## 3 Modified Smith Predictor Control Scheme
### The Disturbance Rejection Controller \(H(s)\)  
The \(H(s)\) controller serves as the disturbance rejection controller, aimed at mitigating and absorbing the impact of persistent perturbations like the wind effects on the \(TRMS\) system [13]. The control scheme of our \(SP\) has two subsystems. This is a special feature that allowed us to design a new controller that can remove the disturbance effect that impacts the system. The new controller is based on adding an inner loop that feedbacks the difference between the process output and the nominal model output. In our control scheme, we define \(\mathbf{H}(s)\) as:  
\[\mathbf{H}(s)=\mathbf{F}(s)\cdot\mathbf{Q}_{0}^{-1}(s) \tag{34}\]  
where \(\mathbf{F}(s)\) is a diagonal transfer function matrix such that \(\mathbf{F}(0)=\mathbf{I}\). A simple choice of \(\mathbf{F}(s)\) is  
\[\mathbf{F}(s)= \begin{bmatrix}F_{1}(s)&0\\ 0&F_{2}(s)\end{bmatrix} ,\quad F_{1}(s)=\frac{1}{(\lambda_{11}\cdot s+1)^{n_{11} }},\quad F_{2}(s)=\frac{1}{(\lambda_{22}\cdot s+1)^{n_{22}}} \tag{35}\]  
where \(n_{11}\) and \(n_{22}\) are chosen as the minima integer values that make the elements of matrix \(H(s)\) proper. Substituting (34) in (33) gives that \[\boldsymbol{\Psi}_{SP-H}(s)=\mathbf{I}_{2}-\mathbf{F}(s)\cdot\mathbf{L}_{0}(s)+ \mathbf{R}(s)\cdot\mathbf{Q}_{0}(s)\cdot(\mathbf{I}_{2}-\mathbf{L}_{0}(s)) \tag{36}\]  
which coincides with \(\boldsymbol{\Psi}_{SP}(s)\) when \(\mathbf{F}(0)=\mathbf{I}\) is set. Comparison of (27) and (36), shows that the difference is that the term \(\mathbf{I}_{2}\) of the first equation becomes \(\mathbf{I}_{2}-\mathbf{F}(s)\cdot\mathbf{L}_{0}(s)\) in the second one. This implies that, if the final value theorem were applied to determine the steady state effect of the disturbance on the output, this term is constant in the first equation while it tends to zero in the second when \(\omega\to 0\). This suggests a better disturbance rejection feature of the \(SP-H\) at low frequencies than that of the \(SP\). The effect of persistent disturbances on the output is reduced if the magnitudes of \(\boldsymbol{\Psi}_{SP}(j\cdot\omega)\) or \(\boldsymbol{\Psi}_{SP-H}(j\cdot\omega)\) are reduced at low frequencies. Then, we define the index  
\[\boldsymbol{\Lambda}(\omega)=\left|\boldsymbol{\Psi}_{SP-H}(j\cdot\omega) \cdot\boldsymbol{\Psi}_{SP}^{-1}(j\cdot\omega)\right| \tag{37}\]  
which allows us to evaluate the disturbance reduction features of our \(SP-H\) compared to the standard \(SP\). Since \(\boldsymbol{\Psi}_{SP-H}(s)\) and \(\boldsymbol{\Psi}_{SP}(s)\) are diagonal, the index \(\boldsymbol{\Lambda}(\omega)\) is composed of two functions:  
\[\Lambda_{1}(\omega)=\left|\frac{\boldsymbol{\Psi}_{SP-H,1,1}(j\cdot\omega)}{ \boldsymbol{\Psi}_{SP,1,1}(j\cdot\omega)}\right| \tag{38}\]  
and  
\[\Lambda_{2}(\omega)=\left|\frac{\boldsymbol{\Psi}_{SP-H,2,2}(j\cdot\omega)}{ \boldsymbol{\Psi}_{SP,2,2}(j\cdot\omega)}\right| \tag{39}\]  
where \(\boldsymbol{\Psi}_{SP-H,i,l}\) and\(\boldsymbol{\Psi}_{SP,i,l}\) represent the \(i\)-row, \(l\)-column element of matrices \(\boldsymbol{\Psi}_{SP-H}(s)\) and \(\boldsymbol{\Psi}_{SP}(s)\), respectively. Comparing expressions (26) and (32), the only difference is the presence of \(\boldsymbol{\Psi}_{SP}\) in the former and \(\boldsymbol{\Psi}_{SP-H}\) in the latter. If \(|\boldsymbol{\Psi}_{SP}(j\cdot\omega)|>|\boldsymbol{\Psi}_{SP-H}(j\cdot \omega)|\), then the effect of the disturbance on the output is greater when using \(SP\) than when using \(SP-H\). Therefore, \(SP-H\) rejects the disturbance better than \(SP\) at frequency \(\omega\). Since these matrices are diagonal, this inequality can be tested for each channel of the multivariable system by comparing the corresponding diagonal elements of the matrices. Translated to the individual diagonal elements, the inequality is equivalent to checking whether the ratio \(\boldsymbol{\Psi}_{SP-H,i,l}/\boldsymbol{\Psi}_{SP,i,j}\) is less than 1. Conversely, if this ratio is greater than 1, then \(SP\) rejects the disturbance better than \(SP-H\) at frequency \(\omega\).