**3. Control schemes of piezoelectric stick-slip actuators**

The piezoelectric stick-slip actuator introduces mechanical structures, such as friction rods and linear cross roller guides on the basis of piezoelectric stacks. A sawtooth wave signal is applied to the piezoelectric stack to achieve a stepping large stroke and high precision motion. In the actual application process, the traditional mechanical structure can no longer meet the demand of real positioning accuracy due to the complex nonlinear effects in the system and the influence of external disturbances. Therefore, intelligent control algorithms combined with computer hardware devices are usually introduced to eliminate or reduce the impact of the above problems on motion accuracy. This section mainly discusses the existing controller design methods from two parts—open-loop control and closed-loop control.

In this paper, control schemes and hardware facilities for piezoelectric stick-slip actuators are briefly described in most of the literature. Depending on whether a closed-loop is formed, the main categories are feedforward control, feedback control, *A Review of Modeling and Control of Piezoelectric Stick-Slip Actuators DOI: http://dx.doi.org/10.5772/intechopen.103838*

and composite control with a combination of feedforward and feedback. The inverse model-based control is mostly feedforward control, which is usually used to compensate for the hysteresis characteristics of piezoelectric stick-slip actuators. The control system is shown in **Figure 5**, *yd* is the expected input, *vinv* is the theoretical input under the expected output value, *y* is the actual output value.

Feedback control is the real-time feedback of the actual measured displacement through data measurement equipment, such as sensors, and the feedback value is one of the inputs of the controller. Feedback control can effectively improve the robustness of the control system, and its control mode is shown in **Figure 6**. Where, *v* is the output of the feedback controller.

In actual control, simple feedforward control and feedback control cannot meet the control demand. Therefore, in most cases, researchers combined the advantages of feedforward control and feedback control, usually the feedforward control and feedback control of the compound control scheme are applied to the piezoelectric stick-slip actuator position tracking control. The control system is shown in **Figure 7**.

#### **3.1 Conventional open-loop control of piezoelectric stick-slip actuators**

Feedforward control is essentially an open-loop control. However, in the overall control of piezoelectric stick-slip actuators, the advantages of this control method are very limited. This has been found by many scholars—since the voltage changes for each step displacement are very fast, there is almost no hysteresis, creep effect, and the use of feed-forward control in this motion mode is not necessary.

**Figure 5.**

*Feedforward control system principle based on inverse model.*

**Figure 6.** *Feedback control system principle.*

**Figure 7.** *Feedforward and feedback compound control system.*

#### **Figure 8.** *(a) A simplified circuit diagram. (b) Experiment effect.*

Feedforward control is often used to improve the quality of the end motion output of piezoelectric stick-slip actuators. Holub et al. compensated for the hysteresis of piezoelectricity by varying the amplitude of the input voltage for position errors and hysteresis modeling [36]. Chang et al. compensate for hysteresis by changing the phase lag of the actuator [37].

Feedforward control of end-effectors is challenging as the control accuracy of feedforward control is heavily dependent on the accuracy of the above model. The main source of difficulty comes from the many problems in the real system, including the hysteresis of the piezoelectric, the creeping nature, and the nonlinearity of the frictional motion, the vibration between the stick and sliding points, the wear of the material between the movers and the stator, and other uncertainties [38].

Another feed-forward control by some scholars-charge control to compensate for hysteresis in the actuator. Špiller et al. developed a hybrid charge-controlled driver that slides by generating a high-voltage asymmetric sawtooth wave and feeding it into a capacitive load to compensate for the piezoelectric hysteresis as well as to achieve fast back-off. A simplified circuit diagram of the piezoelectric driver is shown in **Figure 8a**. This control method combines a charge control scheme with a switch is an effective solution, and the proposed hybrid amplifier has better motion linearity as shown in **Figure 8b** [39].

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

The control of piezoelectric stick-slip actuators is usually divided into one-step control stage and sub-step control stage, also known as step mode and scan mode. Step mode refers to the piezoelectric stick-slip actuator moving forward at a fixed step size and a fixed frequency when it is far from the desired position. Until the error between the actual position and the desired position is less than the single-step displacement of the piezoelectric stick-slip actuator, the precise positioning is realized by controlling the elongation of the piezoelectric stack, which is called the scanning control stage. In the stepping mode, the voltage changes rapidly and there is almost no creep. At the same time, in the stepping mode, the control accuracy is not necessary. Therefore, in the control process of stick-slip motion, there are few overall feedforward control cases. And feedforward control is usually implemented in the scanning control stage. The core component of the piezoelectric stick-slip actuator is the piezoelectric stack, which moves through the inverse piezoelectric effect of the piezoelectric stack. Compared with the motion control of the piezoelectric stick-slip actuator, feedforward motion control of the piezoelectric stack actuator is more mature. Chen of Harbin Institute of Technology defined a new function named mirror function, which connected the dynamic hysteresis model with the classical Preisach model and established a new dynamic hysteresis model to describe the input-output relationship of the piezoelectric actuator under different conditions. On this basis, a feedforward control scheme based on the dynamic hysteresis inverse model is designed [40].

In addition, Ha et al. experimentally identified the hysteretic parameters of the Bouc-Wen model, and on this basis designed a feedforward compensator to compensate for the influence caused by the nonlinearity of the hysteretic effect. Finally, the simulation results of the compensator and the designed voltage waveform are given to realize the feedforward control of the piezoelectric stack [41]. Wei et al. proposed a feedforward controller based on an improved rate-dependent PI hysteresis inverse model, which achieved the expected effect [42]. In recent years, Zhang et al. also proposed a third-order rate-dependent Rayleigh model to describe the hysteresis nonlinearity of piezoelectric stacks. And proposed a feedforward control scheme based on the inverse third-order rate-dependent Rayleigh model, which also verified the effectiveness of the method through experiments [43]. Feedforward control often plays an obvious role in hysteresis compensation. In the future piezoelectric stick-slip drive controller design, the existing piezoelectric stack feedforward control methods can be used for reference to realize the feedforward control in the scanning stage. Simple feedforward control has poor robustness in the application, so most researchers use compound control to improve the control accuracy.

## **3.2 Conventional closed-loop control of piezoelectric stick-slip actuators**

Piezoelectric stick-slip actuators are affected in practice by factors such as environmental vibration and their own nonlinear characteristics, and their controllability becomes poor. Therefore, appropriate control methods are needed for closed-loop control to meet the actual working requirements. Zhong B et al. found that differences in object surface roughness and wear can cause inconsistent velocities during the movement of a piezoelectric stick-slip actuator. Therefore, a dual closed-loop controller for velocity and position was designed to achieve high accuracy positioning. Its principle of double closed-loop control is shown in **Figure 9**. The experimental results show that the standard deviation of the speed of the controller is less than 0.1 mm/s, and the repeated positioning accuracy reaches 80 nm, both of which achieve a good control effect [44]. The design of the controller considers a more accurate speed

**Figure 9.** *Double closed-loop control principle.*

control scheme, which has a high reference value for realizing the fast positioning of piezoelectric stick-slip actuators. Rong et al. introduced strain gauge as positioning sensor of the precision manipulator with piezoelectric stick-slip actuators and developed a displacement prediction method based on this. The feedforward PID control method is used throughout the system to improve the dynamic performance of the system. As shown in **Figure 10a**, it is the simulation of displacement-prediction control positioning performance (target displacement of 5.5 μm). **Figure 10b** shows the comparison of displacement response under the open-loop control and displacement response under the displacement-prediction control (target displacement of 20 μm). It can be seen from the experimental results that the 200 nm steady-state error of the proposed control method is much lower than that of the open-loop control [45]. The commonly used classical control methods are difficult to achieve high control accuracy in practical applications due to the limitations of parameters, weak automatic regulation and poor robustness.

Rakotondrabe et al. designed a micro-positioning device based on a stick-slip actuator. The control process is divided into a step mode and a scan mode, where the scan mode is precisely controlled by a PI controller [46]. Theik et al. used an inertial piezoelectric actuator to suppress the vibration of the hanging handle and designed three controllers—PID manual setting, PID self-setting, and PID-AFC. The best damping effect was achieved by experimentally comparing the PID-AFC controller. When the mass of the inertia block is larger, the vibration damping effect is more obvious [47]. These control methods are developed by the classical control theory in the actual control process of piezoelectric stick-slip actuators.

#### **Figure 10.**

*(a) Simulation of displacement-prediction control positioning performance (target displacement of 20 μm). (b) Comparison between displacement response under the open-loop control and displacement response under the displacement-prediction control (target displacement of 20 μm).*
