## 2 Main Results
### Normal Model  
In general, we consider the following normal SISO non-affine nonlinear system structure:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=F(t,x,u_{fs})+D\end{array} \right. \tag{4}\]  
where \(x=[x_{1},x_{2}]^{T}\in\mathbb{R}^{2}\) is measurable system state, \(F(t,x,u_{fs})\) denotes a known smooth nonlinear function, \(D\) is external disturbance. \(u_{fs}\) is faulted saturation control input which is \(u_{fs}=\rho u_{s}\), and \(\rho\) is an unknown function that satisfies \(0<\rho\leq 1\), \(u_{s}\) is a saturation input, which is shown in the Appendix B\(u\) denotes an actual control input to the system. On account of the limited workspace and security considerations, the states of the system are constrained and need to satisfy: \(|x_{1}|\leq\alpha_{1}\), \(|x_{2}|\leq\alpha_{2}\). **Assumption 1**. : \(F(. )\) _is continuous and \(\frac{\partial F(t,x,u_{fs})}{\partial u_{fs}}\) is bounded._  
To facilitate the design of the controller, a Taylor series expansion is applied to convert the control input in the non-affine nonlinear system of Equation (4) to an explicit expression, resulting in  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f(t,x)+g(t,x)\rho\;u_{s}+d\end{array} \right. \tag{5}\]  
where \(f(t,x)=F(t,x,u_{fs}\xi)-\frac{\partial F(t,x,u_{fs})}{\partial u_{fs}}|_{u_{fs }}u_{fs\xi}\), \(g=\frac{\partial F(t,x,u_{fs})}{\partial u_{fs}}|_{u_{fs\xi}}\), \(d=D+\Delta(. )\) is the compound disturbance, \(\Delta(. )\) is a higher-order term, and \(u_{fs\xi}\) denotes the filtered value of \(u_{fs}\) which is acquired by a robust sliding mode differentiator. \[\dot{u}_{fs\xi}=-\frac{u_{fs\xi}-u_{fs}}{\tau}-\frac{\zeta_{1}(u_{fs\xi}-u_{fs} )}{\left\|u_{fs\xi}-u_{fs}\right\|+\zeta_{2}} \tag{6}\]  
where \(\tau\) is the filter time constant and the two positive parameters \(\zeta_{1}\) and \(\zeta_{2}\) denote the switching gain and the switching rate to regulate the sliding mode, respectively. **Remark 1**. : _From Equation (6), \(u_{fs\xi}\) can approximate \(u_{fs}\) with any small deviation by choosing an appropriate the filter time parameter \(\tau\). If present, the higher-order term \(\Delta(. )\) can tend to zero. Supposing that the \(\Delta(. )\) is small enough, its effect can be compensated in \(d\)._  
To proceed to the following work, the definitions and lemmas are given. **Definition 1**. : _[_31_]_ _If on an open region \(\mathcal{D}\) containing the origin defined about the system \(\dot{x}=f(x)\), a scalar function \(V(x)\) that is continuously differentiable and positive definite has the property \(V(x)\rightarrow\infty\) as \(x\) approaches the boundary of \(\mathcal{D}\) and has positive constant boundedness along with the solution of the system \(\dot{x}=f(x)\) with \(x(0)\in\mathcal{D}\), then \(V(x)\) is known as a barrier Lyapunov function.__In brief, the symmetric barrier Lyapunov function is defined as_  
\[V_{i}=\frac{1}{2}\ln(\frac{k_{ai}^{2}}{k_{ai}^{2}-z_{i}^{2}}) \tag{7}\]  
_If \(z_{i}\) stays within the boundary of \(k_{ai}\), especially since the barrier Lyapunov function tends toward infinity at \(|z_{i}|=k_{ai}\), there can exist an available Lyapunov function in the interval \(|z_{i}|<k_{ai}\)._  
**Lemma 1**. : _[_31_]_ _For any positive constant \(k_{ai}\in\mathbf{R}\), the following equality satisfies \(z_{i}\in\mathbf{R}\) in the set \(|z_{i}|<k_{ai}\):_  
\[\ln\frac{k_{ai}^{2}}{k_{ai}^{2}-z_{i}^{2}}\leq\frac{z_{i}^{2}}{k_{ai}^{2}-z_{i }^{2}} \tag{8}\]  
**Lemma 2**. : _([PERSON]'s inequality) For \(\forall\ x,y\geq 0\), the following inequality holds:_  
\[xy\leq\frac{\epsilon^{p}}{p}x^{p}+\frac{1}{q\epsilon^{q}}y^{q} \tag{9}\]  
_with \(\epsilon>0,p>1,q>1,\frac{1}{p}+\frac{1}{q}=1\). Only if \(x^{p}=y^{q}\), the equals sign holds in Equation (9)._  
**Lemma 3**. : _For any positive constant \(\epsilon>0\) and \(x\in\mathbf{R}\), the following is satisfied:_  
\[0\leq|x|-xtanh(\epsilon x)\leq\iota/\epsilon \tag{10}\]  
_where the normal number \(\iota=e^{-(\iota+1)}\), namely \(\iota\approx 0.2785\)._  
**Lemma 4**.