Memory-Enhanced Neural Network Control of Piezoelectric Actuators With a Rate-Amplitude-Dependent Hysteresis Model

Due to the presence of strong hysteresis nonlinearity, achieving robust and precise control of piezoelectric actuators (PEAs) is highly challenging. In this article, a novel rate-amplitude-dependent asymmetric Prandtl-Ishlinskii (RADAPI) model is proposed for modeling the hysteresis nonlinearity in PEAs and ultimately used for feedforward control based on its inverse model. Then, an uncertainty and disturbance estimator (UDE)-based controller using radial basis function (RBF) neural network is developed to address the issue of integral windup. To overcome the issue of passive knowledge forgetting, the selective memory recursive least squares weight update law is adopted. Moreover, the stability of the closed-loop system is demonstrated. A combined control scheme, incorporating RADAPI hysteresis inverse model feedforward compensation along with RBF-UDE based closed-loop feedback control, is devised to enhance the trajectory tracking accuracy of PEAs. Both theoretical analysis and experimental results are provided to validate the proposed control scheme.

Memory-Enhanced Neural Network Control of Piezoelectric Actuators With a Rate-Amplitude-Dependent Hysteresis Model Ju Zhang , Yiming Fei , Jiangang Li , Senior Member, IEEE, and Yanan Li , Senior Member, IEEE Abstract-Due to the presence of strong hysteresis nonlinearity, achieving robust and precise control of piezoelectric actuators (PEAs) is highly challenging.In this article, a novel rate-amplitude-dependent asymmetric Prandtl-Ishlinskii (RADAPI) model is proposed for modeling the hysteresis nonlinearity in PEAs and ultimately used for feedforward control based on its inverse model.Then, an uncertainty and disturbance estimator (UDE)-based controller using radial basis function (RBF) neural network is developed to address the issue of integral windup.To overcome the issue of passive knowledge forgetting, the selective memory recursive least squares weight update law is adopted.Moreover, the stability of the closed-loop system is demonstrated.A combined control scheme, incorporating RADAPI hysteresis inverse model feedforward compensation along with RBF-UDE based closed-loop feedback control, is devised to enhance the trajectory tracking accuracy of PEAs.Both theoretical analysis and experimental results are provided to validate the proposed control scheme.

I. INTRODUCTION
P IEZOELECTRIC actuators (PEAs) are renowned for gen- erating high-precision, high-resolution linear and rotational movements, making them widely applicable in various micro/nanolevel motion scenarios [1], [2], [3].The popularity of PEAs can be attributed to their inherent advantages, which include rapid responsiveness and large bandwidth [4].Ju Zhang and Jiangang Li are with the School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China (e-mail: 22S153113@stu.hit.edu.cn;jiangang_lee@hit.edu.cn).
Yiming Fei was with the School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China.He is now with the College of Computer Science and Technology, Zhejiang University, Hangzhou 310027, China (e-mail: yimingfei@zju.edu.cn).
Yanan Li is with the Department of Engineering and Design, University of Sussex, BN1 9RH Brighton, U.K. (e-mail: yl557@sussex.ac.uk).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TIE.2024.3376802.
Digital Object Identifier 10.1109/TIE.2024.3376802 While PEAs offer unique advantages compared to other actuators manufactured from smart materials, they often exhibit strong hysteresis effects in their output response [5].This can lead to distortions, oscillations, and even instability in closedloop systems.
Current research on control methods for PEAs focuses on voltage control, which is achieved by modulating voltage signals during the operation of PEAs.According to the presence or absence of output feedback, voltage control can be categorized into open-loop control with feedforward and closed-loop control with feedback.The most widely used open-loop control method is feedforward control based on hysteresis inverse models, which necessitates obtaining the inverse model of the nonlinear hysteresis system a priori.Some phenomenological models have been proposed and widely utilized for modeling the hysteresis in PEAs, such as the Preisach model [6], the Prandtl-Ishlinskii (PI) model [7], the Krasonsel'skii-Prkrovskii model [8], Duhem model [9], and the Bouc-Wen model [10].The PI model has gained widespread application due to its characteristic of being analytically invertible.However, the hysteresis nonlinearity in PEAs displays rate-dependent and asymmetric characteristics, making it difficult for the classic PI model to provide a more precise description.Thus, some researchers proposed enhanced PI model.In [5], the analytical inverse of a generalized PI model was formulated to compensate for the hysteresis nonlinearity of PEAs.A Dynamic Delay PI model was proposed to describe the asymmetrical and dynamic characteristics of PEAs in [11], which utilized delay coefficients to construct a modified Play operator and ultimately succeeded in acquiring an inverse model for the purpose of feedforward compensation in PEAs.To mitigate the hysteresis effect in PEAs, this article introduces a novel rate-amplitude dependent asymmetric PI (RADAPI) model to better represent the characteristics in PEAs.
In practical applications, feedback loops are indispensable to reduce tracking errors associated with modeling inaccuracies, disturbances, uncertainties, and other factors in nonlinear hysteresis systems.A PID sliding mode controller based on radial basis function (RBF) neural network was proposed in [12], leading to more precise tracking.In [13], a strategy combining iterative learning and hysteresis inverse model feedforward compensation was employed, enhancing the robustness of the system and achieving tracking accuracy close to the sensor resolution.
Furthermore, high-frequency or high-bandwidth control of PEAs has been one research hotspot in the past decade.In [14], an inversion-free predictive controller based on a dynamic linearized multilayer feedforward neural network model has been employed for trajectory tracking tasks at several hundred Hertz, yielding satisfactory results.The authors in [15] achieved precise trajectory tracking within the frequency range of up to 1000 Hz using a similar approach.In [16], trajectory tracking results up to 1500 Hz have been reported, demonstrating excellent performance, particularly in fast scanning applications.The aforementioned approaches each possess their own merits, yet they all grapple with tradeoffs between accuracy and design complexity, and the uncertainty and disturbance estimator (UDE)-based controllers are no exception.
The UDE-based control scheme has been introduced to mitigate the impact of uncertainties and disturbances, and developed successful applications in trajectory tracking control tasks for PEAs [17], [18], [19].Nevertheless, the integral action of the filter, combined with the initial large control input, can lead to integral windup in systems with input constraints.In [20], a design of bounded UDE-based controller using specialized error dynamic equations was developed to address the integral windup problem.However, this design resulted in diminishment of tracking performance in the closed-loop system.Improving the UDE-based control scheme while avoiding the integration effects of filters is an important question to address.
Motivated by deterministic learning [21], the RBF neural network is used for online estimation of uncertainties and disturbances, as a replacement for the filter employed in the UDE-based control method.However, the weight update law based on stochastic gradient descent (SGD) in [21] introduces the issue of passive knowledge forgetting.Thus, featured with the mechanism of memory, the selective memory recursive least squares (SMRLS) weight update law is adopted to overcome the issue of passive knowledge forgetting and achieve better inter-task generalization [22].Ultimately a UDE-based controller using RBF neural network based on the SMRLS weight update law is introduced in this article.Following this, in order to enhance tracking precision and disturbance resilience, a combination of the two aforementioned methods is devised, yielding a feedforward-feedback compound control strategy.In summary, this article makes the following contributions.
1) Compared to existing works [5], [23], the RADAPI model takes into account both rate and amplitude dependencies.
With the same number of operators, it has fewer identification parameters and demonstrates higher accuracy in modeling PEAs' hysteresis.
2) The integration of the RBF neural network with the UDE control framework, while ensuring weight convergence and system stability, mitigates integral saturation issue, and eliminates the need for complex filter design.3) This article extends the application of the SMRLS algorithm, previously employed in simulated scenarios [22], to a practical system.While overcoming the issue of passive knowledge forgetting, SMRLS enhances the convergence speed and generalization of the RBF neural network.
The rest of this article is organized as follows.The design of the RADAPI model will be presented in Section II.The design of the UDE-based controller, its integration with the RBF neural network, and the proof of the closed-loop stability will be discussed in Section III.Section IV gives the corresponding experimental results.Finally, Section V concludes this article.

A. Classic PI Model
The classic PI model is a well-known phenomenological model that uses weighted Play operators and linear input functions to describe hysteresis nonlinearity [23].
Analytically, suppose that C m [0, t E ] is the space of the piecewise monotone continuous functions.For any input v(t) ∈ C m [0, t E ], where 0 = t 0 < t 1 < . . .< t N = t E , such that the function v is monotone on each of the subintervals [t i , t i+1 ], the output of the Play operator F r [v](t) with a threshold r for t i < t < t i+1 and 0 ≤ i ≤ N − 1 can be expressed by the following equation [23]: where w(t) is the output of the Play operator.
According to (1), the expression for the classic PI model to describe the hysteresis nonlinearity of PEAs in the continuous time domain can be obtained as follows: where p(r) is an integrable positive density function, which is generally identified from the experimental data for a specific material or actuator; a is a positive constant; h(t) represents the hysteresis output, i.e., the output displacement of PEAs; v(t) represents the input voltage of PEAs.The upper limit of integration R can be chosen as infinity for convenience.
In practical applications, a finite number of Play operators are sufficient for modeling the hysteresis nonlinearity of PEAs in the discrete time domain as follows: where n is the number of the Play operators for modeling and ϕ(r i ) is the weight coefficient corresponding to different thresholds r i .

B. RADAPI Model
The classic PI model struggles to accurately represent hysteresis nonlinearity in PEAs with rate-dependent, amplitudedependent, and asymmetric characteristics.Therefore, some improvements will be made to the classic PI model to reflect the aforementioned characteristics.
Given that PEAs exhibit a positive excitation property, the one-side Play (OSP) operator can be utilized to describe hysteresis nonlinearity in PEAs without computing v(t) + r for each moment, resulting in a reduced computational burden of parameter identification.By replacing the input function v(t) in the rising and falling branches of the OSP operator with two dynamic envelope functions, the modified one-side Play (MOSP) operator can be expressed as follows: where s(t) is the output of the MOSP operator; h l (v, v) and h r (v, v) are dynamic envelope functions that influence the shape of the hysteresis loop.In addition, by reasonably selecting the dynamic envelope functions, the MOSP operator can exhibit asymmetry characteristics.
Given the characteristic of the hysteresis loop width of PEAs increasing with the rate, the dynamic envelope functions can be chosen as follows [24]: where α and β are positive constants; v(t) is the derivative of the input voltage v(t) and is estimated as T , with sampling period T .
Considering that the peak-to-peak amplitude of the output displacement of PEAs decreases with increasing rate, a dynamic density function is introduced as follows [24]: where ρ, λ, and γ are constants to be identified, with γ > 0.
To better represent the asymmetrical nature of hysteresis nonlinearity in PEAs, a cubic polynomial function with strong nonlinear features is introduced.To reduce the number of parameters to be identified, let β = α in (5), resulting in the MOSP operator having only rate-dependent characteristics and no longer exhibiting asymmetry properties.Finally, after introducing amplitude-dependent coefficient functions a(A), b(A), and c(A), the expression for the RADAPI model to describe the hysteresis nonlinearity of PEAs in the discrete time domain can be obtained as follows: where A is the amplitude of the input signal; a(A), b(A), and c(A) are coefficient functions to be identified, which are related to the amplitude of the input signal; ϕ(r i , v) = (r i − r i−1 )p(r i , v) represents the dynamic weighting coefficient corresponding to the threshold r i .
Remark: By prescribing the dynamic density function with a particular form in (6), the necessity for parameter identification for every weighting coefficient is eliminated, resulting in a substantial reduction of computational burden in the identification process.In comparison with existing works such as [24], the RADAPI model adopts a dynamic density function, as shown in (6), and introduces amplitude-dependent coefficient functions in (7) to better represent the hysteresis in PEAs.

A. Problem Formulation and UDE-Based Controller Design
The response of PEAs exhibits characteristics similar to distributed-parameters systems and can be accurately described using partial differential equations [25].In practical applications, the working frequency of PEAs hardly exceeds their first natural frequency.Therefore, the distributed-parameters nature of PEAs can be neglected, and the model can be simplified as a lumped parameter representation [26].The dynamics of the PEAs can be identified as a mass-spring-damper second-order linear system cascaded with a nonlinear hysteresis function as follows: where t is the time variable; m, b, k, and d represent mass, damping coefficient, stiffness, and piezoelectric coefficient, respectively; v(t) and y(t) represent the input voltage and output displacement of the PEAs, respectively; h(t) represents the hysteresis output.
As shown in Fig. 1, taking into account the influence of uncertainties and disturbances in practical systems, the dynamic model of the PEAs is ultimately represented as follows: where ξ = By defining ] T as the state vector, (9) can be rewritten as follows: where To ensure that the closed-loop system meets the specified control requirements, the following stable reference model ( 11) is selected, which has a controllable canonical form: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where represents the reference state vector and is also the desired trajectory; y m (t) is the reference model output, and c(t) is the piecewise continuous and uniformly bounded control input for the reference system.The control objective is to design the control signal v(t) in such a way that the tracking error e(t) = [e 1 (t) e 2 (t)] T = x m (t) − x(t) asymptotically converges to zero, satisfying the desired error dynamics equation as follows: where K is the error feedback gain matrix.
To facilitate the design of the UDE-based control law, we will rewrite the first term of ( 10) and the lumped vector u d (v(t)) can be expressed as In traditional UDE-based controller for PEAs [19], a strictly proper stable filter G f (s) with unit gain and zero phase shift in the spectrum range of u d (v(t)) and zero gain elsewhere is used to estimate the lumped vector as follows: where * represents the convolution operator, and g f (t) is the impulse response of the filter G f (s).
As derived in [19], the UDE-based control law for PEAs is ultimately as follows:

B. UDE-Based Controller Design Using RBF Neural Network
To address the issue of integral windup and circumvent the need for intricate filter design, this article will use the RBF neural network to replace the filter in the traditional UDE-based controller as the estimator for uncertainties and disturbances.
While omitting the independent variable time t, the dynamics of the PEAs proposed in (10) will be rewritten as follows: Combining (14) and the relevant matrix definitions, we can rewrite the lumped vector as follows: where f 1 = 0 and the actual uncertainties and disturbances term (17), the actual dynamics of the PEAs can be obtained as follows: To achieve the trajectory tracking task of the PEAs, consider the reference model as follows: where is the reference trajectory, which is uniformly bounded and within a compact set R 2 ; f d (x d ) : R 2 → R is a known smooth function a priori.Define the composite tracking error as follows: where z = [z 1 , z 2 ] T ∈ R 2 is the error state vector; K 1 is a proportional control gain.Considering the dynamics (19), the reference model (20), and the definition of the composite tracking error (21), the open-loop error dynamics can be expressed as follows: Then, an ideal feedback linearization control law can be designed as follows: where K 2 is a proportional control gain, and p(x) = ) is an unknown nonlinear function that includes uncertainties and disturbances.According to Remark 3 and Section IV-B in [27], we can consider using p(x d ) instead of p(x) in ( 23) to obtain a new control law as follows: which represents the ideal control variable after using p(x d ) to replace p(x).Lemma 1: The UDE-based control law ( 16) has the same form as the feedback linearization control law (24).
Proof: Without the filter, the UDE-based control law ( 16) can be rewritten as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
which can be further rewritten as follows: If we consider e 1 = z 1 , e 2 = ż1 , and change the symbol of the coefficients, the above equation becomes where 27) is of the same form as (24), and the lemma is proven.
From Lemma 1, the relevant content derived above can serve as the foundation for weight update law and stability analysis.In order to enable the RBF neural network to estimate the unknown nonlinear function p(x d ) and thus realize the control law (24), the output of the RBF neural network, Ŵ T S(x d ), is considered as a replacement for p(x d ) to obtain the real control law where Ŵ ∈ R N is an estimated value of the optimal weight vector W * .In reference to [28, Section 5.1], we can derive the weight update law based on SGD as follows: where W = W * − Ŵ represents the weight estimation error; Γ ∈ R N ×N is a positive definite symmetric matrix, and σ is a positive constant.In order to achieve faster convergence and overcome the issue of passive knowledge forgetting, the SMRLS weight update law is introduced as follows [22]: where S a (k) is the recorded regressor vector before sampling time k and a = P S a (F a − Ŵ T S a ) satisfying a < * a with F a being the latest record of F (F = ηz 2 + Ŵ T S(x d ) is the estimated desired output of the RBF neural network) within the current partition before the current time is the continuous representation of a (k + 1).
In SMRLS weight update law, the feature space of the RBF neural network is normalized and uniformly discretized into a finite number of partitions.During the online learning stage, each partition continuously aggregates all its internal samples into a single integrated sample.

C. Stability Analysis
Assumption 1: The persistent excitation (PE) condition of the regressor subvector S ζ (Z) is satisfied with any input trajectory of the RBF neural network whose duration is finite.
With Assumption 1, only the PE condition of S ζ (x d ) is guaranteed.Through elementary row transformation, Ŵ and S(x d ) can be transformed into the following form: where Q ζ ∈ R N ×N is an orthogonal constant matrix, and the subscript ζ represents the neurons which are far from the trajectory of x d and barely affect the approximation.Therefore, the following equations are obtained: where Substitute (33) into (32) and the following weight update law is obtained: where is a positive definite matrix.According to the localized approximation property of the RBF neural network, the values of Sζ(x d ) are close to zero during the learning stage.Therefore, we have where W ∈ R N ζ satisfying W < * W is a small filtered approximation error, and Defining the function estimation error as p2 (x, x d ) = p(x) − p(x d ), the output of the RBF neural network can be expressed as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

= p(x)
where p1 (x d ) is a bounded estimation error that satisfies | p1 (x d )| < * p1 , and p (x, x d ) = p1 (x d ) + p2 (x, x d ) is the composite estimation error.Therefore, the closed-loop dynamics of the RBF-UDE based system is obtained where Theorem 1: Consider the nominal system of (37) as follows: With the PE condition of S ζ (x d ) and sufficiently large control gains η, k 1 and k 2 , (38) is exponentially stable.
Proof: Apply the linear transformation to (38) and we obtain where Thus, we can guarantee that Q is positive definite by choosing where c 1 > ω 2 n and c 2 > 0 are constants.Consider the following Lyapunov function candidate: and V1 along (38) satisfies where σ > 0 is an estimate of the maximum eigenvalue of Ṗ −1 ζ .
Let A = w Ẇζ and ( 40) is decomposed into where From [29, Lemma 1], the nominal system Ȧ = BA is exponentially stable.And from [30,Th. 4.12], there exists a positive definite matrix Γ such that Γ T B + B T Γ = −D where D is a given positive definite matrix.Consider the following Lyapunov function candidate: Thus, V2 along (45) satisfies where λ 0 is an estimate of the upper bound of . Consider the following Lyapunov function candidate: and V3 along (40) satisfies By choosing π = 2λ 2 0 λ min (Q)λ min (D) , we have Consider ( 41) and (50), and we increase the gains η, k 1 , and k 2 such that Substitute (51) to (50) and we obtain The theorem is proved.Notably, the proof generally follows the steps of Lemma 4 in [22].Theorem 2: Consider the system (37), with the PE condition of S ζ (x d ) and properly designed control gains η, k 1 and k 2 , z and Wζ will exponentially converge to small neighborhoods around zero.
Proof: According to Theorem 1, the nominal part of (37) is exponentially stable.
According to [22, Theorem 2], the system (37) is semiglobally exponentially stable with properly designed control gains.Specifically, z and Wζ will exponentially converge to small neighborhoods around zero.The theorem is proved.

A. Experimental Setup
The experimental setup comprises several components, including PEA, piezoelectric controller, real-time simulation controller, and computer, as shown in Fig. 2. Specifically, the PEA utilized in this system is a cylindrical low-voltage PEA (model number PSt150/10/60VS15) manufactured by Harbin Core Tomorrow Science and Technology Company Ltd.The physical parameters of the PEA are listed in Table I.The piezoelectric controller used is the E53.B servo controller equipped with an strain sauge displacement sensor with a sensitivity of 6 µm/V and measurement noise of 0.05 µm.Finally, the real-time simulation controller is a DS1103 PPC Controller Board manufactured by dSPACE.

B. System Identification
1) Performance Index: In this article, the root mean square error (RMSE) and the maximum absolute error (MAXE) are used as the evaluation criteria for the subsequent experiments.Their definitions are as follows: where L represents the total number of data points; y i corresponds to the experimental data at time i, and ŷi represents the model output or expected output at time i. 2) Hysteresis Modeling: To obtain the specific form of the amplitude-dependent coefficient functions in the proposed RADAPI model, parameter identification for the RADAPI model is performed using the particle swarm    III, and Table IV.d) Use the model output h(t) and actual displacement y(t) as the input and output, respectively, for the second-order linear system identification using the MATLAB System Identification Toolbox.e) Eventually, the damping ratio ξ is determined to be 0.2707, and the natural frequency ω n is found to be 2325.1 rd/s.

C. UDE-Based Control for PEAs Using RBF Neural Network in Trajectory Tracking
With the input signal v(t) = 0.5sin(20πt − π 2 ) + 0.5, the experiment for the trajectory tracking task of the PEAs using an RBF-UDE based controller based on SMRLS weight update law is conducted.When our PEA operates under precise control strategies, the output displacement (the digital signal output from the sensor) is six times the input voltage (the digital signal input to Dspace).Therefore, the actual desired trajectory is v(t) = 3sin(20πt − π 2 ) + 3. The PID controller (serving as an error filter) parameters are set as K p = 0.54, K i = 400, and K d = 0.0001.The positions of hidden nodes are distributed with a [0, 1] lattice distribution, as shown in Fig. 4(a).The initial covariance matrix is chosen as P 0 = 1 × 10 6 , with the feature space partition of 400 × 400 in a uniform distribution.The receptive field width is set to 0.2, and the learning rate is 0.01.The learning stage of the RBF-UDE based controller shows the tracking error and weight convergence situation in Fig. 4(b) and (c).In the knowledge reuse stage, the weight coefficients are fixed to the values obtained at 30 s into the learning stage.The tracking error during the knowledge reuse stage is depicted in Fig. 4(d).
From Fig. 4, it is evident that the RBF-UDE based controller achieves a high level of precision, reaching the level of measurement noise (0.05 µm).This suggests that the RBF-UDE based control method exhibits excellent trajectory-tracking performance.

D. Performance Under Different Weight Update Laws
To comprehensively assess the impact of the SMRLS weight update law compared to the SGD weight update law on learning performance, with the input signal v(t) = 0.5sin(2πt − π 2 ) + 0.5, the experiment for the trajectory tracking task of the PEAs using an RBF-UDE based controller based on different weight update laws is conducted.As depicted in Fig. 5, the closed-loop system converged at 1 s under the SMRLS weight update law, while under the SGD weight update law, it did not fully converge even at 10 s.This indicates that the SMRLS weight update law exhibits faster convergence compared to the SGD weight update law.
To validate the generalization capability of RBF-UDE based controller under different weight update laws, a random nonuniform rational B-splines (NURBS) reference trajectory ϕ A with Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.x d1 , x d2 ∈ [0, 1] is adopted in the learning stage, as shown in Fig. 6(a).In Fig. 6(b), the feature distribution of ϕ A is not uniform, which is very suitable for verifying the nonlinear representation ability of RBF neural networks under different weight update laws.We applied the weights at the 50 s in the learning stage to the tracking task of another random NURBS trajectory ϕ B , and the tracking errors are shown in Fig. 6(c).The results indicate that the control method based on SMRLS has a stronger generalization capability.

E. Comparison With Different Control Schemes
Existing literature substantiates the presence of an inverse model for a hysteresis model in the form of the PI model, exhibiting a structural congruence with the original one [24].Expanding the aforementioned concept to the RADAPI model, it is evident that the form of the inverse model for the RADAPI model remains the RADAPI model itself.Therefore, by utilizing the input-output of the RADAPI model as the output-input for its inverse model and employing the same parameter identification methodology, we can obtain the RADAPI inverse model to serve as a feedforward compensator.
In this section, the RBF-UDE based controller, the RBF-UDE based controller with a RADAPI inverse model compensator (RADAPI-RBF-UDE), and the third-order integral sliding mode controller (3-ISMC) [31] will be considered with experimental results presented for the purpose of comparison.The control diagram of RADAPI-RBF-UDE is illustrated as shown in Fig. 7.
The input signal used in this experiment is v(t) = 0.5sin(2πf t − π 2 ) + 0.5 with frequency change range f = [1,5,10,20,30,50, 100]Hz.The tracking errors and hysteresis loops for the three control schemes at a frequency of 1 Hz are shown in Fig. 8

TABLE V PERFORMANCE INDEX FOR DIFFERENT CONTROL SCHEMES AT DIFFERENT FREQUENCIES
the three control schemes at different frequencies are presented in Table V.
The results indicate that the 3-ISMC control scheme performs the worst in terms of trajectory tracking, exhibiting a more noticeable closed-loop hysteresis, with a significant increase in error at higher frequencies.Among the three control schemes, RADAPI-RBF-UDE significantly outperforms the others in terms of RMSE and MAXE.When considering the average values, its RMSE is reduced by 44.0% compared to RBF-UDE and by 67.5% compared to three-ISMC.MAXE is reduced by 25.4% compared to RBF-UDE and by 62.1% compared to 3-ISMC.

V. CONCLUSION
The main contributions of this article lie in the development of the highly accurate RADAPI model and the innovative integration of RBF neural networks with the traditional UDE-based control method.This approach overcomes issues such as heavy reliance on actual system characteristics, limited design margins, and integral windup.Experimental results demonstrated that the RADAPI-RBF-UDE control scheme employed in the trajectory tracking of the PEAs exhibited superior performance.Moreover, this work utilized a novel weight update law to address the phenomenon of passive knowledge forgetting.Nevertheless, the closed-loop stability of the system was influenced by the convergence properties of the RBF neural network and may need to be addressed in future work.His current research interests include reinforcement learning, cognitive neuroscience, neural network control, system identification, and motion control.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Manuscript received 29
November 2023; revised 31 January 2024 and 23 February 2024; accepted 5 March 2024.Date of publication 26 March 2024; date of current version 7 August 2024.This work was supported by the National Natural Science Foundation of China under Grant U1913213.(Corresponding authors: Jiangang Li; Yanan Li.)

k m and ω n = b 2 1
km represent the damping ratio and natural frequency, respectively; H[v](t) = dv(t) + h(t) is the rewritten hysteresis output and d 1 (t) represents bounded disturbances and uncertainties.
optimization (PSO) algorithm.This involves applying the PSO algorithm to identify the parameters based on input signals of different amplitudes but the same frequency.The final expression for the coefficient functions in the RADAPI model is as follows: a(A) = − 0.04111 A −4.323 b(A) = − 0.08818 A 2 + 0.8012A + 5.525 c(A) = − 0.02654 A 3 + 0.1767 A 2 + 0.2684A + 0.03005.(54) To validate the modeling accuracy of the RADAPI model, the modified PI (MPI) model, the generalized PI (GPI) model, and the RADAPI model are employed to model the hysteresis nonlinearity of the PEAs.The parameter identification results for the RADAPI model are presented in Tabel II.The modeling results are shown in Fig. 3, Table

3 )
Linear Dynamics: Choosing the RADAPI model for the hysteresis modeling of the PEAs, the linear dynamics of the PEAs can be identified through the following steps.a) Utilizing the PSO algorithm, we can identify the parameters of the RADAPI model under a sinusoidal input signal with an amplitude of 0.5 V, frequency of 1 Hz, and phase shift of -0.5π.b) Generate a sinusoidal sweep signal ranging from 1 to 100 Hz as the input for the identified RADAPI model and obtain the model output h(t).c) Take the sinusoidal sweep signal generated in the previous step as the input to the PEA in an openloop state and obtain the actual displacement y(t).

Fig. 4 .
Fig. 4. (a) Lattice distribution.(b) Tracking error in the learning stage.(c) Weight convergence situation in the learning stage.(d) Tracking error in the knowledge reuse stage.
(a) and (b), respectively.Performance indices for

Ju
Zhang received the B.S. degree, in 2022, from the Harbin Institute of Technology, Shenzhen, China, where he is currently working toward the M.S. degree, all in control engineering.His research interests include hysteresis nonlinearity and robust control.Yiming Fei received the B.Eng. degree in automation and M.Eng.degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2020 and 2023, respectively.He is currently working toward the Ph.D. degree in computer technology with Zhejiang University, Hangzhou, China.

TABLE I PHYSICAL
PARAMETERS OF PEA

TABLE IV PERFORMANCE
INDEX UNDER DIFFERENT AMPLITUDES