## 4 Simulation Studies  
The proposed design approach is now applied to the attitude stabilization problem of a spacecraft. Consider the attitude control model of a spacecraft shown in [35,44], which has the same form as (1) with \(n=m=3\). The system states of this model are the three Euler angles \((\phi,\theta,\psi)\) and their derivatives \((\dot{\phi},\dot{\theta},\dot{\psi})\), i.e., \(\mathbf{x}_{1}=(x_{11},x_{12},x_{13})^{T}=(\phi,\theta,\psi)^{T}\) and \(\mathbf{x}_{2}=(x_{21},x_{22},x_{23})^{T}=(\dot{\phi},\dot{\theta},\dot{ \psi})^{T}\). Moreover, the drift term \(\mathbf{f}(\mathbf{x},t)\) is time-invariant and \(G(\mathbf{x},t)\) is a constant matrix [44]; thus, \(\mathbf{f}(\mathbf{x},t)\) and \(G(\mathbf{x},t)\) are briefly denoted by \(\mathbf{f}(\mathbf{x})=(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),f_{3}(\mathbf{x }))^{T}\) and \(G\), respectively, while having the following form [35]:  
\[f_{1}(\mathbf{x}) =\omega_{0}x_{23}cx_{13}cx_{12}-\omega_{0}x_{22}sx_{13}sx_{12}+ \frac{I_{y}-I_{z}}{I_{x}}\left[x_{22}x_{23}+\omega_{0}x_{22}cx_{11}sx_{13}sx_{ 12}\right.\] \[+\omega_{0}x_{22}cx_{13}sx_{11}+\omega_{0}x_{23}cx_{13}cx_{11}+ \frac{1}{2}\omega_{0}^{2}s(2x_{13})c^{2}x_{11}sx_{12}\] \[+\frac{1}{2}w_{0}^{2}c^{2}x_{13}s(2x_{11})-\omega_{0}x_{23}sx_{13 }sx_{12}sx_{11}-\frac{1}{2}\omega_{0}^{2}s^{2}x_{12}s^{2}x_{13}s(2x_{11})\] \[-\frac{1}{2}\omega_{0}^{2}s(2x_{13})sx_{12}s^{2}x_{11}-\frac{3}{2 }\omega_{0}^{2}c^{2}x_{12}s(2x_{11})\left.\frac{}{}\right]\]\[f_{2}(\mathbf{x}) = \omega_{0}x_{23}sx_{13}cx_{11}+\omega_{0}x_{21}cx_{13}sx_{11}+\omega_ {0}x_{23}cx_{13}sx_{12}sx_{11}\] \[+\omega_{0}x_{22}sx_{13}cx_{12}sx_{11}+\omega_{0}x_{21}sx_{13}sx_{12 }cx_{11}+\frac{I_{z}-I_{x}}{I_{y}}\left[x_{21}x_{23}\right.\] \[+\omega_{0}x_{21}cx_{11}sx_{13}sx_{12}+\omega_{0}x_{21}cx_{13}sx_{ 11}-\omega_{0}x_{23}sx_{13}cx_{12}\] \[-\frac{1}{2}\omega_{0}^{2}s(2x_{12})s^{2}x_{13}cx_{11}-\frac{1}{2 }w_{0}^{2}cx_{12}sx_{11}s(2x_{13})+\frac{3}{2}\omega_{0}^{2}s(2x_{12})cx_{11} \left.\frac{}{}\right]\] \[f_{3}(\mathbf{x}) = \omega_{0}x_{21}sx_{11}sx_{13}sx_{12}-\omega_{0}x_{23}cx_{11}cx_ {13}sx_{12}-\omega_{0}x_{22}cx_{11}sx_{13}cx_{12}\] \[+\omega_{0}x_{23}sx_{13}sx_{11}-\omega_{0}x_{21}cx_{13}cx_{11}+ \frac{I_{x}-I_{y}}{I_{z}}\left[ \begin{array}{c}x_{21}x_{22}\\ \end{array} \right.\] \[+\omega_{0}x_{21}cx_{13}cx_{11}-\omega_{0}x_{21}sx_{13}sx_{12}sx_{ 11}-\omega_{0}x_{22}sx_{13}cx_{12}\] \[-\frac{1}{2}\omega_{0}^{2}s(2x_{13})cx_{12}cx_{11}+\frac{1}{2}w_ {0}^{2}s^{2}x_{13}sx_{11}s(2x_{12})-\frac{3}{2}\omega_{0}^{2}s(2x_{12})sx_{11} \left.\frac{}{}\right]\] \[G = \left( \begin{array}{ccc}0.67&0.67&0.67\\ 0.69&-0.69&-0.69\\ 0.28&0.28&-0.28\end{array} \right).\]  
Here, \(I_{x}=2000\)\(N\cdot m\cdot sec^{2}\), \(I_{y}=400\)\(N\cdot m\cdot sec^{2}\), and \(I_{z}=2000\)\(N\cdot m\cdot sec^{2}\) are the inertias of the coordinate axes, \(\omega_{0}=1.0312\times 10^{-3}\) rad/sec denotes the orbital rate, and \(s(\cdot)\) and \(c(\cdot)\) represent the sine and cosine functions, respectively. Additionally, we also assume that the attitude model suffers from the following discontinuous disturbances  
\[\mathbf{d}(\mathbf{x},t)=\left\{ \begin{array}{ll}(\sin(t),\cos(2t),\sin(3t))^ {T}&\text{if}&t\leq 0.4\;\;sec\\ 0.1(\cos(3t),\sin(t),\cos(2t))^{T}&\text{if}&t>0.4\;\;sec\end{array} \right..\]  
For demonstration, the parameters \(\kappa_{1}\) and \(\kappa_{2}\) are selected as \(\kappa_{1}=-2/15\) and \(\kappa_{2}=2/15\), respectively. With these settings, the settling-time estimate is \(T_{max}=35\) and the associated gain matrices \(\mathcal{L}_{1}(\mathbf{x})\), \(\mathcal{L}_{2}(\mathbf{x})\), and \(\Phi(\mathbf{x})\) can be determined accordingly. The simulation results shown in Figures 1 and 2 are conducted for the initial state \(\mathbf{x}(0)=(0.5,0.3,-0.48,-1.9,-1.2,2)^{T}\).