This paper discusses the stabilising problems of a nonlinear and multivariate underactuated systems, also known as Inverted Pendulum (IP) systems. The attitude control of a nonlinear IP system is thus considered. One of the most important techniques, a Linear Quadratic Regulator (LQR), is used in this paper to design a controller for an IP system. The structure of LQR parameters is based on weight matrices, and the adjustment of the weight matrices parameters is the most challenging part of this, being normally computed by repeating trials and adjusting based on successful or unsuccessful outcomes. The difficulty in guessing the optimum matrices and achieving automatic adjustment of the weighting matrices is mitigated by the introduction of a metaheuristic algorithm to investigate the optimal solution; this is based on a swarm of particles moving in a virtual space. The proposed approach successfully stabilised the IP system in the upright position and removed perturbations; the simulated results thus showed satisfactory performance of the suggested control method.

This paper discusses the design and implementation of multiple optimisation algorithms for tuning a PID controller. A metaheuristic algorithm known as particle swarm optimisation (PSO) is used together with convex optimisation techniques, and the validity of the proposed algorithm is examined by comparing its performance with the performance of the classic PSO-PID and the well-known Ziegler–Nichols PID (ZN-PID). To obtain useful comparisons, a non-linear system, the air levitation system (AL), was utilised, controlled using a PID with three tuning strategies: the modified particle swarm technique (M-PSO-PID), the classic particle swarm (C-PSO-PID), and the Ziegler-Nichols method. The performance of the controllers was monitored through a cost function, the Integral Absolute Error (IAE). A disturbance was also imposed on the AL system to test the performance and the robustness of the algorithm under different conditions. The proposed algorithm is designed to determine and set the parameters for the M-PSO-PID controller utilising stabilising regions Kp, Ki, and Kd which form a convex problem. The PSO technique is then used to search for the optimal values inside the convex area. The simulation results for the control system show that the M-PSO-PID offered good closed-loop performance with advantages over other algorithms with regard to the system settling time and rise time. The results thus indicate the supremacy of the proposed algorithm.