Indrawanto, Indra Agung Ariwi Saputro, Vani Virdyawan, Tegoeh Tjahjowidodo

Corresponding email: indrawanto@itb.ac.id

Corresponding email: indrawanto@itb.ac.id

**Published at : ** 10 Jul 2024

**Volume :** **IJtech**
Vol 15, No 4 (2024)

**DOI :** https://doi.org/10.14716/ijtech.v15i4.6605

Indrawanto, Saputro, I.A.A., Virdyawan, V., Tjahjowidodo, T., 2024. Design, Manufacture and Control of a Multi-layer Piezoelectric Actuator.

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Indrawanto | Production Engineering Research Group, Faculty of Mechanical and Aerospace Engineering, Institut Teknologi Bandung, Jl. Ganesa 10, Bandung 40132, Indonesia |

Indra Agung Ariwi Saputro | Mechanical Engineering Study Program, Faculty of Mechanical and Aerospace Engineering, Institut Teknologi Bandung, Jl. Ganesa 10, Bandung 40132, Indonesia |

Vani Virdyawan | Production Engineering Research Group, Faculty of Mechanical and Aerospace Engineering, Institut Teknologi Bandung, Jl. Ganesa 10, Bandung 40132, Indonesia |

Tegoeh Tjahjowidodo | KU Leuven, Department of Mechanical Engineering, Jan Pieter de Nayerlaan 5, 2860 Sint-Katelijne-Waver, Belgium |

Abstract

A piezoelectric-based
micro motion actuator is typically
used in micro-scale
movement technologies, with the actuator developed to
deliver very
small movements and high resolution for motion within several micrometer
ranges. However, a significant challenge from the strong, nonlinear hysteresis
arises affects the piezoelectric materials joining input voltage to output
movement, which deteriorates the accuracy of the actuator and causes
instability in a closed-loop system. To obtain high precision, accuracy and
reduced nonlinear effects, piezoelectric actuators must be controlled with
hysteresis compensation. Therefore, this research developed a
piezoelectric-based microactuator system with a control scheme based on PID
(proportional-integral-derivative) combined with the inverse hysteresis model
implemented to compensate for the actuator's hysteresis. Furthermore, a
Modified Prandtl-Ishlinskii (MPI) model was used to capture the hysteresis phenomenon, where its
parameters were obtained through a system identification process. The inverse
model of the hysteresis was then used to
generate
feedforward signals in the control system. The results showed that the control
scheme is able to provide an accurate motion due to the
decrease in hysteresis compensation signals from 4.87 to 0.97 . The closed
loop control system consisting of the PID control and hysteresis compensation
further improved the accuracy of the piezoelectric actuator and reduced the
error down to 0.41 .

Hysteresis; Micro motion actuators; Modified Prandtl-Ishlinskii model; PID control; Piezoelectric

Introduction

A
micro motion actuator is a device capable of generating microscale movements at
an accuracy of 0.001 mm, even extending to nanometres. This technology is
widely applied in tools that require movement with exceptionally high precision
and accuracy, such as micro robots *et al.*, 2020)

Currently,
the types of actuators used to achieve precise and accurate movements
incorporate active or smart materials such as piezoelectric *et al.*, 2014)*et al.*, 2018), *et al*, 2019)*et al.*, 2011; Heywang,
Lubitz and Wersing, 2008)*et al.*, 2022).

Several mechanisms have been developed to
enable both planar and rotary motions using piezoelectric actuator. *et
al.*, (2004)

In some applications, a direct piezo
actuation is required, and a typical example is a micro-macro manipulator such
as the one designed for in-vitro intracytoplasmic sperm injection *et
al.*, 2020)

This paper focused on a detailed design of
a direct piezoelectric-based actuator system, with a feedback controller and
model-based feedforward compensation to counteract the hysteresis phenomenon.
The validation test results of the piezoelectric actuator, and the control law
were presented through numerical simulations and experimental trials. In
conclusion, it examined the main advantages of the developed actuator, with
focus on the internally equipped position sensor, which significantly enhanced
the performance.

The research was carried out in four
phases shown in Figure 1, with the first focused on determining the
specifications and design requirements of the actuator, taking into account
certain limitations to ensure manufacturability with available materials. The
actuator range of motion was set to a maximum of 50
and
an error of less than 2
.
Subsequently, the design comprising both mechanical and electrical components,
were developed based on predetermined specifications. The actuator displacement
was measured using a calibrated load cell, which evaluated the models in
piezoelectric crystals with significant impact on accuracy at the micro-scale.
The results obtained served as a basis for developing the hysteresis
compensator model.

**Figure ****1** Phases in the development of the
piezoelectric-based actuator system

The
second phase focused on the manufacturing of the mechanical and electronical
components, as well as designing the structure of the position control system,
which integrated feedback and feedforward control strategies, using the
hysteresis model examined in the first phase.

The
third phase also known as the identification phase focused on distinguishing
the dynamic parameters of the mechanical system, and hysteresis model,
including testing the electronic circuit. The results obtained were then used
to adjust and optimize the parameters of the feedback and feedforward control
gains.

The
fourth phase centered on testing the performance of the developed system, by
using a reciprocating trajectory to investigate the effectiveness of the
control system. In addition, this phase was completed by proving the
satisfactory performance of the developed piezoelectric actuator system.

Experimental Methods

Hysteresis modelling was carried out to capture the characteristic behaviors of piezoelectric systems. In addition, through mathematical modeling, hysteresis can be accurately represented, enabling the development of compensatory strategies to reduce the effect. In this context, three hysteresis models, namely the Bouc-Wen, Prandtl-Ishlinskii (P-I) and Modified Prandtl-Ishlinskii(MPI), was discussed in the following sub-sections.

*2.1. Bouc-Wen Model*

The Bouc-Wen model was widely used to characterize hysteresis in piezoelectric systems. This model was expressed in a unified form as stated in Equations 1 and 2 (Lin and Yang, 2006; Gomis-Bellmunt et al, 2009; Zhu and Wang, 2012),

where y(t) is the output of the piezoelectric actuator displacement, m, b and k are mass, damping, and spring constant, respectively. In addition, u is the input voltage, d is the ratio of the linear force constant to the input voltage, and h is the force with hysteresis. The values of and n are shape factors tuned for the hysteresis model. One advantage of the Bouc-Wen model is that it uses only a few parameters however, the traditional one is only suitable for symmetrical hysteresis forms (Ha *et al*, 2006; Wang and Zhu, 2011).

*1.2. **Prandtl-Ishlinskii *(*P-I*)* Model*

The Prandtl-Ishlinskii (P-I) model uses a combination of backlash operators to form a hysteresis profile, as shown in Figure 2. Despite the similarity to the Bouc-Wen model, the P-I model lacks the ability to capture asymmetric hysteresis.

Figure 2 Illustration of backlash, (a) backlash operator with weight/slope, (b) physical example of backlash in mechanical systems, (c) Simulink^{®}

The P-I model shown in Figures 2(a) and (b), included the backlash operator (play) as stated in Equation 3 (Xu and Li, 2010) ,

where is the input,

where is the initial condition of the output, in addition the backlash operator has two parameters, where is the difference between the forward and backward paths and is the slope between input and output.

The output of the P-I model is a combination of several backlash operators multiplied by the weight values. Furthermore, the value determines the slope of the backlash, while the output of the P-I model is stated in Equation 5 (Xie *et al.*, 2018) ,

P-I model, proven to effectively capture hysteresis behavior, lacks the capability to distinguish the direction of motion, as stated in Equation 3. However, it is only effective for modeling symmetrical non-local memory hysteresis.

*2.3. Modified Prandtl-Ishlinskii (MPI) Model*

The Prandtl-Ishlinskii (P-I) model is only effective for representing symmetrical hysteresis. To address this shortcoming, the modified Prandtl-Ishlinskii (MPI) model was developed by Kuhnen (2003) . Both models are similar, except that a dead-zone operator is integrated at the P-I output. A combination of the backlash and dead-zone operators, enables the MPI to model asymmetric hysteresis. The formular for determining the dead-zone operator is stated in Equation 6 (Xie *et al.*, 2018) , while the threshold is shown in Figure 3.

Based on Equation 6, the model output from the combined dead-zone operators multiplied by weights is stated in Equation 7. In addition, the structure of the MPI model is shown in Figure 3(d).

Figure 3 Dead zone with threshold: (a) negative {d<0}, (b) without dead zone {d=0}, and (c) positive {d>0}, (d) the MPI (Modified Prandtl-Ishilinskii) model is composed of combination of several backlash and dead zone operators

The MPI model is formed from a series combination of backlash and dead zone operators. By substituting Equation 5 in 7, the resulting equation for the model output is stated in Equation 8,

where

The proposed modification of the P-I model effectively captures the asymmetric hysteresis phenomenon. This improvement required additional parameters, potentially leading to a longer computational process. The use of a more elemental model increases the number of parameters to be optimized. Therefore, the trade-off between the model complexity and effectiveness needs to be carefully considered.

Actuator Design

The research developed a multilayer piezoelectric actuator, with the constituent components shown in Figure 4(a). Additionally, Figure 4(b) presents the manufactured parts, while Figure 5 illustrates the corresponding piezoelectric actuator driver circuit. In addition, the manufactured parts and the corresponding piezoelectric actuator driver circuit are shown in Figures 4(b), and 5.

The actuator design parameters obtained from the component measurements shown in Figure 4(b), are summarized in Table 1. The experimental set-up, comprising the piezoelectric actuator, a loadcell used as a displacement sensor, step-up module, driver, and data acquisition card are shown in Figure 6. To enable the usage as a position sensor, the loadcell is calibrated by applying force to the tips and measuring both the displacement and output voltage. The results obtained led to the establishment of a relationship between the loadcell output voltage and the displacement of the tip.

Figure 4(a) The piezoelectric actuator design contains (1) actuator rod, (2) front chassis, (3) spring, (4) piezoelectric crystal, (5) rear chassis, (6) adjustment bolt, (7) washers, (8) lock nut; (b) Piezoelectric actuator components

Figure 5 The piezoelectric actuator driver circuit

Table 1 Actuator's parameters

Figure 6 (a) Block diagram of the experimental setup, (b) The experimental setup to evaluate piezoelectric actuator performance. A load cell is used to measure the actuator displacement

The dynamics of a piezoelectric actuator can be modeled using a second order system as stated in Equation 1. By assuming the hysterical part and addressing the compensated phenomenon separately as discussed in Section 6, the actuator dynamic model is stated in Equation 9,

Based on Equation 9, the transfer function between the output and is stated in Equation 10,

is stated in Equation 10,

Using the parameters given in Table 1, the transfer function Equation 10 can be stated as 11,

The transfer function in Equation 11 is therefore used to assist in the design of the linear feedback controller.

**Estimation
of the Hysteresis Parameters**

A sinusoidal signal with decreasing amplitude was fed to the
driver circuit at low frequency to estimate the P-I and MPI hysteresis model
parameters. Initially, the signal had an amplitude of 150V, covering the
typical range of actuations in the system. Input at low frequencies puts the
piezoelectric in a quasi-static state, minimizing the impact of actuator
dynamics on hysteresis, including the rate-dependent effect (Qin, Zhao, and Zhou, 2017; Zhu and Rui, 2016) .
The frequency of the input signal used for the parameter estimation is 0.1 Hz,
and the voltage measurement is stated in Equation 12,

The estimation process was carried out based
on Levenberg-Marquardt algorithm using the Simulink^{®} parameter
estimation feature that minimizes the quadratic cost in Equation 13 (MathWorks, 2018) .

The Simulink^{®}
models for the P-I and MPI models are shown in Figures 7(a) and (b),
respectively.

The estimated parameters for both P-I and MPI models, comprising a total
of 15 elementary models, where each element consists of two and four parameters
respectively, are shown in Table 2.

**Hysteresis
Model Validation**