^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We develop convergence criteria of an iterative learning control on the whole desired trajectory to obtain the hysteresis-compensating feedforward input in hysteretic systems. In the analysis, the Prandtl-Ishlinskii model is utilized to capture the nonlinear behavior in piezoelectric actuators. Finally, we apply the control algorithm to an experimental piezoelectric actuator and conclude that the tracking error is reduced to 0.15% of the total displacement, which is approximately the noise level of the sensor measurement.

Piezoelectric actuators (PEAs) have been widely used in nanopositioning systems due to their fast response and nanometer scale resolution [

The main challenge in iterative approaches for hysteretic systems is to assure convergence of the iterative algorithm. Leang and Devasia divide a general desired trajectory into some monotonicity partition [

The remainder of this paper is organized as follows. First, we state the problem in the next section. Afterwards, we briefly review the Prandtl-Ishlinskii model in the context of this work and prove convergence of the ILC algorithm we designed. Finally, we implement the ILC algorithm on experimental stage and show our experimental results and conclusions.

Consider a hysteretic system of the following form:

The ILC scheme.

In this paper,

An operator

Let the operator

We use the mathematical induction to prove this assertion. First, we prove

Let the operator

See [

Consider a system of the form

To prove the convergence of the ILC algorithm, we show contraction of the input (

The Prandtl-Ishlinskii (PI) model can be used to capture the rate-independent hysteresis nonlinearity in piezoelectric actuators. In this section, the PI model is presented.

The PI model utilizes the play or stop operators and a density function to characterize the hysteresis behavior. The hysteresis play operator is illustrated in Figure

Play operator.

For any

See [

If

Consider

The PI model assumes that the output

If

Consider

Consider a hysteretic system of the form

The proof is identical to the proof of Theorem

The ILC algorithm is applied on an experimental piezoelectric actuator PST150/7/40VS12, which is a preloaded PZT from Piezomechanik in Germany. The natural frequency of the actuator is 20 kHz. The actuator provides a maximum displacement of 40

Structure of the experimental setup.

We apply the ILC algorithm to track a sinusoidal trajectory

Experiment results.

Displacement versus time

Input versus time

Figure

In this paper, we designed an ILC algorithm to compensate for hysteresis-caused tracking error in piezoelectric actuators and proved convergence of this algorithm on the whole tracking trajectory. Experiments were carried out to verify the effectiveness of the ILC algorithm. The experimental results show that the tracking error can be reduced to the noise level of the sensor measurements.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (no. 61174044).