Vibrational energy is also widely found in buildings, human bodies, and vehicles. In order to collect a large amount of mechanical energy, researchers adopted a cantilever energy collector. Then they developed piezoelectric nano powered shoes with piezoelectric plates embedded in the sole, as shown in Figure 1(a) [1] [32] .

For energy harvesting, researchers use PEAs to capture mechanical energy from living and natural environments and convert it into electrical energy. Because of conventional wind, hydroelectric and solar power, such technological devices are large, expensive and unsuitable for electronic power supplies. Existing chemical batteries provide unsustainable power and are difficult to recycle. To this end, researchers have developed friction nanogenerators suitable for recovering mechanical energy from human movement and life. As shown in Figure 1(b), wearable electronic products made of multi-layer flexible piezoelectric materials worn by humans will produce friction when they move [6] . And friction nanogenerators are also self-contained power sources for wearable electronics [33] .

The environmentally friendly wind energy is abundantly available in nature. Traditional wind conversion devices are not suitable for power micro devices due to their large volume and elevated cost. For researchers, piezoelectric harvesters have been developed to convert wind energy into electricity through mechanical structures [5] . To that end, researchers have developed piezoelectric wind harvesters, which converts wind-to-electricity energy. Figure 1(c) illustrates the fundamental principle of the piezoelectric wind energy harvester [34] . Researchers will utilize wind turbines to harness natural wind energy and convert it into kinetic energy for rotational motion. When a windmill blade contacts a piezoelectric cantilever, it bends the cantilever and creates vibrations and voltages.

Recently, with the development of interactive human-machine interfaces in MEMS, it is important to control the machine more accurately and collect the feedback information of the machine. Researchers have developed bending angle piezoelectric sensors made of flexible piezoelectric materials. As shown in Figure 1(d), sensors based on flexible piezoelectric materials are embedded in the manipulator, which can transmit commands and feedback to the manipulator’s motion state in real-time in the two-way electronic system of human-machine interaction [7] .

In order to enrich the recreational life of the people by recording sound, Edison [35] invented the phonograph. As shown in Figure 1(e), researchers used ageing-resistant piezoelectric single crystals for phonograph pickup. When the record is flipped, the stylus produces different deformations depending on the texture of the record, which then generate different currents that pass through the microphone to produce sound waves [36] .

In the information age, researchers have been devoted to rapid data transmission. The currently available frequency-stable lasers with narrow linewidth exhibit exceptional properties and enjoy widespread utilization. However, to additionally achieve the requirements of narrower linewidth and faster frequency control, researchers have developed a modulator using lithium niobate chip. As shown in Figure 1(f), the lithium niobate external modulator developed by researchers has less noise and lower voltage than the conventional modulator [9] [37] . MEMS are widely used in human production and life to improve their integration and miniaturization. The researchers used quartz crystal tuning forks as piezoelectric resonators to time MEMS. As shown in Figure 1(g), the electronic control unit adopts a tuning fork crystal resonator to time and generate clock frequency signals [38] . Robots have experienced significant growth in recent years. However, their effectiveness is limited in certain specialized situations such as disaster site search and rescue, infrastructure monitoring, and reconnaissance missions. In addition, the robot is limited by traditional actuators’ size and manufacturing process. Therefore, the field of microrobots in MEMS has become a research hotspot [39] . As shown in Figure 1(h), researchers use piezoelectric materials as fly wings [11] . The piezoelectric material vibrates fast at high frequency and outputs enough vibration force to meet the requirements of the miniature robotic fly wing.

As medical research deepens and more advanced medical devices become available, the level of awareness of malignant tumour-related diseases among medical workers is also improving. Current techniques for inducing apoptosis of tumour cells by reactive oxygen species are highly dependent on oxygen in the tumour microenvironment. Therefore, how to generate oxygen in the tumour microenvironment is key to efficient apoptosis of tumour cells. To induce specific cell apoptosis in cancer therapy, barium titanate nanoparticles have been developed based on ultrasonic catalysis and hydrolysis. As shown in Figure 1(i), ultrasonic piezoelectric catalysis is mainly used in the medical field, and ultrasonic stress-induced asymmetric piezoelectric catalysis of BaO₃Ti nanoparticles is used to treat hypoxic tumours [21] [40] .

The internal complexity of living organisms limits the diagnosis and treatment of patients in modern medicine. Researchers have developed ultrasound detectors for medical imaging to assist doctors in diagnosing diseases. Next, the working principle of ultrasonic PEA is introduced. A sound wave is a form of energy transfer of an object in a mechanical vibrational state. The piezoelectric ultrasound transducer exploits the acoustic transmission properties of the object in the mechanical vibrational state to realize the mutual conversion of mechanical and electrical energy [41] . As shown in Figure 1(j), the principle of a high-frequency ultrasonic instrument is to generate a high-frequency vibration wave by using the piezoelectric effect and then analyse the received vibration wave by using inverse piezoelectric effect and combine it with image processing to obtain the internal detection map of the object [23] . The piezoelectric materials that generate vibration are mainly KNbO₃-based single crystals [42] .

Human health and longevity are limited by heart disease, which accounts for one in three deaths worldwide. The most direct and effective way to treat patients with heart disease is with an electronic device implanted in the heart. However, when the battery of an implantable electronic device is depleted, the surgical replacement of the battery poses a significant risk and financial burden to the patient. As a result, researchers have used flexible piezoelectric materials and biodegradable materials to develop self-powered heart monitoring devices attached to human blood vessels [43] . It converts vibrational energy generated by the human body into PEAs and continuously stores it as electrical energy to power heart monitoring devices. As shown in Figure 1(k), the heart detector can monitor the heart status in real-time [44] . The heart detector is mainly attached to the human heart by biodegradable flexible piezoelectric materials and is self-powered in the human body.

Deaf individuals experience challenges in communication due to the impairment or absence of cochlear hair cells. Existing cochlear implants struggle to recognize hearing in multiple sound sources. To this end, researchers have developed flexible, high-precision, high-sensitivity sound detection and recognition sensors [45] . As shown in Figure 1(l), there is a flexible piezoelectric sensor in the acoustic sounder, which is mainly composed of flexible poly (vinylidenefluoride-co-trifluoroethylene) (PVDF-TrFE) piezoelectric sheets [25] [46] .

In addition to energy harvesting, biomedicine, MEMS, and PEAs are widely utilized in aerospace, precision machining, optical manipulation, and numerous more advanced technologies. With the extensive range of applications of PEAs in precision engineering, engineers and technicians are demanding higher accuracy in PEAs modelling [47] [48] . The distinct characteristics of PEAs vary according to their types. To understand more about PEAs, the following sections further describe typical PEAs.

As depicted in Figure 2(a), conventional PEAs can be categorized into five types. The joint construction of unimorph PEAs involves the insertion of square, circular, annular or cantilevered unimorphs into a multilayer conducting metal electrode.

Different actuators may undergo shrinkage, expansion or bending in their designated driving directions due to the varying orientations of electric field and polarization, as illustrated in Figure 2(b). Bimorph actuators are composed of two layers of piezoelectric material, each potentially attached to a metal gasket depending on the situation. Consequently, bimorph actuators can also be stretched/contracted or bent.

In Figure 2(c), the piezoelectric tube actuator is formed by longitudinally polarizing a thin cylindrical piezoelectric material and bonding it to an electrode layer. Moreover, the piezoelectric tube actuators can be driven in axial/radial or transverse directions.

In Figure 2(d), the multilayer piezoelectric stack actuator drivers employ multiple layers of piezoelectric material stacked on top of each other, with appropriate insulation separating the electrodes. Each layer is solely composed of piezoelectric material, and an electrode layer of the same polarity is connected to the external electrode. The driving range of these drivers varies from a few micrometres to tens of micrometres, while their driving force ranges from hundreds to thousands of newtons [67] . By applying different electric fields to the various piezoelectric layers, multilayer piezoelectric stack actuators can achieve more degrees of freedom and allow for longitudinal or shear displacements [68] .

In Figure 2(e), the amplified PEA consists of a bending hinge or compliant mechanism with stacked piezoelectric materials, depending on the design [69] . The elastic deformations generated by the amplified PEAs enable amplified motion to be achieved.

In Figure 2(f), the piezoelectric stepper actuator is capable of producing continuous linear or rotational motion that satisfies the requirements for extensive range and high precision in manipulator applications [60] . Based on either a stack PEA or an amplified PEA as the power source, researchers have proposed an inchworm-inspired PEA that utilizes the crawling principle for clamping feed stepping [70] . To achieve precise control over a wide range, they employ the inchworm-inspired PEA featuring an alternating gripping and driving mechanism. The most commonly utilized types include thrust-type actuators with fixed positions and walking-type actuators with moving positions [71] .

In Figure 2(g), the inertial PEAs, also referred to as stick-slip actuators, are depicted. are typically driven by sawtooth waves [72] . These actuators harness the energy generated by the deformation of piezoelectric materials through inertial and frictional forces. Based on different driving modes, researchers have classified them into two types: impact-driven and friction-driven. The former utilizes inertia to displace loads with the aid of static friction force and rapid contraction between loads [73] . The latter, known as a stick-slip PEA, generates its driving force by utilizing the difference in friction during stick-slip motion and rapid contraction [74] .

The utilization of impact-type load inertia deviates from the primary characteristics of friction-type as depicted in Figure 2(h). The ultrasonic PEA is primarily distinguished by its employment of high-frequency ultrasonic resonance drive, which endows it with exceptional speed and a broad range of continuous motion. The ultrasonic PEA propels the piezoelectric material to generate high-frequency vibrations through voltage waveforms of travelling and standing waves, which in turn drives the elastic stator on the piezoelectric material to produce an elliptical trajectory. Ultimately, this stator conducts high-frequency excitation ultrasonic waves to either a slider or rotor. The aforementioned process facilitates the implementation of linear and rotational motion with the structure of the piezoelectric ultrasonic actuator [75] .

In Figure 2(i), the MDOF PEA can be connected in series or parallel with a single degree of freedom PEA, stacked PEA, or amplified PEA at a perpendicular angle to each other. Then, depending on the required degree of freedom, a hinge or flexible mechanism is designed to enable MDOF linear or rotational motion. Due to the different actuation modes, MDOF PEAs can be classified into two types: direct actuation and step actuation [76] [77] . The direct-drive MDOF PEAs offer high accuracy as an advantage, yet their range of motion is limited due to the utilization of multi-layer PEAs.

In Figure 2(j), inchworm and ultrasonic actuators are commonly used in series or parallel to design multiple degrees of freedom (MDOF) stepping PEAs, which enable a more comprehensive range of motion. The accuracy and travel range of MDOF stepper PEAs depend on the type, precision, and arrangement of the PEAs [78] .

The typical actuator of this section is used as the foundation mechanism in a typical application shown in Figure 1. In designing a PEA, the first step is determining the type of piezoelectric material and using a set of constants to assess whether the chosen piezoelectric material can achieve the desired performance. Second, researchers design the mechanism of the PEA and then study the modelling of the PEA [79] . Researchers have sought to better utilize piezoelectric materials in the fields of sensing and micromanipulation. Numerous studies have been conducted on the modelling of PEAs. However, the hysteresis nonlinearity of PEAs limits their application in precision engineering [1] [80] .

After F. Preisach [87] proposed the Preisach model, Krasnosel’skii [88] developed it into a pure mathematical model, and Mayergoyz [89] further generalized it based on its characteristics. The Preisach model is derived from the superposition of relay operators, and the resulting curve can be utilized to depict the hysteresis phenomenon exhibited by PEAs. The nonideal relay operator formula is as follows.

y ( x ) = { 1 if   x ≥ β 0 if   x ≤ α k if   α < x < β (1)

This definition of the hysteron indicates that the current value of the complete hysteresis loop is contingent upon the historical record of variable input. The Relay operator of the convention Preisach model is shown in Figure 5. Because the history value of the hysteresis is nonlinear or the value of the previous position impacts the next output value, the local memory property. Furthermore, when A < x 0 < x < x 1 < C , the displacement is output according to the path of the red line segment, and the nonideal relay operator also has the local memory property.

The convention Preisach model is represented by N relay operators with weights in parallel [92] [93] . The formula is as follows.

y ( t ) = ∬ α ≥ β μ ( α , β ) R α β x ( t ) d α d β (2)

One of the most straightforward approaches to comprehending the Preisach model is through a geometric interpretation assignment to the coordinate plane ( α , β ) . On this plane, each point ( α i , β i ) is associated wit a specific relay hysteron R α i , R β i .

Each relay can be represented on this so-called Preisach plane with its values [94] . As shown in Figure 6 various trajectories can be obtained by stacking different weights with different rely operators. Then the curve is fitted to simulate the nonlinear hysteresis behaviour of the PEAs.

Due to the complexity of solving the double integral in the Preisach model, KP model was developed by Bank et al. [96] The KP operator was established based on reliability and a superposition method was employed instead of integration. The KP operator is depicted in Figure 7 [97] .

The KP model is expressed as follows [100] [101] [102] [103] .

x ( t ) = ∫ S [ k p ( v , ξ S ) ] ( t ) d μ ( s ) (3)

where k p ( v , ξ S ) expresses the KP operator as follows.

[ k p ( v , ξ S ) ] ( t ) = { min ( ξ S , r ( v − s 1 ) ) v ˙ ( t ) < 0 max ( ξ S , r ( v − s 2 ) ) v ˙ ( t ) > 0 k p 0 v ˙ ( t ) = 0 (4)

where v ( t ) is the input voltage, x ( t ) is displacement output, S is the integral domain.

PI model is an improvement of Prandtl and Krasnosel ‘skii et al. [104] [105] based on Preisach model. PI model is characterized by the inclusion of play operator or stop operator. As shown in Figure 8, the PI model principle approximates the mass-spring system. Play operator is defined by.

y r ( t ) = H r [ v ( t ) , y 0 ] = max { v ( t ) − r , min { v ( t ) + r , y r ( t − ) } } (5)

where H r is the play operator and the threshold is r, y r ( t − ) is the output at

the last moment; v ( t ) is the input; y r ( t − ) is the output; initial value is y 0 . PI model is expressed as a weighted sum of finite play operators.

So it is as follows.

y ( t ) = ω T H r [ v ( t ) , y 0 ] = ∑ i = 0 n ω i max { v ( t ) − r i , min { v ( t ) + r i , y i ( t − ) } } (6)

where ω T is the weight and value are nonnegative, the threshold must be 0 = r 0 < r 1 < ⋯ < r n < v max . Observe Figure 9, and the same can be done by using the stop operator, this paper will not repeat it here.

Researchers have developed polynomial models based on the control accuracy of PEAs in various application scenarios. Although there is no universal formula,

hysteresis models using polynomials are all composed of linear functions to obtain polynomials [110] [111] [112] [113] . Examples of such polynomial models include.

y ¯ i = β 0 + β 1 x i + β 2 x i 2 + ⋯ + β p x i p + ε i ; i = 1 , n (7)

Wen [114] proposed the Bouc-Wen model based on the differential equation proposed by Bouc [115] [116] .

The Bouc-Wen model is expressed as follows.

F ( x , t ) = α k x ( t ) + ( 1 − α ) k z ( t ) (8)

z ˙ = A x ˙ − β | x ˙ | | z | n − 1 z − λ x ˙ | z | n (9)

where α is the ratio of the anterior and posterior terms and α ∈ ( 0 , 1 ) , A, β and λ are dimensionless parameter; F ( x , t ) is restoring force; x is the displacement; z is the output.

Coleman and Hodgdon [117] refined Duhem’s [118] differential equation, resulting in the Duhem model which is expressed mathematically as follows.

d w d t = α | d v d t | [ f ( v ) − w ] + d v d t g ( v ) f ˙ ( x ) > g ( x ) > α e α x ∫ x ∞ | f ˙ ( ε ) − g ( ε ) | e − α ε d ε (10)

where f ( v ) and g ( v ) are a given continuous function, The former is an odd function that increases monotonically and is piecewise smooth, and the latter is a piecewise smooth continuous even function. Both of them are bounded in their

domain. f ˙ ( x ) > g ( x ) > α e α x ∫ x ∞ | f ˙ ( ε ) − g ( ε ) | e − α ε d ε is true for any x > 0 . α is a constant, v is the input and w is the output.

Su et al. [119] proposed a dynamic hysteresis backlash model based on first-order differential equation. The expression of the Backlash-like model is as follows.

p ( v ( t ) ) = { c ( v ( t ) − B ) v ˙ ( t ) > 0 , w ( t ) = c ( v ( t ) − B ) c ( v ( t ) + B ) v ˙ ( t ) < 0 , w ( t ) = c ( v ( t ) + B ) w ( t _ ) v ˙ ( t ) = 0 (11)

where v ( t ) is the model input, w ( t ) is the model output, c > 0 , B > 0 .

Dahl [120] proposed the Dahl model based on friction phenomenon as follows [121] .

d F d t = δ | 1 − F F c sgn ( x ˙ ) | i sgn ( 1 − F F c sgn ( x ˙ ) ) (12)

where F is the static friction, Fc is the sliding friction, x is the output displacement. The slope of the force curve under F = 0 is denoted by δ , and i denotes the different kinds of piezoelectric materials. Both δ and i should be set to 1 when researchers study the hysteresis behaviour of piezoelectric materials.

The Dahl model is as follows.

d F d t = d x d t − F F c | d x d t | (13)

d 2 x d t 2 + γ d x d t + k n x = k v x − k l F (14)

where v stands for the input voltage, and x stands for the output displacement. k l and Fc determine the shape of the hysteresis loop. The remaining unknowns are model parameters that need to be identified. Because of the existence of differential equations in the Dahl model, the control method can be designed simply in the Dahl model [81] [122] [123] .

The Ikuta model is a sublayer model [126] proposed by Ikuta et al. [127] for SMA. As shown in Figure 10, the researchers found that the temperature-martensite fraction exhibits hysteresis loops during the heating and cooling of SMA materials. The Ikuta model uses an exponential function to fit the phase transition in the temperature change process. Equation (17) provides the phase fraction during heating. The phase fraction of SMA materials during heating is shown in Equation (20), and the cooling process is the same [128] .

The Maxwell model is originally proposed by James C. Maxwell [130] . The hysteresis behaviour of the unloaded PEA can be simulated by an equivalent circuit consisting of n serially connected charge-saturated capacitors (CSC), as illustrated in Figure 11. Thus, the governing equation of a PEA is as follow.

q ˙ ( t ) = { q ˙ ( t ) | q i ( t ) | < Q i 0 | q i ( t ) | ≥ Q i (15)

q ( t ) = C i u i ( t ) (16)

u ( t ) = ∑ i n u i ( t ) (17)

q ( t ) = T x ( t ) (18)

where q i and u i are the charge stored in the charge-saturated capacitor C S C i and the voltage applied on it, C i and Q i are its capacitance and capability to store charge, x and q are the PEA’s output displacement and the charge stored in it, and T is the electro-mechanical transfer ratio from effective displacement x to charge q.

Jiles and Atherton [124] present the ferromagnet hysteresis model in mathematical form based on the concept of domain wall motion. Chen et al. [131] found that the modified Jiles-Atherton model can simulate permanent magnets. The

modified Jiles-Atherton model can accurately and stably describe the dynamic magnetization behaviour of magnetic particles [132] . Classical Jiles-Atherton model equations as follow [133] .

M r e v = c ( M a n − M i r r ) (19)

where M is magnetisation, M r e v is reversible and M i r r is irreversible components. M a n is an hysteretic magnetisation and based on the Langevin function they get the following.

M a n = M s [ coth ( H e a − a H e ) ] (20)

With,

d M i r r d H e = M a n − M i r r k δ (21)

where H e = H + α M is the effective field experienced by the magnetic domains. The model parameters are M s , a, α , c and k. δ is a directional parameter assuming the value +1 if d H / d t > 0 , otherwise the value is −1. Jiles-Atherton model can be written as follow.

d M d H = ( 1 − c ) ( d M i r r / d H e ) + c ( d M a n / d H e ) 1 − α ( 1 − c ) ( d M i r r / d H e ) − α c ( d M a n / d H e ) (22)

Compared with other models, neural network models have the characteristics of self-learning, self-correction, and high-precision approximation and are widely used in nonlinear system identification and controller design [135] . Because the neural network model can only describe one-to-one mapping [136] , and the piezoelectric transducer has multi-value properties and memory properties, the neural network model cannot be directly used to describe the hysteresis behaviour of the piezoelectric transducer. Generally, the composite algorithm combining neural networks and other methods describes the hysteresis nonlinearity [137] . In addition, Quondam-Antonio et al. [138] used neural networks to model the hysteresis behaviour-related characteristics and were able to reproduce the evolution of the magnetization process under arbitrary excitation waveforms.

Some scholars used elliptic curves to describe the hysteresis characteristics of piezoelectric materials because the hysteresis loop is very similar to an ellipse. The hysteresis curve includes a rising and falling curve, similar to a part of an elliptic curve, so Zou et al. [139] use the polar elliptic curve method for modelling. The general equation of an ellipse is:

A x 2 + B x y + C y 2 + D x + E y + F = 0 (23)

Define x ′ = x − x 0 , y ′ = y − y 0 , Then the elliptic equation can be rewritten as.

A ( x − x 0 ) 2 + B ( x − x 0 ) ( y − y 0 ) + C ( y − y 0 ) 2 + f = 0 (24)

So the solution is:

A x ′ 2 + B x ′ y ′ + C y ′ + f = 0 (25)

The standard equation is:

x ˜ 2 a 2 + y ˜ 2 b 2 = 1 (26)

The relationship between x ˜ and x ′ as follows.

{ x ˜ = x ′ cos θ − y ′ sin θ y ˜ = x ′ sin θ − y ′ cos θ (27)

It can be obtained by combining Equations (25) and (26).

x ′ cos θ − y ′ sin θ a 2 + x ′ sin θ − y ′ cos θ b 2 (28)

Each parameter in the equation is obtained by comparing Equation (12) with Equation (17).

{ A = a 2 sin θ + b 2 cos θ B = 2 ( a 2 + b 2 ) sin θ cos θ C = a 2 cos 2 θ + b 2 sin 2 θ D = − a 2 b 2 (29)

In this section, typical hysteresis models are briefly introduced and classified. To further investigate the hysteresis model, the hysteresis modelling based on the typical model and other hysteresis models are introduced in the following subsections.

There are many cases where the direct IOHM is employed. In 2012, Guo et al. [164] utilized an enhanced PI model and aimed at the asymmetric inverse hysteresis effect and subsequently employed the adaptive particle swarm optimization (PSO) algorithm to solve the inverse hysteresis model of real-time control. In the same year, Qin et al. [151] found that rate-independent PI model inversion parsing is desirable They employed the inverse PI model as a feedforward controller for a piezoelectric actuated compliant mechanism, obtaining the inverse PI model directly from experimental data. This method avoids solving the inverse of the PI model and is also effective for the rate-dependent PI model. In

2017, Ko et al. [165] proposed a generalized PI model-based inverse feedforward compensation approach for addressing asymmetric hysteresis nonlinearity. In 2020, Lallart et al. [166] proposed a system-level inverse method to simulate the quasi-static hysteresis of a PEA and demonstrated it on a piezoelectric transducer. This approach for solving the direct inverse model of the direct hysteresis model is well-suited for implementation in embedded control systems. In 2021, Qin et al. [154] achieved high-precision hysteresis compensation over a more comprehensive frequency range. They used the direct inverse modelling method to build a multilayer feedforward neural network inverse model as a feedforward lag compensator. In the same period, Zhang et al. [167] aimed to enhance the precision of piezoelectric fast steering mirror’s swing. They proposed the Hammerstein dynamic inverse hysteresis model for PEAs. In addition to this, the authors employed the generalized Bouc-Wen inverse model to characterize static nonlinearity and utilized the autoregressive exogenous model to capture rate dependence. The hybrid model parameters are then identified through employment of the adaptive PSO algorithm. This method avoids the complex inverse process of the Bouc-Wen model and is suitable for rate-dependent and asymmetric hysteresis behaviour. In 2022, Nguyen Ngoc Son et al. [168] proposed neuroevolutionary adaptive sliding mode control for solving piezoelectric ceramic actuators’ nonlinearity. They utilized the neuroevolution model to identify the inverse hysteresis model of piezoelectric ceramic actuators, and employed both differential evolution algorithm and Jaya algorithm for global and local optimization respectively. In the same period, Nie et al. [85] proposed a rate-dependent asymmetric hysteresis model based on the PI model. They introduced the dynamic envelope function year into the Play operator, which has a simple structure and few parameters.

In 2006, to solve the problem that there is no inverse of the PI operator when the PI model is not positive definite, Tan et al. [169] extended the PI operator and then calculated the inverse hysteresis PI model for feedforward controller. As early as 2009, Yang et al. [170] adopted Preisach and BP neural networks and used a linear interpolation algorithm. In 2016, the identification of at least ten parameters is required for the PI model, so Gan et al. [171] combined quadratic polynomials and linear equations to identify only four parameters and solve the PEA hysteresis model. Finally, the IOHM is obtained and applied to real-time hybrid control. In 2018, Janaideh et al. [172] proposed a solution to address the saturation and asymmetric energy phenomena observed in PEA under high-frequency excitation and numerous inputs. With the rate-dependent PI model and a dead-zone operator as feedforward compensator, an inverse model for real-time compensation is established to effectively suppress asymmetric nonlinear hysteresis. In 2019, Tao et al. [173] order to solve the problem that the nano-positioning platform’s positioning accuracy is subject to hysteresis effects. The hysteresis is modeled using a Gaussian process, and the parameters are determined through the integration of a differential evolution algorithm and Bayesian inference. The outputs based on the Gaussian process hysteresis model are then interleaved to obtain the inverse model as a feedforward compensator. This method is suitable for nonlinear, memory, and rate-dependent. hysteresis modelling. In 2020, Janaideh et al. [174] to mitigate the impact of temperature on the precise positioning of piezoelectric tube actuators. The simultaneous modelling of hysteresis nonlinearity and temperature effects is undertaken. Two different temperature-dependent Prandtl-Ishlinskii models are proposed, which are solved by MATLAB-Simulink and Grey Wolf optimization algorithms, respectively. The inverse model of the temperature-dependent Prandtl-Ishlinskii model is derived and utilized as a feedforward controller to compensate for the hysteresis nonlinearity exhibited by the piezoelectric tubular actuator. To improve the classical Preisach model, The dynamic hysteresis model has been presented by Chen et al. [145] . It is based on the classical Preisach model, and the dynamic term is introduced and solved numerically. The dynamic inverse model is ultimately implemented as the feedforward controller. In 2020, to address the limitation of the PI model in describing asymmetric hysteresis, Wang et al. [175] proposed the polynomial-modified Prandtl-Ishlinskii (PMPI) model that can describe both asymmetric and symmetric hysteresis. They used a modified-play operator and memoryless polynomial to form the PMPI model. They used a differential evolution algorithm and a simplex algorithm to identify the parameters of the PMPI model. Global and local optima are then obtained to search the hysteresis model in the design into an inverse model feedforward compensator. In 2021, The piezoelectric sensor-actuator was designed by Shan et al. [176] and applied to a micro-high precision integrated system. They proposed nonlinear and inverse models based on dynamic hysteresis using quasi-static models and linear transfer functions. Based on classical hysteresis modelling, it is typically necessary to optimize the algorithm within the model in order to enhance its modelling accuracy. Hysteresis modelling based on the hybrid model is usually divided into direct IOHM and indirect IOHM. The former solves the inverse model of the hybrid model as a feedforward controller, followed by the series hysteresis model. Its characteristic is that the choice of the hysteresis model must be able to solve the inverse analysis, the inverse hysteresis accuracy depends on the hysteresis model accuracy, and the structure is simple. The latter is to solve the direct inverse model using the known output input data as a feedforward compensator and then concatenate the hysteresis models. Its characteristics are that it can be modelled based on irreversible hysteresis, the direct inverse hysteresis model is not associated with the hysteresis model, and it has many parameters. In the last decade, due to the diversity of ideas for PEA hysteresis modelling.

The hysteresis models in the current literature include not only single and multiple models based on classical approaches [177] [178] , but also those based on

the other models [179] [180] . There are also hybrid models that combine the classical models with the other models [181] .

The model parameters can be estimated using both nonparametric and parametric identification methods. Nonparametric identification refers to designing the recognizer to mimic the behaviour of the real system in order to minimize error, while parameter identification involves estimation through optimization tools [109] . The original input and output data of the PEA serve as known variables in solving for the parameters using the aforementioned method.

The original input data is fed into the hysteresis model to obtain the output, which is then compared with the original data output to determine modelling accuracy. At this point, hysteresis modelling is completed. In addition to the improvement of hardware equipment accuracy, the following methods are commonly employed in hysteresis modelling to enhance its accuracy.

1) Design an enhanced classical model by refining the fundamental computational units within the classical models.

2) Develop improved algorithms for identifying unknown parameters in the classical models.

3) Explore hybridization of the classical models with the other model using varying weights.