## 2 Twin Rotor \(Mimo\) System
### Dynamics of the \(Trms\)  
The \(TRMS\) system composed of two rotors--a main rotor and a tail rotor--is subject to forces such as gravity, thrust, centrifugal force, friction, and disturbance torque, all of which influence the motion of the propellers. To counteract these forces, control inputs are applied via \(DC\) motors. In this setup, only the pitch and yaw angles are measured outputs, and maintaining their stability is the primary objective of the controller design. With respect to the mechanical unit, the following nonlinear momentum equation can be derived for the pitch motion (vertical plane) of the \(TRMS\):  
\[I_{1}\cdot\ddot{\psi}=M_{1}-M_{FG}-M_{B,\psi}-M_{G} \tag{1}\]  
where:  
* \(\ddot{\psi}\): pitch acceleration.
* \(I_{1}\): moment of inertia of vertical plane.
* \(M_{1}=a_{1}\cdot\tau_{1}^{2}+b_{1}\cdot\tau_{1}\): nonlinear static characteristic.
* \(\tau_{1}\): the momentum of the main rotor.
* \(a_{1}\), \(b_{2}\): constants.
* \(M_{FG}=M_{g}\cdot sin(\psi)\): gravity momentum.
* \(M_{g}\): gravity momentum
* \(\psi\): pitch angle.
* \(M_{B,\psi}=B_{1\psi}\cdot\dot{\psi}+B_{2,\psi}\cdot sin(2\psi)\cdot\dot{\psi} ^{2}\): friction forces momentum.
* \(\dot{\psi}\): pitch velocity.
* \(\dot{\psi}\): yaw velocity.
* \(\dot{\psi}\): yaw acceleration.
* \(B_{1\psi}\), \(B_{2\psi}\): constants.
* \(M_{G}=K_{gy}\cdot M_{1}\cdot\dot{\phi}\cdot cos(\psi)\): gyroscopic momentum.
* \(K_{gy}\): gyroscopic momentum of parameter.
* \(\phi\): yaw angle.  
The motors and the electrical control circuits mounted on the main and tail rotors are approximated by first-order transfer functions. Accordingly, the rotor momentum of the motor mounted in the main rotor (1), expressed in the Laplace domain, is given by:  
\[\tau_{1}(s)=\frac{K_{1}}{T_{11}\cdot s+T_{10}}\cdot u_{1}(s) \tag{2}\]  
where \(K_{1}\) is the gain of the rotor, \(T_{11}\) and \(T_{10}\) are coefficients that define the main rotor denominator, and \(u_{1}\) is the voltage (V) applied on the main rotor.  
The momentum equation of the \(TRMS\) in the horizontal plane (yaw motion) is:  
\[I_{2}\cdot\ddot{\psi}=M_{2}-M_{B,\phi}-M_{R} \tag{3}\]  
where:  
* \(\ddot{\psi}\): yaw acceleration.
* \(\phi\): yaw angle.
* \(I_{2}\): moment of inertia of horizontal plane.
* \(M_{2}=a_{2}\cdot\tau_{2}^{2}+b_{2}\cdot\tau_{2}\): nonlinear static characteristic.
* \(\tau_{2}\): the momentum of the tail rotor.
* \(a_{2}\), \(b_{2}\): constants.
* \(M_{B\phi}=B_{1\phi}\cdot\dot{\phi}\): friction forces momentum.
* \(B_{1\phi}\): a constant.
* \(\dot{\phi}\): yaw velocity.  
* \(M_{r}=\frac{K_{c}(T_{0};s+1)}{T_{p}\cdot s+1}\cdot M_{1}\): cross-reaction momentum.
* \(K_{c}\): cross-reaction momentum gain.
* \(T_{0}\), \(T_{p}\): constants.  
After taking the inverse Laplace transform of equation \(M_{r}\) and applying approximations to simplify the transfer function, we consider the case where the dynamic response dominates, i.e., \(T_{0}\)\(\otimes\)\(1\) and \(T_{p}\)\(s\)\(\gg\)\(1\). Under this assumption, the terms \((T_{0}\)\(s+1)\) and \((T_{p}\)\(s+1)\) can be approximated as \(T_{0}\)\(s\) and \(T_{p}\)\(s\), respectively. This yields the simplified relationship:  
\[M_{R}=\frac{K_{c}\cdot T_{0}}{T_{p}}\cdot M_{1} \tag{4}\]  
The rotor momentum of the motor mounted in the tail rotor (3) expressed in the Laplace domain is given as:  
\[\tau_{2}(s)=\frac{K_{2}}{T_{21}.s+T_{20}}.u_{2}(s) \tag{5}\]  
where \(K_{2}\) is the gain of the tail rotor, \(T_{21}\) and \(T_{22}\) are the coefficients that define the tail rotor denominators, and \(u_{2}(s)\) is the voltage (V) applied on the tail rotor.  
The values of the parameters of the previous equations are given in Table 1.