Nonlinear System Identification Methodology Research Articles

Control-Enabling Adaptive Nonlinear System Identification

In this article, we rely on the theoretical foundations of radial basis function neural networks to form an adaptive parameter estimation problem, which we solve using the recently introduced prescribed performance control methodology. Such combination results in a compact user-configurable adaptive nonlinear system identification methodology that can be used to retrieve the open-loop nonlinear plant dynamics in any compact region of interest.

Study of nonlinear effects in a bolted joint using the base excitation as an input

In this paper, the nonlinearity detection, characterization and identification of a bolted beam assembly is presented. The methodology utilized in this work is based on the force reconstruction using the base excitation as an input and the nonlinear system identification. The force reconstruction procedure was carried out by exciting the bolted beam assembly at very low excitation with fully tightened bolt condition. The nonlinear effect in the bolted beam assembly was induced by reducing the bolt clamping loads. A collection of frequency response functions (FRFs) are shown at different clamping loads to detect and characterize the nonlinearities. Once the nonlinearities are detected and characterized, the restoring force surface method using the reconstructed force was used to identify the nonlinear parameters in the modal space. Four different base excitation (energy) levels with three different tightening torques were considered in the tests in order to study the energy dependence of the damping nonlinearities. In all the cases, the nonlinear system identification methodology employed was successful in identifying the damping and stiffness nonlinearities.

Experimental system identification of the dynamics of a vibro-impact beam with a view towards structural health monitoring and damage detection

We perform nonlinear system identification (NSI) on the acceleration signals that were experimentally measured at ten, almost evenly spaced positions along a cantilever beam undergoing vibro-impacts between two rigid stops with clearances. Our goal is to characterize the nonlinear dynamics due to vibro-impacts with a view toward structural health monitoring (SHM) and damage detection (DD). The NSI methodology is based on the correspondence between analytical and empirical slow-flow dynamics, with the first step requiring empirical mode decomposition (EMD) analysis of the measured time series leading to sets of intrinsic modal oscillators (IMOs) governing the vibro-impact dynamics at different time scales. By comparing the spatiotemporal variations of the nonlinear modal interactions (and hence the IMOs), we examine how vibro-impacts influence the low- and high-frequency modes in global and local senses. In applications of the NSI results to SHM/DD, we calculate typical measures such as the modal assurance criterion (MAC) and the coordinate modal assurance criterion (COMAC) by extracting information about the mode shape functions from the spatiotemporal IMO solutions. Whereas the MAC provides a global aspect of damage occurrence (i.e., which modes are more affected by induced defects), the COMAC can narrow down the damage locations (i.e., where in the structure defects exist that yield low correlation values in specific modes).

Nonlinear system identification of frictional effects in a beam with a bolted joint connection

We perform nonlinear system identification (NSI) of the effects of frictional connections in the dynamics of a bolted beam assembly. The methodology utilized in this work combines experimental measurements with slow-flow dynamic analysis and empirical mode decomposition, and reconstructs the dynamics through reduced-order models. These are in the form of single-degree-of-freedom linear oscillators (termed intrinsic modal oscillators—IMOs) with forcing terms derived directly from the experimental measurements through slow-flow analysis. The derived reduced order models are capable of reproducing the measured dynamics, whereas the forcing terms provide important information about nonlinear damping effects. The NSI methodology is applied to model nonlinear friction effects in a bolted beam assembly. A ‘monolithic’ beam with identical geometric and material properties is also tested for comparison. Three different forcing (energy) levels were considered in the tests in order to study the energy-dependencies of the damping nonlinearities induced in the beam from the bolted joint. In all cases, the NSI methodology employed was successful in identifying the damping nonlinearities, their spatial distributions and their effects of the vibration modes of the structural component.

A new evolving compact optimised Takagi–Sugeno fuzzy model and its application to nonlinear system identification

A new encoding scheme is presented for a fuzzy-based nonlinear system identification methodology, using the subtractive clustering and non-dominated sorting genetic algorithm. The proposed method consists of two parts. The first part is related to the selection of most relevant or influencing inputs to the system and the second one is related to the tuning of fuzzy rules and parameters of the membership functions. The main purpose of the proposed scheme is to reduce the complexity and increase the accuracy of the model. In particular, three objectives are considered in the process of optimisation, namely, the number of inputs, number of rules and the root mean square of the modelling error. The performance of the developed method is validated by identifying the Box–Jenkins nonlinear benchmark system, and to the modelling of the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper. The latter is also a challenging problem due to the inherent hysteretic and highly nonlinear dynamics of the MR damper. It is shown that the developed evolving Takagi–Sugeno (T–S) fuzzy model can identify and grasp the nonlinear dynamics of both systems very well, while a small number of inputs and fuzzy rules are required for this purpose.

Current efforts towards a non-linear system identification methodology of broad applicability

A review of current efforts towards developing a non-linear system identification (NSI) methodology of broad applicability [ 1 – 4 ] is provided in this article. NSI possess distinct challenges, since, even the task of identifying a set of (linearized) modal matrices modified (‘perturbed’) by non-linear corrections might be an oversimplification of the problem. In that context, the integration of diverse analytical, computational, and post-processing methods, such as slow flow constructions, empirical mode decompositions, and wavelet/Hilbert transforms to formulate a methodology that holds promise of broad availability, especially to systems with non-smooth non-linearities such as clearances, dry friction and vibro-impacts is proposed. In particular, the proposed methodology accounts for the fact that, typically, non-linear systems are energy- and initial condition-dependent, and has both global and local components. In the global aspect of NSI, the dynamics is represented in a frequency–energy plot (FEP), whereas in the local aspect of the methodology, sets of intrinsic modal oscillators are constructed to model specific non-linear transitions on the FEP. The similarity of the proposed methodology to linear experimental modal analysis is discussed, open questions are outlined, and some applications providing a first demonstration of the discussed concepts and techniques are provided.

Reduced Order Nonlinear System Identification Methodology

A new method is presented which enables the identification of a reduced order nonlinear ordinary differential equation (ODE) which can be used to model the behavior of nonlinear fluid dynamics. The method uses a harmonic balance technique and proper orthogonal decomposition to compute reduced order training datawhich is then used to compute the unknown coefficients of the nonlinear ODE. The method is used to compute the Euler compressible flow solutions for the supercritical two-dimensional NLR-7301 airfoil undergoing both small and large pitch oscillationsatthreedifferentreducedfrequenciesandataMachnumberof0.764.Steadyanddynamicliftcoefficient datacomputedusingathreeequationreducedordersystemidentificationmodelcomparedwellwithdatacomputed using the full CFD harmonic balance solution. The system identification model accurately predicted a nonlinear trend in the lift coefficient results (steady and dynamic) for pitch oscillation magnitudes greater than 1 deg. Overall the reduction in the number of nonlinear differential equations was 5 orders of magnitude which corresponded to a 3 order of magnitude reduction in the total computational time.

Nonlinear Model for Sub- and Superharmonic Motions of a MDOF Moored Structure, Part 1—System Identification

In this first part of a two-part study, the general nonlinear system identification methodology developed earlier by the authors for a single-degree-of-freedom (SDOF) system using the reverse-multi-input/single-output (R-MI/SO) technique is extended to a multi-degree-of-freedom (MDOF), sub-merged, moored structure with surge and heave motions. The physical nonlinear MDOF system model and the formulation of the R-MI/SO system-identification technique are presented. The corresponding numerical algorithm is then developed and applied to the experimental data of the MDOF system using only the subharmonic motion responses to identify the system parameters. The resulting model is then employed in Part 2 for a detailed analysis of both the sub and superharmonic dynamic behavior of the MDOF experimental system and a comparison of the MDOF response results and observations with those of the corresponding SDOF system examined earlier by the authors.

A METHODOLOGY FOR IDENTIFICATION OF NARMAX MODELS APPLIED TO DIESEL ENGINES

In this paper a nonlinear system identification methodology based on a polynomial NARMAX model representation is considered. Algorithms for structure selection and parameter estimation are presented and evaluated. The goal of the procedure is to provide a nonlinear model characterized by a low complexity and that can be efficiently used in industrial applications. The methodology is illustrated by means of an automotive case study namely a variable geometry turbocharged diesel engine. The nonlinear model representing the relation between the variable geometry turbine command and the intake manifold air pressure is identified from data and validated.

Nonlinear System Identification of Systems with Periodic Limit-Cycle Response

A nonlinear system identification methodology based on the principle of harmonic balance and bifurcation theory techniques like center manifold analysis and normal form reduction, is presented for multi-degree-of-freedom systems. The methodology, called Bifurcation Theory System IDentification, (BiTSID), is a general procedure for any nonlinear system that exhibits periodic limit cycle response and can be used to capture the bifurcation behavior of the nonlinear systems. The BiTSID methodology is demonstrated on an experimental system single-degree-of-freedom system that deals with self-excited motions of a fluid-structure system with a sub-critical Hopf bifurcation. It is shown that BiTSID performs excellently in capturing the stable and unstable limit cycles within the experimental regime. Its performance outside the experimental regime is also studied. The application of BiTSID to experimental multi-degree-of-freedom systems has also been very successful. However in this study only the results of the single-degree-of-freedom system are presented.

Nonlinear System Identification of Multi-Degree-of-Freedom Systems

A nonlinear system identification methodology based on theprinciple of harmonic balance is extended tomulti-degree-of-freedom systems. The methodology, called HarmonicBalance Nonlinearity IDentification (HBNID), is then used toidentify two theoretical two-degree-of-freedom models and anexperimental single-degree-of freedom system. The three modelsand experiments deal with self-excited motions of afluid-structure system with a subcritical Hopf bifurcation. Theperformance of HBNID in capturing the stable and unstable limitcycles in the global bifurcation behavior of these systems is alsostudied. It is found that if the model structure is well known,HBNID performs well in capturing the unknown parameters. If themodel structure is not well known, however, HBNID captures thestable limit cycle but not the unstable limit cycle.

Non-linear system identification using an artificial immune system

In this paper, a novel non-linear system identification methodology is developed employing the features of the artificial immune system. A simplified incremental approach integrated with the maximum entropy principle and an instantaneous feedback mechanism is proposed to reorganize the system's parameters simultaneously. To verify and demonstrate the effectiveness of the proposed algorithm, a simulation example on a two-link robot was studied. This algorithm can achieve robustness and efficiency in identifying complex non-linear systems. The simulation results show that the identified immune models are robust to noise and various uncertainties in the robot dynamics.