## 4 Discussion of GFPS Scheme  
To illustrate the effectiveness of the GFPS scheme proposed in the previous sections on the basis of the Laplace transform method, the following chaotic systems with uncertain parameters, which have been considered in [37; 38], are discussed:  
\[ \begin{split}\dot{x}&=f(x)+\Phi(x)\Theta,\\ \dot{y}&=g(y)+\Psi(y)\Omega+U,\end{split}  \tag{22}\]  
where \(x,y\in R^{n}\) are the state vectors; \(\Phi(x):R^{n}\to R^{n\times p}\), \(\Psi(y):R^{n}\to R^{n\times q}\), \(\Theta\in R^{p}\) and \(\Omega\in R^{q}\) represent the vectors of uncertain parameters; and \(U\in R^{n}\) is a controller. For simplicity, \(f(0)+\Phi(0)\Theta=g(0)+\Psi(0)\Omega=0\) hold for any values of \(\Theta\) and \(\Omega\). Define the error vector of GFPS as  
\[e=y-h(x)x,\]  
where \(h(x)=diag\{h_{1}(x),h_{2}(x),\cdots,h_{n}(x)\}\), \(h_{i}(x):R^{n}\to R\) (\(i=1,2,\cdots,n\)) are scaling function factors that compose the scaling function matrix \(h(x)\). According to [38], GFPS in (22) can be achieved by the controller and the parameter update laws as follows:  
\[ \begin{split}& U=J(f(x)+\Phi(x)\hat{\Theta})-g(y)-\Psi(y)\hat{ \Omega}+Ke,\\ &\dot{\Theta}=-(J\Phi(x))^{T}e-(\hat{\Theta}-\Theta),\\ &\dot{\Omega}=\Psi^{T}(y)e-(\hat{\Omega}-\Omega),\end{split}  \tag{23}\]where \(J=d(h(x)x)/dx\), \(\hat{\Theta}\) and \(\hat{\Omega}\) are estimated values of the uncertain parameters in Equation (22); \(K\) is a \(n\times n\) matrix, which can be decomposed into \(K=K_{1}+K_{2}\), in which \(K_{1}\) and \(K_{2}\) satisfy the following assumptions:  
\[ \begin{split}(i)& K_{1}^{T}=-K_{1},\\ (ii)& K_{2}=diag\{b_{1},b_{2},\cdots,b_{n}\},\;b_{i} <0,\;(i=1,2,\cdots,n).\end{split}  \tag{24}\]  
From the first condition in condition (24), all real parts of the eigenvalues of \(K_{1}\) are zero since it is a real antisymmetric matrix. Then, all eigenvalues of matrix \(K\) (\(K=K_{1}+K_{2}\)) must have negative real parts if condition (24) is satisfied. From Theorem 3 in Section 3, GFPS in system (22) appears due to the controller and parameter update laws presented in Equation (23) if all eigenvalues of the matrix  
\[ \begin{bmatrix}K&0&0\\ 0&-I_{p}&0\\ 0&0&-I_{q}\end{bmatrix} , \tag{25}\]  
have negative real parts, in which \(I_{p}\) and \(I_{q}\) are the \(p\times p\) and \(q\times q\) real identity matrices, respectively. Obviously, the above GFPS condition implies that all eigenvalues of matrix \(K\) should have negative real parts. **Remark 1**. : _The GFPS condition derived in [38] is a special condition of Theorem 3 in Section 3._  
An example is given below to demonstrate that the condition (24) is not necessary for GFPS occurring in system (22) under the controller and parameter update laws in (23). Consider the hyperchaotic Chen system and hyperchaotic [PERSON] system as the drive and response systems, respectively [38]. \[ \begin{split}\dot{x}_{1}&=-ax_{1}+ax_{2}+x_{4},\\ \dot{x}_{2}&=dx_{1}+cx_{2}-x_{1}x_{3},\\ \dot{x}_{3}&=-bx_{3}+x_{1}x_{2},\\ \dot{x}_{4}&=rx_{4}+x_{2}x_{3},\end{split}  \tag{26}\]  
and  
\[ \begin{split}\dot{y}_{1}&=-a_{1}y_{1}+a_{1}y_{2}+y_{4 }+u_{1},\\ \dot{y}_{2}&=b_{1}y_{1}-y_{2}-y_{1}y_{3}+u_{2},\\ \dot{y}_{3}&=-c_{1}y_{3}+y_{1}y_{2}+u_{3},\\ \dot{y}_{4}&=d_{1}y_{4}-y_{1}y_{3}+u_{4},\end{split}  \tag{27}\]  
where \(a,b,c,d,r,a_{1},b_{1},c_{1}\) and \(d_{1}\) are uncertain parameters to be identified. The true values of the uncertain parameters of systems (26) and (27) are chosen as \(a=35\), \(b=3\), \(c=12\), \(d=7\), \(r=0.08\), \(a_{1}=10\), \(b_{1}=28\), \(c_{1}=8/3\) and \(d_{1}=1.3\). In addition, we choose the scaling function matrix as \(h(x)=diag\{h_{1}(x_{1}),h_{2}(x_{2}),h_{3}(x_{3}),h_{4}(x_{4})\}\), where  
\[ \begin{split} h_{1}(x_{1})&=2\sin(x_{1}+1)+1,\\ h_{2}(x_{2})&=3\sin(-2x_{2}+3)+1,\\ h_{3}(x_{3})&=-\sin(3x_{3}-1)+1,\\ h_{4}(x_{4})&=-2\sin(-x_{4}+4)+1,\end{split}  \tag{28}\]Then, one can find that \(J=d(h(x)x)/dx=diag\{J_{1},J_{2},J_{3},J_{4}\}\), where  
\[