Piezo-actuated Nanopositioning Stage Research Articles

Design of General Parametric Repetitive Control Using IIR Filter With Application to Piezo-Actuated Nanopositioning Stages

The achievable performance with repetitive control is limited due to its inherent sensitivity to the frequency shift away from the intended periodic frequencies and the undesired gain amplification of the aperiodic disturbances. To address these limitations, the paper proposed a general parametric repetitive control (GPRC) method based on the IIR filter with the features of low-order and excellent magnitude responses to result in better tracking performance in diverse applications. By analyzing its sensitivity function, it is found that the design of GPRC can be converted to the explicit parametric design of an IIR high pass filter. The controller design process and the stability analysis are presented in detail. To show the effectiveness of GPRC, comparative experiments are conducted via tracking sinusoids, triangular trajectories and other complex trajectories with multi-frequency components. The experimental results show that, in contrast with the conventional repetitive control (CRC) and a modified repetitive control (MRC), the GPRC exhibits excellent robustness against the frequency shift and advanced performance at the aperiodic frequencies. The tracking results of the sinusoids show that the maximum tracking error obtained with GPRC for a frequency shift of 3 Hz decreases from 0.0259 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> (CRC) and 0.2207 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> (MRC) to 0.0101 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> at the nominal frequency of 1000 Hz, demonstrating the merits of the proposed GPRC. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —To enable automation systems, one of the crucial requirements is to track repetitive references with high precision. Although the normal repetitive control (RC) based schemes are successfully applied to improve the tracking accuracy of periodic trajectories, the existing RC schemes suffer from the problems of lower robustness against frequency shift and the unwanted gain amplification at the aperiodic frequencies due to Bode’s sensitivity integral. To overcome this problem, this paper proposes a novel general parametric repetitive control (GPRC) method via characterizing the loop properties quantitatively based on the IIR high pass filter design. Focusing on specific issues, the detailed variations to handle the errors at only the odd-and even-harmonics are also demonstrated. This framework leads to a flexible solution in practical implementations. The experimental validation on a piezo-actuated nanopositioning stage is comparatively presented in terms of tracking accuracy and rejection ability of the gain amplification at the aperiodic frequencies. With its flexibility and effectiveness, the proposed GPRC can be easily implemented in diverse applications.

A piezo-actuated nanopositioning stage based on spatial parasitic motion principle

This paper presents the configuration, design, and characteristic analysis of a novel two-degree-of-freedom stick-slip nanopositioning stage based on the spatial parasitic motion principle. The planar-spatial evolution process is proposed to develop a compliant tripod mechanism by which the spatial parasitic motion can be generated for dual-axis actuation. An X-shaped hinge configuration is designed by freedom and constraint space analysis and introduced to the mechanism to improve the loading capacity of the nanopositioning stage. The actuation principle for driving along X- and Y-directions forward and backward is provided. The chain-based compliance matrix method is adopted for kinematic and static modeling of the compliant tripod mechanism. The dominant parameters are determined based on sensitivity analysis, and the performance of the stage is further studied by finite element method. The nanopositioning stage prototype is fabricated and assembled, and experimental investigations are conducted. The proposed nanopositioning stage has maximum velocities of 0.77 mm/s and 1.03 mm/s along X- and Y-direction, and a loading capacity of 5 kg can be achieved. Furthermore, the coarse-fine positioning experiments are carried out, and the positioning resolution is measured to be 18 nm.

High-Bandwidth Tracking Control of Piezoactuated Nanopositioning Stages via Active Modal Control

Due to the lightly damped resonance and intrinsic nonlinearities, it is difficult for the piezoactuated nanopositioning stage to realize high-bandwidth and high-accuracy control. To handle these limitations, in this work, a dual-loop control scheme based on state-feedback-based modal method is designed to both actively damp and stiffen the resonant mode and to suppress the effects of nonlinearities of the piezoactuated nanopositioning stage. In this scheme, the state-feedback-based modal controller is first designed in the inner loop to enlarge both the damping ratio and natural frequency of the first resonant mode. Then, a proportional–integral (PI) controller is utilized in the outer loop for eliminating the tracking errors caused by other disturbances and nonlinearities including hysteresis and creep. To maximize the control bandwidth of system under the proposed dual-loop scheme, an optimization method is thus proposed for simultaneously tuning the parameters of the inner and the outer loop controllers. Finally, to validate the proposed dual-loop control scheme, comparative experiments are carried out on a piezoactuated nanopositioning stage. Results demonstrate that the proposed control scheme improves the bandwidth of the system from 497 Hz (with PI control) and 1543 Hz (with a commonly used positive acceleration, velocity, and position damping control and a PI controller) to 6546 Hz, which is 664 Hz larger than the first resonant frequency of the original system, validating the effectiveness of the proposed dual-loop scheme on high-bandwidth control. Note to Practitioners—The demand of high-bandwidth and high-accuracy piezoactuated nanopositioning stages increases rapidly. However, the lightly damped resonance of the mechanism and the intrinsic nonlinearities of the piezoelectric actuator limit the tracking performance of the stage. A dual-loop control structure is adopted in this work to improve the tracking performance of the nanopositioning stage. Different from most of the vibration control methods proposed in the literature which aimed only at improving the damping ratio, a state-feedback-based modal controller is designed in the inner-loop for improving both the damping ratio and the stiffness of the system. This task is realized by re-placing the resonant poles of the system to the optimized location. The outer-loop controller adopts the high-gain PI control for eliminating the tracking errors. More importantly, in order to realize the high-bandwidth and high-accuracy control, a numerical optimization method is proposed for simultaneously tuning the parameters of the controllers in inner and outer loops. The controller design is simple, and it can be applied to other systems with second or higher order in which the first resonant mode dominates the system dynamics.

Robust high-bandwidth control of nano-positioning stages with Kalman filter based extended state observer and H∞ control.

The achievable performance of the piezo-actuated nano-positioning stages is severely limited by the intrinsic nonlinearities of the actuators, the lightly damped resonant mode of the flexure-hinge mechanism, and the external disturbances. To overcome all these limitations, this paper presents a novel robust dual-loop control scheme with a Kalman filter-based extended state observer and H∞ control for nano-positioning stages to implement high-bandwidth tracking operations. In this scheme, the extended state observer (ESO) is first developed and assisted by the identified system model to estimate both the system states and total disturbances, where the estimated disturbance is compensated by the direct feedback. In particular, to further improve the estimation performance, the Kalman filter is thus incorporated into the ESO to optimize the observer gain. Then, the state-feedback-based inner-loop controller is designed via the pole-placement method to damp the resonant mode of nano-positioning stages. Finally, a H∞ robust controller is adopted in the outer-loop to eliminate the influence resulting from the external disturbances, nonlinearities, and unmodeled dynamics on tracking operations. To validate the effectiveness of the proposed approach, comparative experiments are conducted on a piezo-actuated nano-positioning stage. Experimental results demonstrate that the proposed control scheme improves the control bandwidth of the system from 3.6 kHz (the stand-alone H∞ controller) to 5.52 kHz, which is 93.5% of the first resonant frequency of the original system. Moreover, it shows excellent robustness against the variation of system dynamics due to the change in the mounted mass and the external disturbances.

Signal Transformed Internal Model Control for Non-raster Scanning of Piezo-actuated Nanopositioning Stages

This paper proposes a new signal transformed internal model control (STIMC) scheme for non-raster scanning patterns of piezo-actuated nanopositioning stages. To smooth the scanning signals superimposed with a ramp or time-varying amplitudes, a signal transformation operator is calculated to convert the references into standard harmonic waveforms. An inverse transformation operator is then added in the control loop to generate the driving signals. For the contained internal model control (IMC) design, an H∞ mixed-sensitivity method is utilized for the first order internal mode synthesis. A second and a third internal modes are included in the IMC for alleviating residual high-frequency errors resulted from the nonlinearity of hysteresis. To verify the proposed STIMC scheme, comparative experiments with conventional IMC are conducted based on a nanopositioning platform. Results prove that the STIMC is effective on non-raster signals’ tracking. A same tracking precision for Lissajous scanning can be obtained by STIMC compared with IMC. An improvement of larger than 50% and 80% of root-mean-square errors can be obtained for cycloid and spiral scanning patterns, respectively.

Disturbance observer-based model prediction control with real-time modified reference for a piezo-actuated nanopositioning stage

Piezo-actuated micro-/nanopositioning systems have been widely employed in diverse high-precision positioning applications. However, the inherent hysteresis nonlinearity seriously deteriorates the tracking performance of piezo-actuated stages. This paper presents the design, analysis, and validation of a novel control scheme termed model prediction control (MPC) with real-time modified reference based on disturbance observer (DOB) to suppress the hysteresis nonlinearity and model uncertainty, in which the nonlinear effects are treated as an unknown disturbance to the system. In order to remove the most of the interference and diminish the effect of noise, a DOB is designed for the non-minimum phase (NMP) system. Then the difference between the actual displacement and the output of the nominal model termed the residual error is estimated and used to modify the reference in real time for a better performance. By the proposed method, the model of the inherent hysteresis is not required and the controller is established based on the identified nominal model. Its effectiveness is validated through experimental investigations on a commercial nanopositioner. Experimental results show that the proposed method can improve the tracking performance of the piezo-actuated stage, as compared with the traditional MPC and DOB-based MPC.

Fractional repetitive control of nanopositioning stages for tracking high-frequency periodic inputs with nonsynchronized sampling.

The repetitive control (RC) has been employed for high-speed tracking control of nanopositioning stages due to its abilities of precisely tracking periodic trajectories and rejecting periodic disturbances. However, in digital implementation, the sampling frequency should be integer multiple of the tracking frequency of the desired periodic trajectory. Otherwise, the rounding error would result in a significant degradation of the tracking performance, especially for the case of high input frequencies. To mitigate this rounding effect, the fractional repetitive control (FRC) technique is introduced to control the nanopositioning stage so as to precisely track high-frequency periodic inputs without imposing constraints on the sampling frequency of the digital control system. The complete procedure of controller design and implementation is presented. The techniques to deal with the problems of non-minimum phase system and fractional delay points number are described in detail. The proposed FRC is plugged into the proportional-integral control, and implemented on a custom-built piezo-actuated nanopositioning stage. Validation experiments are conducted, and the results show that the tracking errors caused by the rounding effect in the traditional RC approach are almost completely eliminated, when tracking sinusoidal waveforms with frequencies from 1000 Hz to 1587.3 Hz under the sampling frequency of 50 kHz, where the fractional parts being rounded vary from 0 to 0.5.

A Robust Resonant Controller for High-Speed Scanning of Nanopositioners: Design and Implementation

This brief presents a novel damping control scheme for piezoactuated nanopositioning platforms with robust resonant control (RRC). The RRC is developed to attenuate the resonant-vibrational modes of the lightly damped dynamics of the stage in an inner positive feedback loop. The parameters in the proposed RRC are determined through an analytical approach. Indeed, the controller gains constrained by both the robustness and the damping ratio of the inner loop are tuned based on the small gain theory. Then, a high gain integral tracking controller is applied in the outer loop to minimize the tracking errors due to unmodeled nonlinearity and uncertainties. To validate the effectiveness of the proposed RRC, comparative experiments with conventional positive position feedback (PPF) and integral resonant control (IRC) are conducted on a piezoactuated nanopositioning stage. Results demonstrate that the proposed RRC improves the closed-loop bandwidth from 67 Hz with PPF and 135 Hz with IRC to 176 Hz. Moreover, better robustness against load variations with a range of 0–1000-g loading under 0–20-Hz input raster scanning signals are obtained by RRC compared with PPF as well as IRC.

An Efficient Identification Method for Dynamic Systems With Coupled Hysteresis and Linear Dynamics: Application to Piezoelectric-Actuated Nanopositioning Stages

The coupling between hysteresis and linear dynamics remains a challenge for modeling and control of piezo-actuated nanopositioning stages. A decoupling identification approach is proposed in this paper. The system is first modeled by a Bouc–Wen hysteresis model cascading a linear dynamic part. Then, a proposed direct integration method with high computational efficiency is used for the identification of the Bouc–Wen model. With the identified hysteresis parameters, a harmonic linearization method is subsequently used to estimate the actual input to the linear dynamics. Experiment results on a two degree-of-freedom nanopositioning stage demonstrate that both the rate-dependence and amplitude-dependence resulted from the coupling between the hysteresis and linear dynamics can be well predicted by the proposed approach. Compared with existing methods, the proposed approach shows advantages in terms of both model accuracy and identification efficiency.

A Model Prediction Control Design for Inverse Multiplicative Structure Based Feedforward Hysteresis Compensation of a Piezo Nanopositioning Stage

The inherent hysteresis nonlinearity of piezoelectric actuators seriously deteriorates the tracking performance of piezo-actuated nanopositioning stage, especially in large stroke applications. Usually, the model of piezo-actuated stage is given by cascading a rate-independent hysteresis submodel with a linear dynamics submodel. This paper develops a composite model predictive control (MPC) with feedforward hysteresis compensation based on the inverse multiplicative structure. The feedforward controller has the merit of non-inverse requirement. The linear MPC is utilized as a feedback controller with the feature of simple solution for the feedforward compensation system. Experimental tracking results of sinusoidal signals at different frequencies as well as complex signals show that the proposed method can improve the tracking performance of the piezo-actuated stage, verifying its effectiveness.

Fuzzy Adaptive Sliding Mode Control for the Precision Position of Piezo-Actuated Nano Positioning Stage

This paper concerns hysteresis and creep of a piezo-actuated nano positioning stage. The hysteresis and creep, considered as bounded disturbance or uncertainty, are suppressed without a nonlinear model. An improved Fuzzy Adaptive Sliding Mode Control (FASMC) with a Proportional-Integral-Derivative sliding surface is designed to cancel both hysteresis and creep. However, the constant slopes of the sliding function may increase oscillations. Some variable gains with adaptive rules are introduced to overcome this drawback by changing the sliding function values online. Fuzzy control is applied to tune the switching control part to improve performance. Furthermore, an adaptation law is used to approximate the optimal value of the switching control. The stability of the sliding mode control law is proved in the sense of Lyapunov stability theorem. To eliminate chattering and obtain a smooth signal, the switching control is modified and a smooth function is introduced to substitute the signum function. An anti-saturation control is introduced to keep the input voltage within safety scope. Experimental results show that FASMC can achieve faster response for step input and sinusoid signal. Both hysteresis and creep of the piezoelectric actuator are suppressed by the proposed FASMC. Therefore, the FASMC can reduce the tracking errors of the piezo-actuated stage.

Positive acceleration, velocity and position feedback based damping control approach for piezo-actuated nanopositioning stages

This paper proposes a new damping control approach with positive acceleration, velocity and position feedback (PAVPF) scheme for piezo-actuated nanopositioning stages to implement high-bandwidth operation. To achieve this objective, the intrinsic hysteresis nonlinearity of the piezoelectric actuator is firstly handled by a feedforward compensator with a modified Prandtl–Ishlinskii model. Afterwards, the PAVPF controller with the pole-placement method is implemented to suppress the lightly damped resonant mode of the hysteresis compensated system. With the PAVPF controller, the poles of the damped system in a third-model can be placed to arbitrary positions with an analytical method. Finally, for accurately tracking a predefined trajectory, a high-gain proportional-integral (PI) controller is designed, which could deal with the disturbance and the unmodeled dynamics. For verifying the proposed PAVPF-based control approach, comparative experiments with positive velocity and position feedback controller and with PI controller are conducted on a piezo-actuated nanopositioning stage. Experimental results demonstrate that the developed control approach with PAVPF controller is effective on damping control and improves the control bandwidth of the conventional PI controller from 111 Hz to 766 Hz, which leads to the significant increase of the tracking speed.

Damping Control of Piezo-Actuated Nanopositioning Stages With Recursive Delayed Position Feedback

This paper presents a new damping control scheme for piezo-actuated nanopositioning stages with recursive delayed position feedback (RDPF). The RDPF is proposed to attenuate the resonant mode of the nanopositioning stage in the inner feedback loop, which results in a neutral-type time-delay system. To realize the pole placement of this system, a new numerical integration method is proposed to determine the rightmost pole and select the parameters of the RDPF. Then, a high-gain proportional-integral (PI) controller is designed in the outer loop to minimize the tracking errors caused by the hysteresis nonlinearity and modeling uncertainties. To validate the effectiveness of the proposed approach, comparative experiments are conducted on a piezo-actuated nanopositioning stage. Experimental results demonstrate that the proposed approach improves the control bandwidth of the system from 32.5 Hz (with the PI controller) and 687 Hz (with the conventional delayed position feedback based controller) to 793 Hz.

Odd-harmonic repetitive control for high-speed raster scanning of piezo-actuated nanopositioning stages with hysteresis nonlinearity

In raster scanning applications of atomic force microscopies, precisely tracking periodic triangular trajectories is the major objective of nanopositioning stages. Considering the fact of periodic operations, the repetitive control technique becomes promising and has been recently developed to reduce tracking errors. In our new experiments, it is found that, with triangular reference input, the hysteresis nonlinearity mainly affects the system at the odd harmonics of the input signal. In this sense, an odd-harmonic repetitive control (ORC) strategy is proposed to handle the hysteresis nonlinearity, with the hysteresis treated as the odd-harmonic periodic disturbance. Therefore, it avoids the modeling and inverting of the complex hysteresis nonlinearity. Another benefit of the developed ORC strategy is that it can also account for the tracking errors caused by the linear dynamics effect. To verify the effectiveness of the ORC strategy, real-time experiments are performed on a custom-built piezo-actuated nanopositioning stage. Experimental results show that the developed ORC strategy achieves precise tracking of 1562.5-Hz triangular trajectory with the hysteresis nonlinearity mitigated to a negligible level, which demonstrates the feasibility and effectiveness of the developed ORC strategy on hysteresis compensation during high-speed raster scanning.

Modeling and Control of Piezo-Actuated Nanopositioning Stages: A Survey

Piezo-actuated stages have become more and more promising in nanopositioning applications due to the excellent advantages of the fast response time, large mechanical force, and extremely fine resolution. Modeling and control are critical to achieve objectives for high-precision motion. However, piezo-actuated stages themselves suffer from the inherent drawbacks produced by the inherent creep and hysteresis nonlinearities and vibration caused by the lightly damped resonant dynamics, which make modeling and control of such systems challenging. To address these challenges, various techniques have been reported in the literature. This paper surveys and discusses the progresses of different modeling and control approaches for piezo-actuated nanopositioning stages and highlights new opportunities for the extended studies.

Modeling and compensating the dynamic hysteresis of piezoelectric actuators via a modified rate-dependent Prandtl–Ishlinskii model

This paper presents a modified rate-dependent Prandtl–Ishlinskii (MRPI) model for the description and compensation of the rate-dependent asymmetric hysteresis in piezoelectric actuators. Different from the commonly used approach with dynamic weights or dynamic thresholds, the MRPI model is formulated by employing dynamic envelope functions into the play operators, while the weights and thresholds of the play operators are still static. By this way, the developed MRPI model has a relatively simple mathematic format with fewer parameters and easier parameter identification process. The benefit for the developed MRPI model also lies in the fact that the existing control approaches can be directly adopted with the MRPI model for hysteresis compensation in real-time applications. To validate the proposed model, an open-loop tracking controller and a closed-loop tracking controller are developed based on a dynamic hysteresis compensator, which is directly constructed with the MRPI model. Comparative experiments are carried out on a piezo-actuated nanopositioning stage. The experimental results demonstrate the effectiveness and superiority of the controllers based on the developed MRPI model compared to the controllers based on the rate-independent P–I model and the rate-dependent P–I model with dynamic weighting functions.

High-Bandwidth Control of Nanopositioning Stages via an Inner-Loop Delayed Position Feedback

This paper presents a novel high-bandwidth control approach for piezo-actuated nanopositioning stages. A delayed position feedback (DPF) controller is first developed in the inner loop to damp the resonant mode of piezo-actuated stages. A generalized Runge–Kutta method (GRKM) is proposed to determine the parameters of the DPF controller with pole placement. The benefit of the DPF for active damping is its simple structure and ease of implementation. Then, a high-gain proportional-integral (PI) controller is designed in the outer loop to deal with the hysteresis nonlinearity, disturbance and modeling errors. The stability of the control system is analyzed via a graphical method. Finally, experiments are conducted to demonstrate the effectiveness and superiority of the proposed approach in terms of tracking accuracy at high speed as compared to the PI controller.

A rate-dependent Prandtl-Ishlinskii model for piezoelectric actuators using the dynamic envelope function based play operator

In this paper, a novel rate-dependent Prandtl-Ishlinskii (P-I) model is proposed to characterize the rate-dependent hysteresis nonlinearity of piezoelectric actuators. The new model is based on a modified rate-dependent play operator, in which a dynamic envelope function is introduced to replace the input function of the classical play operator. Moreover, a dynamic density function is utilized in the proposed P-I model. The parameters of the proposed model are identified by a modified particle swarm optimization algorithm. Finally, experiments are conducted on a piezo-actuated nanopositioning stage to validate the proposed P-I model under the sinusoidal inputs. The experimental results show that the developed rate-dependent P-I model precisely characterize the rate-dependent hysteresis loops up to 1000 Hz.

An experimental comparison of proportional-integral, sliding mode, and robust adaptive control for piezo-actuated nanopositioning stages

This paper presents a comparative study of the proportional-integral (PI) control, sliding mode control (SMC), and robust adaptive control (RAC) for applications to piezo-actuated nanopositioning stages without the inverse hysteresis construction. For a fair comparison, the control parameters of the SMC and RAC are selected on the basis of the well-tuned parameters of the PI controller under same desired trajectories and sampling frequencies. The comparative results show that the RAC improves the tracking performance by 17 and 37 times than the PI controller in terms of the maximum tracking error e(m) and the root mean tracking error e(rms), respectively, while the RAC improves the tracking performance by 7 and 9 times than the SMC in terms of e(m) and e(rms), respectively.

High-bandwidth tracking control of piezo-actuated nanopositioning stages using closed-loop input shaper

An integrated control strategy for piezo-actuated nanopositioning stages is proposed in this paper. The aim is to achieve high-speed and high-precision tracking control of nanopositioning stages. For this purpose, a direct inverse compensation method is firstly applied to eliminate the hysteresis nonlinearity without involving inverse model calculation. Then, an inside-the-loop input shaper is designed to suppress the vibration of the compensated system. A Smith predictor is introduced to prevent the potential closed-loop instability caused by the time delay of the inside-the-loop input shaper. Finally, a high-gain feedback controller is employed to handle the disturbances and modeling errors. To demonstrate the effectiveness of the proposed control method, comparative experiments are carried out on a piezoelectric actuated stage. The results show that the proposed control approach increases the tracking bandwidth of the stage from 22.6Hz to 510Hz.