Identification of discontinuous nonlinear systems via a multivariate Padé approach
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, and can be expressed as,
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,
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 which are connected via six nonlinear elements 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 and respectively. The magnitude of the system masses are, , , . 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.
References (29)
Parametric identification of nonlinear dynamic systems
Comput. Struct.
(1985)- et al.
Parametric time-domain methods for the identification of vibrating structures – a critical comparison and assessment
Mech. Syst. Signal Process.
(2001) - et al.
Parametric identification of a time-varying structure based on vector vibration response measurements
Inverse Problems
Mech. Syst. Signal Process.
(2009) - et al.
Past present and future of nonlinear system identification in structural dynamics
Mech. Syst. Signal Process.
(2006) - et al.
Development of data-based model-free representation of non-conservative dissipative systems
Nonlinear Dynamic Stability of Nonconservative Dissipative Systems
Int. J. Non-Linear Mech.
(2007) - et al.
Identification and prediction of stochastic dynamical systems in a polynomial chaos basis
Computational Methods in Stochastic Mechanics and Reliability Analysis
Comput. Methods Appl. Mech. Eng.
(2005) - et al.
On the identification of non-linear mechanical systems using orthogonal functions
Int. J. Non-Linear Mech.
(2004) - et al.
Proper orthogonal decomposition for model updating of non-linear mechanical systems
Mech. Syst. Signal Process.
(2001) - et al.
Development of adaptive modeling techniques for non-linear hysteretic systems
Int. J. Non-Linear Mech.
(2002) - et al.
On-line identification of non-linear hysteretic structures using an adaptive tracking technique
Themes in Non-linear Stochastic Dynamics
Int. J. Non-Linear Mech.
(2004)
Padé–Legendre approximants for uncertainty analysis with discontinuous response surfaces
J. Comput. Phys.
Multivariate Padé-approximants
J. Math. Anal. Appl.
How well can the concept of Padé approximant be generalized to the multivariate case?
J. Comput. Appl. Math.
Generalized multivariate Padé approximants
J. Approx. Theory
Cited by (2)
A multivariate nonlinear dynamic programming of female figures in Qing dynasty literature
2024, Applied Mathematics and Nonlinear SciencesNonlinear updating method: a review
2017, Journal of the Brazilian Society of Mechanical Sciences and Engineering