# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 4 Simulation Results  
### Simulation Setup  
To verify the improved performance of the proposed FTC, the simulation results are conducted in the EHA system by using MATLAB/Simulink. Two desired trajectories, i.e., the variable frequency sinusoidal (VFS) signals and a ramp signal, are described as  
\[\begin{array}{l}\zeta_{1d}=10+10(sin(\pi t-\pi/2)+cos(0.5\pi t))(\text{mm})\\ \zeta_{1d}=10t(\text{mm})\end{array} \tag{48}\]  
The values of the EHA system parameters are given on the basis of data reported in Ref. [21], as listed in Table 1.  
For comparison, other control strategies are given to validate the advantages of the proposed FTC method in the simulation.  
(1) DBC: This is the direct backstepping controller without two DOs. The control law is designed by Ref. [41]:  
\[\begin{array}{l}e_{1}=\zeta_{1}-\zeta_{1d};\zeta_{2d}=\dot{\zeta}_{1d}-k_{1} e_{1};\\ e_{2}=\zeta_{2}-\zeta_{2d};\zeta_{3d}=m\dot{x}_{2d}-f_{2}-e_{1}-k_{2}e_{2};\\ e_{3}=\zeta_{3}-\zeta_{3d};u=\frac{1}{g_{3}}(\dot{\zeta}_{3d}-f_{3}-k_{3}e_{3}- e_{2});\end{array} \tag{49}\]  
where \(k_{1},\ k_{2},\ and\ k_{3}\) represent the positive gain parameters.  
(2) BSMCDO: This is the traditional backstepping sliding-mode controller with two DOs [42]. The control law is described by:  
\[\begin{array}{l}s=m\dot{e}_{1}+\eta e_{1}\\ \zeta_{3d}=-k_{1}\dot{e}_{1}+m\ddot{\zeta}_{1d}-f_{2}-k_{2}s-\hat{\sigma}_{1}\\ e_{3}=\zeta_{3}-\zeta_{3d}\\ u=-\frac{1}{g_{3}}\Big{(}f_{3}-\dot{\zeta}_{3d}+\hat{\sigma}_{2}+k_{3}e_{3}+s \Big{)}\end{array} \tag{50}\]  
where \(k_{i}\), \(i=1,\ 2,\ 3\) are the positive constant.  
(3) Proposed controller: This strategy is a nonsingular terminal sliding-mode optimized backstepping control (NTSMOBC) with two DOs.  
The following control gains are carefully regulated and tuned via the trial and error method and chosen as: \(k_{1}=50\); \(\gamma_{1}=20\); \(\gamma_{2}=1.5\); \(\lambda_{1}=3\); \(\lambda_{2}=5\); \(\eta=10\). Meanwhile, the parameters of the optimal force controller are chosen as \(k_{3}=25\); \(\beta_{a}=5.5\); \(\beta_{c}=23\). There exist two RBF NNs in the control EHA system. The critic NN part contains 65 nodes with a width equal to 0.5 and centers \(\mu_{i}=2.5+0.4i\) (\(i=1\div 65\)) evenly spaced in the interval \([-2.5,\ 2.5]\). The actor NN part contains 25 nodes with a width equal to 0.5 and centers \(\mu_{i}=2.5+0.4i\) (\(i=1\div 25\)) evenly spaced in the interval \([-2.5,\ 2.5]\). The initial conditions of the critic NN, actor NN are selected as \(\hat{\phi}_{c}(0)=[0.04\,\dots\,0.04]^{T}\), \(\hat{\phi}_{a}(0)=[4\,\dots\,4]^{T}\), respectively. The parameters for the DO gains are set as \(\kappa_{1}=60\); \(\kappa_{2}=5.5\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**System Parameter** & **Value** & **Unit** \\ \hline \(\beta_{e}\) & \(1.25\times 10^{3}\) & MPa \\ \(k_{s}\) & \(3.2\times 10^{-8}\) & m\({}^{3}\)/s/V/Pa\({}^{-1/2}\) \\ \(k_{b}\) & 450 & Ns/m \\ \(m\) & 4.5 & kg \\ \(A\) & \(4\times 10^{-4}\) & m\({}^{2}\) \\ \(P_{s}\) & 16 & MPa \\ \(V_{c}\) & \(6\times 10^{-5}\) & m\({}^{3}\) \\ \(P_{r}\) & 0.1 & MPa \\ \hline \hline \end{tabular}
\end{table}
Table 1: EHA parameters.  
In fair comparison, the control parameters of DBC, BSMCDO controller are the same as the corresponding parameters in the proposed controller, e.g., same value of \(k_{1}\); \(k_{2}=\gamma_{1}\); \(k_{3}\); \(\eta\); \(\kappa_{1}\); \(\kappa_{2}\).  
To assess the qualification of the proposed control algorithm in the face of the actuator fault, the various operational conditions are provided during 30 s. In the first condition, the system operates in normal condition with lumped disturbances. In the second condition, the internal leakage fault occurs after 10 s.