Iterative Learning Control of Hysteresis in Piezoelectric Actuators

We develop convergence criteria of an iterative learning control on the whole desired trajectory to obtain the hysteresiscompensating 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.


Introduction
Piezoelectric actuators (PEAs) have been widely used in nanopositioning systems due to their fast response and nanometer scale resolution [1][2][3].However, the hysteresis existing in PEAs can greatly limit system performance [4,5].Control of hysteretic system is an important area of control system research and a challenging problem [6][7][8][9].Research on feedback and model-based feedforward control has been studied to achieve relatively high-precision positioning [10][11][12][13].Iterative methods can be used to improve the positioning performance if the positioning application is repetitive.Therefore, many researchers study the iterative and adaptive control methods to minimize the adverse effect of hysteresis [14][15][16][17][18].
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 [15,16].Afterwards, they prove the convergence of iterative learning control (ILC) algorithm on each single branch.In this paper, we study the design of (ILC) algorithm to compensate for hysteresis-caused error in PEAs.The main contribution of our work is proving convergence of ILC algorithm on whole tracking trajectory.
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.

Problem Statement
Consider a hysteretic system of the following form: where V() ∈ R is the input, () ∈ R is the output, and  denotes the hysteresis function R → R. For a given desired trajectory   () defined on the finite time interval  ∈ [0, ], the objective is to find an input V  () by way of the following iterative learning control (ILC) algorithm: where   () =   ()−  (),  is a constant (to be determined), and V  () and   () are the input and output at the th iteration, respectively.Figure 1   a feedforward control input V  () → V  () in the ‖ ⋅ ‖ ∞ norm sense, where In this paper, [0, ] is used to denote the space of continuous functions on  ∈ [0, ], and   [0, ] denotes the space of continuous monotone functions on  ∈ [0, ].

The Prandtl-Ishlinskii Hysteresis Model
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 2, while its detailed formulations have been presented in [20].For a given input V() ∈   [0, ], the play operator   [V]() with threshold  is defined by with where  is initial value of the operator   , and Proof.See [20] Section 2.3.
Proof.The proof is identical to the proof of Theorem 4.  2) is chosen to be  = 0.5 and the initial input V 0 () = 5sin((1/2) − (/2)) + 5.The experimental results are shown in Figure 4. Figure 4(a) shows the results of the ILC algorithm to track the desired trajectory, and the feedforward input of the ILC algorithm is depicted in Figure 4(b).The maximum error at the th iteration ‖  ()‖ ∞ is shown in Figure 4(c), and it demonstrated that convergence of ILC algorithm is achieved.Figure 4(d) shows the tracking error at the 30th iteration.The maximum error is approximately 0.015 m, which is 0.15% of the total displacement range.

Conclusions
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.

Figure 3 :
Figure 3: Structure of the experimental setup.