Tracking control of piezoelectric actuator using adaptive model
- Tran Vu Minh†^{1}Email author,
- Nguyen Manh Linh†^{2} and
- Xinkai Chen^{3}
Received: 23 January 2016
Accepted: 18 April 2016
Published: 10 May 2016
Abstract
Piezoelectric actuators (PEAs) have been widely used in micro- and nanopositioning applications due to their fine resolution, rapid responses, and large actuating forces. However, a major deficiency of PEAs is that their accuracy is seriously limited by hysteresis. This paper presents adaptive model predictive control technique for reducing hysteresis in PEAs based on autoregressive exogenous model. Experimental results show the effectiveness of the proposed method.
Keywords
Background
The use of piezoelectric actuator (PEA) has become very popular recently for a wide range of applications, including atomic force microscopes [1–3], adaptive optics [4], computer components [5], machine tools [6], aviation [7], internal combustion engines [8], micromanipulators [9] due to their subnanometer resolution, large actuating force, and rapid response. However, PEA exhibits hysteresis behavior in their response to an applied electrical energy. This leads to problems of inaccuracy, instability, and restricted system performance.
The control of PEA has been extensively studied recently. Ge and Jouaneh [10] discuss a comparison between a feedforward control, a regular PID control, and a PID feedback control with Preisach hysteresis. In this research, the nonlinear dynamics of piezoelectric actuator is first linearized and then reformulated the problem into a disturbance decoupling problem. In [11], an explicit inversion of Prandtl–Ishlinskii model is used to control a piezoelectric actuator. Webb et al. [12] proposed an adaptive hysteresis inverse cascade with the system, so that the system becomes a linear structure with uncertainties. Another adaptive control approach is fused with the Prandtl–Ishlinskii model without constructing a hysteresis inverse, since the inverse is usually difficult to be obtained [13]. In this concept, the implicit inversion of Prandtl–Ishlinskii model is developed and is associated with an adaptive control scheme. A new perfect inverse function of the hysteresis (which is described by Bouc–Wen model) is constructed and used to cancel the hysteresis effects in adaptive backstepping control design [14].
In this paper, the dynamics of the piezoelectric actuator is identified as a linear model with unknown parameters. These parameters will be updated online by using least square method. Then, a model predictive controller using estimated parameters is designed to achieve the desired control behavior. The experimental results show the effectiveness of the proposed method.
This paper is organized as follows. In “Modeling method” section, the adaptive model of PEA is given. In “Controlling method” section, the model predictive control design is presented. The experimental results are shown in “Result” section. “Discussion” section will conclude this paper.
Modeling method
The parameters a _{1}, a _{2}, b _{1}, b _{2} are unknown.
Controlling method
For simplicity, denote \( M = \left[ {\begin{array}{*{20}c} 0\quad & \quad1 &\quad 0 \\ { - \hat{\theta }_{2} (k)} \quad& { - \hat{\theta }_{1} (k)}\quad & {\hat{\theta }_{3} (k) + \hat{\theta }_{4} (k)} \\ 0\quad & \quad0 &\quad 1 \\ \end{array} } \right] \), \( N = \left[ {\begin{array}{*{20}c} 0 \\ {\hat{\theta }_{3} (k)} \\ 1 \\ \end{array} } \right] \) and \( Q = \left[ {\begin{array}{*{20}c} 0 \quad& 1 \quad & 0 \\ \end{array} } \right] \).
Result
Experimental setting parameters
N _{ p } | ω(i) | ρ(i) | λ | \( \hat{\theta }\left( 0 \right) \) | Offset (V) | Δt (ms) | |
---|---|---|---|---|---|---|---|
y _{ d1}(k) | 3 | 1 | 0.1 | 0.1 | 0.2 | 30 | 0.5 |
y _{ d2}(k) | 3 | 1 | 0.1 | 0.1 | 0.2 | 30 | 0.5 |
Discussion
This paper has discussed the adaptive model predictive control for piezoelectric actuators, where the model of PEA is regarded as linear model. The unknown parameters in the model are estimated online. The proposed method shows its effectiveness in tracking performance. Moreover, it is simple and easy to be implemented. In the future, we will try to employ the proposed method to control piezo-actuated systems with load.
Notes
Declarations
Authors’ contributions
The contribution of this paper is that the positioning performance of PEA with nonlinearity hysteresis phenomenon can be achieved by fusing model predictive control with adaptive linear model. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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