## 3 Modified Smith Predictor Control Scheme
### Design of the Main Controller  
The main controller is composed of a regulator \(\mathbf{R}(s)\) and a decoupler \(\mathbf{D}(s)\). The design procedures for these two components are outlined next. 1. Design of the Decoupler \(\mathbf{D}(s)\) The decoupler splits the control structure into two subsystems: one controlling the vertical plane and other controlling the horizontal plane. It eliminates the interactions between \(u_{1}\) and \(y_{2}\) and between \(u_{2}\) and \(y_{1}\), allowing each output to be independently regulated by its corresponding input. A simplified decoupling structure, aligned with this tuning approach [34] is employed: \[\mathbf{D}(\mathbf{s})= \begin{bmatrix}1&-\frac{G_{0,12}(s)}{G_{0,11}(s)}\\ -\frac{G_{0,21}(s)}{G_{0,22}(s)}&1\end{bmatrix} = \begin{bmatrix}1&0\\ \frac{-17s^{3}-102s^{2}-85s}{3.6s^{3}+21.27s^{2}+16.37s}&1\end{bmatrix} \] (9) where \(G_{0,ij}(s)\) are the elements of the nominal transfer function matrix \(\mathbf{G}_{0}(s)\). Then, we define \[\mathbf{Q}(s)=\mathbf{G}(s)\cdot\mathbf{D}(s)=diag(Q_{1}(s),Q_{2}(s))\] (10) which, in general, is not a diagonal matrix. In the case of the nominal process, we have that \[\mathbf{Q}_{0}(s)=\mathbf{G}_{0}(s)\cdot\mathbf{D}(s)=diag(Q_{0,1}(s),Q_{0,2} (s))\] (11) where \(Q_{0,1}(s)=G_{0,11}(s)\) and \(Q_{0,2}(s)=G_{0,22}(s)\). Taking into account the commutative property of the product of diagonal matrices, we have that \[\mathbf{G}_{0}(s)\cdot\mathbf{L}_{0}(s)\cdot\mathbf{D}(s)=\mathbf{Q}_{0}(s) \cdot e^{-L_{0}s}\] (12) where \(L_{0}\) is the nominal time delay. 2. Design of the Regulator \(\mathbf{R}(s)\) \(PID\) controllers are designed in the diagonal elements of \(\mathbf{R}(s)\): \[R_{i}(s)=K_{P,i}\bigg{(}1+\frac{1}{T_{I,i}\cdot s}+T_{D,i}\cdot s\bigg{)}\ \,\ \ \ i=1,2\] (13) using the nominal process, i.e, \(Q_{0}(s)\) and \(L_{0}(s)\). Figure 2: Proposed multivariable control scheme: Modified Smith Predictor (SP) controller for the Twin Rotor MIMO System (\(TRMS\)). The structure includes dynamic decoupling, disturbance compensation, and delay handling for coordinated control of pitch and yaw angles. Frequency domain techniques are often used to design these controllers. In particular, they are designed using the phase margin \(\theta_{m}\) and the gain margin \(A_{m}\), according to [7]:  
\[Q_{0,i}(j\cdot\omega_{p,i})\cdot e^{-j\cdot\omega_{p,i}\cdot L_{0}}\cdot\left[K_{ P,i}+j\cdot\left(K_{P,i}\cdot T_{D,i}\cdot\omega_{p,i}-\frac{K_{P,i}}{T_{I,i} \cdot\omega_{p,i}}\right)\right]=-\frac{1}{A_{m,i}}, \tag{14}\]  
\[Q_{i}(j\cdot\omega_{g,i})\cdot e^{-j\cdot\omega_{g,i}\cdot L_{0}}\cdot\left[K_ {P,i}+j\left(K_{P,i}\cdot T_{D,i}\cdot\omega_{g,i}-\frac{K_{P,i}}{T_{I,i}\cdot \omega_{g,i}}\right)\right]=-e^{-j\cdot\theta_{m,i}}. \tag{15}\]  
The parameters of the controllers are defined as follows:  
\[K_{P,i} =Re\left(\frac{-1}{A_{m,i}\cdot Q_{i}(\mathfrak{j}\cdot\omega_{ p,i})\cdot e^{-j\cdot\omega_{p,i}\cdot L_{0}}}\right)\quad, \tag{16}\] \[\frac{K_{P,i}}{T_{I,i}} =\left(X_{p,i}\cdot\omega_{g,i}-X_{g,i}\cdot\omega_{p,j}\right) \cdot\left(\frac{\omega_{p,i}}{\omega_{g,i}}-\frac{\omega_{g,i}}{\omega_{p,i}} \right)^{-1}\quad,\] (17) \[\frac{K_{P,i}}{T_{D,i}} =\left(\frac{X_{p,i}}{\omega_{g,i}}-\frac{X_{g,i}}{X_{p,i}}\right) \cdot\left(\frac{\omega_{p,i}}{\omega_{g,i}}-\frac{\omega_{g,i}}{\omega_{p,i}} \right)^{-1} \tag{18}\]  
where:  
\[\mathbf{Q}_{0}(s)=\mathbf{G}_{0}(s)\cdot\mathbf{D}(s)=diag(Q_{0,1}(s),Q_{0,2} (s))\]0 \[\mathbf{Q}_{0}(s)=\mathbf{G}_{0}(s)\cdot\mathbf{D}(s)=diag(Q_{0,1}(s),Q_{0,2} (s))\]1  
The specifications of the \(R_{1}(s)\) controller for the vertical plane are as follows:  
Phase margin: \(\phi=40^{\circ}\). Phase crossover frequency: \(w_{g}=0.12\) rad/s. Gain crossover frequency: \(w_{p}=2.2\) rad/s. The specifications of the \(R_{2}(s)\) controller for the horizontal plane are as follows:  
Phase margin: \(\phi=40^{\circ}\). Phase crossover frequency: \(w_{g}=0.12\) rad/s. Gain crossover frequency: \(w_{p}=2.8\) rad/s.