## 4 Simulation Results  
To confirm the effectiveness of the proposed \(SP-H\) controller, the linearized model of the feedback instrument \(TRMS\) 33-220 was implemented in Simulink (Matlab). The step responses of this control scheme are presented in this section and compared to those of the conventional \(SP\) controller, emphasizing both set-point tracking and disturbance rejection. Both \(SP\) control schemes were applied to the linearized model of the \(TRMS\) given in Equation (8). The amplitudes of the reference step inputs were set to 0.4 rad for the vertical plane and 1.4 rad for the horizontal plane. A step disturbance of \(-0.3\) rad was introduced at 80 s to simulate an external perturbation. In the control structure implemented in the simulation, the main controller parameters from Equation (21) were used to command the \(TRMS\) system. Additionally, the decoupler parameters from Equation (9) were applied to separate the system into two subsystems: vertical and horizontal. Furthermore, the disturbance rejection controller parameters from Equation (41) were utilized to enhance the system's performance. The responses of the control schemes applied to the \(TRMS\) system in both the vertical and horizontal planes are illustrated in Figures 4 and 5. Figure 4 presents the responses of both the \(SP\) and \(SP-H\) controllers on the vertical plane, while both controllers exhibit similar behavior in response to a reference change, they differ in their transient response when preventing external disturbances. Similarly, Figure 5 shows the responses of the same control schemes on the horizontal plane. Although the controllers demonstrate comparable behavior in terms of reference tracking, they display noticeable differences  
Figure 3: Comparison of robustness functions \(\Lambda_{1}(\omega)\) and \(\Lambda_{2}(\omega)\). in transient performance during disturbance rejection. The control signals applied to the vertical and horizontal planes are displayed in Figures 4a and 5a, respectively. The classical \(SP\) lacks a dedicated component for disturbance rejection; although it can attenuate disturbances, its response is slower compared to the proposed scheme. In the proposed \(SP-H\) approach, the controller \(H(s)\) significantly improves disturbance rejection capabilities. The simulation results outlined in Figures 4b and 5b indicate that \(SP-H\) exhibits superior disturbance rejection performance, providing improved transient behavior, in contrast to the conventional \(SP\) scheme. Figure 4: Simulated command step and disturbance step responses in the vertical plane using \(SP\) and \(SP-H\) techniques : (**a**) control signals; (**b**) system responses including both controller outputs to the step reference and to the applied disturbance. Figure 5: Simulated command step and disturbance step responses in the horizontal plane using \(SP\) and \(SP-H\) techniques: (**a**) control signals; (**b**) system responses including both controller outputs to the step reference and to the applied disturbance. To evaluate the performance of our control system under varying operating conditions, we conducted simulations with \(\pm 30\%\) variations in all \(TRMS\) parameters, except for the time delay. The time delay was excluded because the \(SP\) scheme is sensitive to mismatches in time delay. However, it can be estimated in real time, allowing the controller's time delay parameter to be retuned dynamically (e.g., [36]). Figures 4 and 5 present the simulation results, which demonstrate that (1) both \(SP-H\) and \(SP\) exhibit a wide range of stability robustness (tolerating more than 30% parametric variation); (2) the time responses to step-point tracking and step disturbances of both control systems show relatively small variations despite the parameter changes; (3) \(SP-H\) rejects disturbances significantly more effectively than \(SP\); and (4) the time responses to step-point tracking and step disturbances of \(SP-H\) are more robust than those of \(SP\), as they remain more consistent under parameter variations.