overline{B}\right)\left(I_{m}-K_{c}D_{c}\right)\left(I_{m}- \left(I_{m}-K_{c}D_{c}\right)^{-1}\overline{\Delta}_{0}\right)\end{array}  \tag{10}\]  
provided that \(\overline{D}\) and \(\overline{K}\) are such that \(\left(I_{m}-K_{d}D_{d}\right)^{-1}=\left(I_{m}-K_{c}D_{c}-\overline{\Delta}_{ 0}\right)^{-1}\) exists (note that this always holds if \(\overline{D}=0_{p\times m}\) and \(\overline{K}=0_{m\times p}\) from Assumption 1), where:  
\[\overline{\Delta}_{0}=K_{d}D_{d}-K_{c}D_{c}=\overline{K}\left(\overline{D}-D_ {c}\right)-K_{c}\overline{D}=\left(\overline{K}-K_{c}\right)\overline{D}- \overline{K}D_{c} \tag{11}\]  
\[\overline{\Delta}_{1}=\overline{K}\left(C_{c}-\overline{C}\right)+K_{c} \overline{C}=\left(K_{c}-\overline{K}\right)\overline{C}+\overline{K}C_{c} \tag{12}\]  
\[\overline{\Delta}_{2}=\left[I_{m}-\left(I_{m}-K_{c}D_{c}\right)^{-1}\overline {\Delta}_{0}\right]^{-1}-I_{m} \tag{13}\]  
and note that  
\[\|\overline{\Delta}_{0}\|\leq\varepsilon\left[\|K_{c}\|\overline{\sigma}_{D} +\varepsilon\overline{\sigma}_{D}\overline{\sigma}_{K}+\|D_{c}\|\overline{ \sigma}_{K}\right] \tag{14}\]  
\[\|\overline{\Delta}_{1}\|\leq\varepsilon\left[\|K_{c}\|\overline{\sigma}_{C} +\varepsilon\overline{\sigma}_{C}\overline{\sigma}_{K}+\|C_{c}\|\overline{ \sigma}_{K}\right] \tag{15}\]  
and one gets from [PERSON]'s Perturbation Lemma [7] that  
\[\|\overline{\Delta}_{2}\|\leq 1+\frac{1}{1-\|(I_{m}-K_{c}D_{c})^{-1}\overline{ \Delta}_{0}\|}\leq 1+\frac{1}{1-\|(I_{m}-K_{c}D_{c})^{-1}\|\|\overline{\Delta}_{0}\|} \tag{16}\]  
provided that \(\|\overline{\Delta}_{0}\|<\overline{\delta}_{0}=1/\|(I_{m}-K_{c}D_{c})^{-1}\|\). Equivalently, if  
\[q(\varepsilon)=\overline{\sigma}_{D}\overline{\sigma}_{k}\varepsilon^{2}+( \overline{\sigma}_{D}\|K_{c}\|+\overline{\sigma}_{K}\|D_{c}\|)\varepsilon-1/ \|(I_{m}-K_{c}D_{c})^{-1}\|<0 \tag{17}\]  
since \(q(\varepsilon)\) is a convex parabola with zeros \(\varepsilon_{1}<0\) and \(\overline{\varepsilon}=\varepsilon_{2}>0\), Equation (17) holds, guaranteeing that \(\|\overline{\Delta}_{0}\|<\overline{\delta}_{0}\), if \(\varepsilon\in[0\,,\,\overline{\varepsilon})\), where:  
\[\overline{\varepsilon}=\frac{\sqrt{(\overline{\sigma}_{D}\|K_{c}\|+\overline{ \sigma}_{K}\|D_{c}\|)^{2}+4\overline{\sigma}_{D}\overline{\sigma}_{K}/\|(I_{m }-K_{c}D_{c})^{-1}\|-(\overline{\sigma}_{D}\|K_{c}\|+\overline{\sigma}_{K}\|D_ {c}\|)}}{2\overline{\sigma}_{D}\overline{\sigma}_{K}} \tag{18}\]  
Before continuing with the calculations, we give the following auxiliary result:  
**Lemma B1. **_If \((I_{m}-K_{c}D_{c})\) is non-singular with \(\|K_{c}D_{c}\|_{2}<1\) and \(\|\overline{\Delta}_{0}\|<\overline{\delta}_{0}=1/\|(I_{m}-K_{c}D_{c})^{-1}\|\), equivalently if \(\varepsilon\in[0\,,\,\overline{\varepsilon})\), with \(\overline{\varepsilon}\) defined in Equation (18), then \(\|\overline{\Delta}_{2}\|\leq C\|\overline{\Delta}_{0}\|<1\) with a norm-dependent real constant \(C\geq\frac{1}{2\lambda_{0}}\) if \(\varepsilon\in[0\,,\,\overline{\varepsilon}_{1})\) with_  
\[\overline{\varepsilon}_{1}=\frac{\|(I_{m}-K_{c}D_{c})^{-1}\|}{2[\|K_{c}\| \overline{\sigma}_{D}+\varepsilon\overline{\sigma}_{D}\overline{\sigma}_{K}+\|D _{c}\|\overline{\sigma}_{K}]} \tag{19}\]  
**Proof.