Elsevier

Journal of Computational Physics

Volume 306, 1 February 2016, Pages 520-545
Journal of Computational Physics

Identification of discontinuous nonlinear systems via a multivariate Padé approach

https://doi.org/10.1016/j.jcp.2015.11.051Get rights and content

Abstract

We present a nonlinear system identification technique based on multi-dimensional rational polynomials. A multi-dimensional Padé–Legendre approximation is developed to circumvent challenges in dealing with sharp shocks. The purpose of this paper is to investigate the accuracy of such approximations for identification of various nonlinear systems, particularly systems with a non-smooth response surface. This identification approach utilizes the generalized form of a Padé–Legendre approximation for studying multivariable functions. In the studied problems, the nonlinearity is a function of state variables (displacement and velocity), which requires multi-dimensional formulation. Furthermore, a spatial filter is applied to minimize the effects of the singular points in the applicable rational function of the response surface. This study presents different types of nonlinearities including smooth, irregular, and hysteretic functions, in order to demonstrate the performance of the approach under different conditions. In order to study the robustness of the method in comparison to other identification techniques based on plain polynomial representation, a nonlinear system with a sharp discontinuous restoring force surface is considered. The performance of both approaches is investigated for different degrees of “sharpness”. In addition, the accuracy of the identified models to represent the nonlinear system is verified by comparing the output of the system (computed on the basis of the identified model) from data sets corresponding to different excitations than those used for identification purposes. It is shown that the proposed approach provides a robust identification technique for a broad class of highly-nonlinear systems, and it is particularly advantageous to use when dealing with systems incorporating discontinuous properties.

Introduction

The identification of highly nonlinear dynamical systems has presented a challenging problem for researchers over many years. Different methodologies have been employed to represent the nonlinearity of dynamical systems. Parametric identification, with parameters representing the characteristics of the nonlinearity in the system, has had widespread use in the literature [1], [2], [3]. Due to the lack of prior knowledge about the nonlinear mechanism underlying the majority of practical identification problems, nonparametric studies based on information about the state variables have proven to be uniquely suitable in many settings [4], [5], [6], [7]. In such nonparametric approaches, the nonlinearity is characterized as a function of the state variables. While various functional representations can be assumed by employing different basis functions, the accuracy of representation of the objective complex nonlinear system is highly dependent on the selection of the basis functions. Orthogonal polynomials have been mainly used as a robust basis to represent smooth nonlinearities in many studies [8], [9], [10], [11], [12], [13], [14]. In spite of smooth nonlinearities, if the problem of interest involves sharp interfaces or discontinuities, standard polynomial representations may not yield sufficient computational robustness. Moreover, accurate identification of nonlinear systems with hysteretic behavior has always been a challenging problem that has received considerable attention in the literature [15], [16], [17], [18], and thus an efficient robust approach is needed that can be utilized for various nonlinear cases.

The identification method in this study is based on the use of rational functions, namely the Padé–Legendre representation, which is more appropriate than standard polynomial expansion for representing functions with strange local features. In Padé approximations, a function is represented by a rational function. Both the numerator and denominator are represented as a sum of a finite number of orthogonal polynomials [19], [20]. The proposed approach is very well suited for nonlinear identification problems, since it does not require a priori knowledge of a discontinuity location, and is only based on the available point values of a response surface. In the presented problems, the nonlinear functions consist of two variables, displacement and velocity, and consequently a homogeneous multi-dimensional formulation has been adopted [21], [22], [23]. To alleviate the effect of the Gibbs phenomenon and singular points in the rational function representation, a filtering procedure which is the subject of much active research [19], [21], [24] is used. The accuracy of the applied method in identifying either a polynomial-type of nonlinearities with smooth surfaces, or a non-polynomial-type, is verified in different examples. The variety of nonlinear features used in this study to demonstrate the utility and accuracy of the method under discussion (such as Duffing, Van der Pol, Bilinear Hysteretic, piece-wise linear, etc.) are characteristics that are widely encountered, at different scales, in many situations that appear in the broad Applied Mechanics field. Representative problem include the nonlinear behavior of MEMS, nonlinear dampers, nonlinear aero-elastic interaction problems, behavior of cracked reinforced concrete structures, plastic deformations of steel joints, magnetic bearings, etc. The study also validates the reconstructed surface of the nonlinear functions through testing with different sets of excitations.

The main focus of this study is to investigate the accuracy of a proposed multi-dimensional variant of the Padé–Legendre approximation in identifying different types of dynamical systems and to apply this approach to more complicated hysteretic systems and nonlinear systems with restoring forces that exhibit pronounced localized behavior. In order to show the robustness of this approach, results are compared with other available identification techniques.

This paper is organized as follows: Section 2 reviews one-dimensional and multi-dimensional Padé–Legendre approximations. An error study for the multi-dimensional Padé–Legendre approximation is also presented. The different steps involved in the identification with the proposed approach are explained through an illustrative example. In Section 3, the identification and validation of different types of smooth nonlinearities are presented. Section 4 contains the results of identification and validation of a bilinear hysteretic system. In Section 5, results of a nonlinear model with a discontinuous restoring force surface are compared with those from other identification technique based on standard orthogonal polynomials. Section 6 discusses the identification results of a three degrees of freedom system obtained via available samples of state space. Finally, in Section 7 a discussion is presented that summarizes the findings from this investigation.

Section snippets

Method description

In this study, use is made of the Padé–Legendre approximation based on the ratio of two Legendre polynomial expansions. The applied method in this study is the generalization of one-dimensional approximation. For the sake of completeness, the one-dimensional approximation is briefly explained in this section.

Legendre polynomials are solutions of the Legendre differential equation,ddx[(1x2)ddxPn(x)]+n(n+1)Pn(x)=0, and can be expressed as,Pn(x)=12nn!dndxn[(x21)n].

The significant property of

Identification and validation of smooth nonlinearities

For the sake of completeness, the performance of the proposed method in identifying linear systems and smooth nonlinearities such as the Duffing and Van der Pol oscillator is investigated in this section. In each case, the restoring force equation is introduced in the system, and the response of the system when subjected to a random force is obtained by solving the governing differential equation for a one-degree-of-freedom system. The identification process for the examples in this section is

Hysteretic system

The identification of hysteretic systems has always been a challenging problem due to the fact that the function is not exactly representable by polynomials. Satisfactory results, however, can be obtained by polynomial-type approximations [4], [6], [15]. In this section, a bilinear hysteretic oscillator with viscous damping is selected to demonstrate the ability of the method in representing non-smooth nonlinear systems. The schematic diagram of the displacement–force for a bilinear hysteretic

Comparison with different identification techniques

As stated before, several techniques have been previously employed by other investigators in order to identify the nonlinearity in a system. One of the non-parametric techniques that has been used extensively in the literature is the functional representation of a nonlinear system by employing plain orthogonal polynomials [6]. The restoring force in this approach is represented by orthogonal polynomials such as Chebyshev polynomials, which are expressed as,Rˆ(x,x˙)=i=0mj=0ncijTi(x)Tj(x˙),

Identification of a 3-DOF system

In this section, we present results of a three degrees of freedom system as shown in Fig. 26. This system consists of three masses m1,m2,m3 which are connected via six nonlinear elements g1,,g6 anchored to an interface at three locations, thus resulting in a redundant system with 3-DOF. The absolute displacement of each mass and the three excitation forces acting on the system are denoted by xi and fi (i=1,2,3) respectively. The magnitude of the system masses are, m1=1, m2=1.2, m3=1.5. The

Summary and conclusions

This study presents a novel Padé–Legendre approximation technique for identifying severely nonlinear systems. The approach is based on the use of rational functions, which provides an accurate representation of the restoring force in a nonlinear system with sharp discontinuities. For the problems of interest, a general multi-dimensional type of approximation, along with a spatial filter dependent on the denominator of the rational function, is employed. In each test case the system is

Acknowledgements

The authors would like to thank Prof. Roger G. Ghanem for insightful comments and discussions regarding the multivariate Padé approximation. The senior author appreciates the financial support from the USC Graduate School through a Provost's Doctoral Fellowship. He would also like to thank Dr. Emily M. Anton for her critical reading of an earlier version of the manuscript.

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