a MOE Engineering Research Center of Autonomous Intelligent Unmanned Systems, School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China
b Department of Mechanical Engineering, University of Victoria, Victoria, BC V8W 2Y2, Canada
c State Key Laboratory of Intelligent Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
This study establishes a high-speed nano-positioning stage composed of a symmetrically driven structure with multiple parallel-bonded thin piezoelectric ceramic layers capable of performing micro- or nano-scale manipulations. Accordingly, a neural-network-based switching output regulation controller (NN-SORC) was developed to compensate for the associated hysteresis nonlinearity. To address the challenges of slow floating-point computation speeds and low compilation efficiency, a closed-loop control system with a field-programmable gate array-central processing unit (FPGA-CPU) dual-layer data-processing framework was developed. A feedback linearization method was designed to linearize the hysteresis nonlinearity of the framework, resulting in a switching-tracking error system. With the assistance of Lyapunov theory and an average dwell time technique, sufficient conditions were derived to ensure the asymptotic stability of the NN-SORC governing closed-loop system using the switching reference signals often encountered in realistic micro-/nano-scale detection and manufacturing processes. Finally, extensive comparative experiments were conducted to verify the effectiveness and superiority of the proposed NN-SORC scheme.
The recent years have witnessed a tremendous development in nano-positioning stages owing to their widespread application in various nano-scale detection and manufacturing processes, such as atomic force microscopes (AFMs) [1], fast tool servo (FTS) [2], three-dimensional (3D) printing machines [3], and ultra-precision machine tools [4]. However, for conventional motors, achieving high-speed and high-precision nano-scale positioning remains challenging. Therefore, numerous recent studies [[5], [6], [7], [8]] have utilized piezoelectric ceramics as executors, namely piezoelectric actuators (PEAs), for nano-positioning stages. However, the positioning accuracy of the nano-positioning stages is inevitably influenced by the hysteresis nonlinearity effect of the voltage-driven PEAs. As a remedy, some scholars have established simplified feedforward hysteresis control systems that describe the piezoelectric-driven nano-positioning stages by sequentially combining the hysteresis nonlinearity with linear dynamics models, accordingly designing inverse hysteresis models as feedforward controllers to eliminate the complex hysteresis in nano-positioning stages. So far, there are two mainstream types of hysteresis models: operator-based models such as Preisach [9], Prandtl-Ishlinskii [10], and Krasnosel’skii-Pokrovskii [11], and differential-equation-based models including backlash-like [12] and Bouc-Wen [13]. Because most of these models can only describe hysteresis along a single axis, approximating the cross-coupling effect in multi-axis cases remains a dilemma [14]. To address this challenge, most existing studies have focused on decoupling the cross-coupling, aiming to facilitate the design of feedforward controllers. For example, a data-driven cross-axis that influences the compensation scheme was developed to mitigate the cross-coupling effects of AFMs [15]. A novel, adaptive, reliable controller was developed based on a fuzzy backstepping protocol [16]. Furthermore, a proportional resonant controller that compensates for the baseline tracking controller at different frequencies was designed [17]. An adaptive global sliding-mode control scheme using an integral extended-state observer technique was devised to avoid sensitivity to external perturbations during the reaching phase [18]. A fast nonsingular terminal sliding-mode controller with an embedded inverse hysteresis model was also proposed to guarantee global convergence with a high bandwidth [19].
However, the tracking precision of such inverse model-based compensation control highly depends on the accuracy of the identified hysteresis model, and obtaining a theoretical inverse for such complex hysteresis nonlinearities is challenging. Therefore, in recent years, efforts have been increasingly devoted to a more promising research direction, namely, treating hysteresis as an external disturbance and transforming hysteresis compensation into an anti-disturbance problem. Thus, a series of feedback controllers can be devised to eliminate the influence of external disturbances, thereby compensating for hysteresis. As a pioneering work along this research line, a disturbance-observer-based controller was employed to estimate and compensate for hysteresis nonlinearity [20]. Furthermore, based on the active disturbance rejection control (ADRC) technique, a current-cycle iterative learning control scheme was developed in Ref. [21] by transforming a PEA into a second-order disturbance-based system to compensate for both hysteretic nonlinearities and system uncertainties. In Ref. [22], a data-driven iterative parameter-tuning ADRC scheme was developed to improve the high-bandwidth signal tracking efficiency. Along this line of research, a reinforcement learning-based ADRC scheme was presented with adaptively tuned parameters by the Q-learning method to adapt to variational reference signals [23]. A novel hierarchical anti-disturbance control strategy equipped with both a feedforward compensator and a disturbance observer was proposed in Ref. [24] to alleviate the bandwidth constraint of the disturbance observer imposed by mechanical resonance. An online identification and neural-network (NN)-based iterative learning control scheme was proposed in Ref. [25] to minimize the repetitive tracking errors caused by hysteresis nonlinearity and model perturbations. A deep parallel model-based model predictive control (MPC) method was designed in Ref. [26] to mitigate the influence of hysteresis nonlinearity.
The switching control strategy offers an effective way to improve the performance, adaptability, and robustness of complex systems subjected to external disturbances, uncertainties, and nonlinearities [27]. In a pioneering study, a switching control method was used to resolve the estimated speed error problem and guarantee the stability of systems with a speed-sensorless induction motor driving control system [28]. In Ref. [29], a dynamic zero-moment-point-based adaptive switching control strategy was devised to promote the controllability of a versatile hip exoskeleton under hybrid locomotion. Subsequently, a unified switching control framework was established to improve the performance of the continuous robot-assisted training [30]. Despite the contributions made to the development of high-speed/high-precision nano-positioning techniques by these advanced controller research contributions, unsolved challenges remain, as outlined below:
(1) High-efficiency computations with high-precision floating-point numbers place significant demands on the hardware capabilities of nano-positioning control systems.
(2) Compensating for reference rate-dependent hysteresis adaptively can yield high tracking precision for switching reference signals.
To this end, a parallel-bonded differential-driven nano-positioning stage combined with capacitive displacement sensors and voltage amplifiers is established in this study. A field-programmable gate array (FPGA) dual-layer data processing module is embedded in the control system to facilitate high-efficiency data acquisition, processing, compilation, and high-precision control signal calculation. In addition, an NN-based switching output regulation controller (NN-SORC) is developed to adaptively compensate for the hysteresis nonlinearities for switching reference signals, and an NN is employed for tuning the adaptive controller parameter. Finally, sufficient conditions are derived to guarantee the asymptotic convergence of the closed-loop nano-positioning control system governed by the proposed NN-SORC.
In brief, the contributions of the present study are two-fold:
(1) Establishing an FPGA-based high-efficiency closed-loop nano-positioning control system with a parallel-bonded differential-driven structure, enabling 10 µm stroke with 140 Hz bandwidth.
(2) Developing an NN-SORC for high-speed/high-precision switching reference-signal tracking, as often encountered in realistic micro-/nano-scale detection and manufacturing processes, thereby deriving sufficient conditions to guarantee the asymptotical stability of the closed-loop nano-positioning control system.
Notation: ℕ denotes the positive integer set, $\mathbb{R}$ is the set of real numbers; AT is the transpose of matrix A; $\left|\mathit{A}\right|$ is the Euclidean norm of matrix A; $\ddot{x}$ is the double differential form of variable x; $\dot{x}$ is the differential form of variable x. A function f1:[0,∞) → [0,∞) is a $\mathcal{K}$ -class function if it is continuous, strictly increasing, and satisfies f1(0) = 0. A function f2:[0,∞) → [0,∞) is a ${\mathcal{K}}_{\mathrm{\infty }}$ -class function if it is continuous, strictly increasing, and satisfies f2(0) = 0 and $\underset{s\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{f}_{2}\left(s\right)=\mathrm{\infty }$ . “◦” denotes composition of functions.
2. System description
2.1. Hardware architecture of the nano-positioning platform
The experimental platform for the nano-positioning stage is shown in Fig. 1. This stage is driven by a stacked PEA consisting of multiple parallel-bonded thin piezoelectric ceramic layers. This design enables the actuator (Fig. 1(c)) to withstand large axial compression loads and accordingly generate proper thrust. The stage employs a differential-driven module, which has a symmetrically driven structure, that is, two stacked PEAs installed on each side of the stage, respectively, working in a “push-and-pull” way to drive the displacement of the stage. As shown in Fig. 1, PK2JQP1-stacked (18 µm) PEA of Thorlabs (USA), with dimensions of 3.0 mm × 3.0 mm × 20.0 mm was selected. The standard preload force and maximum output force were 785 and 1960 N, respectively. In the differentially driven mode, the input signal to the PEA consists of a varying voltage signal vax for the X-axis and vay for the Y-axis with the same amplitude yet opposite phases and a constant bias voltage signal Vb. To avoid depolarization, the instantaneous input voltage to each stacked PEA was always positive and did not exceed the maximum voltage Vmax, that is, 0 < Vb ± vax < Vmax, 0 < Vb ± vay < Vmax with Vmax = 150 V. Additionally, a capacitive displacement sensor (MicroSense 6810, with a 6502 probe, MicroSense, LLC, USA) was adopted to detect the displacement along each axis within the 10 kHz measurement bandwidth and ±100 µm stroke. Because PEAs require high driving voltages, a voltage amplifier (PD200, PiezoDrive, Australia) with a gain factor of 20 was employed to increase the output signal and filter the voltage analog signal. A Performance real-time target machine (Intel Core i7 4.2 GHz, 4 core, 4 GB random access memory (RAM), Speedgoat GmbH, Switzerland) and an upper computer (Intel Core i7 8700 central processing unit (CPU), 3.20 GHz, 6 core, 16 GB RAM) were used for data acquisition and design of the controller.
The architecture of the nano-positioning stage, including both top and oblique side views, along with the key mechanical parameters, is shown in Fig. 2 [5]. The driving mechanism of the PEA is also illustrated, highlighting its structural design and mechanical principles. The structure utilizes a differential actuation technique, whereby each axis is equipped with two opposing actuators that work differentially to enable bilateral motion. This approach offers the dual advantage of enhancing both the stroke and linearity of the actuation mechanism in contrast to traditional single-sided actuation systems [6]. The stroke of our nano-positioning stage was confined to 10 µm because of the constraints of flexible hinge size and driving force limitations of the PEAs. The stroke can be significantly increased with a larger hinge size or driving force.
The closed-loop control system architecture of the nano-positioning stage is shown in Fig. 3. The tracking error ${\stackrel{\sim }{x}}_{1}$ between the reference xr(t) and measured output signal x is detected by the capacitive sensor and fed into NN-SORC, where NN is trained to adaptively tune the key control parameter κ1. Subsequently, the control signal vi calculated by NN-SORC is utilized to drive the stacked PEAs, generating a preferable displacement x of the stage through the mechanical structure of the stage (Figs. 1(b) and (c)). Thereafter, the displacement x is detected by the capacitive sensor and fed back to NN-SORC, forming a feedback loop.
2.2. FPGA-CPU dual-layer data processing module
As shown in Fig. 4, the NN-SORC requires extensive computations involving high-precision floating-point numbers; therefore, an FPGA is employed in the data processing system. We selected the FPGA IO334 module, whose core is a Xilinx Kintex-7 XC7K325T equipped with a 16-channel, 16-bit high-precision analog-to-digital converter (ADC) and a digital-to-analog converter (DAC). The NN-SORC depends on a Speedgoat real-time target machine running in the Simulink real-time environment.
Although FPGA can efficiently process large amounts of data, during the control signal calculation process two issues emerged:
(1) The code of the stage was produced using Simulink in MATLAB, and the C-coding executable program for the CPU was automatically generated using Simulink Coder. By contrast, the executable language for the FPGA was generated as a hardware description language (HDL) by the HDL Coder. Therefore, the compilation time for the CPU was short (1-2 min), whereas the FPGA compilation process required more than 40 min, resulting in incompatible compilation periods for the FPGA update.
(2) Introducing external parameter-tuning interfaces that are intractable makes real-time data flow monitoring difficult.
To address these two issues, we developed an FPGA-CPU dual-layer data processing module, as shown in Fig. 4(a). In the internal layer, the FPGA converts input and output signals, implements control algorithms, and generates signals. In the external layer, the CPU sets the parameters, monitors real-time status, and calls the control toolbox and does not require high efficiency. Using this design, the proposed system achieves a sampling computation frequency of up to 10 MHz in the internal layer and 100 kHz in the external layer. Fig. 4(b) illustrates the data flow architecture of the designed controller, highlighting its integration and operation. The FPGA handles the conversion of external signals through its DAC and ADC modules in performing initial data processing. High-speed data exchange between the FPGA and CPU is realized via direct memory access (DMA), ensuring efficient communication. In addition, the real-time target machine interfaces with FPGA using AXI4-Stream for seamless data exchange.
2.3. Electromechanical dynamics model of the PEA of the nano-positioning stage
The electromechanical model of the PEA is divided into n segments according to the magnitude of the system output, each approximated by [23,31]
where ${v}_{i}^{\sigma \left(t\right)}\in {\mathbb{R}}^{2}$ (i = 1, 2) is the control input of X-axis (i = 1) or control input of Y-axis (i = 2); $x\in {\mathbb{R}}^{2}$ is the control output; and the operating sequence is scheduled using the switching signal below.
Here, the function $\sigma \left(t\right)\in \left\{\mathrm{1,2},\dots,n\right\}$, $n\in \mathbb{N}$ ; the parameter set boundary values δl ($l\in \left\{\mathrm{1,2},\dots,n\right\}$ ) are randomly set. Switching instant tk is defined as
where ${\stackrel{\sim }{x}}_{1}\in \mathbb{R}$ and ${\stackrel{\sim }{x}}_{2}\in \mathbb{R}$ are the state variables, ${\kappa }_{1}={\mathit{\omega }}^{\text{T}}\varphi \left({\stackrel{\sim }{x}}_{2}\left(t\right)\right)$ is the output of the NN, ω is the weight parameter vector of the NN, as shown in Fig. 3, and $\varphi \left({\stackrel{\sim }{x}}_{2}\left(t\right)\right)=1/\left(1+{\text{e}}^{{\stackrel{\sim }{x}}_{2}\left(t\right)}\right)$ is the activation function of the NN. Then, the closed-loop switched system becomes
where ${v}_{i}^{\sigma \left(t\right)}\left(t\right)={f}^{\sigma \left(t\right)}\left(x\left(t\right),\dot{x}\left(t\right),{x}_{\text{r}}\left(t\right),{\dot{x}}_{\text{r}}\left(t\right),{\ddot{x}}_{\text{r}}\left(t\right),\mathit{\omega }\right)$ is the NN-SORC law, which is designed later. In this study, we focus on the associated signal-tracking control problem, formulated as follows. For simplicity, we use “$x,\dot{x},{x}_{\text{r}},{\dot{x}}_{\text{r}},{\ddot{x}}_{\text{r}},{\stackrel{\sim }{x}}_{1},{\stackrel{\sim }{x}}_{2},{v}_{i}$ ” instead of “$x\left(t\right),\dot{x}\left(t\right),{x}_{\text{r}}\left(t\right),{\dot{x}}_{\text{r}}\left(t\right),{\ddot{x}}_{\text{r}}\left(t\right),{\stackrel{\sim }{x}}_{1}\left(t\right),{\stackrel{\sim }{x}}_{2}\left(t\right),{v}_{i}\left(t\right)$ ” in the following context.
Problem 1
For a switched PEA system governed by Eq. (12), a control protocol is designed as follows:
To facilitate the subsequent derivation, the following assumption and definition are necessary.
Assumption 1
The reference signal xr in Problem 1 is second-order differentiable.
Definition 1
[33] For a switching signal σ(t) and any instants T > t > 0, let Nσ(t, T) be the switching numbers over the time interval (t, T). If there exist N0 ≥ 0, τa > 0 such that Nσ(t, T) ≤ N0 + (t − T)/τa, then, τa is called the average dwell time (ADT) [33], and N0 is the chatter bound.
Definition 2
[34] For the closed-loop switched system Eq. (12), if all solutions ${\stackrel{\sim }{x}}_{2}\left(t\right)$ of the system starting from arbitrary initial condition ${\stackrel{\sim }{x}}_{2}\left({t}_{0}\right)$ satisfy ϛ$∥{\stackrel{\sim }{x}}_{2}\left(t\right)∥\le ϛ{\text{e}}^{-b\left(t-{t}_{0}\right)}∥{\stackrel{\sim }{x}}_{2}\left({t}_{0}\right)∥$ ($\forall t\ge {t}_{0}$ ), then for constants ς ≥ 1 and b > 0, the system Eq. (12) is globally exponentially stable (GES) under the switching signal σ(t).
Definition 3
[35] Let V(x) be a continuously differentiable function such that
where ${\gamma }_{1},{\gamma }_{2}\in {\mathcal{K}}_{\infty }$, $\beta \in \mathcal{K}$, and W(x) is a continuous positive definite function. Then, the system $\dot{x}=f\left(t,x,{v}_{i}\right)$ is input-to-state stability (ISS) with $\gamma ={\gamma }_{1}^{-1}\circ {\gamma }_{2}\circ \delta $ .
Then, the main analytical result is shown as follows.
Theorem 1
Consider the closed-loop switched system government by Eq. (12), an NN-SORC protocol is designed by
For $\sigma \left(t\right)\in \{2,\dots,n\}$, $\forall t\in \left\{{t}_{k-1},{t}_{k}\right\}$, following the same derivation as for Eqs. (23), (24), (25), (26), (27), (28), we have
It follows from Eq. (17) that ${V}^{\sigma \left(t\right)}\left({t}_{k-1}\right)\le \mu {V}^{\sigma \left({t}_{k-1}^{-}\right)}\left({t}_{k-1}^{-}\right)$, which leads to
This implies that there exists a ${\mathcal{K}}_{\infty }$ -class function ${h}_{1}\left({∥{\stackrel{\sim }{x}}_{1}∥}^{2}\right)$ and ${h}_{2}\left({∥{\stackrel{\sim }{x}}_{1}∥}^{2}\right)$, and a $\mathcal{K}$ -class function $\delta \left(∥{\stackrel{\sim }{x}}_{2}∥\right)=∥{\stackrel{\sim }{x}}_{2}∥/\left(-\beta {\mathit{\omega }}^{\text{T}}\varphi \left({\stackrel{\sim }{x}}_{2}\right)\right)$ with an appropriate h1 and h2 that satisfy Definition 3. Therefore, state ${\stackrel{\sim }{x}}_{1}$ is an ISS with $\gamma \left(r\right)=r\sqrt{{h}_{2}/{h}_{1}}/\left(-\beta {\mathit{\omega }}^{\text{T}}\varphi \left({\stackrel{\sim }{x}}_{2}\right)\right)$ and $r=\underset{0\le \tau \le t}{\mathrm{s}\mathrm{u}\mathrm{p}}∥{\stackrel{\sim }{x}}_{2}∥$ . It follows from Eqs. (34), (38) that
Problem 1 is thus solved, which completes the proof.
Now, we relax Assumption 1 and require only the reference signal xr to be continuous. When the reference signal xr is not second-order differentiable, it can handle the reference signal xr in segments, ensuring that each segment is second-order differentiable.
where tp ($p\in \left[\mathrm{0,1},\dots,n\right]$, $n\in \mathbb{N}$ ) denotes the switching instant of ${x}_{\text{r}}^{t}\left(t\right)$ . Let ${\stackrel{\sim }{t}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\underset{p=\mathrm{1,2},\dots,n}{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{{t}^{p}-{t}^{p-1}\right\}$ denotes the minimum dwell time of the reference signal xr. Next, we present the following corollary to guarantee that NN-SORC tracks switching references asymptotically.
Corollary 1
For switching reference signal ${x}_{\text{r}}^{t}\left(t\right)$, considering the closed-loop switched system Eq. (12) under the NN-SORC law Eq. (16), if
$ϛ>\eta $
where ϛ=eN0lnμ and η > 0 are the expected tracking error parameters, then if the minimum dwell time ${\stackrel{\sim }{t}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ of the reference signal satisfies
It follows from Eq. (18) that lnμ/τa − 2κ2 < 0, whereas Eq. (41) leads to that Tres > 0. Then, condition Eq. (42) guarantees that the error system in Eq. (12) is asymptotically stable with the switching reference ${x}_{\text{r}}^{t}\left(t\right)$, which completes the proof.
4. Experimental results
The NN-SORC Eq. (16) was implemented on a nano-positioning stage platform, as shown in Fig. 1, to track the frequency-switching reference signals. The displacement was measured by a MicroSense 6810 capacitive displacement sensor and subsequently fed to the Speedgoat Performance real-time target machine, where NN-SORC Eq. (16) is embedded in the FPGA chip to yield a feedback control signal. Using the PD200 voltage amplifier, the control signal is amplified 20 times to drive the PEA to regulate the displacement of the nano-positioning stage, resulting in an entire feedback loop.
The experiments were conducted along the Y-axis (as shown in Fig. 4) of the nano-positioning stage with a sampling period Ts = 5 × 10−7 s. The parameters of Eq. (8) were identified using the MATLAB system identification toolbox as m1 = m2 = 1, ${d}_{\text{c}}^{1}$ = 4.779, ${d}_{\text{c}}^{2}$ = 46.44, ${s}_{\text{c}}^{1}$ = 6568, ${s}_{\text{c}}^{2}$ = 8665, ${P}_{\text{em}}^{1}{g}^{1}$ = 12 930, ${P}_{\text{em}}^{2}{g}^{2}$ = 24 060, ${H}_{\text{q}}^{1}$ = ${H}_{\text{q}}^{2}$ = 0. The switching signal was set to
with n = 2 and xmax = 10 µm indicating the physical limitations of the platform. According to Theorem 1 and Corollary 1, the parameters of NN-SORC were set as μ = 1, κ2 = 5, N0 = 1.1. Then the remaining parameters can be calculated as ς = 1.1, τa = 0.05 > lnμ/2κ2 = 0.00953, ${\stackrel{\sim }{t}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ = 0.04 s ≥ 0.005 s, and η = 0.02. To demonstrate the superiority of the proposed NN-SORC, we compared it with a proportional-integral-derivative (PID) controller (with finely tuned PID parameters kP = 1.2, kI = 1700) and a well-accepted Prandtl-Ishlinskii inverse protocol [36], which combines the classic inverse Prandtl-Ishlinskii model [37] as the hysteresis compensator in the feedforward loop and a proportional plus integral controller in the feedback loop.
The corresponding parameters are as follows: The number of play operators was 10; the thresholds of these 10 operators were −5.0929, −3.8091, 0.2045, −0.0067, 0.0472, 0.2438, −0.0406, 0.2467, 1.2695, and −45.2277; and the associate weights were 0, 0.0491, 0.2438, 0.5775, 0.8396, 1.1037, 1.3551, 1.6173, 1.8256, and 1.9268; kP_inverse = 0.4, kI_inverse = 1300. Specifically, a set of frequency-switching cosinusoidal signals xr(t) = 4 + 4cos(2πft) (µm) (f = 5, 10, 20 Hz) and frequency-switching triangular wave signals ${x}_{\text{r}}\left(t\right)=4+\left(32/{\mathrm{\pi }}^{2}\right){\sum }_{n=\mathrm{1,3},5,\dots }^{\infty }\left\{\left[{\left(-1\right)}^{\left(n-1\right)/2}/{n}^{2}\right]\mathrm{c}\mathrm{o}\mathrm{s}\left(2\mathrm{\pi }ft\right)\right\}$ (f = 5, 10, 20 Hz). The root mean square error (RMSE) was used to evaluate modeling performance.
where x(k) and xr(k) denote the measured displacement and reference at the kth sampling instant, respectively, and tmax denotes the entire evolution time.
Fig. 5, Fig. 6 show the displacement tracking output and error curve, respectively, for the cosinusoidal and triangular reference signals with frequencies of 5, 10, and 20 Hz. In the intervals [0,0.4], [0,0.8], and [0.8,1.2] s. The reference signal is fixed, and all three methods (PID, Prandtl-Ishlinskii inverse, and NN-SORC) can achieve cosinusoidal and triangular wave reference tracking. However, as shown in Table 2, the tracking error (RMSE) for the cosinusoidal signal of NN-SORC is 59% and 40% smaller than those of the PID and Prandtl-Ishlinskii inverse methods at 5 Hz; 61% and 45% smaller at 10 Hz; and 58% and 43% smaller at 20 Hz, respectively. For the triangular wave references, the NN-SORC is 46% and 28% smaller than PID and Prandtl-Ishlinskii inverse at 5 Hz; 56% and 38% smaller at 10 Hz; and 55% and 38% smaller at 10 Hz, respectively. As shown in the frequency spectrum comparison in Fig. 6, the tracking error of the NN-SORC is remarkably lower than those of the PID and Prandtl-Ishlinskii inverse within the bandwidth of [1,80] Hz, demonstrating the superiority of the NN-SORC at the most frequently used operational frequencies along with the feasibility of Theorem 1. More significantly, upon switching the reference signal, stable tracking control can still be achieved as long as the switching frequency satisfies constraints Eqs. (41), (42), verifying the validity of Corollary 1.
5. Conclusions
In this study, a high-speed nano-positioning stage was established with multiple parallel bonded thin piezoelectric ceramic layers. An FPGA-based dual-layer data framework was built to improve both computation speed and compilation efficiency. Moreover, a feedback linearization technique was adopted to convert hysteresis nonlinearity into a switched linear model. Accordingly, an NN-SORC method was proposed for tracking the variational reference signals often encountered in realistic micro- and nano-scale detection and manufacturing processes. Sufficient conditions were derived to guarantee the asymptotic stability of the associated closed-loop system. Finally, the effectiveness and superiority of the proposed NN-SORC were verified through extensive switching signal tracking experiments. In this study, we designed a closed-loop single-axis control system for nano-positioning stages to meet the desired performance requirements. However, modeling and controlling the dual-axis coupling effects of the nano-positioning stage present challenges. In the future, we will focus on expanding and refining our method to address the complexities associated with dual-axis coupling control and further improve the performance of more general nano-positioning systems.
CRediTauthorshipcontributionstatement
Hongwei Sun: Writing - original draft, Visualization, Validation, Software, Methodology, Formal analysis, Data curation. Ning Xing: Writing - original draft, Validation, Software, Methodology, Formal analysis. Jiayu Zou: Writing - original draft, Validation, Methodology, Investigation. Yuqi Rong: Validation. Yang Shi: Writing - review & editing. Han Ding: Writing - review & editing. Hai-Tao Zhang: Writing - review & editing, Project administration, Funding acquisition, Formal analysis.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by the National Key Research and Development Program of China (2022ZD0119601), the National Natural Science Foundation of China (52188102, 62225306, and U2141235), and the Guangdong Basic and Applied Research Foundation (2022B1515120069).
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