# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 3 Multi-Switching Combination Synchronization Scheme  
Multi-switching combination synchronization among three non-identical fractional-order delayed systems is investigated in this section. The two drive systems are  
\[ \begin{split} D^{a}x(t)&=x(t)+x(t-\tau)+A(x(t),x(t-\tau) ),\\ x(t)&=x(0),t\in[-\tau,0],\end{split}  \tag{9}\]\[D^{\alpha}v(t) =y(t)+y(t-\tau)+B(y(t),y(t-\tau)), \tag{10}\] \[y(t) =y(0),t\in[-\tau,0].\]  
The response system is  
\[D^{\alpha}z(t) =z(t)+z(t-\tau)+C(z(t),z(t-\tau))+U, \tag{11}\] \[z(t) =z(0),t\in[-\tau,0],\]  
in which, \(\alpha\in(0,1)\) is the fractional order, \(\tau>0\) is the time-delay, \(U=(U_{1},\,...,\,U_{n})\) is the controller vector, \(x=(x_{1},\,...,\,x_{n})^{T}\in R^{n}\), \(y=(y_{1},\,...,\,y_{n})^{T}\in R^{n}\) and \(z=(z_{1},\,...,\,z_{n})^{T}\in R^{n}\) are state vectors, and \(A:R^{2n}\to R^{n}\), \(B:R^{2n}\to R^{n}\) and \(C:R^{2n}\to R^{n}\) are continuous vector functions. Define the error state as \(e_{klm}=f_{k}z_{k}-g_{l}x_{l}-h_{m}y_{m}(k,l,m=1,...,n)\). Then, we have the error state vector  
\[e(t)=Fz-Gx-Hy, \tag{12}\]  
where \(e(t)\) is the vector form of \(e_{klm}\), \(F=diag\{f_{1},f_{2},\ldots,f_{n}\}\in R^{n\times n}\), \(G=diag\{g_{1},g_{2},\ldots,g_{n}\}\in R^{n\times n}\) and \(H=diag\{h_{1},h_{2},\ldots,h_{n}\}\in R^{n\times n}\) are real scaling matrix. Accordingly, \(e_{klm}(t-\tau)=f_{k}z_{k}(t-\tau)-g_{l}x_{l}(t-\tau)-h_{m}y_{m}(t-\tau)\). Definition 1 ([27]): The systems in Equations (9) and (10) and the system in Equation (11) are defined to be multi-switching combination synchronization if \(F,\,G,\,H\) are non-zeros, and \(k\
eq l\
eq m\), \(k=l\
eq m\), \(k\
eq l=m\), \(k=m\
eq l\), such that:  
\[\lim_{t\to+\infty}\parallel e(t)\parallel=\lim_{t\to+\infty}\parallel Fz-Gx-Hy \parallel=0 \tag{13}\]  
where \(\parallel.\parallel\) represents the matrix norm. Remark 1: If \(k=l=m\), the systems in Equations (9) and (10) and the system in Equation (11) are defined to be combination synchronization [22]. Remark 2: If the scaling matrix \(F\
eq 0\), \(G=0\) or \(H=0\), the multi-switching combination synchronization mentioned above is simplified to multi-switching hybrid projective synchronization. From the systems in Equations (9)-(11), we have the error system as follows  
\[D^{\alpha}e(t)=FD^{\alpha}z(t)-GD^{\alpha}x(t)-HD^{\alpha}y(t) \tag{14}\]  
To achieve multi-switching combination synchronization among the above systems, a non-linear controller is constructed:  
\[U=\tilde{K}e(t)+GA(x(t),x(t-\tau))+HB(y(t),y(t-\tau))-FC(z(t),z(t-\tau)), \tag{15}\]  
where \(\tilde{K}=K-I\), \(I\) is an n-dimensional unit matrix, \(K=diag\{k_{1},k_{2},\ldots,k_{n}\}\) is a feedback gain matrix. Substituting the systems in Equations (9)-(11) and (15) into the system in Equation (14), we have  
\[D^{\alpha}e(t)=(\tilde{k}+I)e(t)+e(t-\tau)=Ke(t)+e(t-\tau). \tag{16}\]  
Thus, the multi-switching combination synchronization between the systems in Equations (9) and (10) and the system in Equation (11) is changed into the analysis of the asymptotical stability of the system in Equation (16). In light of Corollary 1, a sufficient condition to achieve multi-switching combination synchronization between the systems in Equations (9) and (10) and the system in Equation (11) is obtained as follows. Proposition 1: _Multi-switching combination synchronization between the systems in Equations (9) and (10) and the system in Equation (11) can be achieved if there exists a matrix \(K=diag\{k_{1},k_{2},\ldots,k_{n}\}\) in the system in Equation (16) such that \(k_{i}<-1/\sin(\alpha\pi/2),\ (i=1,2,\ldots,n)\)._  
Proof: For the system in Equation (16), \(A=K+I\) is the coefficient matrix. Since \(k_{i}<-1/\sin(\alpha\pi/2),\alpha\in(0,1)\), the eigenvalues of \(A\) are \(\lambda_{i}=k_{i}+1<0\), \((i=1,2,\ldots,n)\). Then, \(|\arg(\lambda)|>\pi/2>\alpha\pi/2\) holds.