## 4 Fixed-Time NITSM Controller Formulation
### Fixed-Time Stability Analysis  
Subsequently, we will provide the main result regarding the stability proof of the spin recovery system (10), as stated in Theorem 1.  
**Theorem 1.**_For system (10), if the proposed fixed-time NITSM is chosen as (13), the spin recovery algorithm is designed as (16), and the reaching law is given by (15), it is concluded that._  
_(1) The ITSM (13) converges to \(\mathbf{s}=0\) in a fixed time._  
_(2) The tracking error vector in \(\phi\), \(\theta\), \(\dot{\psi}\), and \(\dot{H}\) converges to the origin in a fixed time._  
_Two applications of Lemma 2 yield that the convergent time is bounded by:_  
\[T<T_{\max}=T_{1}+T_{2} \tag{17}\]  
_where \(T_{1}=\frac{1}{\mu_{1}}\frac{2}{1-p}+\frac{1}{\mu_{2}}\frac{2}{q-1}\) and \(T_{2}=\frac{1}{\mu_{3}}\frac{2}{1-m}+\frac{1}{\mu_{4}}\frac{2}{n-1}\)._  
**Proof of Theorem 1.** (1) The quadratic Lyapunov function, expressed through sliding functions, is expressed as follows:  
\[V_{1}=\frac{1}{2}\mathbf{s}^{T}\mathbf{s}>0 \tag{18}\]  
Taking the time derivative of \(V_{1}\) along the dynamics (10) yields  
\[\dot{V}_{1}=\mathbf{s}^{T}\dot{\mathbf{s}}=-\mathbf{s}^{T}\mathbf{k}_{1}sig^{p}(\mathbf{s})-\mathbf{s} ^{T}\mathbf{k}_{2}sig^{q}(\mathbf{s}) \tag{19}\]  
This is equivalent to  
\[\dot{V}_{1}\leq-\mu_{1}V_{1}^{\frac{p+1}{2}}-\mu_{2}V_{1}^{\frac{q+1}{2}} \tag{20}\]  
where \(\mu_{1}=\min\left\{2^{\frac{p+1}{2}}k_{1i}\right\}\) and \(\mu_{2}=\min\left\{2^{\frac{q+1}{2}}k_{2i}\right\},i=1\cdots 4\).  
Referring to Lemma 1, the spin recovery system (10) reaches \(\mathbf{s}=0\) in a fixed time, satisfying:  
\[T_{1}=\frac{1}{\mu_{1}}\frac{2}{1-p}+\frac{1}{\mu_{2}}\frac{2}{q-1} \tag{21}\](2) As soon as the system's trajectories reach \(\mathbf{s}=0\), we have  
\[\mathbf{s}=\mathbf{e}+\mathbf{C}_{1}\int sig^{m}(\mathbf{e})dt+\mathbf{C}_{2}\int sig^{n}(\mathbf{e})dt=0 \tag{22}\]  
By defining a Lyapunov function \(V_{2}=\frac{1}{2}\mathbf{e}^{T}\mathbf{e}\), \(\dot{V}_{2}\) is given by  
\[\dot{V}_{2}=\mathbf{e}^{T}(-\mathbf{C}_{1}sig^{m}(\mathbf{e})-\mathbf{C}_{2}sig^{n}(\mathbf{e})) \tag{23}\]  
which is equivalent to the following:  
\[\dot{V}_{2}=-2^{\frac{m+1}{2}}\sum_{i=1}^{4}c_{1i}|e_{i}|^{m+1}-2^{\frac{m+1}{ 2}}\sum_{i=1}^{4}c_{2i}|e_{i}|^{n+1}\ \leq-\mu_{3}V_{2}^{\frac{m+1}{2}}-\mu_{4}V_{2}^{\frac{m+1}{2}} \tag{24}\]  
where \(\mu_{3}=\min\Bigl{\{}2^{\frac{m+1}{2}}c_{1i}\Bigr{\}}\) and \(\mu_{4}=\min\Bigl{\{}2^{\frac{m+1}{2}}c_{2i}\Bigr{\}},i=1\cdots 4\). \(\square\)  
By using Lemma 1 again, it is deduced that \(\mathbf{e}\) reaches the origin in a fixed time satisfying  
\[T_{2}=\frac{1}{\mu_{3}}\frac{2}{1-m}+\frac{1}{\mu_{4}}\frac{2}{n-1} \tag{25}\]  
Finally, the system state reaches the origin in a fixed time that is obtained by Equation (17).  
**Remark 1**.: The control variables, namely \(\phi_{\
u}\), \(\theta\), \(\dot{\psi}\), and \(\dot{H}\), of the spin recovery system (10) can uniformly converge to their desired values within a fixed time T. To decrease the convergence time, appropriate adjustments to the gain parameters of the controller are required. However, the convergence time is also subject to saturation limits, and the control saturation constraint presents an unavoidable challenge during the spin recovery process [29].  
To offer a clearer summation of the control design procedure, a block diagram illustrating the control architecture is presented in Figure 1.  
**Remark 2**.: _The variables highlighted in red in Figure 1 are regarded as the input states to design controller._  
Figure 1: The proposed fixed-time NITSM spin recovery control architecture.