## 5 The Design of Simple Parameter Update Laws  
According to the GFPS condition given in Theorem 1 in Section 3, simple parameter update laws can be designed for the estimation of the uncertain parameters in system (1). From \(M\) matrix defined in Equation (13), \(g_{1}(\hat{\phi}-\phi,e_{1})\) and \(g_{2}(\hat{\psi}-\psi,e_{1})\) in Equation (3) can be selected so that the eigenvalues of matrix \(M\) are independent of \(\phi\) and \(\psi\). Consider that \(g_{1}(\hat{\phi}-\phi,e_{1})\) and \(g_{2}(\hat{\psi}-\psi,e_{1})\) in Equation (3) have the following form:  
\[ \begin{split}& g_{1}(\hat{\phi}-\phi,\delta_{1})=\gamma_{11}(\hat{ \phi}-\phi)+\gamma_{12}\Gamma_{1}(\delta_{1}),\\ & g_{2}(\hat{\psi}-\psi,e_{1})=\gamma_{21}(\hat{\psi}-\psi)+\gamma _{22}\Gamma_{2}(\delta_{1}),\end{split}  \tag{36}\]  
where \(\gamma_{11},\gamma_{12},\gamma_{21},\gamma_{22}\) are control parameters; \(\Gamma_{1,2}(\delta_{1})\) are the functions of \(\delta_{1}\left.\frac{\partial\Gamma_{1,2}(\delta_{1})}{\partial\delta_{1}} \right|_{\delta_{1}=0}=0\). With parameter update laws (36), matrix \(M\) in Equation (13) becomes  
\[M=\left[ \begin{array}{ccc}kI_{n}&0&0\\ 0&\gamma_{11}I_{m}&0\\ 0&0&\gamma_{21}I_{l}\end{array} \right], \tag{37}\]  
where \(I_{n}\), \(I_{m}\) and \(I_{l}\) represent \(n\times n\), \(m\times m\) and \(l\times l\) real identity matrices, respectively. All eigenvalues of matrix (37) have negative real parts as long as matrices \(k\), \(\gamma_{11}\) and \(\gamma_{21}\) have no eigenvalues with non-negative real parts. For the sake of simplicity, one can choose all \(k\), \(\gamma_{11}\) and \(\gamma_{21}\) to be scalars. Therefore, GFPS in system (1) can be achieved by the controller and parameter update laws presented in Equations (3) and (36) if \(k\), \(\gamma_{11},\gamma_{21}<0\). To illustrate the effectiveness of the parameter update laws (36), drive-response systems constructed with the Lorenz and Chen systems are selected to perform numerical simulations. The Lorenz chaotic system [49], as the drive system, can be described by the following equations:  
\[ \begin{split}\dot{x}_{1}&=\sigma(x_{2}-x_{1}),\\ \dot{x}_{2}&=rx_{1}-x_{2}-x_{1}x_{3},\\ \dot{x}_{3}&=x_{1}x_{2}-\beta x_{3},\end{split}  \tag{38}\]  
where \(x_{1},x_{2},x_{3}\) are state variables, and \(\sigma,r,\beta\) are uncertain parameters to be identified. In particular, when \(\sigma=10\), \(r=28\) and \(\beta=8/3\), system (38) displays a chaotic attractor. Figure 4: The estimates of the uncertain parameters of systems (26) and (27) for GFPS with the scaling function matrix \(h(x)=diag\{2 sin(x_{1}+1)+1,3 sin(-2x_{2}+3)+1,-sin(3x_{3}-1)+1,-2 sin(-x_{4}+4)+1\}\) and \(K\) matrix (35). (**a**) uncertain parameters in system (26) (**b**) uncertain parameters in system (27). The Chen chaotic system [50], as the response system, is given as below:  
\[ \begin{split}\dot{y}_{1}&=a(y_{2}-y_{1})+u_{1},\\ \dot{y}_{2}&=(c-a)y_{1}-y_{1}y_{3}+cy_{2}+u_{2},\\ \dot{y}_{3}&=y_{1}y_{2}-by_{3}+u_{3},\end{split}  \tag{39}\]  
where \(y_{1},y_{2},y_{3}\) are state variables, \(a,b,c\) are uncertain parameters to be estimated and \(u_{1},u_{2},u_{3}\) are the control laws to be designed. System (39) exhibits chaotic dynamics when \(a=35\), \(b=3\) and \(c=28\). The scaling function matrix between systems (38) and (39) is chosen as \(h(x)=diag\{2 sin(x_{1}+1)+1,3 sin(-2x_{2}+3)+1,-sin(3x_{3}-1)+1\}\), and one has  
\[ \begin{split} J=\frac{d(h(x)x)}{dx}=diag\{& 2\cos(x_{1}+1)x_{1}+2 \sin(x_{1}+1)+1,\\ &-6\cos(2x_{2}-3)x_{2}-3\sin(2x_{2}-3)+1,\\ &-3\cos(3x_{3}-1)x_{3}-\sin(3x_{3}-1)+1\}.\end{split} \]  
According to Equations (3) and (36), the controller \(u_{1},u_{2},u_{3}\) and the parameter update laws can be given by  
\[