# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 2 Preliminaries  
Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator \({}_{a}D_{t}^{r}\), which is defined as  
\[{}_{a}D_{t}^{r}=\left\{\begin{array}{l l}\frac{d^{r}}{dt^{r}}&:r>0,\\ 1&:r=0,\\ \int_{a}(d\tau)^{-r}&:r<0.\end{array}\right. \tag{1}\]  
There are several different definitions for the fractional-order differential operator [34]. Because the Caputo definition is easy to understand and is frequently used in the literature, we apply this definition in this paper, which is  
\[{}_{a}D_{t}^{r}f(t)=\frac{1}{\Gamma(n-r)}\int_{a}^{t}\frac{f(\tau)}{t-\tau}^{ r-n+1}d\tau \tag{2}\]  
where \(1<r<n\).  
**Lemma 1** ([35]).: _Suppose \(f(t)\in C_{a}^{\alpha}([a,b]),D_{a}^{\alpha}f(t)\in C_{a}^{\beta}([a,b]), \alpha>0,\beta>0,m-1<\beta<m,n-1<\alpha<n\), then_  
\[{}_{a}D_{t}^{\beta}\left(\begin{array}{l}C_{a}^{\alpha}D_{t}^{\alpha}f(t) \end{array}\right)=_{a}^{C}D_{t}^{\alpha}(_{a}^{C}D_{t}^{\beta}f(t))=_{a}^{C}D _{t}^{\alpha+\beta}f(t) \tag{3}\]  
**Lemma 2** ([35]).: _Consider_  
\[{}^{C}_{a}D^{a}_{t}x^{n}=\frac{\Gamma(n+1)x^{n-a}\,^{C}}{\Gamma(n+1-\alpha)\,_{a} }D^{a}_{t}x \tag{4}\]  
**Lemma 3** ([35]).: _Let \(\phi(t)\in R\) be a continuous and derivable function. Then, for any time instant \(t\geq t_{0}\)_  
\[\frac{1}{2}D^{q}\phi^{2}(t)=\phi(t)D^{q}\phi(t),\;\forall q\in(0,1) \tag{5}\]  
Given the following n-dimensional linear fractional-order system with multiple time-delays [36]:  
\[\left\{\begin{aligned} D^{q_{1}}y_{1}(t)& =\alpha_{11}y_{1}(t-\tau_{11})+\alpha_{12}y_{2}(t-\tau_{12})\\ &+\cdots+\alpha_{1n}y_{n}(t-\tau_{1n}),\\ D^{q_{2}}y_{2}(t)&=\alpha_{21}y_{1}(t-\tau_{21})+ \alpha_{22}y_{2}(t-\tau_{22})\\ &+\cdots+\alpha_{2n}y_{n}(t-\tau_{2n}),\\ \vdots\\ D^{q_{n}}y_{n}(t)&=\alpha_{n1}y_{1}(t-\tau_{n1})+ \alpha_{n2}y_{2}(t-\tau_{n2})\\ &+\cdots+\alpha_{nn}y_{n}(t-\tau_{nn}),\end{aligned}\right. \tag{6}\]  
where \(q_{i}\in(0,1)\) is the fractional-derivative order, \(y_{i}(t)\) is the state, and \(\tau_{ij}>0\) is the time-delay, the initial value \(y_{i}(t)=\phi_{i}(t)\) is given by \(-max\tau_{ij}=-\tau_{max}\leq t\leq 0\), \(A=[a_{ij}]\in R_{n\times n}\) is the coefficient matrix.  
Performing Laplace transform on the system in Equation (6) yields  
\[\Delta(s)\cdot Y(s)=b(s), \tag{7}\]  
where \(Y(s)=(Y_{1}(s)\), \(Y_{2}(s)\),..., \(Y_{n}(s))^{T}\) is the Laplace transform of \(y(t)=(y_{1}(t)\), \(y_{2}(t)\),..., \(y_{n}(t))^{T}\), \(b(s)=(b_{1}(s)\), \(b_{2}(s)\),..., \(b_{n}(s))^{T}\) is the remaining non-linear part, the characteristic matrix of the system in Equation (6) is  
\[\Delta(s)=\left(\begin{array}{cccc}s^{q_{1}}-\alpha_{11}e^{-s\tau_{11}}&- \alpha_{12}e^{-s\tau_{12}}&\cdots&-\alpha_{1n}e^{-s\tau_{1n}}\\ -\alpha_{21}e^{-s\tau_{21}}&s^{q_{2}}-\alpha_{22}e^{-s\tau_{22}}&\cdots&- \alpha_{2n}e^{-s\tau_{2n}}\\ \vdots&\vdots&\ddots&\vdots\\ -\alpha_{n1}e^{-s\tau_{n1}}&-\alpha_{n2}e^{-s\tau_{n2}}&\cdots&s^{q_{n}}- \alpha_{nn}e^{-s\tau_{nn}}\end{array}\right). \tag{8}\]  
**Theorem 1** ([36]).: _If all the roots of the characteristic equation \(\det(\Delta(s))=0\) have negative real parts, then the zero solution of the system in Equation (6) is Lyapunov globally asymptotically stable._  
**Corollary 1** ([36]).: _If \(q_{1}=q_{2}=\cdots=q_{n}=\beta\in(0,1)\), all the eigenvalues \(\lambda\) of the coefficient matrix \(A\) satisfy \(|\arg(\lambda)|>\beta\tau/2\), and the characteristic equation \(\det(\Delta(s))=0\) has no purely imaginary roots for any \(\tau_{ij}>0\), \(i,j=1,2,\ldots,n\), then the zero solution of the system in Equation (6) is Lyapunov globally asymptotically stable._