## 3 Modified Smith Predictor Control Scheme
### Modified Smith Predictor  
The innovative structure, based on a modified Smith predictor for controlling the \(TRMS\), introduces a compensator \(\mathbf{H}(s)\), whose output is inserted between the \(\mathbf{R}(s)\) and the decoupler \(\mathbf{D}(s)\). Placing H(s) at this point allows it to operate on the decoupled control signals, effectively mitigating external disturbances and modeling uncertainties in both the vertical and horizontal channels. This configuration ensures that disturbance compensation is performed before the cross-coupling dynamics, handled by D(s), are reintroduced, thereby improving the overall robustness and performance of the \(TRMS\) control system. This structural modification is represented by the blue lines in Figure 2. The compensator is integrated with the original \(SP\) architecture to generate a modified control signal. It will be shown that, for the nominal process, this block affects only the disturbance response while leaving the set-point tracking performance unchanged. Therefore, it can be used reduce the impact of disturbances on the system. The proposed control system, hereafter referred as \(SP-H\), was initially designed for a heating furnace modeled as a \(SISO\) second-order process with a time delay [31]. It was later adapted to control a marginally stable \(DC\) motor (i.e., with a pole at the origin) [32]. In the present work, we have redesigned the control system to handle a high-order \(MIMO\) system along with time delays, which characterizes the \(TRMS\). Additionally, we examined the benefits of this new \(SP\) structure over the standard one in mitigating the effects of wind disturbances on the \(TRMS\). The transfer function matrices of the \(TRMS\) with the \(SP-H\) control scheme are expressed as:\[\mathbf{Y}_{SP-H}(s)=\mathbf{Y}_{r_{SP-H}}(s)+\mathbf{Y}_{d_{SP-H}}(s) \tag{28}\]  
\[\mathbf{Y}_{r_{SP-H}}(s) =\left[\mathbf{I}_{2}+\mathbf{Q}(s)\cdot\mathbf{L}(s)\cdot\left[ \mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q}_{0}(s)-(\mathbf{R}(s)+\mathbf{H}(s ))\cdot\mathbf{Q}_{0}(s)\cdot\mathbf{L}_{0}(s)\right]^{-1}\right.\] \[\left.\cdot(\mathbf{R}(s)+\mathbf{H}(s))\right]\cdot\mathbf{Q}(s) \cdot\mathbf{L}(s)\cdot\left[\mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q}_{0}(s )-(\mathbf{R}(s)+\mathbf{H}(s))\cdot\mathbf{Q}_{0}(s)\cdot\mathbf{L}_{0}(s) \right]^{-1}\] \[\left.\cdot\mathbf{R}(s)\cdot\mathbf{Y}^{*}(s)\right. \tag{29}\]  
\[\mathbf{Y}_{d_{SP-H}}(s) =\left[\mathbf{I}_{2}+\mathbf{Q}(s)\cdot\mathbf{L}(s)\cdot\left[ \mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q}_{0}(s)-(\mathbf{R}(s)+\mathbf{H}( s))\cdot\mathbf{Q}_{0}(s)\cdot\mathbf{L}_{0}(s)\right]^{-1}\right.\] \[\left.\cdot(\mathbf{R}(s)+\mathbf{H}(s))\right]\cdot\mathbf{Q}(s )\cdot\mathbf{D}^{-1}(s)\cdot\mathbf{Z}(s). \tag{30}\]  
In the nominal situation, the above equations become:  
\[\mathbf{Y}_{r_{SP-H}}^{n}(s)=\left[\mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q} _{0}(s)\right]^{-1}\cdot\mathbf{Q}_{0}(s)\cdot\mathbf{L}_{0}(s)\cdot\mathbf{R} (s)\cdot\mathbf{Y}^{*}(s) \tag{31}\]  
\[\mathbf{Y}_{d_{SP-H}}^{n}(s)=\left[\mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q }_{0}(s)\right]^{-1}\cdot\boldsymbol{\
abla}_{SP-H}(s)\cdot\mathbf{Q}_{0}(s) \cdot\mathbf{D}^{-1}(s)\cdot\mathbf{Z}(s) \tag{32}\]  
where  
\[\boldsymbol{\
abla}_{SP-H}(s)=\mathbf{I}_{2}+\mathbf{R}(s)\cdot\mathbf{Q}_{0}( s)-(\mathbf{R}(s)+\mathbf{H}(s))\cdot\mathbf{Q}_{0}(s)\cdot\mathbf{L}_{0}(s) \tag{33}\]  
Transfer function (31) demonstrates that the set-point tracking performance is influenced solely by \(\mathbf{R}(s)\). In contrast, transfer function (32), (33) is affected by both \(\mathbf{R}(s)\) and \(\mathbf{H}(s)\). Therefore, a careful design of \(\mathbf{H}(s)\) can mitigate the impact of the disturbance \(\mathbf{Z}(s)\) on the output \(\mathbf{Y}(s)\), which would otherwise be present in the standard \(SP\) configuration.