## 2 Twin Rotor \(Mimo\) System
### Nonlinear Dynamic Model  
The following state space model depicts the complete dynamics of the nonlinear \(TRMS\):  
\begin{table}
\begin{tabular}{l r} \hline
**Parameter** & **Value** \\ \hline Inertia moment relative to the vertical plane (\(I_{1}\)) & \(6.8\times 10^{-2}\) (kg \(\cdot\) m\({}^{2}\)) \\ Inertia moment relative to the horizontal plane (\(I_{2}\)) & \(2\times 10^{-2}\) (kg \(\cdot\) m\({}^{2}\)) \\ Characteristic parameter (\(a_{1}\)) & 0.0135 \\ Characteristic parameter (\(b_{1}\)) & 0.0924 \\ Characteristic parameter (\(a_{2}\)) & 0.02 \\ Characteristic parameter (\(b_{2}\)) & 0.09 \\ Gravity momentum (\(M_{g}\)) & 0.32 (N \(\cdot\) m) \\ Effect of friction on momentum (\(B_{1\varphi}\)) & \(6\times 10^{-3}\) (N \(\cdot\) m \(\cdot\) s/rad) \\ Effect of friction on momentum (\(B_{2\varphi}\)) & \(1\times 10^{-3}\) (N \(\cdot\) m \(\cdot\) s/rad) \\ Effect of friction on momentum (\(B_{1\varphi}\)) & \(1\times 10^{-1}\) (N \(\cdot\) m \(\cdot\) s/rad) \\ Effect of friction on momentum (\(B_{2\varphi}\)) & \(1\times 10^{-2}\) (N \(\cdot\) m \(\cdot\) s/rad) \\ Gyroscopic momentum of parameter (\(K_{gy}\)) & 0.05 (s/rad) \\ Main rotor thrust gain (\(K_{1}\)) & 1.1 \\ Tail rotor thrust gain (\(K_{2}\)) & 0.8 \\ Denominator of the main rotor (\(T_{11}\)) & 1.1 \\ Denominator of the main rotor (\(T_{10}\)) & 1 \\ Denominator of the tail rotor (\(T_{21}\)) & 1 \\ Denominator of the tail rotor (\(T_{20}\)) & 1 \\ Cross-momentum effect (\(T_{p}\)) & 2 \\ Cross-momentum effect (\(T_{0}\)) & 3.5 \\ Cross-coupled momentum gain (\(K_{c}\)) & \(-0.2\) \\ \hline \end{tabular}
\end{table}
Table 1: The parameters of the \(TRMS\).  
\[\dot{x_{1}} =x_{2},\] \[\dot{x_{2}} =\frac{a_{1}}{I_{1}}x_{5}^{2}+\frac{b_{1}}{I_{1}}x_{5}-\frac{M_{g}}{ I_{1}}sin(x_{1})-\frac{B_{1\cdot\psi}}{I_{1}}x_{2}-\frac{B_{2\psi}}{I_{1}}x_{4}^{2 }\cdot sin(2x_{1})-\frac{K_{gy}}{I_{1}}\cdot cos(x_{1})x_{3}(a_{1}\cdot x_{5}^{2 }+b_{1}\cdot x_{5}),\] \[\dot{x_{3}} =x_{4},\] \[\dot{x_{4}} =\frac{a_{2}}{I_{2}}x_{6}^{2}+\frac{b_{2}}{I_{2}}x_{6}-\frac{B_{1 \cdot\psi}}{I_{2}}\cdot x_{4}-\frac{K_{c}\cdot T_{0}}{T_{p}\cdot I_{2}}\cdot( a_{1}\cdot x_{5}^{2}+b_{1}\cdot x_{5}),\] \[\dot{x_{5}} =-\frac{T_{10}}{T_{11}}\cdot x_{5}+\frac{k_{1}}{T_{11}}\cdot u_{ 1},\] \[\dot{x_{6}} =-\frac{T_{20}}{T_{21}}\cdot x_{6}+\frac{k_{2}}{T_{21}}\cdot u_{ 2}.\]  
We define the state and input vectors as \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{6}]^{T}\) and \(\mathbf{u}=[u_{1},u_{2}]^{T}\), respectively, where \({}^{(T)}\) means a matrix transpose. Each state represents the following:  
\[x_{1} =\psi:\text{ Pitch angle.}\] \[x_{2} =\psi:\text{ Pitch velocity.}\] \[x_{3} =\phi:\text{ Yaw angle.}\] \[x_{4} =\phi:\text{ Yaw velocity.}\] \[x_{5} =\tau_{1}:\text{ Momentum of main rotor.}\] \[x_{6} =\tau_{2}:\text{ Momentum of tail rotor.}\]  
To obtain the transfer functions of a linearized model of the \(TRMS\), we first calculate the equilibrium point \(\mathbf{x}_{0}=[x_{01},x_{02},x_{03},x_{04},x_{05},x_{06}]^{T}\) under the assumption of input values \(u_{1}=0\), \(u_{2}=0\). The resulting value is \(\mathbf{x}_{0}=[k\cdot\pi,0,0,0,0,0]^{T}\), where \(k\) is an integer.