## 3 Sliding Surface and Control Design
### Sliding-Mode Control Design  
After this novel SS is obtained through the above off-line design process, an adaptive fault-tolerant sliding-mode control is developed. Considering the system (1) and (2) and the approximating optimal control policy of (19), the sliding-mode variable can be expressed as  
\[s=x_{2}-\hat{x}_{2}^{*}(x_{1}). \tag{30}\]  
Note that the control \(\hat{x}_{2}^{*}(x_{1})\) is differential according to (19), and the factory is that \(\hat{W}_{c}\) is a constant vector after convergence. Therefore, the derivative of the sliding-mode variable is written as  
\[\dot{s} =\hat{x}_{2}-\hat{\hat{x}}_{2}^{*}(x_{1})\] \[=f_{2}(x)+g_{2}(x)(u+f_{a})-\hat{\hat{x}}_{2}^{*}(x_{1}). \tag{31}\]  
The novel SMC design with the RADP-based SS and the adaptive fault compensation law is designed as  
\[u=g_{2}^{+}(x)(\hat{\hat{x}}_{2}^{*}(x_{1})-f_{2}(x)-K_{s}(sgn(s)))-\hat{f}_{a}, \tag{32}\]and  
\[\hat{f}_{a}=K_{f}g_{2}^{T}(x)s, \tag{33}\]  
respectively, where \(K_{s},K_{f}>0\) are the control gain and learning rate of the adaptive law and \(\hat{f}_{a}\) is estimation of actuator faults. Define the fault estimation error as \(\hat{f}_{a}=f_{a}-\hat{f}_{a}\). **Theorem 1**. : _Considering the mismatched uncertain and faulty system described by (1) and (2) under the proposed RADP-based sliding-mode variable (6), the control policy (32) and adaptive fault compensation law (33), the fault estimation error is bounded and the closed-loop system dynamics is UUB._  
Proof of Theorem 1.: Choose the following Lyapunov function candidate  
\[V(t)=\frac{1}{2}s^{T}s+\frac{1}{2K_{f}}\hat{f}_{a}^{T}\hat{f}_{a}. \tag{34}\]  
The derivative of the Lyapunov function candidate (34) with respect to time along the sliding mode variable dynamics is  
\[\dot{V}(t) =s^{T}\dot{s}-\frac{1}{K_{f}}\hat{f}_{a}^{T}\hat{f}_{a}\] \[=s^{T}(f_{2}(x)+g_{2}(x)(u+f_{a})-\hat{x}_{2}^{*}(x_{1}))\] \[-\frac{1}{K_{f}}\hat{f}_{a}^{T}\hat{f}_{a}. \tag{35}\]  
Substituting the control policy (32) and the actuator fault compensation law (33) into above formula (35), we obtain  
\[\dot{V}(t) =-K_{s}\|s\|+s^{T}g_{2}(x)\hat{f}_{a}-\frac{1}{K_{f}}\hat{f}_{a}^ {T}\hat{f}_{a}\] \[\dot{s} =\hat{x}_{2}-\hat{\hat{x}}_{2}^{*}(x_{1})\]0 \[\dot{s} =\hat{x}_{2}-\hat{\hat{x}}_{2}^{*}(x_{1})\]1  
According to the above derivation process, we could find that all signals of closed-loop system are bounded. Based on the [PERSON] theorem and Lemma 2, the sliding mode exists and the states of the whole system are uniformly ultimately bounded. From the above derivation, it can be seen that the reaching time of the SMC can be estimated by \(t_{r}\leq\frac{2V^{1/2}(0)}{\alpha}\), where \(t_{r}\) is the reaching time, \(V(0)\) is the initial value of the Lyapunov function candidate, and \(\alpha\) is a positive design parameter. Therefore, the corresponding time can be reduced by adjusting the parameter \(\alpha\). The proof ends here. Novelty and comparison: How to make the designed sliding surface (SS) insensitive to the mismatched uncertainties is crucial for the SMC design and the SMC-based FTC. In [4], a mismatched disturbance is estimated by a disturbance observer, and then the obtained estimation signal of this disturbance is used to design the SS. The disturbance observer makes the system more complex, which increases the difficulty of controller design. However, our designed SS can directly deal with this problem by the optimal control method without designing any estimator. Even though finding the solution of optimal control is tricky, we can solve it directly using the ADP method off-line. In [10], the integral SMC (ISMC) law is divided into two parts, one is a continuous part, and the other is a discontinuous part. The continuous part of control is used to reject the mismatched uncertainties and consequently the SS is insensitive to the mismatched uncertainties.