Modeling Of Nonlinear Dynamic Systems Research Articles

Data-driven modeling of interrelated dynamical systems

Non-linear dynamical systems describe numerous real-world phenomena, ranging from the weather, to financial markets and disease progression. Individual systems may share substantial common information, for example patients’ anatomy. Lately, deep-learning has emerged as a leading method for data-driven modeling of non-linear dynamical systems. Yet, despite recent breakthroughs, prior works largely ignored the existence of shared information between different systems. However, such cases are quite common, for example, in medicine: we may wish to have a patient-specific model for some disease, but the data collected from a single patient is usually too small to train a deep-learning model. Hence, we must properly utilize data gathered from other patients. Here, we explicitly consider such cases by jointly modeling multiple systems. We show that the current single-system models consistently fail when trying to learn simultaneously from multiple systems. We suggest a framework for jointly approximating the Koopman operators of multiple systems, while intrinsically exploiting common information. We demonstrate how we can adapt to a new system using order-of-magnitude less new data and show the superiority of our model over competing methods, in terms of both forecasting ability and statistical fidelity, across chaotic, cardiac, and climate systems.

Volterra Black-Box Models Identification Methods: Direct Collocation vs. Least Squares

The Volterra integral-functional series is the classic approach for nonlinear black box dynamical system modeling. It is widely employed in many domains including radiophysics, aerodynamics, electronic and electrical engineering and many others. Identifying the time-varying functional parameters, also known as Volterra kernels, poses a difficulty due to the curse of dimensionality. This refers to the exponential growth in the number of model parameters as the complexity of the input-output response increases. The least squares method (LSM) is widely acknowledged as the standard approach for tackling the issue of identifying parameters. Unfortunately, the LSM suffers with many drawbacks such as the sensitivity to outliers causing biased estimation, multicollinearity, overfitting and inefficiency with large datasets. This paper presents an alternative approach based on direct estimation of the Volterra kernels using the collocation method. Two model examples are studied. It is found that the collocation method presents a promising alternative for optimization, surpassing the traditional least squares method when it comes to the Volterra kernels identification including the case when input and output signals suffer from considerable measurement errors.

Neuro-adaptive non-singular terminal sliding mode control for distributed fixed-time synchronization of higher-order uncertain multi-agent nonlinear systems

This paper presents a novel design for a distributed fixed-time synchronization controller based on neuro-adaptive non-singular terminal sliding mode control for higher-order multi-agent nonlinear systems. In this study, a distributed control strategy is employed in which networked nonlinear systems, referred to as nodes, communicate through a predefined network topology. Within this framework, a leader-follower configuration is adopted, designating one node as the leader and the others as follower agents. Notably, the dynamics of the agents are assumed to be unknown, leading to the development of a generalized uncertain higher-order nonlinear dynamical system model that accounts for the disturbed dynamics of all agents, including the matched bounded uncertainty with an upper algebraic bound. The control design leverages radial basis function neural networks for approximation, effectively addressing unknown nonlinear terms, such as drift terms and input channels. Importantly, the proposed technique successfully mitigated matched-type uncertainty, resulting in rapid sliding-mode enforcement with reduced chattering and stress. This innovative approach establishes fixed-time synchronization by creating a sliding mode with respect to the sliding surface of networked synchronization mismatches. To underscore the effectiveness of this approach, a simulation example that vividly illustrates its captivating properties and contributions is presented.

Two-Dimensional Transient Cycle Decomposition and Reduction (CDR) for Data Driven Nonlinear Dynamic System Modeling

Two-Dimensional Transient Cycle Decomposition and Reduction (CDR) for Data Driven Nonlinear Dynamic System Modeling

Parameter estimation and modeling of nonlinear dynamical systems based on Runge–Kutta physics-informed neural network

Parameter estimation and modeling of nonlinear dynamical systems based on Runge–Kutta physics-informed neural network

Discovering conservation laws using optimal transport and manifold learning

Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify, making it hard to analyze their dynamics and build stable predictive models. Current approaches for discovering conservation laws often depend on detailed dynamical information or rely on black box parametric deep learning methods. We instead reformulate this task as a manifold learning problem and propose a non-parametric approach for discovering conserved quantities. We test this new approach on a variety of physical systems and demonstrate that our method is able to both identify the number of conserved quantities and extract their values. Using tools from optimal transport theory and manifold learning, our proposed method provides a direct geometric approach to identifying conservation laws that is both robust and interpretable without requiring an explicit model of the system nor accurate time information.

Training recurrent neural networks by sequential least squares and the alternating direction method of multipliers

This paper proposes a novel algorithm for training recurrent neural network models of nonlinear dynamical systems from an input/output training dataset. Arbitrary convex and twice-differentiable loss functions and regularization terms are handled by sequential least squares and either a line-search (LS) or a trust-region method of Levenberg–Marquardt (LM) type for ensuring convergence. In addition, to handle non-smooth regularization terms such as ℓ1, ℓ0, and group-Lasso regularizers, as well as to impose possibly non-convex constraints such as integer and mixed-integer constraints, we combine sequential least squares with the alternating direction method of multipliers (ADMM). We call the resulting algorithm NAILS (nonconvex ADMM iterations and least squares) in the case line search (LS) is used, or NAILM if a trust-region method (LM) is employed instead. The training method, which is also applicable to feedforward neural networks as a special case, is tested in three nonlinear system identification problems.

GLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

MULTI-MOTOR DRIVE NON-PARAMETRIC BLACK-BOX FUZZY MODEL

The multi-motor drives, the typical examples of which are continuous lines (CL) for tension processing of various materials are complex and coupled MIMO higher-order nonlinear systems, which parameters are difficult to identify. The article focuses on a method for the design of a non-parametric black-box fuzzy model of a continuous production line with emphasis on minimal knowledge on the modelled system. The proposed fuzzy model structure is based on the state space representation of the dynamic system in discrete form, which only requires a suitable set of information on its input/output relations. The properties of the proposed CL fuzzy model were verified by experimental measurements on a CL laboratory model. The obtained results have confirmed the rightness and effectiveness of the fuzzy model design method that can be applied not only in the field of industry technologies with CL but also in modeling and control of nonlinear dynamic systems.

A quadratic decoder approach to nonintrusive reduced‐order modeling of nonlinear dynamical systems

AbstractLinear projection schemes like Proper Orthogonal Decomposition can efficiently reduce the dimensions of dynamical systems but are naturally limited, e.g., for convection‐dominated problems. Nonlinear approaches have shown to outperform linear methods in terms of dimension reduction versus accuracy but, typically, come with a large computational overhead. In this work, we consider a quadratic reduction scheme which induces nonlinear structures that are well accessible to tensorized linear algebra routines. We discuss that nonintrusive approaches can be used to simultaneously reduce the complexity in the equations and propose an operator inference formulation that respects dynamics on nonlinear manifolds.

Fuzzy Graph Modeling and Clustering Analysis of Nonlinear Dynamical Systems

Analyzing stability and designing control strategies for interconnected nonlinear dynamical systems poses mathematical challenges due to combinatorial growth in complexity. This paper develops a methodology integrating fuzzy graph theory with spectral clustering techniques to enable tractable certification of stability properties. Fuzzy similarity relations between state variables model imprecise couplings within the nonlinear dynamics. Representing these relations through graph adjacency matrices facilitates partitioning strongly connected states using spectral algorithms. Stability of the overall fuzzy state graph is inferred from the spectral radii of decoupled fuzzy subgraphs. The graph-theoretic abstraction provides a coarse-grained lens into nonlinear stability properties while circumventing computational barriers. The fuzzy graph modeling and spectral partitioning pipeline ultimately streamlines control synthesis targeting local clustered subdynamics. Case studies on power network stabilization, swarm navigation, and process operations showcase scalable applications to high-dimensional complex systems. The integrated fuzzy graph and spectral clustering approach provides a systematic toolkit for analysis and control of heavily interconnected nonlinear dynamical systems across engineering domains.

Production Change Optimization Model of Nonlinear Supply Chain System under Emergencies

Aiming at the problem that the upstream manufacturer cannot accurately formulate the production plan after the link of the nonlinear supply chain system changes under emergencies, an optimization model of production change in a nonlinear supply chain system under emergencies is designed. Firstly, based on the structural characteristics of the supply chain system and the logical relationship between production, sales, and storage parameters, a three-level single-chain nonlinear supply chain dynamic system model containing producers, sellers, and retailers was established based on the introduction of nonlinear parameters. Secondly, the radial basis function (RBF) neural network and improved fast variable power convergence law were introduced to improve the traditional sliding mode control, and the improved adaptive sliding mode control is proposed so that it can have a good control effect on the unknown nonlinear supply chain system. Finally, based on the numerical assumptions, the constructed optimization model was parameterized and simulated for comparison experiments. The simulation results show that the optimized model can reduce the adjustment time by 37.50% and inventory fluctuation by 42.97%, respectively, compared with the traditional sliding mode control, while helping the supply chain system to return the smooth operation after the change within 5 days.

Respiratory Diseases Prediction from a Novel Chaotic System

Pandemics can have a significant impact on international health systems. Researchers have found that there is a correlation between weather conditions and respiratory diseases. This paper focuses on the non-linear analysis of respiratory diseases and their relationship to weather conditions. Chaos events may appear random, but they may actually have underlying patterns. Edward Lorenz referred to this phenomenon in the context of weather conditions as the butterfly effect. This inspired us to define a chaotic system that could capture the properties of respiratory diseases. The chaotic analysis was performed and was related to the difference in the daily number of cases received from real data. Stability analysis was conducted to determine the stability of the system and it was found that the new chaotic system was unstable. Lyapunov exponent analysis was performed and found that the new chaotic system had Lyapunov exponents of (+, 0, -, -). A dynamic neural architecture for input-output modeling of nonlinear dynamic systems was developed to analyze the findings from the chaotic system and real data. A NARX network with inputs (maximum temperature, pressure, and humidity) and one output was used to to overcome any delay effects and analyze derived variables and real data (patients number). Upon solving the system equations, it was found that the correlation between the daily predicted number of patients and the solution of the new chaotic equation was 90.16%. In the future, this equation could be implemented in a real-time warning system for use by national health services.

A deep generative framework for data-driven surrogate modeling and visualization of parameterized nonlinear dynamical systems

A deep generative framework for data-driven surrogate modeling and visualization of parameterized nonlinear dynamical systems

Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability

Elimination of unknowns in a system of differential equations is often required when analyzing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from the literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available online as a Julia package.

Recovery and adjustment trajectories among Hurricane Florence survivors: Analysis utilizing nonlinear dynamic system modeling.

Self-regulation shift theory (SRST) argues that most individuals are able to successfully recover from trauma via engagement in self-regulation processes as well as the effective utilization of internal and environmental resources. However, a minority of individuals may instead experience a self-determination violation as a result of their self-regulatory capacity being overwhelmed. This self-determination violation is marked by chaotic and shifting adjustment, maladaptive regulation attempts, and, ultimately, a shift to an impaired self-state and the development of persistent psychopathology, such as posttraumatic stress disorder (PTSD). The current study utilized nonlinear dynamic system (NDS) analysis to identify adjustment trajectory dynamics among rural hurricane survivors in North Carolina (N = 131) who completed daily ecological momentary assessments (EMAs) regarding their distress (i.e., negative mood and PTSD symptoms), regulation efforts (e.g., coping), and appraisals (e.g., coping self-efficacy) over a 6-week period. Four adjustment trajectories were identified, including two largely adaptive trajectories (69.0% and 5.7%), a less stable adjustment trajectory (6.9%), and a fourth trajectory (18.4%) marked by shifting adjustment states and more frequent maladaptive regulation and negative appraisals, suggesting possible self-determination violation. Consistent with this possibility, this final trajectory was also associated with more severe PTSD symptoms relative to the other three trajectories at enrollment and 6-month follow-up. Future work should utilize NDS to model posttrauma adjustment dynamics from within a SRST framework to identify patterns of positive and negative adjustment dynamics at different time points in the trauma recovery process.

Neural network modeling and dynamic behavior prediction of nonlinear dynamic systems

Neural network modeling and dynamic behavior prediction of nonlinear dynamic systems

Model-Predictive Control Design for Power System Oscillation Damping via Excitation – A Data-Driven Approach

This paper presents a novel power system oscillation damping controller design based on a data-driven model-predictive control (MPC) approach. The system dynamics are extracted from the measurements and applied to the synthesis of control actions in an equation-free manner, without requiring explicit knowledge of the underlying system model. Dynamic mode decomposition with control (DMDc), originating from the Koopman operator theory, connects the collected data with the nonlinear dynamical system modeling. The discrepancy between the model identified by DMDc and the real plant is labeled as a total disturbance. It is estimated in real time by an extended state observer (ESO) and mitigated via a linear MPC, assuming the estimate to be constant over the prediction horizon. The overall control framework is referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ESO-Koopman-MPC (EKM)</i> . Two <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">EKM</i> -based oscillation damping controllers are proposed - a standalone PSS and an integrated AVR+PSS. The controller operation is illustrated on the single machine infinite bus system (SMIB), Kundur two-area system and IEEE 39-bus system, and the performance is compared with conventional power system stabilizers (CPSSs) under various test scenarios.

Existence and uniqueness of the solution to a second-order nonlinear dynamical system model with an unbounded variable and a discontinuous input

We consider an initial value problem for a second-order nonlinear dynamical system that allows discontinuous inputs. In this problem, one state variable is restricted in a closed and bounded interval while the other state variable is unlimited. The existence and uniqueness of the solution to this problem is studied in a time interval determined by two adjacent instants at which the input may be discontinuous. Some results are obtained on the maximal interval of existence of the unique solution in this time interval. The results are then extended to the entire time domain. A robot example is given to verify our results.

On the universal transformation of data-driven models to control systems

The advances in data science and machine learning have resulted in significant improvements regarding the modeling and simulation of nonlinear dynamical systems. It is nowadays possible to make accurate predictions of complex systems such as the weather, disease models or the stock market. Predictive methods are often advertised to be useful for control, but the specifics are frequently left unanswered due to the higher system complexity, the requirement of larger data sets and an increased modeling effort. In other words, surrogate modeling for autonomous systems is much easier than for control systems. In this paper we present the framework QuaSiModO (Quantization-Simulation-Modeling-Optimization) to transform arbitrary predictive models into control systems and thus render the tremendous advances in data-driven surrogate modeling accessible for control. Our main contribution is that we trade control efficiency by autonomizing the dynamics – which yields mixed-integer control problems – to gain access to arbitrary, ready-to-use autonomous surrogate modeling techniques. We then recover the complexity of the original problem by leveraging recent results from mixed-integer optimization. The advantages of QuaSiModO are a linear increase in data requirements with respect to the control dimension, performance guarantees that rely exclusively on the accuracy of the predictive model in use, and little prior knowledge requirements in control theory to solve complex control problems.