Optimal design and experimental validation of a two-loop fractional-order controller for a linearized 2-DOF helicopter system

Abstract

This paper addresses the problem of reference tracking in nonlinear multivariable systems by proposing an optimal two-loop controller design. The methodology begins with input-output linearization and decoupling via state feedback, transforming the original two-input two-output (TITO) nonlinear system into decoupled second-order single-input single-output (SISO) subsystems, which correspond to standard double-integrator systems. For each subsystem, a dual-loop control structure is implemented: the inner loop employs a classical proportional–derivative (PD) controller to stabilize the fast dynamics, while the outer loop uses a fractional-order integral (FI) compensator to enhance steady-state performance and robustness. Controller parameters are optimally tuned using a novel analytical approach based on Bode’s ideal transfer function as an open-loop reference model, in combination with the generic particle swarm optimization (GPSO) algorithm. The proposed method minimizes the mean square tracking error while satisfying input constraints. Comparative evaluations with other optimally tuned fractional-order controllers, including fractional-order PI (FPI), fractional-order PID (FOPID), and fractional-order Tilt–Integral–Derivative (FTID), indicate that the proposed optimal PD + $I^{\alpha }$ controller achieves substantially lower overshoot and tracking error, while offering markedly higher phase and gain margins. Although the rise and settling times are longer, this trade-off prioritizes robustness and smooth transient response over aggressive dynamics, which is particularly desirable for safety-critical systems. Simulation and experimental validations conducted on a 2-degree-of-freedom (2-DOF) helicopter system confirm precise trajectory tracking, effective disturbance rejection, and enhanced robustness against model uncertainties. These results demonstrate that the proposed approach provides an improved compromise between transient performance and robustness, making it particularly suitable for safety-critical multivariable nonlinear systems.