Scenario-Based Robust Tuning and Generalization Analysis of PID and Fractional-Order PID Controllers for a Nonlinear Cart-Inverted Pendulum Under Parametric and Disturbance Uncertainty

Abstract

Robust controller tuning is essential for nonlinear systems operating under plant uncertainty and external disturbances. This study proposes a scenario-based framework for tuning PID and FOPID controllers for a nonlinear cart-inverted-pendulum system, using separate training and test scenarios to assess generalization. Five training scenarios incorporating parametric variations, disturbances, and noise are used during optimization, while four unseen scenarios are reserved for evaluation. The Slime Mold Algorithm (SMA), Artificial Hummingbird Algorithm (AHA), and Grey Wolf Optimizer (GWO) are compared under identical computational budgets and initial populations. The objective combines weighted integral of time-weighted absolute error (ITAE) measures with a standard-deviation penalty to promote consistent performance across scenarios. Robust tuning reduces the mean unseen test cost of PID controllers by approximately 20% compared with nominal tuning. Its effect is more pronounced for FOPID controllers: nominal tuning causes instability in several unseen cases, whereas robust tuning eliminates failures in the test set. Increasing the robustness penalty reduces the generalization gap by about 17% for PID and 46% for FOPID. Across fifteen unseen plant configurations, the robust-tuned FOPID-AHA controller remains stable, while the nominal counterpart fails under the two most severe combined-stress conditions. These results show that scenario-based tuning improves controller reliability beyond a single nominal operating point.