Adaptive Hybrid Metaheuristic Optimization for FOPID-Based Controller-Energy CODESIGN in Nonlinear Underactuated Systems

Nonlinear underactuated systems pose significant challenges for controller design owing to their instability, nonlinear dynamics, strong state coupling, and sensitivity to parameter variations. This paper presents an adaptive hybrid optimization framework for controller-energy CODESIGN in nonlinear systems. The term ‘adaptive’ refers to the runtime adjustment of the search strategy rather than to an adaptive control law: the FOPID controller structure and parameters are fixed during closed-loop operation. The framework consists of two phases: a calibration stage focused on control accuracy and a CODESIGN stage that incorporates an energy-related term through a weighted objective. A conditional intensification mechanism supports the search under stagnation or near the end of the optimization process. Unlike simple sequential hybridization, the proposed Hybrid AHA-SMA (HAHASMA) activates the Slime Mould Algorithm (SMA) only when additional local refinement is required, whereas the Artificial Hummingbird Algorithm (AHA) remains the main global search engine. The method is evaluated on a cart-inverted pendulum system with a fractional-order proportional-integral-derivative (FOPID) controller under external disturbance and stochastic input noise, using a fixed function-evaluation budget and a same-start initialization protocol. Across 20 benchmark functions, HAHASMA obtained the best Friedman average rank (1.35). The calibration stage yielded the lowest run-to-run variability (std: 0.240 versus 30.574–164.522). In the CODESIGN stage, increasing $\beta $ from 0.00 to 0.60 reduced the composite objective from about 1.00 to 0.710 while the normalized control-quality constraint remained within the prescribed tolerance.