Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties

This paper employs the Direct Synthesis approach to present an analytical methodology for designing fractional-order controllers, aiming to balance simplicity and robustness for practical industrial implementation. Although significant progress has been made in developing fractional-order PID controllers, the advancement of Direct Synthesis controllers has been comparatively slower. This study underscores the importance of further research on these controllers and the need for innovative approaches to enhance parameter adjustment. The proposed methodology is based on the fractional “second-order” transfer function and the solution of two equations derived from four key specifications: overshoot, settling time, and the frequency and magnitude of disturbance rejection. Additionally, the fractional order should be chosen as close as possible to 1, ensuring practical implementation and minimizing the system’s sensitivity to parameter variations. The resulting controller demonstrates strong robustness against plant parameter variations, input noise, and disturbances while achieving shorter settling times and lower overshoot. It outperforms fractional-order PID and ID controllers optimized numerically and surpasses integer-order phase lead-lag compensators designed analytically. The validation process involved Monte Carlo simulations and Kruskal–Wallis statistical analysis on a complex system characterized by closely spaced poles and significant parametric variations. Furthermore, the proposed controller effectively reduces the integral of the control signal (control effort), enhancing energy efficiency.