## 1 Introduction  
Flexible structure based positioning is generally achieved using motors to expand the range of strokes and applications. Examples include the grabbing motion of flexible robot manipulators and the lifting operations of cranes. Due to their poor damping, the coupling of the entire structures, and various nonlinear effects, the rotary motions of flexible structures produce complex vibrations that can severely affect the precision of positioning. A popular and effective research approach to address this issue in recent years has been the use of piezoelectric materials as actuators and sensors in conjunction with active suppression control algorithms to create smart active suppression and control designs for structures [1; 2; 3]. Motor-driven flexible structures generally have two control objectives: (1) motor tracking control; and (2) active suppression control of vibrations in flexible structures. Most of these control systems are multi-input-multi-output (MIMO) systems in which achieving the high-performance motion control of these two objectives makes for a certain level of difficulty in controller design. However, discussion regarding these systems usually focuses on vibration-suppression control for the flexible structures. Thus, the majority of existing literature examines the effectiveness of various control methods in active vibration suppression [4; 5; 6; 7; 8]. [PERSON] et al. [4] employed proportional-derivative (PD) control to actively suppress the vibrations in a single-axis flexible robot manipulator. [PERSON] et al. [5] used positive position feedback (PPF) control for the active suppression of vibrations in a flexible robot manipulator with one degree of freedom. [PERSON] et al. [6] combined fuzzy control with PD control and input shaping control for the tracking and control of deflection in flexible robot manipulators. [PERSON] et al. [7] combined fuzzy controller design with H-infinity control to enhance system robustness in reducing vibrations in a single-axis flexible robot manipulator during positioning. [PERSON] and [PERSON] [8] combined a flexible beam with a beam-cart system to investigate vibration responses during cart motion. They also employed piezoelectric transducers and adaptive neuro-fuzzy control for the active suppression of vibrations in the flexible beam. Aside from the common single-axis flexible robot manipulators, flexible structures with two or more degrees of freedom enable greater work space and more diverse actions. However, coupling effects and nonlinearities greatly complicate vibration behavior [9; 10; 11; 12; 13]. [PERSON] et al. [10] took into account the coupling effects between the axes in the design of a fuzzy controller aimed at reducing vibrations in a dual-link flexible robot manipulator. [PERSON] et al. [11] designed a piezoelectric vibration absorber for a beam-cart-seesaw system and suppressed vibration responses in the system by pairing passive mechanical components with active piezoelectric proportional-integral-derivative (PID) feedback control. [PERSON] and [PERSON] [12] designed a piezoelectric truss structure driven by two motors to enable rotation in two directions. They combined neural-fuzzy control with a genetic algorithm to achieve active vibration suppression control; however, the system was limited to static positioning and did not take into account the vibration responses created by dynamic tracking [13], [PERSON] et al. established a mathematical model to facilitate modal analysis of the large flexible structure in [12]. They also applied a hybrid PD/repetitive control framework [14] for the active suppression of vibrations produced in a thin plate undergoing periodic movement. Their simulation results demonstrated the effectiveness of a hybrid repetitive control (RC) algorithm in suppressing vibration responses in a motor-driven structure.