## 2 Problem Statement
: _If Assumption 1 holds, then there exists a centralized stabilizing controller gain \(K_{c}\in\mathbf{R}^{m\times p}\) such that the matrices \((I_{m}-K_{c}D_{c})\) and \((I_{p}-D_{c}K_{c})\) are non-singular, thus the closed-loop centralized control system is solvable and given by:_  
\[\dot{x}_{c}(t)=\Big{(}A_{c}+B_{c}(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c}\Big{)}x_{c} (t);\;x_{c}(0)=x_{c0} \tag{7}\]  
\[y_{c}(t)=\Big{(}I_{p}-D_{c}K_{c}\Big{)}^{-1}C_{c}x_{c}(t) \tag{8}\]  
_and asymptotically stable for any given \(x_{c0}\in\mathbf{R}^{n}\) under the generated control law:_  
\[u_{c}(t)=(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c}x_{c}(t) \tag{9}\]  
_that is, the polynomial \(p(s)=\det\Big{(}sI_{n}-A_{c}-B_{c}(I_{m}-K_{c}D_{c})^{-1}K_{c}C_{c}\Big{)}\) is Hurwitz._  
**Proof. ** The replacement of Equation (3) into Equations (1) and (2) yields Equation (7)-(9). Since the (1) and (2) is linear output stabilizable, a stabilizing controller gain \(K_{c}\in\mathbf{R}^{m\times p}\) has to exist such that (7)-(9) are solvable and the closed-loop dynamics is stable. \(\Box\)  
**Assumption 2**. : _The system in Equations (4) and (5) is linear output-feedback stabilizable via some decentralized control law (Equation (6))._  
In the same way as Proposition 1, we get the following result:  
**Proposition 2**. : _If Assumption 2 holds, then there exists a decentralized stabilizing controller gain \(K_{d}\in\mathbf{R}^{m\times p}\) such that the matrices \((I_{m}-K_{d}D_{d})\) and \(\big{(}I_{p}-D_{d}K_{d}\big{)}\) are non-singular, thus the closed-loop decentralized control system is solvable and given by:_  
\[\dot{x}_{d}(t)=\Big{(}A_{d}+B_{d}(I_{m}-K_{d}D_{d})^{-1}K_{d}C_{d}\Big{)}x_{ d}(t);\;x_{d}(0)=x_{d0} \tag{10}\]  
\[y_{c}(t)=C_{c}x_{c}(t)+D_{c}u_{c}(t) \tag{2}\]0  
_and asymptotically stable for any given \(x_{d0}\in\mathbf{R}^{n}\) under the control law:_  
\[y_{c}(t)=C_{c}x_{c}(t)+D_{c}u_{c}(t) \tag{2}\]1  
_that is, the polynomial \(p(s)=\det\Big{(}sI_{n}-A_{d}-B_{d}(I_{m}-K_{d}D_{d})^{-1}K_{d}C_{d}\Big{)}\) is Hurwitz._  
**Proposition 3**. : _Assume that \(A_{d}=A_{c}\), \(B_{d}=B_{c}\), \(C_{d}=C_{c}\) and \(D_{d}=D_{c}\), and that the system in Equations (1) and (2) is not linear output-feedback stabilizable via some centralized control law (Equation (3)). Then, it is not stabilizable under any linear output-feedback decentralized control (Equation (6)) either._  
**Proof. ** Obviously, if there is no completely free-design matrix \(K_{c}\) that stabilizes Equations (1) and (2), then there is no \(K_{d}\) with at least a forced zero off-diagonal entry that stabilizes it since \(K_{d}\) has extra design constraints related to \(K_{c}\). \(\Box\)  
It can be pointed out that decentralized control has also been proved to be useful in applications. For instance, an integral-based event-triggered asymptotic stabilization of a continuous-time linear system is studied in [17] by considering actuator saturation and observer-based output feedback are considered. In the proposed scheme, the sensors and actuators are implemented in a decentralized manner and a type of Zeno-free decentralized integral-based event condition is designed to guarantee the asymptotic stability of the closed-loop systems. On the other hand, two decentralized fuzzy logic-based control schemes with a high-penetration non-storage wind-diesel system are studied in [18] for small power system with high-penetration wind farms. In addition, several examples concerning decentralized control are described in [4] to illustrate the theoretical design analysis. A typical described case is that of tandems of electrical power system with a tandem disposal on the same river which are not physically nearly allocated.