## 1 Introduction
The H\({}_{\infty}\) controller seeks to minimize the mixed-sensitivity cost function, while the \(\mu\)-synthesis controller implements the D-K iteration method to find the stabilizing gain that minimizes the upper structured singular value of the system. The simulation results show that the \(\mu\)-synthesis control system has a more robust performance. In (2019, 2020), [PERSON] et al. proposed a method for enhancing the H\({}_{\infty}\) and \(\mu\)-synthesis control systems' robustness and performance for the triple inverted pendulum, by incorporating PSO and the Gazelle Optimization Algorithm (GOA), respectively, into the controllers' design. It has been shown that utilizing the optimization in control system design allows the possibility of balancing robustness and performance aspects within the system. Meta-heuristic optimization algorithms are known to be well-suited for a wide range of complex optimization problems. Their key power lies in their ability to handle large search spaces, constraints, and multi-objective cost functions. The following studies are recommended to researchers interested in the field of meta-heuristic optimization applications. In fact, various meta-heuristic optimization algorithms (such as PSO [26; 27; 28], Ant Colony Optimization [29], the Bees Algorithm [30], Grey Wolf Optimization [31], the Whale Optimization Algorithm [32], Sunflower Optimization [33], the Gorilla Troops Algorithm [34], and the Chimp Optimization algorithm [35]) have been applied successfully in many engineering problems. In addition, it is also noteworthy that [PERSON] et al. [36] tuned the high-gain observer and controller gains to enhance the position and velocity perturbation attenuation for inverted pendulums using a genetic optimizer. The suggested method resulted in better perturbation attenuation compared to simplex and Bat optimizers. As for robots, [PERSON] [37] applied the Bat Algorithm and the Modified Bat Algorithm to minimize both tracking error and control energy consumption by optimizing the control gain. [PERSON] et al. [38] also managed the minimization of the tracking error and the control effort for the experimental two-tank system through applying the genetic algorithm for the tuning of the PID and Fractional Order PID (FOPID) controller gains. The tuned FOPID control system performed better than the tuned PID control system in terms of overshoot, settling time, and control signal smoothness. The literature review shows a gap in a fully robust analysis of the triple inverted pendulum which considers all the possible sources of uncertainties in the system. To fill the gap, this paper contributes from three important perspectives:  
* Formulating a hybrid H\({}_{\infty}\)/SMC design for a triple inverted pendulum system that is robust to parameter variations, external disturbances, and measurement noise;
* Utilizing PSO to tune parameters in the control action based on the Integral Time Absolute Error (ITAE) performance index of the three controlled variable errors and the Integral Square Control Signal (ISCS) performance index of the two actuators' torques;
* Evaluating the robustness, performance, and efficiency of the proposed optimized control system in different cases. The rest of the paper introduces the mathematical model of the triple inverted pendulum system in Section 2, presents the PSO algorithm in Section 3, develops the hybrid H\({}_{\infty}\)/SMC design and optimization in Section 4, and tests the control system response to different cases in Section 5. Finally, it discusses the outcomes of the developed control system and the most noticeable findings, and suggests directions for future work in Section 6.