This paper presents the trajectory tracking approach of a piezoelectric actuator using an iterative learning control (ILC) scheme based on B-spline network (BSN) filtering. The ILC scheme adopts a state-compensated iterative learning formula, which compensates for the state difference between two consecutive iterations in order that the iterative learning can learn from the tracking errors of the previous iteration effectively. The BSN is used to attenuate the noises and retrieve the signals of the tracking errors for the ILC. The BSN serves as a unique filter which generally does not have zero-phase responses. Design details on the ILC scheme using BSN filtering are discussed in the paper. Extensive experiments of tracking two desired trajectories for a piezoelectric actuator are presented. The experimental results show that the state-compensated ILC scheme using BSN filtering can achieve fast error convergence and keep small steady-state tracking errors close to the system noise level. This research thus relaxes the restriction of the zero-phase criterion commonly applied to the ILC filtering in the literature.
Owing to the advantages of infinite small displacement, high stiffness, and wide bandwidth, piezoelectric actuators have been commonly used for precision positioning in various engineering fields, such as positioning of diamond machine tools, positioning of optical lenses, and positioning of masks in semiconductor manufacturing. However, the piezoelectric actuators are nonlinear devices, and can be characterized by certain hysteresis phenomena between their input voltages and output displacements. Various control schemes have been proposed in the literature to compensate for the nonlinear behaviors of the actuators. For example, there were schemes using the Preisach model-based control [1] and the neural sliding-mode control [2].
Differing from other endeavors of control complexities, some studies initiated the simpler approach using the iterative learning control (ILC) [3]. Certain degrees of success were achieved in tracking control of piezoelectric actuators by using the P-type ILC, the current-error-assisted D-type ILC, the model-assisted ILC, and the state-compensated ILC schemes [4], [5], [6], [7] and [8]. The state-compensated ILC (SCILC) proposed by the authors [8] is a novel scheme, which enhances the traditional ILC by adding a state compensation enhancement to the iterative learning. In [8], a zero-phase (ZP) finite impulse response (FIR) filter was used to filter the learnable errors for the ILC. The major goal of this paper, on the contrary, is to relax the restriction of ZP filtering by adopting the B-spline network (BSN) for error filtering. The BSN consists of a number of B-spline basis functions to realize an input–output mapping. The mapped output is a linear combination of these basis functions. The BSN structure has been proposed in the learning feed-forward control (LFFC) [9], [10], [11], [12], [13], [14] and [15] to enhance the performance of feedback control systems. The BSN in the LFFC is tuned by the feedback control effort of the previous iteration to generate a feed-forward control effort in the current iteration. The feed-forward control effort is thus modified iteratively in order that the system output will approach the desired trajectory eventually. The BSN is basically a function approximator [15], and the tuning mechanism of the BSN is in essence a general non-ZP filter [10]. The BSN is adopted in this research to filter the learnable errors of the previous iteration without a zero phase in general. The filtered errors are then used for learning in the current iteration.
The remainder of the paper is organized as follows. Section 2 describes the experimental setup. Section 3 introduces the SCILC. In Section 4, the BSN filtering is discussed. In Section 5, a convergence analysis is performed for the SCILC. Section 6 addresses the time-frequency analysis of tracking errors. Section 7 presents the controller design details and the experimental results. Finally, Section 8 concludes this paper.
This research applied the BSN filtering to a state-compensated ILC scheme for trajectory tracking of a piezoelectric actuator. The ILC filtering in the literature is usually based on a finite impulse response (FIR) or infinite impulse response (IIR) design which aims at sharp cutoff and zero-phase filtering. On the contrary, the BSN filtering is apart from such FIR or IIR design approach in terms of the cutoff sharpness and the zero-phase response. Nevertheless, as revealed by the experimental results for the piezoelectric actuator, the SCILC scheme using BSN filtering could still achieve fast error convergence and small steady-state errors in tracking the desired trajectories for wide ranges of frequencies and amplitudes. The BSN filtering, therefore, relaxes the filter design criterion of zero-phase delay traditionally adopted in the ILC literature. This paper also presents the detailed procedures of the controller design for the SCILC scheme based on BSN filtering. The convergence bandwidth analysis of the system using the BSN-based SCILC scheme is discussed theoretically and then applied to the controller design. The transfer function of the BSN filter is also derived. The formula of the transfer function is then employed to determine the cutoff frequency and the support width of the filter, with the aid of the time-frequency analysis of the tracking errors using the Wigner–Ville distribution method. The clearly described procedures of the controller design are appealing for other applications using the SCILC scheme based on BSN filtering.
