Design of novel exponential PDN controller via quadratic interpolation optimiser for nonlinear and unstable ball and beam system

Abstract

This study presents a novel exponential proportional-derivative controller with filter (exp-PDN) for stabilising the nonlinear and underactuated ball and beam system. Unlike conventional PID-based approaches, the proposed controller removes the integral term, resulting in faster transient responses and improved robustness. It incorporates nonlinear exponential shaping of both the error and its derivative, along with a filtered derivative path for enhanced noise handling. A custom multi-objective cost function, comprising the squared error, settling time, and percent overshoot, is proposed to evaluate control performance. The quadratic interpolation optimiser (QIO), a recently developed metaheuristic based on analytical interpolation, is employed to optimise the controller parameters. To validate its effectiveness, the exp-PDN controller is compared against five state-of-the-art metaheuristic algorithms: QIO, spider wasp optimiser, komodo mlipir algorithm, golden eagle optimiser, and slime mould algorithm. The QIO-optimised exp-PDN achieves the best performance, with the lowest cost value (0.3211), minimal overshoot (5.52%), fast rise time (0.97 s), and smallest steady-state error (4.1643 x 10- 4). Further comparisons with QIO-optimised phase-lead and PID-with-filter controllers demonstrate the superiority of the proposed method in both transient and steady-state behaviour. In summary, this work advances the control of nonlinear unstable systems by delivering a structurally simple yet highly responsive control architecture. The combination of dual-channel exponential shaping and efficient metaheuristic optimisation results in state-of-the-art closed-loop performance, highlighting the practical value of the proposed exp-PDN framework for real-world control applications.