Robust Local Stabilization of the Moore–Spiegel Chaotic Oscillator via Grey Wolf Optimizer-Based Pole Placement

In this paper, the chaotic nature of the Moore–Spiegel system is first confirmed at T = 36 and R = 100 through a detailed dynamical analysis, and then an optimization-based state-feedback controller is developed for its local stabilization. To stabilize the system, a single-input state-feedback controller is designed, and the closed-loop pole locations are optimized by the Grey Wolf Optimizer (GWO). The optimization objective is defined as a composite performance index that combines the integral of time-weighted absolute error (ITAE), the integral of squared error (ISE), and additional terms penalizing control energy, overshoot, and settling time. To improve robustness, the optimization is carried out using a minimax formulation over three different initial conditions, while four control-theoretic inequality constraints are incorporated through a penalty-based approach. The results of 30 independent runs show that GWO, PSO, and GA all converge to the same well-damped pole configuration (ζ=0.707), yielding the optimal feedback gain (K_opt=[18.9901, 100.3393, 8.0550]). Compared with a conventional pole-placement design, the proposed approach reduces ISE by 83.9%, IAE by 16.5%, overshoot by 79.6%, and control energy by 94.8%. These improvements are obtained at the cost of a longer settling time and a higher ITAE than the baseline design, a deliberate trade-off that follows from the lower-gain, well-damped pole configuration. Moreover, the closed-loop system remains stable under (±10%) parametric uncertainty in (T) and (R), with a maximum ITAE deviation below 35.1%, and also preserves stability in the presence of measurement noise up to 10%. These findings show that GWO algorithm optimization provides an effective, robust framework for the local stabilization of chaotic systems.