**3.3 Intelligent control of piezoelectric stick-slip actuators**

By introducing intelligent control algorithms, such as sliding mode control algorithm and neural network algorithm, the self-adjusting control of piezoelectric stickslip actuator is realized. Closed-loop control with feedback is a common control mode of piezoelectric stick-slip actuators, which can effectively compensate for the effects of hysteresis nonlinearity, complex friction relations and external interference on the positioning accuracy, and improve the robustness of the controller. The closed-loop control is mainly divided into two kinds of closed-loop control. The first closed-loop control is the voltage amplitude of the driving signal, which adjusts the single-step size of the piezoelectric stick-slip actuator by controlling the voltage amplitude. The other is the control of the driving signal frequency. By adjusting the frequency of the piezoelectric stick-slip actuator, the speed of the piezoelectric stick-slip actuator can be controlled. Cao et al. proposed a sliding mode control method based on linear autoregressive proportional integral-differential. It can solve the problem that the hysteretic characteristics of piezoelectric stacks in piezoelectric stick-slip actuators and the nonlinear friction relationship between end-effector and workbench affect the control effect. Firstly, an ARX model of the system is designed, and its state space description is obtained. Then, the sliding mode control is introduced, and PID control is introduced as the frequency switching controller in the sliding mode control, so that the error tends to zero, to achieve better speed control [48].

In addition to introducing the inherent mathematical model into the controller, the controller can also be designed by introducing the neural network algorithm to online model identification. Cheng et al. proposed a neural network-based controller to reduce the effect of complex nonlinearities between the end-effector and the driving object. The structure block diagram of the overall controller is shown in **Figure 11**. The control paradigm of piezoelectric stick-slip actuators is usually divided into two phases—the one-step control phase and the sub-step control phase. In the one-step control phase, when the error between the desired position and the actual position is less than the maximum single-step displacement length by continuous sawtooth wave

#### **Figure 11.**

*The schematic of the overall controller in the sub-step control phase—the desired reference ref*ð Þ *tk of the endeffector; the real displacement yef*ð Þ *tk of the end-effector; the estimated displacement yef*ð Þ *tk of the end-effector; the desired displacement y*ð Þ *tk of the driving object/PEA; the input voltage v t*ð Þ*<sup>k</sup> applied to the PEA; the real displacement y t*ð Þ*<sup>k</sup> of the driving object/PEA.*

excitation, the controller switches to the sub-step control phase. In the experiment, the steady-state tracking error is kept within 50 nm, realizing ultra-precise motion control at the nanometer level [49].

Oubellil et al. applied proportional control to the macro motion control of nanorobots based on the piezoelectric stick-slip motion principle. Under macro motion control, the amplitude and frequency of the sawtooth wave voltage signal are adjusted by proportional control. When switching to scan control mode, Hammerstein dynamic model based on the PI hysteresis model is established, and then H∞ robust control scheme based on the model is designed. The hybrid stepper/scan controller can effectively meet the stability, robustness, hysteresis, and accuracy of multi-target nanorobots [50]. Oubellil et al. also applied piezoelectric stick-slip actuators to the nanorobot system of fast scanning probe microscope. To meet the requirements of fast scanning in closed-loop bandwidth and vibration reduction, the uncertain model of the piezoelectric actuator was defined by the multi-linear approximation method. A 2- DOF H∞ control scheme is designed to provide robust performance for the positioning of the nanorobot system. The fast and accurate positioning of the piezoelectric stick-slip actuator is realized [51].

In addition to its characteristics, the model of piezoelectric stick-slip actuators can also absorb the modeling mode of the piezoelectric stack. In a sense, due to the coupling relationship of the structure, the model can be regarded as an inclusion relation. The control mode of piezoelectric stick-slip actuators can also be the control mode of the piezoelectric stack, such as model-based feedforward control, inversion of control, sliding mode control method, active disturbance rejection control and some intelligent control methods can be applied to the precise control of piezoelectric stickslip actuators. The research on piezoelectric stack also has reference significance in the precise control of piezoelectric stick-slip actuators.

Sliding mode controller often appears in the control of piezoelectric stack actuators nonlinear system [52, 53]. It is an effective and simple method to deal with the defects and uncertainties of a nonlinear system model. Sliding mode control is not dependent on an accurate mathematical model, which makes it popular in nonlinear system control of piezoelectric actuators. Li et al. proposed a sliding mode controller with disturbance estimation is designed for piezoelectric actuators. The Bouc-Wen model is chosen to describe the input and output relations of the piezoelectric actuator, and a particle swarm optimization algorithm is used for real-time identification of the model parameters. Considering the external and own uncertain disturbances, adaptive control rules are introduced to change the controller parameters. Experimental results show that the proposed controller can significantly improve the transient response speed of the system [54]. Mishra et al. designed a new continuous third-order sliding mode robust control scheme for the hinged piezoelectric actuator. To ensure the overall stability of the closed-loop system, a disturbance estimator was designed to counteract the effects of external disturbances and nonlinearities [55]. Xu Q et al. proposed an enhanced model predictive discrete sliding mode control (MPDSMC) with proportional-integral (PI) sliding mode function and a novel continuous third-order integral terminal sliding mode control (3-ITSMC) strategy [56, 57].

Because of the hysteresis nonlinearity of the piezoelectric actuator and the existence of system vibration and external disturbance, the robustness of the controller is usually required. Wei et al. proposed a variable bandwidth active disturbance rejection control method for piezoelectric actuators. The control method of the nanopositioning system is based on a cascade model of the hysteresis model and the system structure.

## *A Review of Modeling and Control of Piezoelectric Stick-Slip Actuators DOI: http://dx.doi.org/10.5772/intechopen.103838*

Information about all uncertainties and disturbances excluded items in the model is estimated by a time-varying extended state observer (TESO). Afterwards, a variable bandwidth controller based on the control error is designed. Its control system is shown in **Figure 12**. *z*1, *z*2, *z*3are the states of time-varying extended state observer, *b*0 is adjustable coefficient, and *d* is a disturbance. A series of experiments show that the proposed controller has a higher response speed and stronger anti-interference ability than the traditional active disturbance rejection controller [58].

Neural networks are widely used in the design of adaptive controllers for nonlinear systems because of their strong self-learning ability. In view of the system uncertainty and hysteresis nonlinearity of piezoelectric actuator, Li et al. proposed a neural network self-tuning control method. Two nonlinear function variables about hysteresis output are established and two neural networks are introduced to identify the two hysteresis function variables on line, respectively. Experiments verify that the neural network self-tuning controller has a good track tracking effect [59]. Napole et al. proposed a new method combining super torsion algorithm (STA) and artificial neural network (ANN) to improve the tracking accuracy of high voltage stack actuator [60]. Lin et al. proposed a dynamic Petri fuzzy cerebellar (DPFC) model joint controller for magnetic levitation system (MLS) and two-axis piezoelectric ceramic motor (LPCM) drive system, which is used to control the position of MLS metal ball and track tracking of the two-axis LPCM drive system. The experimental results also show that this method can obtain a high-precision trajectory tracking response [61].

The neural network has a strong self-learning ability and can approach complex nonlinear functions. It plays an important role in the design of the piezoelectric stick-slip actuators controller. In addition to neural network and sliding mode control, data-driven model-free adaptive control is also suitable for systems with model uncertainty. Model-free adaptive control (MFAC) as a typical data-driven control method, this method was proposed in Mr. Hou Z's doctoral thesis in 1994 [62]. In the past two decades, both the continuous development and improvement of theoretical achievements, and the successful practical application in the fields of motor control, chemical industry, machinery and so on, have made MFAC become a new control theory with a systematic and rigorous framework. As for the application of modelfree control in the piezoelectric stack, Muhammad designed a data-driven feedforward controller and feedback controller. To avoid chattering caused by noise and affect the convergence of the learning process, several rules about parameters are also proposed. The experimental results show that the controller can realize highprecision position tracking at low frequency [63].

**Figure 12.** *The variable bandwidth active disturbance rejection control.*
