A novel fractional-order model and controller for vibration suppression in flexible smart beam

Vibration suppression represents an important research topic due to the occurrence of this phenomenon in multiple domains of life. In airplane wings, vibration can cause discomfort and can even lead to system failure. One of the most frequently used means of studying vibrations in airplane wings is through the use of dedicated flexible beams, equipped with sensing and actuating mechanisms powered by suitable control algorithms. In order to optimally reject these vibrations by means of closed-loop control strategies, the availability of a model is required. So far, the modeling of these smart flexible beams has been limited to deliver integer order transfer functions models. This paper, however, describes the mathematical framework used to derive a fractional-order impedance lumped model for capturing frequency response of a flexible beam system exposed to a multisine excitation. The theoretical foundation stems from fractional calculus applied in combination with transmission line theory and wave equations. The simplified model reduces to a minimal number of parameters when converging to a limit value. It is shown that the fractional-order model outperforms an integer order model of the smart beam. Based on this novel fractional-order model, a fractional-order PDμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {PD}^\mu $$\end{document} controller is then tuned. The controller design is based on shaping the frequency response of the closed-loop system such that the resonant peak is reduced in comparison to the uncompensated smart beam system and disturbances are rejected. Experimental results, considering a custom-built smart beam system, are provided, considering both passive and active control situations, showing that a significant improvement in the closed-loop behavior is obtained using the proposed controller. Comparisons with a fractional-order PDμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {PD}^\mu $$\end{document} controller, tuned according to classical open-loop frequency domain design specifications, are provided. The experimental results show that the proposed tuning technique leads to similar results as the classical approach. Thus, the proposed method is a viable alternative, being based on closed-loop specifications, which is more intuitive for practitioners.