## 2 System Statement
### Modeling of Mode-1 in Normal Conditions
#### 2.1.1 Modeling of EHA Component  
It is assumed that the brushless DC motor is used in this system. The motor armature and torque balance equations can be given as following, respectively:  
\[\left\{\begin{array}{l}u_{\text{eha}}=\text{C}_{\text{e}}\omega_{\text{eha}}+ \text{L}_{\text{e}}\frac{d\dot{x}_{i}}{dt}+\text{R}_{\text{e}}i_{\text{eha}}\\ \text{K}_{\text{m}}i_{\text{eha}}=T_{\text{eha}}+\text{J}_{\text{m}}\frac{d \omega_{\text{eha}}}{dt}+\text{B}_{\text{me}}\omega_{\text{eha}}\end{array}\right. \tag{1}\]  
where \(u_{\text{eha}}\), \(i_{\text{eha}}\) and \(\omega_{\text{eha}}\) are the input voltage, output current and speed of the motor, respectively; \(\text{C}_{\text{e}}\) is the back EMF coefficient; \(\text{K}_{\text{m}}\), \(\text{L}_{\text{e}}\) and \(\text{R}_{\text{e}}\) are the electromagnetic torque constant, inductance and resistance of the motor respectively; \(T_{\text{eha}}\) is the output toque of the motor; \(\text{B}_{\text{me}}\) is the simplified equivalent damping coefficient of the motor; \(\text{J}_{\text{m}}\) is the total moment of inertia of motor and pump.  
In the pump-cylinder modeling link, the pump efficiency is assumed as 100%, and there is no pressure loss between the input and output of the cylinder, and the following pressure and flow equation can be given as:  
\[\left\{\begin{array}{l}q_{a}=q_{1},q_{b}=q_{2}\\ p_{a}=p_{1},p_{b}=p_{2}\end{array}\right. \tag{2}\]  
where \(p_{1}\) and \(q_{1}\) are the pressure and flow of the cylinder inlet, \(p_{2}\) and \(q_{2}\) are the pressure and flow of the cylinder outlet; \(p_{a}\) and \(q_{a}\) are the pressure and flow of the pump outlet, \(p_{b}\) and \(q_{b}\) are the pressure and flow of the pump inlet.  
Since the motor output toque can be regarded as the pump input toque, and the pump efficiency is assumed as 100%, that means \(q_{a}=q_{b}\), and the following torque balance equation can be given as:  
\[T_{\text{eha}}=\text{V}_{\text{P}}(p_{a}-p_{b})=\text{V}_{\text{P}}P_{\text{ eha}} \tag{3}\]  
where \(\text{V}_{\text{P}}\) is the pump output of EHA, \(P_{\text{eha}}\) is the cylinder pressure.  
The equation relating the motor speed \(\omega_{\text{eha}}\) to the system velocity \(\dot{x}_{\text{eha}}\) and the EHA cylinder pressure \(P_{\text{eha}}\) is given as:  
\[\text{V}_{\text{P}}\omega_{\text{eha}}\text{=A}_{\text{eha}}\dot{x}_{\text{eha }}+\frac{\text{V}_{\text{eha}}}{4\text{E}_{\text{eha}}}\dot{p}_{\text{eha}}+ \text{C}_{\text{ehal}}P_{\text{eha}} \tag{4}\]  
where \(\text{A}_{\text{eha}}\) is area of EHA hydraulic cylinder; \(\text{V}_{\text{eha}}\) is volume of EHA hydraulic cylinder; \(\text{E}_{\text{eha}}\) is the volumetric modulus of elasticity; \(\text{C}_{\text{ehal}}\) is leakage coefficient of EHA hydraulic cylinder.  
In the final output force link, the external load on the cylinder, \(F_{\text{eha}}\), consists of three components: inertia and damping loads of EHA, inertia load of the control surface as well as EMA, and external disturbance, \(F_{\text{L}}\), caused by gust, is given as:  
\[\text{A}_{\text{eha}}P_{\text{eha}}=F_{\text{eha}}=(\text{m}_{\text{eha}}+ \text{m}_{\text{ema}}+\text{m}_{\text{d}})\ddot{x}_{\text{eha}}+(\text{B}_{ \text{eha}}+\text{B}_{\text{ema}}+\text{B}_{\text{d}})\dot{x}_{\text{eha}}+F_{ \text{L}} \tag{5}\]  
where \(\text{B}_{\text{eha}}\), \(\text{B}_{\text{ema}}\), and \(\text{B}_{\text{d}}\) are equivalent damping parameters of EHA hydraulic cylinder pistons, EMA transmission screw, and the control surface, respectively; \(\text{m}_{\text{eha}}\), \(\text{m}_{\text{ema}}\), \(\text{m}_{\text{d}}\) represent equivalent masses of EHA hydraulic cylinder piston, EMA transmission screw, and the control surface, respectively.