Rate-dependent asymmetric hysteresis modeling and robust adaptive trajectory tracking for piezoelectric micropositioning stages

Hysteresis is an inherent characteristic of piezoelectric materials that can be determined by not only the historical input but also the input signal frequency. Hysteresis severely degrades the positioning precision of piezoelectric micropositioning stages. In this study, the hysteresis characteristics and the excitation frequency effects on the hysteresis behaviors of the piezoelectric micropositioning stage are investigated. Accordingly, a rate-dependent asymmetric hysteresis Prandtl–Ishlinskii (RDAPI) model is developed by introducing a dynamic envelope function into the play operators of the Prandtl–Ishlinskii (PI) model. The RDAPI model uses a relatively simple analytical structure with fewer parameters and then other modified PI models to characterize the rate-dependent and asymmetric hysteresis behavior in piezoelectric micropositioning stages. Considering practical situations with the uncertainties and external disturbances associated with the piezoelectric micropositioning stages, the system dynamics are described using a second-order differential equation. On this basis, a corresponding robust adaptive control method that does not involve the construction of a complex hysteretic inverse model is developed. The Lyapunov analysis method proves the stability of the entire closed-loop control system. Experiments confirm that the proposed RDAPI model achieves a significantly improved accuracy compared with the PI model. Furthermore, compared with the inverse RDAPI model-based feedforward compensation and the inverse RDAPI model-based proportional–integral–derivative control methods, the proposed robust adaptive control strategy exhibits improved tracking performance.

order differential equation. On this basis, a corresponding robust adaptive control method that does not involve the construction of a complex hysteretic inverse model is developed. The Lyapunov analysis method proves the stability of the entire closed-loop control system. Experiments confirm that the proposed RDAPI model achieves a significantly improved accuracy compared with the PI model. Furthermore, compared with the inverse RDAPI model-based feedforward compensation and the inverse RDAPI modelbased proportional-integral-derivative control methods, the proposed robust adaptive control strategy exhibits improved tracking performance.

Symbols b s
Damping coefficient C A Sum of the capacitance of the total piezoelectric ceramics d External disturbance e 1 Tracking error e 2 Intermediate variable Boundary of the nonlinear part αu 2 (t) +

Introduction
Microtechnology and nanotechnology are required in many industries for precision control. The piezoelectric micropositioning stage is one of the most commonly utilized positioning devices and is widely employed in various fields, such as mechanical manufacturing, aerospace technology, and biological engineering [1,2]. The piezoelectric micropositioning stage primarily comprises three parts: a piezoelectric ceramic actuator (PCA), flexure hinge machinery, and a strain gaugetype sensor [3]. The PCA is the core component of a piezoelectric micropositioning stage and affords micro-/nanodisplacement. PCAs have the advantages of a fast response and small volume, and the resolution of a PCA is only restricted by the performance of the controller. When the engineering requires a low frequency and limited motion, PCAs can be reliably controlled using classical control methods. In [4], static linear piezoelectricity dynamics for PCAs were developed. Thus, classical control methods can be applied. However, with increasingly stringent requirements in terms of the driving frequency and motion ranges in high-precision applications, PCAs exhibit complex nonlinear characteristics, particularly hysteresis, which can no longer be ignored. This strong nonlinearity significantly reduces the positioning accuracy or even results in system instability [5][6][7]. The hysteresis of PCAs is caused by the inherent ferroelectric phase transition characteristics in piezoelectric ceramic materials; because of these characteristics, the output displacement is determined by not only the current input and system states but also the historical input. Additionally, the hysteresis behavior can vary with changes in the input signal frequency [8,9]. To address this complex hysteresis nonlinearity, it is essential to establish a highly accurate model.
Hysteresis modeling has attracted significant attention, and several modeling approaches have been proposed. These approaches can be divided into two categories: differential-based and operator-based. Differential-based hysteresis models, such as the Bouc-Wen [10][11][12], Duhem [13,14], and Dahl models [15], are constructed using differential equations. These models have a straightforward architecture with few parameters. However, the limited number of model parameters reduces the modeling accuracy. Operatorbased hysteresis models include the Preisach [16][17][18][19][20], Krasnosel'skii-Pokrovskii [21,22], and Prandtl-Ishlinskii (PI) models [23][24][25][26][27], which can be regarded as a series cumulation of weighted elemental hysteresis operators. The PI model is the most widely used because of its accurate analytical inverse. However, these models assume that hysteresis behavior is rate-independent. Therefore, efforts have been directed toward the development of a rate-dependent hysteresis model. At present, the primary method for establishing rate-dependent hysteresis models is to introduce rate-dependent factors into the model [28,29], e.g., by changing the density weights from static to dynamic [30] and designing rate-dependent operators [31]. These methods usually result in complex mathematical expressions and increase the difficulty of parameter identification. Thus, the rate-dependent hysteresis model requires further investigation.
Another challenge is eliminating the hysteresis nonlinearity of the piezoelectric micropositioning stage to achieve high-precision tracking control is another challenging topic. The feedforward compensation control method based on the hysteresis inverse model is the most widely used approach. In this method, the hysteresis nonlinearity is compensated for by constructing a corresponding inverse hysteresis model of the established hysteresis model [32][33][34][35]. In [36], a feedforward compensation controller was designed using the inverse Bouc-Wen model for a piezoelectric micropositioning stage with multiple degrees of freedom. The compensator combines the inverse multiplication structure with the Bouc-Wen model, thereby avoiding additional calculation of parameters. This type of feedforward compensation controller has a simple structure. However, its performance depends entirely on the accuracy of the established inverse model, and it is difficult to obtain a corresponding inverse model for some hysteresis models. Additionally, the controller cannot cope with environmental perturbations (i.e., temperature changes or external disturbances). For inverse model feedback control, the hysteresis nonlinearity is compensated for by the inverse model compensator, and the feedback loop is then used for ensuring the stability of the system in the presence of disturbances [37][38][39][40]. In [41], an inverse Preisach model was used as a feedforward compensation controller. Furthermore, a proportional-integral-derivative (PID) feedback controller was designed to improve the control accuracy, and it enabled effective control. Nonetheless, the control precision of the inverse model feedback method relies on the accuracy of the inverse model. The noninverse model feedback control method directly incorporates the system hysteresis model into the generated control signal. A nonlinear controller is designed to drive the controlled object and adjusts the control parameters online based on the tracking error to improve the control accuracy [42][43][44][45][46]. This method omits the cumbersome inverse-model establishment process; thus, its performance is unaffected by the inverse-model accuracy [47]. Another advantage is that numerous excellent nonlinear control theories can be applied to nonlinear hysteretic system control. In [48], a dynamic backlash-like hysteresis model was defined without the need to construct an inverse hysteresis model to alleviate the influence of the hysteresis. Moreover, a robust adaptive controller that uses the properties of the hysteresis model was proposed, and this controller achieves a high tracking precision. In [49], an inversion-free predictive controller based on a dynamic linearized multilayer feedforward neural network model was proposed, which can obtain an explicit-form control law without an inverse model. However, it is uncertain whether these schemes can handle external disturbances. Moreover, for piezoelectric micropositioning stages, the classical hysteresis model cannot describe the rate-dependent and asymmetric hysteresis characteristics.
Thus, the effects of the excitation frequency on the hysteresis behaviors of the piezoelectric micropositioning stage were investigated in the present study. A novel rate-dependent asymmetric hysteresis Prandtl-Ishlinskii (RDAPI) model is proposed to describe the asymmetric and rate-dependent hysteresis behaviors that the conventional PI model cannot describe. First, a dynamic envelope function is introduced into the play operators, while the density weights remain static. In this manner, the RDAPI model can describe the hysteresis loops that are affected by not only the historical input but also the input voltage frequency. Second, the proportional input function is replaced with a polynomial input function to describe the asymmetry. On this basis, a robust adaptive control method is presented for the piezoelectric micropositioning stages. Unknown parameters and external disturbances are considered in the dynamic nonlinear system. In contrast to the commonly used approach of constructing a complex hysteretic inverse model, the proposed control method directly uses the established RDAPI model to describe the hysteresis behavior, and the adaptive control laws are designed to eliminate hysteresis nonlinearity and uncertainties online. The main contributions of this study are as follows: • A hysteresis model that can describe rate-dependent and asymmetry hysteresis behaviors is presented for piezoelectric micropositioning stages. • The proposed RDAPI model has few parameters and a simple analytical structure, which makes parameter identification relatively straightforward. • The proposed control method directly uses the established RDAPI model to describe the hysteresis behavior without constructing a complex hysteretic inverse model, and the robust adaptive method enhances the control accuracy and robustness.
The remainder of this paper is organized as follows. Section 2 describes an experiment that was performed to evaluate the hysteresis nonlinearity characteristics of the piezoelectric micropositioning stage. Section 3 presents the electromechanical model of the piezoelectric micropositioning stage and the proposed RDAPI hysteresis model. In Sect. 4, details regarding the robust adaptive controller design process and stability analysis are provided. Experimental studies and comparisons are presented in Sect. 5. Finally, Sect. 6 concludes the paper.   The entire control process of the piezoelectric micropositioning stage experimental system is as follows. First, the control block is set up in MAT-LAB/Simulink, and the control signal is generated through the RTW real-time working environment. The control signal is then translated into an analog voltage signal via a data acquisition card and drives the piezoelectric micropositioning stage such that the stage produces a small displacement. Subsequently, the acquisition card transmits the collected displacement signals back to the host computer to complete the closed loop of the experimental system.

Characterization of hysteresis nonlinearities
Because of energy dissipation, hysteresis occurs between the input voltage and the output displacement in piezoelectric micropositioning stages. The hysteresis characteristics of the piezoelectric micropositioning stage are characterized by applying different input signals. First, the input voltage signal is selected as a sinusoidal signal. Figure 3 presents the voltage-to-displacement hysteresis nonlinearity. It can be observed that the hysteresis causes that the same system input (u 1 , u 2 ) to correspond to different system outputs (y 1 , y 2 ) when the system is in an increasing stateu(t) > 0 or a decreasing stateu(t) < 0. It is thus characterized as "multi-valued." Therefore, the system output is related to not only the current system input but also the historical output, that is, the memory properties. Figure 3 also shows that the increasing and decreasing edges of the hysteresis loop are not centrally symmetric. Obviously, the hysteresis nonlinearity has an asymmetric characteristic. Second, the input voltage signal is selected as a sinusoidal signal with decreasing amplitude. The major hysteresis loops of the piezoelectric micropositioning stage are presented in Fig. 4. When the amplitude of the input voltage changes, the hysteresis loops change accordingly. As the input amplitude increases, the hysteresis loop widens. Additionally, the same input voltage (u) corresponds to five output displacements (y 1 , y 2 , y 3 , y 4 , y 5 ). Finally, a sinusoidal signal with different frequencies is selected as input voltage. As shown in Fig. 5, when the frequency of the input signal increases, the hysteresis loop widens, and the maximum output displacement decreases. Thus, the hysteresis behaviors of the piezoelectric micropositioning stage are also affected by the frequency of the input voltage. A satisfactory hysteresis model must describe all the aforementioned characteristics simultaneously.

Description of piezoelectric micropositioning stage
3.1 Model of the piezoelectric micropositioning stage Figure 6 shows the general model of the piezoelectric micropositioning stage, which includes (a) the electrical part and (b) the mechanical part [3,50]. The notation for the piezoelectric micropositioning stage is presented in the Nomenclature section. According to the aforementioned general model of the piezoelectric micropositioning stage, assuming that R 0 = 0, the complete electrical model can be expressed as follows: where x(t) represents the output displacement of the mechanical part and u(t) represents the control input of the piezoelectric driver. The meanings of other vari-  Table. By implementing a control voltage, charge is supplied for piezoelectric micropositioning stage to generate piezoelectric expansion. Therefore, the charge q(t) can be expressed as a function about control voltage u(t), that is, q(t) = g(u(t)). Then, the term u(t) − H (g(u)) k amp can be represented as a new hysteresis nonlinearity H (u(t)). Define the unknown hysteresis nonlinearities of the piezoelectric micropositioning stage ω ∈ R as which is introduced below. Then, it has Output displacement/µm 1Hz 10Hz 20Hz 50Hz 100Hz

Fig. 5 Hysteresis loops under different input frequencies
According to the physical characteristics of the piezoelectric micropositioning stage, the mechanical dynamics can be represented by a second-order motion model: wherek (3) and (4) represent an analytical model of the piezoelectric micropositioning stage.
By substituting (3) into (4), the overall model of the piezoelectric micropositioning stage can be obtained: where , and d(t) represent the unknown external disturbances.
The control objectives are to design a robust adaptive controller for piezoelectric micropositioning stages with uncertain parameters and external disturbances; to eliminate the effects of hysteresis behavior, such that the output displacement x(t) can accurately track the reference displacement signal x d (t); and to ensure the stability of the closed-loop system. The following assumptions are made.

Assumption 2
The uncertainty parameters in (5) satisfy a i min ≤ a i ≤ a i max , i = 0, 1 and K min ≤ K ≤ K max , where a i min , a i max , K min , and K max are known constants. Remark 1 Assumptions 1 and 3 are commonly used in controller design. The range of the system parameters is specified by Assumption 2, which is reasonable according to prior knowledge regarding piezoelectric micropositioning stages. Assumption 2 is often used in nonlinear controller design.

Hysteresis model for the piezoelectric micropositioning stage
In this study, the PI model was used to describe the hysteresis behavior of the piezoelectric micropositioning stage [21]. The classical PI model is expressed as where R represents the upper integration limit, F r [u](t) represents the classical play operator, r represents the threshold of play operator, p(r ) represents the density function satisfying p(r ) > 0, and p 0 is a constant determined by the density function p(r ). The play operator is a continuous rate-independent hysteresis operator that satisfies where ], and f r : R → R is defined as follows: The classical play operator has rate-independent characteristics. Assume that η(t) is a continuous function of t on [0, t N ]; then, η(t) and the play operator have the following relationship: Hence, the classical play operator cannot cope with the rate-dependent hysteresis characteristics of piezoelectric micropositioning stages. In this study, a dynamic envelope function was introduced into the classical play operator to explain the influence of frequency.
The rate-dependent play operator is expressed as where the dynamic envelope functions h l (u,u) and h r (u,u) can be selected as follows: Here, a, b, c, and d are constants to be identified, which satisfy a = c, b = d.
According to the hysteresis behavior, the hysteresis loop contains two parts: the rising edge and the falling edge. Therefore, two different dynamic input functions, h l (u,u) and h r (u,u), are selected to express the ratedependent play operator. To express the asymmetric hysteresis, a polynomial input function αu 2 (t) + βu(t) is introduced to replace p 0 u(t). Therefore, an improved RDAPI model can be obtained: where α and β are constants to be identified, u(t) represents the system input voltage, p(r ) represents the density function of the rate-dependent PI model, and F h r [u](t) represents the rate-dependent play operator. The integral form of (13) is not easy to realize in practical applications. Therefore, (13) is transformed into a superposition of the n play operators. Subsequently, a discrete RDAPI model expression is obtained via the discrete superposition principle: Compared with previously reported models [28][29][30], the proposed RDAPI model has a simpler analytical format with fewer identification parameters to describe the rate-dependent and asymmetric hysteresis behaviors of piezoelectric micropositioning stages, as explained in Sect. 3.3. According to the system model of the piezoelectric micropositioning stage (5) and the RDAPI model (14), the dynamic equation can be written as follows: Based on (15), the RDAPI model contains two parts: a linear part βu(t) and a nonlinear part For the nonlinear part, the input voltage of the piezoelectric micropositioning stage u is limited, andu exists; therefore, the nonlinear part is bounded. Accordingly, ς > 0 exists, and it satisfies Now, let x = x 1 ,ẋ 1 = x 2 ; then, (15) can be written as ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ẋ where p = 1 K , q = a 0 K , and n = a 1 K .

Identification of model parameters
The parameters to be identified in the RDAPI model include a, b, c, and d in the dynamic input function (12), α and β in the polynomial input function, and the weight parameters p i (i = 1, 2, . . . , n) of the play operators. We denote Z = {a, b, c, d, p 1 , p 2 , p 3 , . . . , p n , α, β} as the parameter vector to be identified in the RDAPI model. The RDAPI model threshold is defined follows: where i = 1, 2, . . . , n, with n being the number of play operators. The choice of n determines the computational burden and the accuracy of the RDAPI model. A larger n generally corresponds to a more accurate hysteresis model. However, the computational burden of parameter identification increase with n. In this study, n was selected as 8 through repeated numerical simulations. Table 1 presents the numbers of identification parameters for different rate-dependent PI models [28][29][30] with n = 8, thereby highlighting the advantage of the proposed RDAPI model. In addition, the modeling performance comparison among [28][29][30] and proposed RDAPI model will be given in Sect. 5.1.
In the identification process, the objective function is selected as where J represents number of the different frequency input signals, N j represents the number of data points of the input signals at different frequencies, and y d (n) and y(n) represent the actual output displacement of the piezoelectric micropositioning stage and the model output displacement under input voltage signals of different frequencies, respectively. An improved genetic algorithm [51] was used to identify the RDAPI model parameters.
First, the cumulative probability Q(k) of the kth individual is defined: (20) where M represents the population size, and F k represents the fitness of the kth individual.
Then, a two-point arithmetic crossover operator is selected. The two chromosome expressions are The offspring resulting from a two-point arithmetic crossover from point i to j is where w k and w k are generated as follows: Here, ϑ ∈ [0, 1] is a random number.
To accelerate the convergence speed of population, the mutation operator in the improved genetic algorithm is as follows: Define the individual code as w = (w 1 , w 2 , . . . , w k , . . . w n ), where w k ∈ [U k min , U k max ] represents the mutant gene. Then, w k mutates to where r 1 ∈ [0, 1] and r 2 ∈ [0, 1] are random numbers. If the variation in w k exceeds the value range, The actual input and output data of the piezoelectric micropositioning stage are used as the training set. The process of identifying the RDAPI model parameters related to the improved genetic algorithm is as follows.
Step 1: Select the fitness function: where φ(i) represents the objective function value of the i th individual.
Step 2: Generate the initial population. Randomly generate an initial population composed of M initial individuals, where each individual code string length l represents the number of parameters to be identified. The iteration starts with the initial population. The initial evolution generation t is set to 0, and the maximum number of iterations is G.
Step 3: According to (28), calculate the fitness value of each individual in the population.
Step 4: Perform the roulette selection according to (20).
Step 5: Perform the crossover operation in accordance with (21)-(23) to update the population.
Step 6: According to (26), apply the mutation operator to the population to obtain the next-generation population.
Step 7: Check whether the identification is complete. If the number of iterations satisfies t < G, return to Step 3 and continue the identification; otherwise, stop the operation and output the optimal solution.
In this study, we selected the piezoelectric micropositioning stage input and output data at frequencies of 1, 10, 20, and 50 Hz as the input for identification. The sampling time was set as T = 0.0001 s. The length of each individual coding string was set as l = 12, which is equal to the number of parameters to be identified. The population number was set as M = 200, and the maximum number of iterations as G = 1400. The identification parameters are presented in Table 2. The experimental results for the identified RDAPI model are presented in Sect. 5.1.
Note To facilitate identification, in the process of identifying parameters, the control input signal and actual output displacement are normalized to the range of 0-1. In actual applications, denormalization is performed on the data.

Robust adaptive controller design
In this section, a robust adaptive controller without constructing a complex hysteretic inverse model is proposed for the piezoelectric micropositioning stages. The proposed control method directly uses the established RDAPI model to describe the hysteresis behavior and design adaptive control laws to eliminate hysteresis nonlinearity and uncertainties online. The robust adaptive control structure diagram is presented in Fig. 7. The robust adaptive control scheme was designed as follows. First, let e 1 be the tracking error of the controlled systems, which is defined as Define an intermediate variable e 2 as where k 1 is a positive design parameter. The time derivative of e 1 iṡ Considering (29), e 2 =ė 1 + k 1 e 1 . Then, The time derivative of e 2 iṡ e 2 = 1 p Then, we obtain The control law is defined as where k 2 is a positive design parameter;p,q, andn represent the estimates of p, q, and n, respectively; the estimation errors are defined asp =p − p,q =q − q, andñ =n − n, respectively; ρ represents the bound of the disturbance d(t); and ς represents the bound of the nonlinear part αu The symbolic function sign (e 2 ) can be expressed as The adaptive update laws are designed as follows: p = −λu eq e 2 (37) n = −γ x 1 e 2 (38) q = −ηx 2 e 2 (39) where λ, γ , and η are the adjusting parameters of the adaptive update laws.
The key conclusion from the foregoing statement is presented in Theorem 1.

Theorem 1
Consider the entire control system, which is composed of the piezoelectric positioning stage system (15), the updated laws (37,39), the actual controller (35), and the Lyapunov function (40). If Assumptions 1-3 are satisfied, all the signals in the closedloop systems are uniformly ultimately bounded, and the tracking error e 1 exhibits asymptotic convergence under properly selected design parameters k 1 , k 2 , λ, γ , and η. Additionally, the output displacement x(t) can accurately track the reference displacement signal x d (t) for all t ≥ 0.
Proof of Theorem 1 Define the following Lyapunov function: The time derivative of V iṡ Substituting (32) and (34) into (41) yieldṡ By substituting the updated laws (37) and (39) into (42), we obtain: Substituting the control law (35) into (43) yieldṡ Considering |e 2 | = e 2 * sign (e 2 ) and |d (t)| ≤ ρ, it follows that, The nonlinear part of the RDAPI model satisfies where k 1 and k 2 are positive design parameters. Thus, V ≤ 0 for all t > 0. Considering that V ≥ 0 anḋ V ≤ 0, it can be concluded that the system is stable. Thus, the closed-loop stability of the designed robust adaptive controller is verified. Equation (46) is solved for any t ≥ 0: According toV ≤ 0, V (t) ≤ V (0). Then, (47) can be written as follows: Clearly, e 1 and e 2 are bounded. We define the following nonnegative continuous function: The derivative of (48) can be obtained as follows: Since e 1 and e 2 are bounded,ė 1 andė 2 exist. Thus, V 1 is uniformly continuous on [0, t]. According to Barbalat's lemma,V 1 → 0 when t → ∞; thus, when t → ∞, the tracking error e 1 is asymptotically convergent. This completes the proof. (47), e 1 and e 2 are bounded. Because the range of output displacement for this piezoelectric micropositioning stage is 0-60 µm, the desired output displacement x d is bounded. Additionally, from e 1 = x 1 − x d , x 1 is bounded. According to (30), e 1 , e 2 , anḋ x d are bounded; hence, x 2 is bounded. Thus, the control law u is bounded.

Remark 2 From
Remark 3 For avoiding the algebraic loops problem, the hysteresis model cannot be directly used in the control law design. To cope with the nonlinear term in the hysteresis model, the robust term ς sign(e 2 ) in (35) is introduced. The bound ς can be obtained according to the hysteresis model. For RDAPI hysteresis model, ς is calculated by . From Sect. 5.1, the proposed RDAPI model describes the hysteresis more accurately than the classical PI model. Then, we can concluded that ς can be obtained more accurately by using the proposed RDAPI model. Consequently, the robust adaptive control method has a better control performance. The comparison of robust adaptive method with PI and RDAPI models is shown in Sect. 5.2.4.

Experimental results
The proposed RDAPI model and the robust adaptive control strategy were tested on piezoelectric micropositioning stages to evaluate their efficiency.

Experimental results for RDAPI model
As mentioned earlier, the present work proposes a new hysteresis model for modeling the rate-dependent asymmetric hysteresis behavior with fewer parameters. To validate modeling performance of the RDAPI model, the classical PI model and modified PI models [28][29][30] that mentioned in Table 1 were selected for comparison. The same identification method as RDAPI model, i.e., improved genetic algorithm, was used for parameter identification. The input voltage signals were a series of sinusoidal signals with different frequencies. Figure 8 presents the modeling performance of the proposed RDAPI model, the classical PI model, and other modified models with input voltage signals having frequency of 1, 10, 20, and 50 Hz, respectively. Figure 9 shows the root-mean-square error (RMSE), the absolute mean error (AME), and the maximum error (ME) values of hysteresis models with different input voltage signal frequencies. When the input signal frequency was 1 Hz, the modeling error of the RDAPI model was close to the other models. When the input voltage signal frequency was 50 Hz, the RMSE was reduced by 47.02% of RDAPI model than classical PI model. The ME of RDAPI model was within 1.38%, whereas that of the classic PI model increased to 3.39% at 50 Hz. The results indicate that at high frequency, the modeling error of the classical PI model is larger than that of the RDAPI model. In addition, at 50 Hz, the proposed RDAPI model has smaller RMSE and AME than models in [28,29]. And compared to [30], the RDAPI model has a smaller RMSE but a larger AME. The proposed RDAPI model has the smallest ME among models in [28][29][30]. It is worth mentioning that, when comparing with models in [28][29][30], the proposed RDAPI model has the fewest parameters.
The experimental results indicate that compared with the classical PI model, the RDAPI model can more accurately describe the nonlinear hysteretic characteristics of the piezoelectric micropositioning stage, particularly at high frequencies. And the RDAPI model also has the better performance than modified models in [28,29]. Though the proposed RDAPI model does not perform as well as the model in [30] regarding AME, it has better RMSE and ME and has fewer parameters.

Experimental results for robust adaptive control
To validate the proposed robust adaptive control strategy, a series of experiments were conducted. The inverse RDAPI model-based feedforward compensation control (IRDAPI-based IFC) and inverse RDAPI model-based PID control (IRDAPI-based PID) methods were used for comparison. In the experiments, different trajectory forms were selected as reference signals, with frequencies ranging from 1 to 50 Hz. The parameters of the controller were selected via a trial-and-error approach. First, the adaptive adjustment parameters λ, η, and γ should be determined. A trade off exists between the tracking error and oscillation. The parameters can be set to small values initially and then gradually increased until a suitable value is reached and selected. Figure 10 presents the tracking performance with different values of λ, η, and γ . As shown, when the selected adaptive adjustment parameters are λ = 1e − 10, η = 1e − 7, and γ = 0.005, the tracking error is small and does not introduce sharp oscillation into the system. Then, by weighing the convergence speed and oscillation, the control gain parameters k 1 and k 2 can be tuned (gradually increased) to a suitable value. The control gain parameters are selected as k 1 = 2000 and k 2 = 0.005. According to the RDAPI model parameters identified in Sect. 3.3, β = 0.6714. According to physical upper bound of the controller input, it can be calculated that the boundary of the nonlinear part ς = αu 2 (t) + n i=1 p i F h r [u] (t) does not exceed 0.45. Therefore, ς = 0.45. According to the performance of the experimental stage and environment, the disturbance (environmental disturbance, tem-perature and humidity changes, etc.) does not exceed 1 µm; therefore, the limit of the disturbance was set as ρ = 1.

Displacement tracking control under sinusoidal reference signals
The desired trajectory is selected as x d (t) = 20 sin(2π f t + 3/2π) + 20, where f represents one of the input voltage signal frequencies (1,10,20,or 50 Hz). The tracking performance is shown in Figs. 11, 12, 13, and 14. The RMSE, AME, and ME were used to quantify the performance of the different control approaches. For an intuitive comparison, the RMSE, AME, and ME were plotted as bar Comparison of modeling errors between various PI models with different input frequencies graphs, as shown in Fig. 15. The proposed robust adaptive controller can remain the RMSE, AME, and ME within 0.3777 µm, 0.3480 µm, and 1.41% at 50 Hz. On the contrary, the RMSE, AME, as well as ME of IRDAPI-based IFC control were produced of 0.7139 µm, 0.6440 µm, and 2.82%, respectively. And those of the IRDAPI-based PID control are 0.6010 µm, 0.5102 µm, and 3.99%, respectively. Apparently, compared with the IRDAPI-based IFC and IRDAPI-based PID controllers, the proposed robust adaptive controller can significantly reduce the AME and the ME. Figures 11c, 12c, 13c, and 14c show the input-output Comparison results for the tracking performance under sinusoidal signals with a frequency of 1 Hz. a Reference and control output displacements; b tracking error; c hysteresis reduction for different controllers relationship of the closed-loop system with different controllers, indicating that the proposed robust adaptive controller can restrain the hysteresis nonlinearity better than IRDAPI-based IFC and IRDAPI-based PID controllers can.

Displacement tracking control under triangular reference signals
A triangular wave signal with an amplitude of 40 µm was selected as the desired trajectory, and the input voltage signal frequencies of 1, 10, 20, and 50 Hz were selected. To dampen the high-frequency modes, we shape the input by flattening the sharp edge of the signal. The tracking performance is shown in Figs. 16, 17, 18, and 19. The experimental results indicate that the proposed control strategy can eliminate the effects of hysteresis behavior at different frequencies more effectively than the IRDAPI-based IFC and IRDAPI-based PID controllers. The RMSE and AME values are presented in Fig. 20. The RMSE of the proposed method was maintained within 0.3084 µm, whereas those of the IRDAPI-based IFC and IRDAPI-based PID methods were 0.6035 and 0.9573 µm, respectively. The AME of the proposed method was within 0.2142 µm, whereas those of the IRDAPI-based IFC and IRDAPI-based PID The ME of the proposed method was within 3.00%, whereas those of the IRDAPI-based IFC and IRDAPI-based PID methods were 4.94% and 4.27%, respectively. As indicated by the results, compared with the IRDAPI-based PID and IRDAPI-based IFC controllers, the proposed control method reduces the maximum tracking error. Also, its RMSE and AME were significantly smaller. Thus, the proposed control strategy achieved better tracking performance than the other methods.

Displacement tracking control under other reference signals
To further verify the effectiveness of the proposed control method, additional types of reference signals were selected as the desired signals. First, a triangular wave with a decreasing amplitude was selected as the reference displacement. The results of the displacement tracking experiment are shown in Fig. 21.
When tracking the first triangular waveform, the proposed controller exhibited a large tracking error. After the robust adaptive controller was adjusted online, the system became stable at the second triangular waveform. Then, a stairstep signal was selected as the reference trajectory. The experimental results are presented in Fig. 22. When the desired displacement changed sharp, the IRDAPI-based PID controller exhibited a large amount of chatter, and the IRDAPI-based IFC controller required a long settling time. In contrast, the proposed controller achieved satisfactory control performance within a shorter settling time. Although the proposed controller exhibited an overshoot. However, compared with that of the IRDAPI-based PID controller, the overshoot was smaller, and compared with that of the IRDAPI-based IFC controller, the convergence was faster. Additionally, the proposed method had the smallest tracking error. A complex sine wave signal with different frequencies and amplitudes was selected as the reference trajectory. The experimental  Fig. 23. The tracking errors for the three types of reference signals are presented in Table 3.

Comparison of PI-based and RDAPI-based robust adaptive controllers
In this part, the control performance of the PI-based and RDAPI-based robust adaptive controllers were compared. The difference of control performance between   respectively. The MEs were 0.07% and 3.00%, respectively. It can be seen that, at low frequency, the PI-based and RDAPI-based robust adaptive controllers have a similar performance. However, in the high frequency, the RDAPI-based controller has a smaller tracking error than the alternative. That is because in low frequency, hysteresis nonlinearity can be described accurately by both PI and RDAPI model. While in high frequency, the hysteresis effect was enhanced, the PI model can no longer describe it precisely, the gain ς of the robust term in the controller was smaller than the actual hysteresis nonlinearity, so that the hysteresis nonlinearity cannot be well compensated. In contrast, the parameter In the foregoing experiments, the proposed robust adaptive controller outperformed both the IRDAPIbased IFC and IRDAPI-based PID control methods. In particular, when the reference trajectory was complex, e.g., harmonics and triangular waves compounded by multiple frequencies, the proposed control method offered significant advantages over the other methods. The experimental results indicate that the online parameter adjustment of the robust adaptive controller plays an important role in ensuring strong robustness and high control accuracy. The results also indicate that the proposed robust adaptive controller is more effective than previously reported controllers in terms of suppressing hysteresis behavior in piezoelectric micropositioning stages under different types of reference displacements.

Conclusion
An RDAPI model was proposed herein for describing the rate-dependent asymmetric hysteresis behaviors of piezoelectric micropositioning stages. The model introduces dynamic envelope functions into play operators to describe hysteresis behavior and replaces the proportional input function with the polynomial input function to describe asymmetry. The model uses few parameters and a simple analytical structure to describe the hysteresis. On this basis, a robust adaptive controller was designed to improve the tracking performance.