Nonlinear System Identification Research Articles

An efficient Volterra series approach for identifying nonlinear system vibration responses to an excitation

This paper focuses on the issue of nonlinear vibration responses identification of nonlinear systems. An efficient algorithm is presented, in which the nonlinear vibration system under studied is decomposed into a multiple-input/single-output (MISO) linear system with a series of power characterized inputs based on Volterra series, and the nonlinear output responses of different orders are identified by taking power spectra operation to the input and output data, revealing the contributions of each order nonlinearity to the output of the system. Compared to the existing method, the input signal of the system required in the presented approach need not remain unchanged in waveform and adjustable in magnitude. To verify the approach, a classic Duffing–Van der Pol oscillator was simulated, obtaining results very close to the fourth order Runge–Kutta method. Finally, an experiment analysis was carried out, in which the vibration transmission properties of a bolt connection were tested when the bolt was tight and loose, respectively. The results proved that the approach is effective in identifying nonlinear vibration frequency responses.

Low-resource dynamic loading identification of nonlinear system using pretraining

To address the challenge of load identification in nonlinear systems, an audio neural network-based method called WaveNet is proposed that leverages its capability to capture long-term dependencies in mechanical systems, enabling accurate load identification. Unlike traditional dynamic load identification methods that often encounter difficulties with matrix solutions, this approach takes advantage of WaveNet’s capabilities, enhancing both accuracy and efficiency. We integrate pre-training and transfer learning techniques to address the data scarcity challenges often encountered in real-world engineering applications. By transferring features across distributed datasets, this method reduces the dependency on single-task data, thereby improving model robustness. The performance of the WaveNet method is rigorously evaluated against traditional benchmarks such as Multilayer Perceptron (MLP) and Convolutional Neural Network (CNN) benchmarks under random load conditions applied to a complex structural framework. The proposed method achieves a root mean squared error (RMSE) of 1.521 and a determination coefficient (R²) of 0.996 in the random load case, demonstrating superior accuracy compared to other approaches. Moreover, its applicability is verified through simulations of both impact load and harmonic load scenarios, showcasing the effectiveness of transfer learning in overcoming domain discrepancies. Finally, the method is tested in random experiments to validate its engineering applicability. The results highlight the significant accuracy improvement in low-resource tasks achieved through pre-training, showcasing the potential and value of the proposed method.

Efficient data-driven nonlinear system identification for structural health monitoring: A proof-of-principle study

Structural health monitoring (SHM) is the process of detecting damage in structures. Typically, the damage-sensitive feature identification is based on fitting some models to the measured system response data. The parameters of these models or the prediction errors associated with these models then become the desired damage-sensitive features. Real-world structures usually exhibit significant nonlinear behaviors. Data-driven nonlinear system identification (NSI) is essential to obtain accurate models of structural dynamics for reliable damage detection. However, on the one hand, the data acquisition of real-world structures for damage detection is usually difficult or scarce. On the other hand, NSI generally needs to solve a nonlinear optimization problem that requires more computational time for convergence than linear system identification. Data efficiency and computational efficiency become two critical challenges in data-driven NSI. To address these challenges, this work presents a proof-of-principle study on developing a meta learning-based method for efficient data-driven NSI. Specifically, the meta learning algorithm leverages a previously collected database of similar but different systems to learn the meta-knowledge about how to identify a new system, enabling a fast identification/modeling with limited data. Numerical simulations are performed to validate the method on fundamental nonlinear dynamical systems widely seen in civil engineering. It is observed that the presented meta learning-based method outperforms the conventional method in terms of both data and computational efficiency. Limitations of this work are further discussed.

Identifying Brain Network Structure for an fMRI Effective Connectivity Study Using the Least Absolute Shrinkage and Selection Operator (LASSO) Method.

Background: Studying causality relationships between different brain regions using the fMRI method has attracted great attention. To investigate causality relationships between different brain regions, we need to identify both the brain network structure and the influence magnitude. Most current methods concentrate on magnitude estimation, but not on identifying the connection or structure of the network. To address this problem, we proposed a nonlinear system identification method, in which a polynomial kernel was adopted to approximate the relation between the system inputs and outputs. However, this method has an overfitting problem for modelling the input-output relation if we apply the method to model the brain network directly. Methods: To overcome this limitation, this study applied the least absolute shrinkage and selection operator (LASSO) model selection method to identify both brain region networks and the connection strength (system coefficients). From these coefficients, the causality influence is derived from the identified structure. The method was verified based on the human visual cortex with phase-encoded designs. The functional data were pre-processed with motion correction. The visual cortex brain regions were defined based on a retinotopic mapping method. An eight-connection visual system network was adopted to validate the method. The proposed method was able to identify both the connected visual networks and associated coefficients from the LASSO model selection. Results: The result showed that this method can be applied to identify both network structures and associated causalities between different brain regions. Conclusions: System identification with LASSO model selection algorithm is a powerful approach for fMRI effective connectivity study.

Data‐driven control and a prey–predator model for sourcing decisions in the low‐carbon intertwined supply chain

AbstractThis paper addresses the challenges of low‐carbon sourcing in intertwined supply chains by proposing a data‐driven control framework and a prey–predator model for sourcing decisions. The objective is to optimize low‐carbon objectives and reduce environmental impact. Existing static models fail to capture the dynamic nature of supply chain systems and overlook the ripple effects when sourcing decisions propagate throughout the interconnected network. To bridge this gap, our study develops a dynamic model that explicitly captures the bullwhip effect and leverages real‐time and historical data. This model conceptualizes suppliers as prey and manufacturers and consumers as predators, employing an ecological analogy to decipher the intricate interactions and dependencies within the supply chain. Through this approach, we identify strategies to promote sustainable practices and motivate suppliers to adopt low‐carbon measures. We assess two data‐driven algorithms, the nonlinear auto‐regressive exogenous (NARX) network and sparse identification of nonlinear dynamic systems with input variables (SINDYc). The results reveal that SINDYc outperforms prediction accuracy and control, offering significant advantages for rapid decision‐making. The study highlights how shifts in market demands and regulatory pressures critically influence the strategies of chemical firms and fertilizer markets. Moreover, it discusses the economic challenges in transitioning from high carbon footprint suppliers (HCFSs) to low carbon footprint suppliers (LCFSs), exacerbated by a notable cost disparity where HCFSs are approximately 30% cheaper. By advancing beyond conventional static models, this research provides a deeper understanding of the environmental impacts and operational dynamics within supply chains, emphasizing the significant “ripple effect” where decisions at one node profoundly affect others within the chain.

Backbone curve orientated parameter identification for systems with coupled nonlinearity

Backbone curve attracts much concern from nonlinear system identification studies because it directly orientates the nonlinearity embedded in a system. Although extensive approaches have been proposed for normal nonlinearities that are symmetric or decoupled in the modal space, it is still challenging to extend the backbone-based identification technique for better applicability. To this end, this paper considers a series of representative 2-degree-of-freedom systems with asymmetric and coupled nonlinearities. First, the expression of the backbone curve, derived by the method of multiple scales (MMS), is augmented from only the driving-point distribution to the transfer-point distribution because of the coupling. Then, a hierarchical decoupling scheme is constructed to decompose the coupling-induced multi-modes in the transfer-point responses, making the backbone extraction technique valid for systems with coupled nonlinearities. Finally, the system parameters are identified by fitting the extended backbone functions to the decomposed samples. Numerical and experimental examples are both provided to demonstrate the detailed operations of the identification algorithm. The results show that by introducing the augmented backbone curves, the system parameters can be identified completely, indicating a successful improvement to the existing backbone-based identification technique.

Assessment of hybrid machine learning models for non-linear system identification of fatigue test rigs

The prediction of system responses for a given fatigue test bench drive signal is a challenging problem, since highly dynamic loads from measurement campaigns must be reproduced accurately. Linear frequency response function models are commonly used for this system identification task, but energy intensive experimental iterations are required to account for system non-linearities. Two novel hybrid modeling strategies are suggested, which augment existing approaches using non-linear Long Short-Term Memory networks. These are trained and deployed on short subsequences of measurement data and recombined using a windowing technique, which enables their application to measurement data with high sampling rates. In addition to fatigue test bench commissioning, these hybrid models can also be employed in the field of virtual sensing. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation. It is shown, that hybrid models can successfully use frequency response function models as a linear baseline estimate, which is further improved by Long Short-Term Memory networks to enable non-linear predictions.

Reduced-order model-inspired experimental identification of damped nonlinear structures

In this work, we address Nonlinear System Identification (NSI) of geometrically nonlinear structures using experimental response data. Specifically we consider nonlinear structures with large inertia effects. A laboratory scale cantilever-type beam structure is considered, which is chosen for its large inertial effects. Free decay data is gathered from the experimental cantilever-type beam system using an image-based measurement technique. A general mathematical model is derived for nonlinear systems with large inertia, by taking inspiration from model reduction methods. Specifically, a reduced-order modelling method, which accounts for the kinetic energy of the modes not included in the reduction basis, is utilised for experimental NSI. Experimental system identification using the ROM-inspired model shows superior accuracy compared to standard stiffness nonlinear models typically used for modelling such systems. We also identify the damping of the structure by projecting the effect of unmodelled modes onto non-conservative dissipative forces in the ROM-inspired model. We show that this addresses issues of poor fit associated with using a linear damping model. This results in a nonlinear damping force in the ROM-inspired model which accounts for the damping effect of the unmodelled modes. These nonconservative effects of unmodelled modes are often neglected, which results in an incorrect damping estimation. A crude Finite-Element (FE) model of the cantilever-type beam is used to generate the nonlinear mapping of nonlinear damping force. The free decay response of the identified nonconservative ROM-inspired model closely matches the measurement decay response. This further validates the accuracy of the ROM-inspired model in NSI.

Exploiting the capacity of deep networks only at the training stage for nonlinear black-box system identification

To benefit from the modeling capacity of deep models in system identification without worrying about inference time, this study presents a novel training strategy that uses deep models only during the training stage. For this purpose, two separate models with different structures and goals are employed. The first one is a deep generative model aiming at modeling the distribution of system output(s), called the teacher model, and the second one is a shallow basis function model, named the student model, fed by system input(s) to predict the system output(s). That means these isolated paths must reach the same ultimate target. As deep models show a great performance in modeling highly nonlinear systems, aligning the representation space learned by these two models makes the student model inherit the teacher model’s approximation power. The proposed objective function consists of the objective of each student and teacher model, adding up with a distance penalty between the learned latent representations. The simulation results on three nonlinear benchmarks show a comparative performance with examined deep architectures applied on the same benchmarks. Algorithmic transparency and structure efficiency are also achieved as byproducts.

Nonlinear system identification via sparse Bayesian regression based on collaborative neurodynamic optimization

Abstract Sparse identification of nonlinear dynamics is a popular approach to system identification. In this approach system identification is reformulated as a sparse regression problem, and the use of a good sparse regression method is crucial. Sparse Bayesian learning based on collaborative neurodynamic optimization is a recent method that consistently produces high-quality solutions. In this article, we extensively assess how this method performs for ordinary differential equation identification. We find that it works very well compared with sparse regression algorithms currently used for this task in terms of the tradeoff between the approximation accuracy and the complexity of the identified system. We also propose a way to substantially reduce the computational complexity of this algorithm compared with its original implementation, thus making it even more practical.

Hybridization of deep learning, nonlinear system identification and ensemble tree intelligence algorithms for pan evaporation estimation

A reliable pan evaporation (Epan) estimation over a daily scale is vital for sustainable water and agriculture management, especially for designing water use allocations, irrigation system management, and demand and utilization assessments. However, designing a reliable computational methodology is a challenging task for Epan estimation due to its stochastic and random characteristics. In this study, the potential of five machine learning (ML) models namely Hammerstein-Weiner (HW), Random Forest (RF), Boosted Regression Tree (BRT), Long Short-Term Memory (LSTM), and Stepwise Linear Regression (SWLR) models were evaluated for Epan modeling in Lake Hawassa catchment, Ethiopia. Afterward, hybrid SWLR-LSTM, SWLR-RF, SWLR-BRT, and SWLR-HW models were developed to benefit from the strength of the non-linear and linear characteristics of the models. The estimation results were evaluated using determination coefficient (R2), Nash-Sutcliffe efficiency (NSE), mean absolute error (MAE) and Root Mean Square Error (RMSE), Index of Agreement (IA) and Kling-Gupta efficiency (KGE). The performance analysis result of individual models showed that the RF model led to more accurate estimation with NSE = 0.938, IA = 0.956, R2 = 0.94, KGE = 0.852, RMSE = 0.483 mm/day and Pbias = -1.193 mm/day during the validation phase. Among the hybrid models, SWLR-RF provides the best predictive performance improving SWLR, HW, LSTM, BRT and RF by 26.563 %, 21.348 %, 15.577 %, 9.091 % and 3.514 %, respectively based on the validation phase NSE value. Generally, the results of the current study demonstrated the capability of deep learning, ensemble tree and hybrid linear-nonlinear models for Epan estimation in the data-scarce catchment.

Nonlinear Multivariable System Identification: A Novel Method Integrating T-S Identification and Multidimensional Membership Functions

In this paper, a new multidimensional Takagi–Sugeno (T-S) identification technique is proposed for multivariable nonlinear systems. In this technique, multidimensional membership functions are designed using concepts from solid mechanics. The design of membership functions is carried out in multidimensional space, defining the principal axes from the eigenvectors of the inertia matrix, and it has the characteristic of dividing the space into regions with the same inertia. These regions are analyzed to define the center of gravity for each rule. Illustrative examples of multivariable nonlinear systems, such as a thermal mixing process and a binary distillation column, are selected to evaluate the effectiveness of the proposed method. The proposed method is compared with traditional T-S identification that uses one-dimensional membership functions and shows a reduction in the relative identification error and the algorithm execution time. Additionally, the proposed method prevents rules from being positioned outside the system’s range, thereby avoiding the generation of unnecessary rules.

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Hermite polynomial based affine projection Blake Zisserman algorithm for identification of robust sparse nonlinear system

Hermite polynomial based affine projection Blake Zisserman algorithm for identification of robust sparse nonlinear system

The filtering-based recursive least squares identification and convergence analysis for nonlinear feedback control systems with coloured noises

The coloured noise is ubiquitous in industrial processes. This paper addresses the identification problem for the nonlinear feedback systems with coloured noise. Firstly, a direct identification scheme based on the least squares principle is developed to estimate the whole parameters of the nonlinear feedback systems and the convergence analysis is carried out through the stochastic stability theory. Secondly, for the purpose of improving the estimation accuracy, a filtering-based identification framework is proposed by constructing a linear filter for filtering the input data, output data and the coloured noise, and the coloured noise is transformed into a white noise. This identification scheme based on the filtering technique can effectively reduce the adverse effects caused by coloured noise and parameter estimation accuracy is enhanced compared with the direct least squares algorithm. Meanwhile, the convergence analysis of the filtering-based identification algorithm is given to provide a theoretical analysis. Finally, the simulation example is carried out by performance test and comparison analysis and simulation results show the effectiveness of the proposed identification methods.

Identification for nonlinear systems modelled by deep long short-term memory networks based Wiener model

This paper is concerned with modeling and identification methodology for practical nonlinear system via deep long short-term memory (DLSTM) networks-based Wiener model. To determine the unknown parameters and simplify parameters identification procedure for the Wiener model, a two-step identification scheme is implemented applying hybrid signals involving separable signal and random data. First, the separable signal to separate the two blocks in Wiener model is imported, then parameters of the linear block is estimated using correlation function-based least squares technique, which handles the issue that the intermediate variable information in Wiener model cannot be measured. Moreover, integrating the advantages of adaptive stochastic gradient descent algorithm and root mean square propagation, an adaptive momentum estimation technique is created to optimize the DLSTM networks parameters based on available random data, which improves the accuracy of identified Wiener model. Compared with the other existing schemes, the superiority of the proposed method in terms of predictive performance and control results are illustrated by permanent magnet synchronous motors.

Przewidywanie atlantyckiej południkowej cyrkulacji z wykorzystaniem nieliniowych metod identyfikacji systemu i modelu NARMAX

The Atlantic Meridional Overturning Circulation (AMOC) plays an important role in the coupled ocean-climate system and in global climate change. The analysis of its own behaviour and the understanding its links to other climate dynamics is of paramount importance today as we encounter an increasing pressure to adapt to climate change. Due to the enormous complexity, it is almost impossible to establish accurate models, purely based on first-principle modelling approaches, that can perfectly represent the relationships between the AMOC and other dynamic climate parameters. Data-based or data-driven modelling methods, can therefore provide an attractive alternative solution. Systematic regular and continuous measurement of the AMOC time series began in April 2004. The main objective of the paper is to use the monthly data of the AMOC measured during April 2004-Febuary 2017, together with the North Atlantic Oscillation (NAO) index, and density anomalies of the Gulf of Mexico, Labrador Sea and Norwegian Sea, measured during the same period, to investigate and understand the quantitative relationship between the AMOC and four drivers (NAO and the three density anomaly variables). In doing so, nonlinear system identification methods and the Nonlinear AutoRegressive Moving Average with Exogenous input (NARMAX) method are employed to develop a quantitative model that relates the AMOC to the four drivers. Experimental results show that the derived nonlinear model skillfully captures and represents the dynamics of the AMOC based on the other four variables. One of the findings from this study is that the use of autoregressive variables can help improve the prediction of the AMOC.

Auxiliary model maximum likelihood gradient‐based iterative identification for feedback nonlinear systems

AbstractThis article considers the iterative identification problems for a class of feedback nonlinear systems with moving average noise. The model contains both the dynamic linear module and the static nonlinear module, which brings challenges to the identification. By utilizing the key term separation technique, the unknown parameters from both linear and nonlinear modules are included in a parameter vector. Furthermore, an auxiliary model maximum likelihood gradient‐based iterative algorithm is derived to estimate the unknown parameters. In addition, an auxiliary model maximum likelihood stochastic gradient algorithm is derived as a comparison. The numerical simulation results indicate that the auxiliary model maximum likelihood gradient‐based iterative algorithm can effectively estimate the parameters of the feedback nonlinear systems and get more accurate parameter estimates than the auxiliary model maximum likelihood stochastic gradient algorithm.

Sigmoid distance metric-based spline adaptive filters for nonlinear adaptive noise cancellation

This article devises a spline adaptive filter (SAF) algorithm with robustness for nonlinear adaptive noise cancellation (NANC), called the SAF-SDM, which is built on a distance-based metric with a sigmoid kernel. The presented SAF-SDM delivers insignificant coefficient updates when the system is subjected to remarkable outliers, which leads to low steady-state errors. Additionally, we thoroughly analyze the mean behaviour and mean-square performance of the SAF-SDM at the theoretical level and validate the analysis results using nonlinear system identification (NSI) simulations. To fulfill the higher computational efficiency when a longer linear filter order is selected in the SAF, an improved SAF-SDM is developed through a frequency domain (FD) filtering program, called the FDSAF-SDM. Then, we provide an analysis of the calculation complexity of the two presented algorithms and compare them with those of several similar algorithms. Finally, simulation and experiment results demonstrate the excellence of the presented algorithms in NSI and NANC.

Online Identification of Nonlinear Systems With Separable Structure.

Separable nonlinear models (SNLMs) are of great importance in system modeling, signal processing, and machine learning because of their flexible structure and excellent description of nonlinear behaviors. The online identification of such models is quite challenging, and previous related work usually ignores the special structure where the estimated parameters can be partitioned into a linear and a nonlinear part. In this brief, we propose an efficient first-order recursive algorithm for SNLMs by introducing the variable projection (VP) step. The proposed algorithm utilizes the recursive least-squares method to eliminate the linear parameters, resulting in a reduced function. Then, the stochastic gradient descent (SGD) algorithm is employed to update the parameters of the reduced function. By considering the tight coupling relationship between linear parameters and nonlinear parameters, the proposed first-order VP algorithm is more efficient and robust than the traditional SGD algorithm and alternating optimization algorithm. More importantly, since the proposed algorithm just uses the first-order information, it is easier to apply it to large-scale models. Numerical results on examples of different sizes confirm the effectiveness and efficiency of the proposed algorithm.