Identification for stable second-order nonlinear systems

A novel deep neural network with the dynamical moving window data for nonlinear system identification

Neural networks can learn the behaviour of nonlinear dynamical systems and achieve high prediction accuracy on test data that is drawn from the same distribution as the training data. However, these models fail to generalise during inference when excited with unseen input trajectories. We tackle this generalisation problem in the context of nonlinear system identification with a system-theoretical approach that ensures input–output stability. We enhance a linear approximation with a recurrent neural network (RNN) that models the residual behaviour to capture complex dynamics and to increase the model's generalisation capabilities. We impose constraints on the learnable parameters to ensure dissipativity, an intrinsic property of most physical systems. This leads to improved generalisation on previously unseen inputs. We evaluate our approach using in-distribution (ID) and out-of-distribution (OOD) data from three different use cases and compare it against non-regularised approaches.

Abstract A novel structure for a Robust Dynamic Neural Network Identifier (RDNNI) based on sliding modes is proposed for the online identification of nonlinear second-order systems. A stability analysis using Lyapunov theory demonstrates that both the neural weight estimation errors and the sliding surface converge to zero in finite. Most of the identification schemes based on dynamic neural networks reported in the literature only assure the asymptotic convergence of the synaptic weights and the identification process. There exist some works which modifies the identifier structure to guarantee the finite time convergence of the identification process. However, all these schemes have the limitation that identification errors converge to a region around the origin because of the upper boundaries introduced by the system disturbances. Compared these types of dynamic neural network identifiers found in the literature, the proposed identifier combines the non-singular terminal sliding mode surface with the synaptic weights’ adaptation laws to guarantee that the identification errors converge to zero in a finite time. To evaluate the performance of the developed structure, a simulation is conducted for the identification of a Duffing oscillator, together with a series of experimental tests to identify the dynamics of depth and yaw in an unmanned underwater vehicle. Finally, a quantitative comparison of the identification performance with a recently introduced State-Input Affine Differential Neural Network (SIADNN) identifier demonstrates the superior performance of the proposed design, both in simulations and in real-time experiments.

Inertial measurement units (IMUs) in low-cost navigation systems suffer from significant drift and noise errors due to sensor biases, scale factor instability, and nonlinear stochastic noise. This paper proposes a hybrid error compensation approach that combines Fast Orthogonal Search (FOS), a nonlinear system identification technique, with deep Long Short-Term Memory (LSTM) neural networks to improve IMU signal accuracy in GNSS-denied navigation. The FOS algorithm efficiently models deterministic error patterns (such as bias drift and scale factor errors) using a small training dataset, while the LSTM learns the IMU's complex time-dependent error dynamics from much longer training data. In the proposed method, FOS is first used to predict the output of a high-end IMU based on that of a low-end IMU, and the trained FOS model is then used to extend the training data for an LSTM-based predictor. We demonstrate the efficacy of this FOS-LSTM hybrid on real vehicular IMU data by training with a limited segment of high-precision reference measurements and testing on extended operation periods. The hybrid model achieves high predictive accuracy for predicting the high-end signal based on the low-end signal, with a mean squared error below 0.1% and yields more stable velocity estimates than models using FOS or LSTM alone. Although long-term position drift is not fully eliminated, the proposed method significantly reduces short-term uncertainty in the inertial solution. These results highlight a promising synergy between model-based system identification and data-driven learning for sensor error calibration in navigation systems. Key contributions include FOS-based pseudo-label bootstrapping for data-efficient LSTM training and a navigation-level evaluation illustrating how signal correction impacts dead reckoning drift.

Filtering-based three-stage Levenberg-Marquardt iterative identification of dual-rate Volterra-Wiener nonlinear systems with colored noise.

A nonlinear inverse system identification for the load region assessment of stiffened plates under short-duration impacts

Identification of nonlinear dynamics from input-output data is crucial in many fields where conventional linear models fail to capture nonlinear dynamics of complex systems. Although recurrent neural network architectures have the potential to deal with these problems, they often face limitations in stability, memory capacity, and convergence efficiency. Recent developments in quantum neural networks (QNNs) offer a promising alternative due to their inherent parallelism and high-dimensional processing power. However, the application of QNNs in dynamic nonlinear modeling is still underexplored, especially with regard to stability-guaranteed learning strategies. To address this gap, a novel Diagonal Recurrent Quantum Neural architecture with Lyapunov Stability (DRQNN-LS) has been developed, which combines the structural simplicity of diagonal recurrent networks harnessing the capabilities of quantum learning algorithms and the mathematical rigor of Lyapunov stability theory. Stable convergence and efficient parameter tuning are ensured by deriving adaptive learning rates through Lyapunov analysis. The proposed model is evaluated through three scenarios: a mathematical nonlinear system, a chaotic Henon map, and a practical DC motor system. Comparative analysis with other models demonstrates the exceptional capabilities of DRQNN-LS in terms of the RMSE, MSE, and FIT metrics. The obtained responses validate the effectiveness and robustness of DRQNN-LS for modeling highly nonlinear and real-world systems.

On continuous-time sparse identification of nonlinear polynomial systems

Multisine input signal design for constrained, “plant-friendly” system identification of nonlinear systems

Helicopter turboshaft modeling via mixtures of experts

Discovering nonlinear PDEs with sparse identification of nonlinear dynamical systems (SINDy) is hindered by high dimensionality, noise, and expensive data acquisition. We propose the greedy sampling neural network for sparse identification of nonlinear PDEs (GN-SINDy), a three–stage framework that integrates strategic sampling, differentiable surrogate modelling, and sparse equation discovery. First, a two–way Q-DEIM–based greedy strategy selects maximally informative space–time samples from snapshot data, drastically reducing data requirements. Second, a deep neural network (DNN) is trained as a differentiable surrogate of the solution field, enabling noise–robust analytic derivatives via automatic differentiation. Third, sparse regression with sparsity–promoting estimators [Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016a). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937; Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614.] is applied to recover the governing PDE. Building on the DeepMoD paradigm, GN-SINDy embeds greedy sampling into data acquisition and stabilises coefficient estimation through neural–enhanced differentiation. We analyze noise robustness, structural stability, and QR–based sampling strategies to guide sampler and hyperparameter selection. Experiments on Burgers', Allen–Cahn, and Korteweg–de Vries equations show that GN-SINDy reliably recovers governing PDEs using under 1 % of the data, outperforming DeepMoD in efficiency, support recovery, and robustness to noise.

Reference evapotranspiration (ET0) is a critical parameter for water resource management and irrigation scheduling. Accurate estimation of ET0 has challenged scientists over the years due to its high sensitivity to climatic variations. Classical methods for estimating ET0 mainly rely on empirical models with a significant number of parameters, which has hampered their use in many cases. Regarding its importance and strong relationship with global food security, this topic has attracted the attention of many researchers. The development of simple models with a low number of parameters or taking advantage of artificial intelligence algorithms has been the aim of different researchers, as it is in this paper, where two approaches are implemented to estimate reference evapotranspiration. The first one is based on the use of classical system identification models, namely linear and nonlinear AutoRegressive models with eXogenous variables (ARX and nonlinear ARX). For the second approach, AI-supported system identification models are used, in which neural networks’ performances are used to develop multilayer and deep neural network models for nonlinear system identification. The four models show a high accuracy, with a system fitting exceeded 98%.

The synapse is the fundamental unit of communication in the nervous system. Determining how information is transferred across the synaptic interface is one of the most complex endeavors in neuroscience, owing to the large number of contributing factors and events. An approach to solving this problem involves collapsing across these complexities to derive concise mathematical formulas that fully capture the governing dynamics of synaptic transmission. We investigated the feasibility of deriving such a formula - an input-output transformation function for the CA3 to CA1 node of the hippocampus - using the Volterra expansion technique for non-linear system identification. The timecourse of the fEPSP in the apical dendrites of mouse brain slices was described with >94% accuracy by a 2nd order equation that captured the linear and non-linear influence of past inputs on current outputs. This function generalized to cases not included in its derivation and uncovered previously undetected timing rules. The basal dendrites expressed a substantially different transfer function and evidence was obtained that, unlike the apical system, a 3rd order system or higher will be needed for complete characterization. At scale, the approach will also provide information needed for the construction of biologically realistic models of brain networks.

This study introduces a novel physics-informed deep learning framework for structural identification of nonlinear systems under seismic excitation. The approach integrates Long Short-Term Memory (LSTM) networks with the governing equations of motion to estimate instantaneous, time-varying modal parameters (natural frequencies, damping ratios, mode shapes) without window-based approximations. A Convolutional Neural Network (CNN) extracts time-frequency features, enhancing the LSTM's ability to capture history-dependent nonlinear behavior. Training employs a composite loss function that enforces physical constraints, ensuring predictions adhere to structural dynamics. Validated using shake-table test data from a full-scale base-isolated building, results demonstrate exceptional accuracy in reconstructing acceleration and displacement responses. The identified time-variant properties enable high-fidelity hysteretic loop reconstruction and tangent stiffness analysis. Sensitivity studies confirm robustness to sensor sparsity and initial conditions. This physics-constrained methodology advances real-time structural health monitoring, offering potential for damage detection and forward response prediction.

Physics-informed exogenous-input embedded LSTM method for nonlinear system identification incorporating unobserved key variable

Abstract Multi-stable vibration energy harvesters (VEHs) are low-energy consumption devices and self-powered core devices for wireless sensor networks. However, it is difficult to model their stochastic differential equations (SDEs) under noise excitation. Traditional sparse regression relies on artificial preset basis functions, which is not suitable for complex nonlinear systems. Aiming at this core problem, this paper introduces an evolutionary symbolic sparse regression (ESSR) method to solve its stochastic dynamic modeling problem. The method realizes the adaptive evolution of the basis function by genetic programming, synchronously models multiple response variables with multi-tree coding, and outputs a high-precision explicit analytical model by iteratively screening the fitting drift term and diffusion term. This method provides reliable support for multi-steady-state VEHs modeling, can promote its application in the optimization of self-powered systems, and also provides technical reference for multi-domain nonlinear system identification.

Accurate parameter identification in nonlinear and chaotic dynamic systems requires optimization algorithms that can reliably balance global exploration and local refinement in complex, multimodal search landscapes. To address this challenge, a modified artificial protozoa optimizer (mAPO) is developed in this study by embedding two complementary mechanisms into the original artificial protozoa optimizer: a probabilistic random learning strategy to enhance population diversity and global search capability, and a Nelder-Mead simplex-based local refinement stage to improve exploitation and fine-scale solution adjustment. The general optimization performance and scalability of the proposed framework are first evaluated using the CEC2017 benchmark suite. Statistical analyses conducted over shifted and rotated, hybrid, and composition functions demonstrate that mAPO achieves improved mean performance and reduced variability compared with the original APO, indicating enhanced robustness in high-dimensional and complex optimization problems. The effectiveness of mAPO is then examined in nonlinear system identification applications involving chaotic dynamics. Offline and online parameter identification experiments are performed on the Rössler chaotic system and a permanent magnet synchronous motor, including scenarios with abrupt parameter variations. Comparative simulations against APO and several state-of-the-art optimizers show that mAPO consistently yields smaller objective function values, more accurate parameter estimates, and superior statistical stability. In the PMSM case, exact parameter reconstruction with zero error is achieved across all independent runs, while rapid and smooth convergence is observed under both static and time-varying conditions.

We design specific neural networks (NNs) for the identification of switching nonlinear systems in the state-space form, which explicitly model the switching behavior and address the inherent coupling between system parameters and switching modes. Such coupling is specifically addressed by leveraging the expectation-maximization (EM) framework. In particular, our technique will combine a moving window approach in the E-step to efficiently estimate the switching sequence, together with an extended Kalman filter (EKF) in the M-step to train the NNs with a local quadratic convergence rate. Extensive numerical simulations, involving both academic examples and a battery charge management system case study, illustrate that our technique outperforms available ones in terms of parameter estimation accuracy, model fitting, and switching sequence identification.

Nonlinear dynamics are inherent in a wide range of industrial processes and mechatronic systems, where impulsive noise—arising from sensor faults, switching transients, or electromagnetic interference—poses serious challenges to accurate measurement, system identification, and subsequent control performance. To address these challenges, this work presents a robust Hammerstein spline adaptive filtering method, termed HSAF-lncosh-MVC, aimed at achieving reliable nonlinear system identification in impulsive environments. The nonlinear module employs an lncosh-based adaptation mechanism that constrains the update direction within a bounded gradient region, thereby ensuring stable parameter estimation. Concurrently, the linear module integrates a maximum versoria criterion (MVC) that suspends weight updates in the presence of pronounced outliers, thus improving resilience against impulsive disturbances. This hybrid strategy of nonlinear and linear components is implemented through stochastic gradient operations without introducing significant additional computational cost. Theoretical analysis establishes the algorithm’s convergence and steady state characteristics. Extensive simulations, covering Gaussian, symmetric α-stable, and Cauchy noise scenarios as well as complex nonlinear dynamical systems, demonstrate that the proposed method achieves accelerated convergence and reduced steady state error relative to conventional Hammerstein spline filters, while maintaining computational efficiency. These results highlight the potential of HSAF-lncosh-MVC as an effective tool for robust signal processing and nonlinear system identification in measurement and instrumentation systems operating under impulsive noise conditions.