TY - JOUR
T1 - Controllable micrometer positioning design of piezoelectric actuators using a robust fuzzy eliminator
AU - Chen, Yung Yue
AU - Lin, Li Kai
AU - Hung, Min Hao
N1 - Funding Information:
This research was sponsored by National Science Council, R.O.C., under the grant MOST 108-2221-E-006 -092 Appendix A Define a min-max performance index J for the micro positioning problem of disturbed piezoelectric actuators as (A1) J E u f w = ∫ 0 t f E T t QE t + u f T t Ru f t − γ 2 w T t w t dt Eq. (A1) can be rewritten as follows (A2) J E u f w = E T 0 PE 0 − E T t f PE t f + 1 γ f Θ ˜ f T 0 Θ ˜ f 0 − 1 γ f Θ ˜ f T t f Θ ˜ f t f 1 γ g Θ ˜ g T 0 Θ ˜ g 0 − 1 γ g Θ ˜ g T t f Θ ˜ g t f + ∫ 0 t f E T t AP + PA T + PB 1 γ 2 I − R − 1 B T P + Q E t + u f T t Ru f t + u f T t B T PE t + E t T PBu f t + E t T PB ξ f + 1 γ f Θ ˜ ̇ f T t Θ ˜ f t + Θ ˜ f T t ξ f T B T PE t + 1 γ f Θ ˜ ̇ f t + E t T PB ξ g + 1 γ g Θ ˜ ̇ g T t Θ ˜ g t + Θ ˜ g T t ξ g T B T PE t + 1 γ g Θ ˜ ̇ g t + w T t B T PE t + E t T PBw t + w T w dt By selecting the adaptive laws Θ ̇ g = − γ g ξ g T x B T PE t and Θ ̇ f = − γ f ξ f T x B T PE t for the fuzzy approximators and A T P + PA + Q + PB 1 γ 2 I − R − 1 B T P = 0 for Eq. (A2) , Eq. (A2) can be simplified as (A3) J E u f w = E T 0 PE 0 − E T t f PE t f + 1 γ f Θ ˜ f T 0 Θ ˜ f 0 − 1 γ f Θ ˜ f T t f Θ ˜ f t f 1 γ g Θ ˜ g T 0 Θ ˜ g 0 − 1 γ g Θ ˜ g T t f Θ ˜ g t f + ∫ 0 t f u f T t Ru f t + u f T t B T PE t + E t T PBu f t + w T t B T PE t + E t T PBw t + w T w dt From Eq. (A3) , a more concise form can be obtained via completing the square (A4) J E u f w = E T 0 PE 0 − E T t f PE t f + 1 γ f Θ ˜ f T 0 Θ ˜ f 0 − 1 γ f Θ ˜ f T t f Θ ˜ f t f 1 γ g Θ ˜ g T 0 Θ ˜ g 0 − 1 γ g Θ ˜ g T t f Θ ˜ g t f + ∫ 0 t f Ru f t + B T PE t T R − 1 Ru f t + B T PE t + w t − 1 γ B T PE t T w t − 1 γ B T PE t + w T w dt The robust compensator u f ( t ) and worst case modelling uncertainties w ( t ) can be selected as follows for minimizing Eq. (A4) (A5) u f t = − R − 1 B T PE t (A6) w t = 1 γ B T PE t Based on the selections of u f ( t ) and w ( t ) of Eqs. (A5) and (A6) , the minimum value of the mini-max performance index J can be obtained (A7) J E u f w = E T 0 PE 0 + 1 γ f Θ ˜ f 0 T Θ ˜ f 0 + 1 γ g Θ ˜ g 0 T Θ ˜ g 0 If E 0 = Θ ˜ f 0 = Θ ˜ g 0 = 0 , Eq. (A7) equals Eq. (19) and Theorem 1 is proven.
Funding Information:
This research was sponsored by National Science Council, R.O.C. under the grant MOST 108-2221-E-006 -092
Publisher Copyright:
© 2019 Elsevier Ltd
PY - 2019/12
Y1 - 2019/12
N2 - To enhance the consistency and micro positioning accuracy of piezoelectric actuators made based on the batch manufacture, in this paper, a robust fuzzy eliminator with abilities to learn the disturbed actuators' models and to mitigate hysteresis is proposed. This proposed robust fuzzy eliminator can convert the error dynamics between the actuator output position and the desired trajectory into a linear disturbed form, and it quickly converge positioning errors. The residual modelling uncertainties attributed to parameter uncertainties are eliminated with a robust compensator exhibiting positioning error fine-tuning capability. From practical tests, this proposed control law depicts the promising positioning performance in the presence of modelling uncertainties (10%, 20%, and 40% modelling uncertainties), and each controlled piezoelectric actuator has an μm positioning accuracy.
AB - To enhance the consistency and micro positioning accuracy of piezoelectric actuators made based on the batch manufacture, in this paper, a robust fuzzy eliminator with abilities to learn the disturbed actuators' models and to mitigate hysteresis is proposed. This proposed robust fuzzy eliminator can convert the error dynamics between the actuator output position and the desired trajectory into a linear disturbed form, and it quickly converge positioning errors. The residual modelling uncertainties attributed to parameter uncertainties are eliminated with a robust compensator exhibiting positioning error fine-tuning capability. From practical tests, this proposed control law depicts the promising positioning performance in the presence of modelling uncertainties (10%, 20%, and 40% modelling uncertainties), and each controlled piezoelectric actuator has an μm positioning accuracy.
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U2 - 10.1016/j.microrel.2019.113497
DO - 10.1016/j.microrel.2019.113497
M3 - Article
AN - SCOPUS:85073555399
VL - 103
JO - Microelectronics Reliability
JF - Microelectronics Reliability
SN - 0026-2714
M1 - 113497
ER -