PRESS-based EFOR algorithm for the dynamic parametrical modeling of nonlinear MDOF systems

doi:10.1007/s11465-017-0459-5

Front. Mech. Eng.

2018, Vol. 13

Issue (3) : 390-400 https://doi.org/10.1007/s11465-017-0459-5

RESEARCH ARTICLE

PRESS-based EFOR algorithm for the dynamic parametrical modeling of nonlinear MDOF systems

Haopeng LIU^1,², Yunpeng ZHU³, Zhong LUO^1,²(

), Qingkai HAN⁴

¹. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
². Key Laboratory of Vibration and Control of Aero-Propulsion Systems (Ministry of Education), Northeastern University, Shenyang 110819, China
³. Department of Automatic Control and System Engineering, Sheffield University, Sheffield S13JD, UK
⁴. School of Mechanical Engineering, Dalian University of Technology, Dalian 116023, China

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Abstract

In response to the identification problem concerning multi-degree of freedom (MDOF) nonlinear systems, this study presents the extended forward orthogonal regression (EFOR) based on predicted residual sums of squares (PRESS) to construct a nonlinear dynamic parametrical model. The proposed parametrical model is based on the non-linear autoregressive with exogenous inputs (NARX) model and aims to explicitly reveal the physical design parameters of the system. The PRESS-based EFOR algorithm is proposed to identify such a model for MDOF systems. By using the algorithm, we built a common-structured model based on the fundamental concept of evaluating its generalization capability through cross-validation. The resulting model aims to prevent over-fitting with poor generalization performance caused by the average error reduction ratio (AERR)-based EFOR algorithm. Then, a functional relationship is established between the coefficients of the terms and the design parameters of the unified model. Moreover, a 5-DOF nonlinear system is taken as a case to illustrate the modeling of the proposed algorithm. Finally, a dynamic parametrical model of a cantilever beam is constructed from experimental data. Results indicate that the dynamic parametrical model of nonlinear systems, which depends on the PRESS-based EFOR, can accurately predict the output response, thus providing a theoretical basis for the optimal design of modeling methods for MDOF nonlinear systems.

Keywords MDOF dynamic parametrical model NARX model PRESS-based EFOR cantilever beam

Corresponding Author(s): Zhong LUO

Just Accepted Date: 19 June 2017 Online First Date: 14 September 2017 Issue Date: 11 June 2018

Cite this article:

Haopeng LIU,Yunpeng ZHU,Zhong LUO, et al. PRESS-based EFOR algorithm for the dynamic parametrical modeling of nonlinear MDOF systems[J]. Front. Mech. Eng., 2018, 13(3): 390-400.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-017-0459-5
https://academic.hep.com.cn/fme/EN/Y2018/V13/I3/390

Fig.1 A SIMO 5-DOF nonlinear system

Fig.2 Input and output signals for 5-DOF nonlinear system, with

c 3 = 5

N·s/m. (a) Input signal corresponding to

c 3 = 5

N·s/m; output signal corresponding to (b) Mass 1, (c) Mass 2, (d) Mass 4, and (e) Mass 5

Items	Coefficients for different data sets
Items	$r = 1, ? k ′ = 1$	$r = 2, ? ? k ′ = 1$	…	$r = 4, ? k ′ = 3$	$r = 5, ? ? k ′ = 3$
$y r, k ′ (t − 4)$	−1.0037	−0.5793	…	$− 0.4284$	$− 0.6599$
$y r, k ′ (t − 3)$	−0.4503	−0.6172	…	$− 0.5227$	$− 0.5805$
$u (t − 2)$	1.5604×10⁻⁷	2.7093×10⁻⁷	…	$2.3781 × 10 - 7$	2.5898×10^–7
$y r, k ′ (t − 5) 2$	−6.4735×10³	1.6380×10³	…	$1.5441 × 103$	–595.5596
$y r, k ′ (t − 1)$	−0.5606	0.3449	…	0.5244	0.1546
$y r, k ′ (t − 5)$	−0.2148	0.7993	…	0.6141	0.5681
$y r, k ′ (t − 2)$	−0.0148	0.5374	…	0.2300	0.4020
$u (t − 1)$	8.6227×10⁻⁹	$1.4004 × 10 − 8$	…	$1.5268 × 10 − 7$	$2.6536 × 10 − 7$
$y r, k ′ (t − 3) 2$	−8.6247×10³	$− 7.9633 × 103$	…	$− 3.2764 × 103$	$− 3.3514 × 103$
$u (t − 4)$	4.7164×10⁻⁸	$− 1.9658 × 10 − 7$	…	$− 1.5217 × 10 − 7$	$1.9945 × 10 − 8$
$y r, k ′ (t − 4) 2$	−1.7444×10⁴	$− 5.0929 × 103$	…	$− 5.4255 × 103$	$− 2.9004 × 103$
$y r, k ′ (t − 3) y r, k ′ (t − 4)$	7.5532×10³	$1.1435 × 104$	…	$9.0404 × 103$	$6.2947 × 103$
$y r, k ′ (t − 2) y r, k ′ (t − 3)$	$− 1.0939 × 104$	$3.0158 × 103$	…	$− 262.0313$	115.0254
$u (t − 1) y r, k ′ (t − 1)$	0.0025	$8.2014 × 10 − 4$	…	$2.2417 × 10 − 4$	$− 0.0011$

Tab.1 Identification results of the using PRESS-based EFOR

Items	m	$β m, n$
Items	m	n=0	n=1	n=2	n=3
$y r, k ′ (t − 4)$	1	$− 0.5119$	0.0022	$− 0.0152$	$− 4.9870 × 10 − 5$
$y r, k ′ (t − 3)$	2	$− 0.5540$	$− 0.0017$	0.0023	$3.2145 × 10 − 5$
$u (t − 2)$	3	$2.8777 × 10 - 7$	$− 1.2005 × 10 − 9$	$− 3.8210 × 10 − 9$	$− 8.3750 × 10 − 12$
$y r, k ′ (t − 5) 2$	4	$2.2553 × 103$	$− 41.2508$	$− 267.4712$	3.7306
$y r, k ′ (t − 1)$	5	0.4662	$8.0458 × 10 − 4$	$− 0.0315$	$7.4240 × 10 − 5$
$y r, k ′ (t − 5)$	6	0.7592	$− 6.3085 × 10 − 4$	$− 0.0281$	$2.0166 × 10 − 4$
$y r, k ′ (t − 2)$	7	0.4043	$7.7886 × 10 − 5$	$− 0.0106$	$1.0270 × 10 − 4$
$u (t − 1)$	8	$2.5045 × 10 - 7$	$− 1.5431 × 10 − 9$	$− 1.1224 × 10 − 8$	$6.5492 × 10 − 11$
$y r, k ′ (t − 3) 2$	9	$− 6.5611 × 10 - 7$	264.7419	$− 49.7505$	$− 19.2054$
$u (t − 4)$	10	$− 1.0255 × 10 - 7$	$7.1959 × 10 − 10$	$2.6353 × 10 − 9$	$− 8.6044 × 10 − 11$
$y r, k ′ (t − 4) 2$	11	$− 2.5302 × 103$	41.1955	$− 485.0265$	$− 0.1359$
$y r, k ′ (t − 3) y r, k ′ (t − 4)$	12	$1.1643 × 104$	$− 260.7791$	$− 140.3549$	16.8510
$y r, k ′ (t − 2) y r, k ′ (t − 3)$	13	$4.7022 × 103$	$− 200.8071$	$− 531.0433$	19.9273
$u (t − 1) y r, k ′ (t − 1)$	14	$8.3216 × 10 -5$	$− 3.8779 × 10 − 5$	$1.2881 × 10 − 4$	$− 8.3858 × 10 − 6$

Tab.2 Estimates for the parameter

β m, n

Fig.3 Comparison of the outputs of the PRESS-based EFOR, AERR-based EFOR, and the corresponding real system.

Fig.4 Experimental set-up. (a) Test rig of the cantilever beam with transducers; (b) LMS (learning management system) test system

Fig.5 Input and output signals for experimental validation. (a) The input signal used for the experiment as well as the output signals corresponding to (b) Transducer 1, (c) Transducer 3, (d) Transducer 4, and (e) Transducer 5

Fig.6 Comparison between the PRESS-based EFOR output and corresponding real measurements

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