## 3 Main Results
: _Define the following error matrices between the centralized and decentralized system parameterizations:_  
\[\widetilde{A}=A_{c}-A_{d}\ ;\ \widetilde{B}=B_{c}-B_{d};\ \widetilde{C}=C_{c}-C_{d};\ \widetilde{D}=D_{c}-D_{d};\ \widetilde{K}=K_{c}-K_{d} \tag{13}\]  
_such that \(\|\widetilde{A}\|\leq\overline{\sigma}_{A}\varepsilon\), \(\|\widetilde{B}\|\leq\overline{\sigma}_{B}\varepsilon\), \(\|\widetilde{C}\|\leq\overline{\sigma}_{c}\varepsilon\), \(\|\widetilde{D}\|\leq\overline{\sigma}_{D}\varepsilon\) and \(\|\widetilde{K}\|\leq\overline{\sigma}_{K}\varepsilon\) for some \(\varepsilon\in\mathbf{R}_{0+}\) and given \(\overline{\sigma}_{A}\), \(\overline{\sigma}_{B}\), \(\overline{\sigma}_{C}\), \(\overline{\sigma}_{D}\in\mathbf{R}_{+}\). Assume that:_  
1. _Assumption 1 holds;_
2. \(K_{c}\in\mathbf{R}^{m\times p}\) _is a centralized linear output-feedback stabilizing controller gain such that the resulting closed-loop system matrix_ \(A_{cc}\in\mathbf{R}^{n\times n}\) _has a stability abscissa_ \((-\rho_{cc})<0\) _and such that_ \(\|K_{c}D_{c}\|_{2}<1\) _(so that_ \((I_{m}-K_{c}D_{c})\) _is non-singular);_
3. \(A_{c}=A_{d}=A\)_; and_
4. _Define_ \(\overline{\varepsilon}^{*}=\min\left(1\,,\,\overline{\varepsilon},\,\overline {\varepsilon}_{1}\,,\,\overline{\varepsilon}_{2}\right)\)_, where:_ \[\overline{\varepsilon}=\frac{\sqrt{\left(\overline{\sigma}_{D}\|K_{c}\|+ \overline{\sigma}_{K}\|D_{c}\|\right)^{2}+4\overline{\sigma}_{D}\overline{ \sigma}_{K}/\|(I_{m}-K_{c}D_{c})^{-1}\|}-(\overline{\sigma}_{D}\|K_{c}\|+ \overline{\sigma}_{K}\|D_{c}\|)}{2\overline{\sigma}_{D}\overline{\sigma}_{K}},\] \[\overline{\varepsilon}_{1}=\frac{\|(I_{m}-K_{c}D_{c})^{-1}\|}{2[\|K_{c}\| \overline{\sigma}_{D}+\varepsilon\overline{\sigma}_{D}\overline{\sigma}_{K}+ \|D_{c}\|\overline{\sigma}_{K}]};\ \overline{\varepsilon}_{2}=1/\left(\overline{a}_{dc} \sup_{\omega\in\mathbf{R}_{0+}}\|(i\omega I_{n}-A_{cc})^{-1}\|_{2}\right)\] _where_ \[\overline{a}_{dc}=(1-\|K_{c}D_{c}\|)^{-1}\times[\overline{\sigma}_{B}\|K_{c}C_ {c}\|+(\,\|B_{c}\|+\|K_{c}C_{c}\|)(\|K_{c}\|\overline{\sigma}_{C}+\|C_{c}\| \overline{\sigma}_{K}+C\,\|K_{c}\|\overline{\sigma}_{D}+\|D_{c}\|\overline{ \sigma}_{K}])],\] _where the non-negative real constant_ \(C\) _is given in Equation (_A17_). Then, the following properties hold:_ **(i)** _If_ \(\overline{\sigma}_{A}=0\) _(that is,_ \(A_{c}=A_{d}\)_), then_ \(A_{dc}\) _is stable and_ \(\varepsilon\in[\,0\,,\,\overline{\varepsilon}^{*}\,]\)_._ **(ii)** _If_ \(\|\widetilde{A}\|_{2}\leq\overline{\sigma}_{A}\varepsilon\)_, then_ \(A_{dc}\) _is stable and_ \(\varepsilon\in[\,0\,,\,\overline{\varepsilon}^{*}\,]\) _where_ \(\overline{\varepsilon}^{*}=\min\left(1\,,\,\overline{\varepsilon},\,\overline{ \varepsilon}_{1}\,,\,\overline{\varepsilon}_{2}^{*}\right)\) _and_ \(\overline{\varepsilon}^{*}_{2}=1/\left((\overline{a}_{dc}+\overline{\sigma}_{A}) \sup_{\omega\in\mathbf{R}_{0+}}\|(i\omega I_{n}-A_{cc})^{-1}\|_{2}\right)\)_._ **(iii)** _If_ \(\widetilde{A}(t)\) _is piecewise continuous and bounded, then Property (ii) holds by replacing_ \(\|\widetilde{A}\|_{2}\leq\overline{\sigma}_{A}\varepsilon\) _by_ \(\sup_{0\leq t<\infty}\|\widetilde{A}(t)\|_{2}\leq\overline{\sigma}_{A}\varepsilon\)_._  
**Remark 3**. : _Some quantified results are given in Lemmas B.2 and B.3 to modify \(\overline{\varepsilon}_{2}\)(and hence \(\overline{\varepsilon}_{2}^{*}\)) in Theorem 2 by considering the second power of \(\varepsilon\) in the calculations of the disturbed parameterization guaranteeing the closed-loop stability in the decentralized case._  
**Remark 4**. : _If the corresponding parametrical error matrices of Equation (13) have some zero off-diagonal entries (or off-diagonal block matrices in the more general case that the system is described by coupled subsystems), then we have at least a partial closed-loop stabilization under decentralized control or, eventually, cut coupled dynamic links to the light of the various Cases 1-5 described after Remark 2 such that closed-loop stability is preserved._  
**Remark 5**.