Original Article

Design of cascaded FOPI-FOPD-BLQG controller tuned with hybrid grasshopper and firefly algorithms for stabilizing cart-inverted pendulum system

Control of Dynamical Systems and Computation Laboratory, Department of Electrical Engineering, Delhi Technological University, Delhi, India

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Received: 29 August 2025
Accepted: 4 February 2026

Abstract

The Cart Inverted Pendulum (CIP) system is a widely studied problem in control theory due to its inherent instability and nonlinearity dynamics, which closely mimic various real-time applications, including segways, robotics, rockets, and missile guiding systems. This requires a controller to maintain the pendulum upright from its inherently stable hanging state and mitigate the oscillations. This paper presents a concept for a hybrid controller that utilizes a cascaded FOPI-FOPD controller and a backstepping linear quadratic Gaussian controller (BLQGC). The proposed controller incorporates both the cart’s position and the pendulum’s angle in its design. However, the existing literature primarily concentrates on the analysis of the pendulum angle alone. Furthermore, the optimal controller gains are tuned with hybrid optimization using the grasshopper and firefly algorithms (GOA-FA). To validate the improved performance of the suggested controller, a comparative transient and frequency domain analysis employing fractional-order PI (FOPI), fractional-order PD (FOPD), and BLQGC approaches is conducted. The results obtained from the proposed system demonstrate a decrease in rising time (t_r) and settling time (t_s), a reduction in steady-state error (e_ss), an enhancement in gain margin and phase margin, and an increase in bandwidth. The system’s stability is confirmed by subjecting it to a 1N impulse disturbance on a stabilized pendulum. Additionally, enhanced resilience is guaranteed in the presence of parametric perturbations affecting the mass of the cart (M), the mass of the pendulum (m), and the length of the pendulum (l), across three distinct scenarios.