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 [1] tried to control a two degree-of-freedom monolithic piezoelectric actuator with a linear resolution to 2 nm. Choi et al. [2] designed a sliding-mode controller for fine motion tracking control of the objective lens in the vertical direction. Lin and Yang [3] employed a PI feedback control associated with feedforward compensation based on the hysteresis observer to compensate the nonlinearity of piezoelectric actuator. Bashash and Jalili [4] proposed a modeling and control methodology for real-time compensation of nonlinearities along with precision trajectory control of piezoelectric actuators in various range of frequency operation. Liaw et al. [5] presented a robust motion tracking control methodology for flexure-based four-bar micro/nano manipulator driven by a piezoelectric actuator. In addition, many control techniques involving feedback and feedforward-feedback features have been proposed to remove the hysteresis-caused tracking errors [6–10]. However, these feedback control techniques do not utilize a precise hysteresis model, in contrast, the feedforward-feedback algorithms deal with the hysteresis and structural dynamics separately. A compensator is added into the feedforward loop to cancel out the undesired effects caused by the hysteresis while designing a feedback controller to control the structural dynamic effects. The feedforward compensator can be used to deal with nonlinear hysteretic effects; however, the effectiveness of the method relies heavily on the accuracy of the inverse hysteresis model.

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 [11–13]. This control scheme is proposed for nonlinear systems with unknown bound time-varying uncertainties. This approach can estimate the system dynamics and uncertainties on-line and no prior knowledge of the system dynamics is required. The uncertainties are represented by using finite linear combinations of basis functions with unknown constant weighting vectors. Appropriate update laws for the weighting vectors can be selected based on the Lyapunov theory to guarantee the system stability. Asymptotic convergence of the output error can be obtained if sufficient number of basis functions is used. The optimal expansion coefficients are also updated on-line and used in constructing the adaptive control input. Here, the functional approximation based adaptive sliding-mode controller is designed for a piezoelectric actuated X–Y table system tracking control.

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 Figure 1 was built for investigating the dynamic system control behavior. Two HR8 motors were used to actuate the X axis and Y axis, respectively. There is a linear response feature between the piezoelectric actuator velocity and the driver control voltage. The actuator and driver can be modeled as a DC motor with internal friction that is driven by a voltage amplifier. The driver generates a 39.6 kHz sinusoidal wave to drive the actuator with an amplitude function when a command voltage within ±10 V is sent to the driver unit. This constant oscillation frequency is generated from the driver unit which was supplied by Nanomotion Limited.

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 Figure 2. The control voltage is calculated on the PC, converted from a digital to an analogue signal by a D/A interface card and sent to the piezoelectric actuator driver unit to actuate the piezoelectric motor. The displacements of this X–Y table are sensing and measured by linear encoders and sent back to the PC via an encoder card for closed-loop control.

To simplify the model description, the system dynamics of the X axis or Y axis can be represented as the following second order model: (1) x ¨ ( t ) = f ( X ,   t ) + b ( t ) u ( t )where x is the displacement of X axis or Y axis stage, f(X,t) is a function of state variables, b(t) is the control gain and u(t) is the control voltage. The function f(X,t) is an unknown time-varying function with unknown variation bound. However, the bound of the unknown function b(t) can be estimated, i.e., b_min≤b(t)≤b_max Define b(t) as: (2) b ( t ) = b m + Δ bwhere b_m is the nominal value and Δb is a bounded uncertainty value: (3) 0 < τ min ≤ Δ b ≤ τ max

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 f(X,t) for releasing the model requirement.

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 M(q) describes the relationship between the voltage input and the sprung mass position output is chosen as an auto-regressive (ARX) model: (4) M ( q ) = B ( q ) A ( q )where A(q) = 1 + a₁q⁻¹ + … + a_na q^−n_z, B(q) = b₁q⁻¹ + … + b_nb q^−n_b and q is the shift operator. To simplify the model description, the second order approximate transfer function models are selected and identified by using the MATLAB’s identification toolbox for X axis and Y axis of the X–Y table, respectively: (5) M x ( q ) = 78.24 q 2 + 0.7202 q + 0.3588 (6) M y ( q ) = 16.8 q 2 + 0.2815 q + 0.4079

The estimation output of these two identified models and the system experimental output with PRBS input voltage excitations are plotted in Figures 3(a) and 3(b) for comparison. The solid line represents the measured system output and the dotted line depicts the simulated model response. It can be observed that the dynamic behavior of the identified model is matched well the piezoelectrically actuated system’s dynamic response. The difference between both response curves is due to the system’s nonlinear, time-varying behavior and hysterisis dynamics effects. Based on these identified models, the unknown time-varying functions can be estimated and used to evaluate the accuracy of the functional approximation scheme. In addition, a model-based sliding-mode controller can be designed based on the estimation models. Its control performance will be compared with that of the proposed FAT controller.

If a piecewise continuous time varying function h(t) satisfies the Dirichlet conditions, it can be transformed into a generalized Fourier series expansion within a time interval [0,T]: (7) h ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos ω n t + b n sin ω n t )where a₀, a_n, and b_n are the Fourier coefficients and ω n = 2 n π T is the frequency of the sinusoidal function. Define: (8) Z ( t ) = [ 1           cos ω 1 t           sin ω 1 t           cos ω 2 t           sin ω 2 t       ⋯       cos ω n t           sin ω n t ] T (9) W = [ a 0         a 1         b 1         a 2         b 2         ⋯         a n         b n ] Tthen Equation (7) can be rewritten as: (10) h ( t ) = W T Z ( t ) + ɛ ( t )where ɛ(t) is the approximation error. When n is large enough, h(t) can be approximated as: (11) h ( t ) ≈ W T Z ( t )

The unknown time-varying function f(X,t) in Equation (1) can be approximated by a linear combination of a finite orthogonal basis functions Z(t) to arbitrarily prescribed accuracy as long as n is large enough: (12) f ( X ,   t ) ≈ W f T Z f ( t )where Z_f(t) is a orthogonal basis function vector and W_f is a weighting coefficient vector. If the number of the basis functions is large enough, Equation (12) can be described as the following approximation form: (13) f ( X ,   t ) = W f T Z f ( t )where Z_f (t) = [Z₁(t) Z₂(t) … Z_n(t)]^T, and W_f = [W₁ W₂ … W_n]^T. This functional approximation in Equation (13) can be used to represent an unknown function with uncertainty and disturbance. The time-varying vector Z_f(t) is a known function and the vector W_f is an unknown regulating constant. A proper Lyapunov function can be selected to find the update laws for these unknown constants based on Lyapunov stability theory.

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, Equations (5) and (6). The control law design and stability analysis for X axis are described in the following sections.

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 λ is chosen as 230 and 220 for the X axis and Y axis, respectively. The robustness parameter η can be estimated based on Equation (28). It is selected as 23,000 and 25,000 for the X axis and Y axis, respectively. In order to improve the control law chattering behavior, the sgn(s) function in Equation (17) is replaced by the saturation function sat(s/ф) with a boundary layer thickness ф = 0.1 and 0.05 for X axis and Y axis, respectively. The nominal value of the control gain b_m is selected as 78.24 for X axis and 16.8 for Y axis, respectively, based on the estimation model, Equations (5) and (6). The weighting matrix Q_f of the Fourier series function coefficients is set as a small constant matrix Q_f = 0.01 [I] to increase the converging speed. The first fifteen terms of the Fourier series functions are chosen as the functional approximation basis functions.

Based on the system identification model, Equations (5) and (6), the parameters A_m = 0.7202 and 0.2815, B_m = 0.3588 and 0.4079, and G_m = 78.24 and 16.8, are employed to design the model based sliding-mode controller for X axis and Y axis, respectively. The converging parameter, η₂, of the sliding surface reaching condition, Equation (38), is chosen as 700 and 800 for X axis and Y axis, respectively. The parameter λ_s which influences the converging slope of the sliding surface was chosen as 70 and 80 for X axis and Y axis, respectively. The values of these control parameters are not critical for practical implementation.

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.