Fractional-Order Dynamics and Control of Rigid–Flexible Coupling Space Structures

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R IGID–FLEXIBLE coupling structures have been widely used in spacecraft with flexible attachments, robotic manipulators, and other flexible mechanical/structural systems [1–3]. Generally, the damping effect generated by the flexible components of these structures is used to improve the overall performance of the systems under dynamic loads by suppressing residual vibration [4]. However, a challenge arises when the precision positioning and motion control of the rigid–flexible coupling structures are attempted, where the residual vibration may occur due to the structural inertia and time delay resulting from the flexible components of structures [5,6]. The problem is further complicated by the existence of viscoelasticity in the flexible components of structures. It is well known that viscoelasticity is characterized by the combination of elastic springs and viscous fluids, which can be representedbyHooke’s elasticityandNewtonianviscosity, respectively. Currently, most of the existing works describe viscoelasticity by the Kelvin–Voigt damping model, which compromises between complicacy and accuracy [7,8]. However, given the rapid development of advanced space materials, the Kelvin–Voigt model was found inadequate in describing composite materials, especially for space composite materials [9]. The need for a more refinedmodel has pushed the development of viscoelasticity. In fact, as early as the 1920s,Nutting [10] found that the viscous deformation of solids, which is always interpreted as the stresshistory under strain, could be best described bya fractional-order power law decay. Nutting’s work attracted little attention due to the mathematical complexity of fractional-order calculus and the associated extensive computational effort, until the pioneeringwork on the fractional-orderMaxwell fluid byCaputo in the 1960s [11], who made the fractional-order calculus applicable for real engineering problems. Since then, it has been proved that fractionalorder calculus can describe the dynamic response of viscoelastic flexible systems more accurately than its classical integer-order counterpart due to its unique time memory capacity [7]. Nowadays, fractional-order calculus has been proved a powerful method in describing the dynamics of various materials, especially viscoelastic materials for vibration control [12]. More details of the related works can be found in [7,12,13]. However, there are no clear criteria for the selection of viscousmodels.The actualmodel used is usuallyoptimized by trial and error for different materials [14]. The control of the rigid–flexible coupling structures is addressed extensively by many control strategies and modeling schemes in literature [15–17].However, the dynamicmodels of theseworkswere generally derived by the Newtonian or Lagrangian mechanics with the Kelvin–Voigt or Rayleigh damping models. Even though the fractional-order viscous model provides an excellent alternative in describing the high stiffness and lightweight compositematerials, the fractional-order control strategy is seldom used in the control of rigid–flexible coupling structures [18–20]. In thisNote,we propose a two-parameter fractional-ordermodel of viscoelastic material. The model is then used to derive the fractionalorder dynamics of rigid–flexible coupling structures to simplify the modeling and controller design for residual vibration suppression. Based on the fractional-order dynamics, a new fractional-order lead– lag compensator design method is developed. Simulation results demonstrate the effectiveness and robustness of the fractional-order dynamics and control for the rigid–flexible coupling structures.