Performance-Oriented Metaheuristic PID Tuning for Parametrically Excited Mathieu Systems

This paper investigates metaheuristic-based tuning of classical PID controllers for the performance-oriented regulation of parametrically excited dynamical systems, with a particular focus on the Mathieu equation as a canonical model of time-periodic instability. Unlike many existing studies that emphasize nonlinear or autonomous dynamics, the work explicitly addresses PID tuning under time-periodic parametric excitation with an emphasis on closed-loop performance and bounded response behavior. Three population-based optimization algorithms–Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), and a hybrid PSO–GWO scheme–are employed to determine PID gain parameters under identical computational budgets. The tuning objective combines the Integral of Time-weighted Absolute Error (ITAE) with the Integral of Squared Overshoot (ISO), enabling a unified assessment of transient performance and overshoot suppression. The controlled Mathieu system is examined across multiple excitation regimes, ranging from stable periodic operation to near-resonant and strongly excited conditions. In addition to the metaheuristic approaches, a deterministic baseline PID controller derived from a nominal LTI approximation is included as a practical reference. All controllers are evaluated on the full time-varying Mathieu dynamics using the same performance index, and control-input saturation tests are conducted to assess behavior under practical actuator constraints. Numerical results demonstrate that all considered metaheuristic methods achieve reliable closed-loop stabilization and maintain bounded oscillatory responses despite strong parametric excitation. In contrast, the deterministic baseline exhibits a pronounced initial overshoot and higher steady-state oscillation amplitudes, resulting in significantly higher performance-index values. While the hybrid PSO–GWO approach occasionally yields smoother transients, it does not provide a consistent performance advantage over standalone PSO or GWO when normalized by computational effort. Overall, the findings indicate that, for low-dimensional PID tuning problems in parametrically excited systems, solution quality is governed primarily by adequate search-space coverage and population diversity rather than algorithmic hybridization. Well-configured single-population metaheuristics therefore offer an effective and practically relevant framework for stability-oriented PID control of time-varying systems subject to parametric excitation.