## 2 System Statement
### Modeling of Mode-1 in Normal Conditions
#### 2.1.2 State Space Form of NT-DRAS in Mode-1  
The state variables are defined as \(\mathbf{x}=\left[x_{\text{eha}},\dot{x}_{\text{eha}},P_{\text{eha}},\omega_{\text{eha}} \right]^{\text{T}}\), by using the Equations (1)-(5), the state space representation of system can be derived as:  
\[\left\{\begin{array}{l}\frac{dx_{\text{eha}}}{dt}=\dot{x}_{\text{eha}}\\ \frac{dx_{\text{eha}}}{dt}=-\frac{\text{B}_{\text{eha}}+\text{B}_{\text{ema}}+ \text{m}_{\text{eha}}}{\text{m}_{\text{eha}}+\text{m}_{\text{ema}}+\text{m}_{ \text{d}}}\dot{x}_{\text{eha}}+\frac{\text{A}_{\text{eha}}}{\text{m}_{\text{eha }}+\text{m}_{\text{ema}}+\text{m}_{\text{d}}}P_{\text{eha}}-\frac{1}{\text{m}_ {\text{eha}}+\text{m}_{\text{ema}}+\text{m}_{\text{d}}}P_{\text{eha}}\\ \frac{dP_{\text{eha}}}{dt}=-\frac{4\text{E}_{\text{eha}}\text{A}_{\text{eha}}} {\text{V}_{\text{eha}}}\dot{x}_{\text{eha}}-\frac{4\text{E}_{\text{eha}}\text {C}_{\text{eha}}}{\text{V}_{\text{eha}}}P_{\text{eha}}+\frac{4\text{E}_{\text {eha}}\text{V}_{\text{eha}}}{\text{V}_{\text{eha}}}\omega_{\text{eha}}\\ \frac{d\omega_{\text{eha}}}{dt}=-\frac{\text{V}_{\text{P}}}{\text{J}_{\text{m} }}P_{\text{eha}}-\frac{\text{B}_{\text{m}}}{\text{J}_{\text{m}}}\omega_{\text {eha}}+\frac{\text{V}_{\text{P}}}{\text{J}_{\text{m}}}\text{R}_{\text{eha}}+ \text{V}_{\text{eha}}.\end{array}\right. \tag{6}\]  
In the state space form, Equation (6) can be represented as:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{G }w(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)\end{array}\right. \tag{7}\]  
where \(\mathbf{u}(t)\) is the system input, \(\mathbf{y}(t)\) is the system output, \(w(t)=F_{\text{L}}\) is unknown disturbance. The state, input, output, and disturbance matrices are presented in Equation (8).  
\[\left\{\begin{array}{l}\mathbf{A}=\left[\begin{array}{cccc}0&1&0&0\\ 0&-\frac{\text{B}_{\text{eha}}+\text{B}_{\text{ema}}+\text{m}_{\text{eha}}}{ \text{m}_{\text{eha}}+\text{m}_{\text{ema}}+\text{m}_{\text{d}}}&\frac{\text{ A}_{\text{eha}}}{\text{m}_{\text{eha}}+\text{m}_{\text{ema}}+\text{m}_{\text{d}}} &0\\ 0&-\frac{4\text{E}_{\text{eha}}\text{A}_{\text{eha}}}{\text{V}_{\text{eha}}}&- \frac{4\text{E}_{\text{eha}}\text{C}_{\text{eha}}}{\text{V}_{\text{eha}}}& \frac{4\text{E}_{\text{eha}}\text{V}_{\text{P}}}{\text{V}_{\text{eha}}}\\ 0&0&-\frac{\text{V}_{\text{P}}}{\text{J}_{\text{m}}}&-\frac{\text{V}_{\text{P}} }{\text{J}_{\text{m}}}\end{array}\right]\\ \mathbf{B}=\left[\begin{array}{cccc}0&0&0&\frac{\text{K}_{\text{m}}}{\text{J}_{ \text{m}}\text{R}_{\text{e}}}\end{array}\right]^{\text{T}}\\ \mathbf{C}=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\\ \mathbf{G}=\left[\begin{array}{cccc}0&-\frac{1}{\text{m}_{\text{eha}}+\text{m}_{ \text{ema}}+\text{m}_{\text{d}}}&0&0\end{array}\right]^{\text{T}}\end{array} \tag{8}\]  
An assumption is made that the above parameters are accurate standard values, so that there is no system uncertainty in the system model represented in Equation (7).  
**Remark 1**.: _In the motor link modeling, the electromagnetic time constant can be represented as \(T_{1}=\text{L}_{\text{e}}/\text{R}_{\text{e}}\), the mechanical time constant can be represented as \(T_{2}=\text{J}_{\text{m}}/\text{B}_{\text{me}}\), since \(T_{1}<<T_{2}\), and considering that the mechanical link is the dominant factor, the motor model can be simplified as a first-order process by ignoring the armature inductance and the back-EMF link:_  
\[\omega_{\text{eha}}=\frac{1}{\text{J}_{\text{m}}\text{s}+\text{B}_{\text{me}}} \left(\frac{\text{K}_{\text{m}}}{\text{R}_{\text{e}}}u_{\text{eha}}-T_{\text{ eha}}\right) \tag{9}\]