## 5 Concluding Remarks  
This paper is devoted to formulating sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The developed stability tests are conceptually simple to evaluate and they rely on the quantification in terms of worst-case norms of interconnection and open-loop system dynamics matrices and the corresponding control gains in the decentralized case compared to the centralized counterpart. The tolerances of the various parametrical matrix errors have been quantified by considering the first or second powers of a small parameter. Such a parameter is a design factor to characterize in the worst-case for the allowed tolerances to the perturbed parameterization norms. Simulated examples are discussed to illustrate the obtained results. The decentralized control design versus its decentralized control counterpart, under eventual output linear feedback, has been studied from the point of view of the amount of information that can be lost or omitted in terms of the total or partial knowledge of the coupled dynamics between subsystems necessary in the decentralized case to keep the closed-loop stability. A foreseen related future work relies on the application of the method to some applied control problems such as consensus protocols under decentralized control and continuous-discrete hybrid controller designs. Author Contributions: Author [PERSON] contributed to the manuscript in the aspects of conceptualization; methodology; formal analysis; investigation; writing original draft preparation; writing review and editing; supervision and fund acquisition. Author [PERSON] contributed to the manuscript in the various aspects of preparation of the numerical experiments; software implementation; visualization and original draft preparation. Funding: This research was co-funded by the Spanish Government and the European Fund of Regional Development FEDER, grant numbers DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE) and the University of the Basque Country UPV /EHU, grant PGC 17/33. Figure 4: Evolution of the state space variables when the decentralized controller \(K_{d}\) is used. **Acknowledgments:** The authors are grateful to the Spanish Government for grants DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE), and to UPV /EHU for grant PGC 17/33. They also thank the referees for their useful comments. **Conflicts of Interest:** The authors declare no conflict of interest. Appendix A _Auxiliary Stability Results on Perturbed Matrices under Constant and Time-Varying Perturbations of Stability Matrices  
**Theorem A1. **_Assume that \(A_{0}\in\mathbf{R}^{n\times n}\) is a stability matrix with stability abscissa \((-\rho_{c})<0\). Then, \(A=A_{0}+\widetilde{A}\) is a stability matrix if any of the subsequent sufficiency-type conditions holds:_  
_(1) The \(H_{\infty}\)-norm of \((sl_{n}-A_{0})^{-1}\widetilde{A}\) satisfies \(\left\|(sl_{n}-A_{0})^{-1}\widetilde{A}\right\|_{\infty}<1\),_  
_(2) \(\left\|\widetilde{A}\right\|_{2}<1/\sup\limits_{\omega\in\mathbf{R}_{0+}}\left\|( i\omega I_{n}-A_{0})^{-1}\right\|_{2}\)._  
_Other alternative sufficiency-type conditions to Conditions 1 and 2 for the stability of \(A\) are:_  
_(3) \(\rho\big{(}A_{0}^{-1}\widetilde{A}\big{)}<1\),_  
_(4) \(\left\|A_{0}^{-1}\widetilde{A}\right\|_{2}<1\),_  
_(a) \(\left\|\widetilde{A}\right\|_{2}<1/\left\|A_{0}^{-1}\right\|_{2}\), that is, \(\lambda_{\max}\big{(}\widetilde{A}^{T}\widetilde{A}\big{)}<\lambda_{\min} \big{(}A_{0}^{T}A_{0}\big{)}\), in the following particular cases:_  
_(a) \(A_{0}<0\) and \(A<-A_{0}\); and_  
_(b) \(A_{0}=\big{(}A_{0_{ij}}\big{)}\in M_{E}^{n\times n}\) and \(\widetilde{A}=\big{(}\widetilde{A}_{ij}\big{)}\) fulfils \(\widetilde{A}_{ij}\geq-A_{0_{ij}}\); \(\forall i,j(\
eq i)\in\widetilde{n}\)._  
**Proof. ** Note that  
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