DOI:10.13465/j.cnki.jvs.2013.13.026 32 13 JOURNAL OF VIBRATION AND SHOCK Vol. 32 No. 13 2013 ARX 1 2 2 1. 400054 2. 230027 ARX ARX ARX TH113 A System identification of a positioning stage based on theoretical modeling and an ARX model WANG Jing-shu 1 GUO Jie 2 ZHU Chang-an 2 1. Chongqing University of Technology School of Mechanical Engineering Chongqing 400054 China 2. Department of Precision Machinery and Precision Instruments University of Science and Technology of China Hefei 230027 China Abstract In order to know the characteristics of a positioning stage a dynamic model with single input and twooutput was developed and the positioning tests were done. Based on the dynamic model and the test data after interpolating and normalizing the unknown parameters were identified with the least square method. Another identification method based on an ARX model was applied to recognize the inside stage system. The results comparison indicated that the ARX model is appropriate to describe the characteristics of the stage system while the dynamic model is meaningful for structural optimization. Key words positioning stage dynamic model system identification ARX STM ARX ARMAX Box- Jenkins 5-6 ARMA ARX 7-9 10-12 13-14 1-4 2012-06 - 13 2012-08 - 07 1988 11 1 1
ARX 67 1 000 Hz 2. 1 1 000 Hz 0. 001 s 2 1 Fig. 1 The structure of the positioning stage 2. 2 2 Fig. 2 The kinetic model of the stage 50 μm 50 μm 0 u x in = x 1 /50 μm x 1 x 2 x out = x 2 /50 μm M m x in k 2 b 2 x out k 1 b 1. Mx 1 = - k 1 x 1 - b 1 x 1 - k 2 u + x 1 - x 2 mx 2 = k 2 u + x 1 - x 2 + b 2 x 1 - x 2 2 1 000 Hz 1 000 Hz 0. 001 s x n T m = T m - t i -1 x t i - t n t i - x n t i -1 i -1 n = 1 2 x n = 1 n = 2 T m = m* 0. 001 t i T m t i - 1 T m x n 3 50 μm
68 2013 32 M m 1 k 1 k 2 b 1 3 3 b 2 1 Tab. 1 The known parameters of the kinetic model M m kg 28. 877 4 14. 404 5 1 000 Hz 4 T i y i i = 1 2 3 n min n i = 0 f T i - y i 2 n T i y i f T - matlab 2 2 Tab. 2 The unknown parameters of the kinetic model k 1 / N m - 1 k 2 / N m - 1 b 1 / Nm s - 1 b 2 / Nm s - 1 1. 336 1 10 6 2. 908 5 10 6 5. 622 10 3 685. 2 ARX ARMAX Box-jenkins 1 ARX ARX A z -1 y k = B z -1 u k - n k + ξ k ξ k n k ARX ξ k = 0 n k = 0 Rissanen [ ( ) ] min V 1 + dlogn N N d V ARX A 10 B 1 3 10 Fig. 3 The identification results of the kinetic model 2
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