What question did this study set out to answer?

This research aims to develop a nonlinear controller framework for the stabilization and tracking of rotary inverted pendulum systems, overcoming challenges of underactuation and instability.

June 20, 2026Open Access

Nonlinear control design for stabilization and tracking of a rotary inverted pendulum

ASArefe ShalbafianHamedan University of Technology FAFarhad AmiriHamedan University of Technology

Key Points

This research aims to develop a nonlinear controller framework for the stabilization and tracking of rotary inverted pendulum systems, overcoming challenges of underactuation and instability.
Developed a second-order sliding mode controller based on a PID sliding surface for rotary inverted pendulum systems.
Applied nonlinear dynamic modeling to improve stability in a wider operational range.
Compared the proposed controller's performance against standard sliding mode control, reinforced PID-SMC, and linear quadratic regulator through simulations.
Mean Square Error (MSE) for arm angle reduced to 0.2229, compared to 0.22346 for reinforced PID-SMC and 0.2250 for SMC.
MSE for pendulum angle decreased by 69.36% compared to reinforced PID-SMC and by substantial margins against other controllers.
Achieved a 41.99% reduction in RMS value of control input compared to reinforced PID-SMC.

Abstract

This paper proposes a nonlinear controller framework for the tracking and stabilization control of rotary inverted pendulum (RIP) systems. These systems present significant control challenges due to their underactuation, high sensitivity, and inherent instability. Most existing studies rely on linearized models of RIP systems. However, these approaches limit the controller’s performance to a small area around the equilibrium point. On the other hand, applying advanced nonlinear control techniques to the RIP underactuated system remains challenging. To overcome these limitations, the second-order sliding mode controller (SOSMC) based on a PID sliding surface is developed and applied to the RIP system. The designed procedure incorporates the nonlinear dynamic model to stabilize the pendulum in a greater region and address the challenges of underactuation. The designed control scheme enhances the system performance in stabilizing the pendulum in the vertical upright position and maintaining its balance when the horizontal arm tracks a desired trajectory. The efficacy of the designed controller is compared with three other control approaches: a standard sliding mode control (SMC), a reinforced PID-SMC, and a linear quadratic regulator (LQR). The simulation results demonstrate that the proposed controller scheme exhibits high-speed tracking performance with reduced overshoot. The results show that the Mean Square Error (MSE) for the arm angle is decreased to 0.2229, compared to 0.22346 for reinforced PID-SMC, 0.2250 for SMC, and 0.2299 for LQR. Moreover, the MSE for the pendulum angle is reduced by approximately 69.36%, 28.6%, and 25.68% compared to the reinforced PID-SMC, SMC, and LQR algorithms, respectively. Furthermore, the designed controller achieves a 41.99%, 10.06%, and 18.33% reduction in the RMS value of the control input compared to the reinforced PID-SMC, SMC, and LQR controllers, respectively. The simulation results reveal that the designed controller achieves superior tracking accuracy, faster response, and reduced overshoot, while maintaining efficient energy consumption.

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