## 3 F-M Synchronization
### Case 2: \(d<m\)  
Let us now assume that the synchronization dimension \(d<m\). We define the vectors \(\dot{Y}_{1}(t)=\left(\dot{y}_{1}\left(t\right),...,\dot{y}_{d}\left(t\right) \right)^{T}\) and \(\dot{Y}_{2}(t)=\left(\dot{y}_{d+1}\left(t\right),...,\dot{y}_{m}\left(t\right) \right)^{T}\) along with matrices  
\[\mathbf{D}_{1}=\left(\begin{array}{cccc}\frac{\partial\mathbf{F}_{1}}{ \partial y_{1}}&\frac{\partial\mathbf{F}_{1}}{\partial y_{2}}&\cdots&\frac{ \partial\mathbf{F}_{1}}{\partial y_{d}}\\ \frac{\partial\mathbf{F}_{2}}{\partial y_{1}}&\frac{\partial\mathbf{F}_{2}}{ \partial y_{2}}&\cdots&\frac{\partial\mathbf{F}_{2}}{\partial y_{d}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\mathbf{F}_{d}}{\partial y_{1}}&\frac{\partial\mathbf{F}_{d}}{ \partial y_{2}}&\cdots&\frac{\partial\mathbf{F}_{d}}{\partial y_{d}}\end{array} \right), \tag{18}\]  
\[\mathbf{D}_{2}=\left(\begin{array}{cccc}\frac{\partial\mathbf{F}_{1}}{ \partial y_{d+1}}&\frac{\partial\mathbf{F}_{1}}{\partial y_{d+2}}&\cdots&\frac {\partial\mathbf{F}_{1}}{\partial y_{m}}\\ \frac{\partial\mathbf{F}_{d+2}}{\partial y_{d+1}}&\frac{\partial\mathbf{F}_{d+ 2}}{\partial y_{d+2}}&\cdots&\frac{\partial\mathbf{F}_{d+2}}{\partial y_{m}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\mathbf{F}_{d}}{\partial y_{d+1}}&\frac{\partial\mathbf{F}_{d}}{ \partial y_{d+2}}&\cdots&\frac{\partial\mathbf{F}_{d}}{\partial y_{m}}\end{array} \right), \tag{19}\]  
and  
\[T=\operatorname{diag}\left(c_{1},c_{2},...,c_{d}\right)e\left(t\right)-\dot{ \mathbf{M}}\left(t\right)X(t)-\mathbf{M}\left(t\right)\dot{X}(t). \tag{20}\]  
The error system (8) may be rearranged to the form  
\[\dot{e}\left(t\right)=-\operatorname{diag}\left(c_{1},c_{2},...,c_{d}\right) e\left(t\right)+\mathbf{D}_{1}\dot{Y}_{1}(t)+\mathbf{D}_{2}\dot{Y}_{2}(t)+T, \tag{21}\]  
where \(c_{i}\) are positive control constants for \(1\leq i\leq d\). We assume that matrix \(\mathbf{D}_{1}\) is invertible and we denote its inverse by \(\mathbf{D}_{1}^{-1}\). The following theorem states the control laws for the \(\mathbf{F}\)-\(\mathbf{M}\) synchronization criterion.  
**Theorem 2**.: _Given the four matrices \(B_{1}=\left(b_{ij}\right)_{d\times m^{\prime}}\), \(B_{2}=\left(b_{ij}\right)_{\left(m-d\right)\times m^{\prime}}\), \(G_{1}=\left(g_{i}\right)_{1\leq i\leq d^{\prime}}\) and \(G_{2}=\left(g_{i}\right)_{d+1\leq i\leq m^{\prime}}\), the master-slave pair (6) is globally \(\mathbf{F}\)-\(\mathbf{M}\) synchronized with dimension \(d\) subject to_  
\[\left(u_{1},u_{2},...,u_{d}\right)^{T}=-B_{1}-G_{1}-J^{1-q}\left(\mathbf{D}_{1 }^{-1}\times T\right), \tag{22}\]  
_and_  
\[\left(u_{d+1},u_{d+2},...,u_{m}\right)^{T}=-B_{2}-G_{2}. \tag{23}\]  
Proof.: Substituting Label (22) and Label (23) in the slave system of (6) yields  
\[\left(D_{t}^{q}y_{1}\left(t\right),...,D_{t}^{q}y_{d}\left(t\right)\right)^{ T}=J^{1-q}\left(-\mathbf{D}_{1}^{-1}\times T\right), \tag{24}\]  
and  
\[D_{t}^{q}y_{i}\left(t\right)=0,\ \ \ i=d+1,...,m, \tag{25}\]  
respectively. The fractional derivative of order \(1-q\) of Label (24) gives  
\[\dot{Y}_{1}(t) =\left(\dot{y}_{1},\dot{y}_{2},...,\dot{y}_{d}\right)^{T}\] \[=D_{t}^{1-q}\left(\left(D_{t}^{q}y_{1}\left(t\right),...,D_{t}^{q }y_{d}\left(t\right)\right)^{T}\right)\] \[=D_{t}^{1-q}J^{1-q}\left(-\mathbf{D}_{1}^{-1}\times T\right)\] \[=-\mathbf{D}_{1}^{-1}\times T. \tag{26}\]  
Doing the same for Label (25) yields  
\[\dot{y}_{i}\left(t\right)=0,\ \ \ i=d+1,...,m. \tag{27}\]  
By evaluating the error (21) for Label (26) and Label (27), we obtain  
\[\dot{e}_{i}\left(t\right)=-c_{i}e_{i}\left(t\right),\ \ 1\leq i\leq d. \tag{28}\]  
It is trivial that all solutions of Label (28) decay to zero as \(t\rightarrow+\infty\) implying that the pair (6) is globally \(\mathbf{F}\)-\(\mathbf{M}\) synchronized with dimension \(d\).