## 1 Introduction  
Vibration control is one of the most important subjects in aeronautics, astronautics, robotics, precision manufacturing, etc. Flexible structures are increasingly applied in aeronautics and astronautics, such as the solar panels of satellites, and their light weight and flexible properties have caused numerous vibration problems because of their low natural frequency. Many accuracy problems have been induced by vibration in the robotics field. For example, if the vibration cannot be well controlled in the process of manufacturing workpieces with a robotic arm, the workpieces and processing tools would be considerably damaged, and product quality would be greatly degraded. In precision manufacturing, vibration will also introduce great machining errors and affect machining accuracy. Among various vibration suppression methods, friction damper is a commonly used tool [1]. The widely used control methods with friction dampers are mainly divided into three types: passive dampers, semi-active dampers, and active dampers [2; 3; 4]. Passive dampers benefit from the fact that the energy dissipated by this type of damper does not change with the size and frequency of the load [5]. This type of damper has been widely used in civil engineering [6; 7; 8], aircraft [9], aeroengine [10; 11; 12], automobile [13; 14], etc. However, the passive friction dampers can only be optimized for a unique given excitation. In order to ameliorate the effect of friction damper under different excitation, semi-active dampers were proposed. With the development of smart materials, some semi-active dampers then emerged. Semi-active damping control can reserve the reliability of passive dampers while presenting a better control effect [15]. Synchronous switch damping based on piezoelectric materials is also a commonly used semi-active damping control method [16]. Magnetorheological (MR) dampers, as a type of semi-active damper, are employed in civil engineering to reduce the absorb seismic energy [7]. However, with the increasing requirements for structural vibration control in different fields such as precision machining, passive and semi-active control methods can no longer fully meet the needs. Active friction control can adjust the friction force according to the functional conditions of the structure, and hence, it can provide a better vibration suppression effect. [PERSON] et al. [17; 18; 19; 20; 21; 22; 23] developed a type of morphing surface whose friction coefficient changes with its wrinkle amplitude. The wrinkle amplitude is controlled by the internal stress mismatch of layered material, and the friction can then be controlled in an active way. [PERSON] [24] developed a metal morphing surface with a varying friction coefficient that depends on the lubrication conditions, pressure, and speed. These studies on realizing the change of friction make friction control possible. Therefore, active friction control is a valuable subject. Many researchers have also taken friction into consideration when designing active control laws. [PERSON] and [PERSON] [25] proposed an active feedback control of a friction-driven oscillator. The feedback control law is specifically dedicated to unstable friction-induced vibrations. [PERSON] and [PERSON] [26] proposed a finite-time robust adaptive sliding mode control of a system with friction compensation, with the stability of the proposed control law proved by the Lyapunov criterion, and the efficiency proved by experience. [PERSON] et al. [27] proposed an adaptive sliding mode-based position tracking control for systems with nonlinear friction compensation. This control law is proved experimentally to have a better performance than the commonly used PI feedback control. In most of these studies, friction is considered to be an item that affects the system performance, and active control force compensation is required. In some other research, semi-active friction is considered with some active control law. [PERSON] et al.