## 1 Introduction  
The piezoelectric-driven fast tool servo (FTS) system [1; 2; 3] is a set of micro-feed servo systems that are commonly used to process and manufacture nanoscale free-form surfaces. It utilizes the inverse piezoelectric effect of piezoelectric ceramic materials [4] to generate a drive force that moves the tool. This system has the advantages of a high-frequency response, high stiffness, and high positioning accuracy [5; 6]. During the FTS machining process, due to the periodicity of the spindle rotation, the FTS system needs to accurately track specific repetitive signals. Disturbance conditions such as changes in the processing load, changes in the external environment, and time delays in data processing pose severe challenges to the motion accuracy of the FTS system, and the motion accuracy determines the processing quality of the FTS system [7; 8]. Therefore, in order to suppress various interferences that affect the tool movement, the fast tool servo system needs to be precisely controlled so that the tool path meets the processing requirements of the workpiece surface accuracy and surface roughness [9]. The motion accuracy of the FTS system is mainly determined by the control algorithm. The commonly used control methods are repetitive control [10; 11], PID control [12], sliding mode control [13], adaptive control [14], etc. Due to its insensitivity to disturbance and good robustness, sliding mode control (SMC) [15; 16] is often applied to deal with system uncertainties and external disturbances, and it has been widely used in the controller design of the FTS system. [PERSON] et al. [17] designed a combined approximation law consisting of an exponential approximation law and a variable rate approximation law, and established a new variable structure sliding mode controller, which effectively reduced the steady-state error of the FTS system. [PERSON] et al. [18] proposed a non-singular terminal sliding mode controller (NTSMC) for the non-axisymmetric aspheric surface machining of the piezoelectric-driven FTS system. Through simulation and experimental examinations, the NTSMC capable of improving tracking error was verified. Due to possible variations in the workpiece shape and material properties during the machining process of the fast tool servo system, the adjustment of parameters for the sliding mode controller is limited. Therefore, adaptive control is needed to continuously adjust the control parameters to accommodate changes in the system. The combination of adaptive control and SMC can continuously extract information about the control during the operation of the FTS system, adjust the control parameters through adaptive algorithms, so that the system automatically works in the optimal operating state or close to the optimal operating state, and is able to compensate for non-linear hysteresis, model uncertainty, and non-periodic disturbances caused by other external disturbances [19]. [PERSON] et al. [20] developed an adaptive terminal sliding mode with online self-adjusting system gain and arrival time-related parameters in order to adapt to system nonlinearity, cutting force disturbance, and fast and accurate tracking of specified tool trajectories required for free-form diamond machining. In FTS systems, periodic disturbance is often caused by cutting force changes, workpiece structural resonance, and other factors, which brings instability and volatility to the machining process. [PERSON] et al. [21] proposed a robust iterative learning control strategy based on adaptive sliding mode control for the purpose of torque ripple in the permanent magnet synchronous motor speed control system and the periodic disturbance of the system. Iterative learning control (ILC) [22; 23] is one of the important methods in the field of learning control, which can gradually suppress these periodic disturbances and make the processing process more stable and controllable. [PERSON] et al.