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Since the piezoelectrically actuated system has nonlinear and time-varying behavior, it is difficult to establish an accurate dynamic model for a model-based sensing and control design. Here, a model-free adaptive sliding controller is proposed to improve the small travel and hysteresis defects of piezoelectrically actuated systems. This sensing and control strategy employs the functional approximation technique (FAT) to establish the unknown function for eliminating the model-based requirement of the sliding-mode control. The piezoelectrically actuated system’s nonlinear functions can be approximated by using the combination of a finite number of weighted Fourier series basis functions. The unknown weighted vector can be estimated by an updating rule. The important advantage of this approach is to achieve the sliding-mode controller design without the system dynamic model requirement. The update laws for the coefficients of the Fourier series functions are derived from a Lyapunov function to guarantee the control system stability. This proposed controller is implemented on a piezoelectrically actuated X-Y table. The dynamic experimental result of this proposed FAT controller is compared with that of a traditional model-based sliding-mode controller to show the performance improvement for the motion tracking performance.

Recently, the micro-positioner has become an important development target for meeting the requirements of the precision industry, such as in the semiconductor manufacturing process, biotechnology processes and opto-electronics systems. Since the piezoelectric actuator has many advantages, such as ultrahigh precision, high resolution, tiny size and quick response speed, it has been widely used as a micro-positioning table actuator in these production areas. Piezoelectric actuators can also be electrically controlled to move with a resolution on the order of nanometers. However, piezoelectric actuators also exhibit undesired serious hysteretic behaviors which limit the performance of piezoelectric actuated systems.

In recent years, different methods were proposed to drive the piezoelectric micropositioning mechanisms and control piezoelectrically actuated systems. Chang and Sun [

Although a piezoelectric actuator has the advantages of high resolution, quick response speed and high power-volume ratio, it has serious hysteresis behavior and small travel defects. Hence, it is difficult to establish an accurate dynamic model for model-based control design. In this paper, a model-free functional approximation based adaptive sliding-mode controller is employed to eliminate this model-based requirement. The functional approximation technique (FAT) was utilized to design an adaptive sliding controller for different nonlinear systems containing time-varying uncertainties [

This paper is organized as follows: Section 2 describes the piezoelectrically actuated X–Y table system structure. The approximate linear model and system identification process are presented in Section 3. This identified model can be used to design the model-based controller for comparison. Section 4 describes the functional approximation technique. The methodology of controller design and the stability analysis are derived in Section 5. Section 6 describes the experimental results of the proposed FAT controller. The experimental results are compared with that of the model-based sliding controller to show the dynamic performance improvement of the proposed FAT controller. Final conclusions are presented in Section 7.

A piezoelectrically actuated X–Y table shown in

A PC-based controller was developed for experiments with this system. The X–Y table has two independent axes, X and Y, actuated by two different piezoelectric actuators. The experimental layout of this positioning system is shown in

To simplify the model description, the system dynamics of the X axis or Y axis can be represented as the following second order model:
_{min}≤_{max} Define _{m}

Since the system dynamics has nonlinear time-varying behavior with unknown uncertainty bounds it is difficult to establish an accurate dynamic model for model-based controller design. Here, the functional approximation technique is employed to approximate this unknown function

In order to evaluate the accuracy of a functional approximation with respect to the nonlinear function of the system, an approximate linear dynamic model for this piezoelectrically actuated X–Y table, instead of the original system model, is identified based on system input-output data. A pseudo-random-binary-sequences (PRBS) signal with appropriate amplitude is chosen as the input signal to excite the piezoelectrically actuated system. The transfer function _{1}^{−1} _{na}^{−nz}, _{1}^{−1} + … + _{nb}^{−nb} and

The estimation output of these two identified models and the system experimental output with PRBS input voltage excitations are plotted in

If a piecewise continuous time varying function _{0}, _{n}_{n}

The unknown time-varying function _{f}_{f}_{f}_{1}(_{2}(_{n}^{T}_{f}_{1} _{2} … _{n}^{T}_{f}_{f}

The objective of this study was to develop a FAT based model-free adaptive sliding-mode controller for a piezoelectrically actuated system. Its control performance is compared with that of a traditional sliding-mode controller based on an identified model,

Due to the piezoelectrically actuated system has nonlinear time-varying dynamics, the functional approximation technique is employed to replace the unknown nonlinear functions for a sliding-mode control design. The system control block diagram of this piezoelectrically actuated X–Y table is shown in _{r}

In order to achieve the sliding surface reaching condition and establish the approximation error compensation, the control law

Here, _{f}_{f}^{n}_{f}^{n}

To prove the stability of this control system and find the update laws for vector _{f}_{f}^{n×n}

Since

The update law for _{f}

Then, the

In order to cover the uncertainty of the unknown function _{max} and _{max} are the maximum values of Δ

That means this control system stability can be guaranteed by using the update laws shown in

The sliding-mode control theory has been widely employed to control nonlinear dynamic systems, especially the systems that have model uncertainties and external disturbances. It employs a discontinuous control effort to drive the system toward a sliding surface, and then switching on that surface. Theoretically, it will gradually approach the control target, the origin of the phase plane [

These system parameters have time-varying behavior. Their variation bounds are assumed as:

The sliding surface for this second order system can be defined as:
_{s}_{s}

The control law _{s}

Substituting

Multiplying both sides of _{s}

If the robustness parameter η_{1} is selected as:

Then the

That means the system stability can be achieved by choosing an appropriate robustness gain constant η_{1}. In addition, the control law _{s}

In order to investigate the performance of the proposed controller, the following experiments were performed. The sampling frequency was chosen as 1,000 Hz. The following control parameters are chosen for the functional approximation based adaptive sliding-mode controller: the sliding surface parameter _{m}_{f}_{f}

Based on the system identification model, _{m}_{m}_{m}_{2}, of the sliding surface reaching condition, _{s}

In this case, a 2 cm diameter circular contour is designed for the two-dimensional motion control. This circular contour can be generated by the accumulation of the angles as a function of time for the X axis and Y axis. The experimental result for the tracking response of the X–Y table is shown in

It can be observed that a good tracking response can be obtained for the X–Y table for the reference circular contour by using the proposed FAT based adaptive sliding controller. The displacements of X axis and Y axis are plotted in

The dashed line shows the reference circular contour signal, the solid line depicts the tracking response by using the proposed FAT based adaptive sliding controller and the dotted line denotes the tracking response by using a traditional model-based sliding-mode controller. The tracking errors of the X axis and Y axis are shown in

In this case, a window contour, shown in

The X axis and Y axis displacements are plotted in

It can be observed that the maximum tracking errors of the X axis are 0.2 mm and 0.013 mm for the model-based sliding-mode controller and the proposed FAT based adaptive sliding controller, respectively. The maximum tracking errors of the Y axis are 0.15 mm and 0.024 mm for the model-based sliding-mode controller and the proposed FAT based adaptive sliding controller, respectively. The root mean square (RMS) values of the tracking error for X axis are 0.0532 mm and 0.0055 mm for the model-based sliding-mode controller and the proposed FAT based adaptive sliding controller, respectively. The RMS values of the tracking error for Y axis are 0.067 mm and 0.0069 mm for the model-based sliding-mode controller and the proposed FAT based adaptive sliding controller, respectively.

The piezoelectric actuating system has non-linear characteristics and time-varying behavior. It is difficult to design a model-based controller for this micro-positioning system. A model-free functional approximation based adaptive sliding controller was developed and successfully employed to control a piezoelectrically actuated X–Y table system. The stability of the proposed controller is guaranteed by means of the Lyapunov theorem. The control performances of the proposed FAT based controller and a model-based sliding-mode controller were compared in this study, too. Only 15 terms of Fourier series functions are used to approximate the nonlinear time-varying function for designing a sliding-mode controller and achieving good control performance. The tracking error can be reduced to less than 0.017 mm and 0.028 mm for the X axis and Y axis with two different tracking trajectories. The tracking error is much better than that of the traditional sliding-mode controller. The proposed approach can thus be effectively applied to control a piezoelectrically actuated system.

Piezoelectrically actuated X–Y table system.

Positioning control system experimental layout.

System identification model output.

System control block diagram.

X–Y table displacement (case A).

The displacements of

The tracking error of

X–Y table displacement (case B).

The displacements of

The tracking error of