Large-scale Nonlinear Dynamical Systems Research Articles

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Extending a sensitivity based algorithm to detect local structural identifiability

Output sensitivities to parameters underly a highly efficient sensitivity based algorithm to compute local structural identifiability of possibly large-scale nonlinear dynamic systems. By means of simple examples, this paper explores exceptional cases where this algorithm fails. That is, if one applies the common definition of local structural identifiability. As also shown in this paper, when applying a closely related definition of identifiability, based on sensitivities, the sensitivity based algorithm always provides the correct answer. The subtle difference between these two definitions, that seems to have been overlooked in the literature, is further explored and explained in this paper. For the common definition of local structural identifiability, an extension of the sensitivity based algorithm is presented that approximately doubles the computation time. This extension is shown to work along non-singular trajectories.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Rapid data-driven model reduction of nonlinear dynamical systems including chemical reaction networks using ℓ1-regularization.

Large-scale nonlinear dynamical systems, such as models of atmospheric hydrodynamics, chemical reaction networks, and electronic circuits, often involve thousands or more interacting components. In order to identify key components in the complex dynamical system as well as to accelerate simulations, model reduction is often desirable. In this work, we develop a new data-driven method utilizing ℓ1-regularization for model reduction of nonlinear dynamical systems, which involves minimal parameterization and has polynomial-time complexity, allowing it to easily handle large-scale systems with as many as thousands of components in a matter of minutes. A primary objective of our model reduction method is interpretability, that is to identify key components of the dynamical system that contribute to behaviors of interest, rather than just finding an efficient projection of the dynamical system onto lower dimensions. Our method produces a family of reduced models that exhibit a trade-off between model complexity and estimation error. We find empirically that our method chooses reduced models with good extrapolation properties, an important consideration in practical applications. The reduction and extrapolation performance of our method are illustrated by applications to the Lorenz model and chemical reaction rate equations, where performance is found to be competitive with or better than state-of-the-art approaches.

Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system’s governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

A scheme for comprehensive computational cost reduction in proper orthogonal decomposition

Abstract This paper addresses the issue of offline and online computational cost reduction of the proper orthogonal decomposition (POD) which is a popular nonlinear model order reduction (MOR) technique. Online computational cost is reduced by using the discrete empirical interpolation method (DEIM), which reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD: this is the POD-DEIM approach. Offline computational cost is reduced by generating an approximate snapshot-ensemble of the nonlinear dynamical system, consequently, completely avoiding the need to simulate the full-order system. Two snapshot ensembles: one of the states and the other of the nonlinear function are obtained by simulating the successive linearization of the original nonlinear system. The proposed technique is applied to two benchmark large-scale nonlinear dynamical systems and clearly demonstrates comprehensive savings in computational cost and time with insignificant or no deterioration in performance.

Systematic Computation of Nonlinear Bilateral Dynamical Systems With a Novel Low-Power Log-Domain Circuit

Simulation of large-scale nonlinear dynamical systems on hardware with a high resemblance to their mathematical equivalents has been always a challenge in engineering. This paper presents a novel current-input current-output circuit supporting a systematic synthesis procedure of log-domain circuits capable of computing bilateral dynamical systems with considerably low power consumption and acceptable precision. Here, the application of the method is demonstrated by synthesizing four different case studies: 1) a relatively complex 2-D nonlinear neuron model; 2) a chaotic 3-D nonlinear dynamical system Lorenz attractor having arbitrary solutions for certain parameters; 3) a 2-D nonlinear Hopf oscillator, including bistability phenomenon sensitive to initial values; and 4) three small neurosynaptic networks comprising three FHN neuron models variously coupled with excitatory and inhibitory synapses. The validity of our approach is verified by nominal and Monte Carlo simulated results with realistic process parameters from the commercially available AMS 0.35- $\mu \text{m}$ technology. The resulting continuous-time, continuous-value, and low-power circuits exhibit various bifurcation phenomena, nominal time-domain responses in good agreement with their mathematical counterparts and fairly acceptable process variation results (less than 5% STD).

Time-varying stiffness method for extracting the frequency–energy dependence in the nonlinear dynamical systems

In this work, a time-varying stiffness method is proposed for extracting highly accurate approximation for the fundamental backbone branches of the frequency–energy plot from the numerical simulation response of the nonlinear dynamical system. The purely nonlinear duffing oscillator with a nonnegative real power restoring force is firstly considered to develop the method, and later the method is applied to linear systems attached with a nonlinear energy sink for more demonstration. The systems of concern are numerically simulated at an arbitrary high level of initial input energy to apply the proposed method. Accordingly, the obtained responses of these systems are employed via the proposed time-varying stiffness method to extract an approximation for the fundamental backbone branches in the frequency–energy plot. The obtained backbones have been found in excellent agreement with the exact backbones of the considered systems. Even though these approximate backbones have been obtained for only one high energy level, they are valid for any other initial energy below that level. In addition, they are not affected by the damping variations in the considered systems. The proposed method is found to be applicable to well approximate the fundamental backbone branches of the large-scale nonlinear dynamical systems. The frequency–amplitude dependences have been also studied here for the considered systems.

Identifiability of large-scale non-linear dynamic network models applied to the ADM1-case study

In this work, both the structural and practical identifiability of the Anaerobic Digestion Model no. 1 (ADM1) is investigated, which serves as a relevant case study of large non-linear dynamic network models. The structural identifiability is investigated using the probabilistic algorithm, adapted to deal with the specifics of the case study (i.e., a large-scale non-linear dynamic system of differential and algebraic equations). The practical identifiability is analyzed using a Monte Carlo parameter estimation procedure for a ‘non-informative’ and ‘informative’ experiment, which are heuristically designed.The model structure of ADM1 has been modified by replacing parameters by parameter combinations, to provide a generally locally structurally identifiable version of ADM1. This means that in an idealized theoretical situation, the parameters can be estimated accurately. Furthermore, the generally positive structural identifiability results can be explained from the large number of interconnections between the states in the network structure. This interconnectivity, however, is also observed in the parameter estimates, making uncorrelated parameter estimations in practice difficult.

Distributed Coordinated Control of Large-Scale Nonlinear Networks

We provide a distributed coordinated approach to the stability analysis and control design of large-scale nonlinear dynamical systems by using a vector Lyapunov functions approach. In this formulation the large-scale system is decomposed into a network of interacting subsystems and the stability of the system is analyzed through a comparison system. However finding such comparison system is not trivial. In this work, we propose a sum-of-squares based completely decentralized approach for computing the comparison systems for networks of nonlinear systems. Moreover, based on the comparison systems, we introduce a distributed optimal control strategy in which the individual subsystems (agents) coordinate with their immediate neighbors to design local control policies that can exponentially stabilize the full system under initial disturbances. We illustrate the control algorithm on a network of interacting Van der Pol systems.

Hopf bifurcation of an (n + 1) -neuron bidirectional associative memory neural network model with delays.

Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n + 1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.

A Decomposition Technique for Nonlinear Dynamical System Analysis

A method for analyzing large-scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented. It is shown that the decomposition approach can be used to scale the Sum of Squares programming framework for nonlinear systems analysis. The method constructs subsystem Lyapunov functions which are used to form a composite Lyapunov function for the whole system. Further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem Lyapunov functions.

SUBSPACE METHOD FOR ANALYZING LARGE-SCALE NONLINEAR STOCHASTIC DYNAMIC SYSTEMS WITH POLYNOMIAL TYPE OF NONLINEARITES

In this paper, the probabilistic solutions of the multi-degree-of-freedom (MDOF) or large-scale stochastic dynamic systems with polynomial type of nonlinearity and excited by Gaussian white noise excitations are obtained and investigated with the subspace method proposed recently by the authors. The space of the state variables of large-scale nonlinear stochastic dynamic (NSD) system excited by white noises is separated into two subspaces. Both sides of the Fokker–Planck–Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure (EPC) method. A simple flexural beam on nonlinear elastic springs is analyzed with the subspace method to show the effectiveness of the subspace-EPC method in this case.

A New Weighted Graph-Based Partitioning Algorithm for Decentralized Nonlinear Model Predictive Control of Large-Scale Systems

ABSTRACT This paper proposes a grouping algorithm for partitioning large-scale nonlinear dynamical systems based on graph theory. The algorithm incorporates a novel scheme to quantify the strengths of graph edges, representing the degree of couplings among the system variables via sensitivity functions. This leads to a weighted graph topology with different weights on the obtained graph edges. An algorithm is then developed to partition systems into some sub-graphs based on the weighted graph. A decentralized nonlinear model predictive control (NMPC) methodology is then formulated for the sub-systems. The overall NMPC design methodology is finally evaluated on a process plant benchmark, consisting of two continuous stirred tank reactors (CSTRs) and a flash separator with a recycle path. A set of tracking and regulatory tests is comparatively conducted exploring the successful performance of the proposed algorithm in the context of the decentralized NMPC methodology with respect to an alternative centralized NMPC control scheme.

Decentralised adaptive robust control of uncertain large-scale non-linear dynamical systems with time-varying delays

The problem of decentralised adaptive robust stabilisation is considered for a class of uncertain large-scale interconnected non-linear systems with time-varying delays. It is assumed that the upper bounds of the uncertainties and interconnection terms are unknown, and that the time-varying delays are any non-negative continuous and bounded functions, and do not require their derivatives to be less than one. In particular, it is only required that the non-linear interconnection terms, which can also include time-varying delays, are bounded in any non-negative non-linear functions, which are not required to be known for the system designer. For such a class of uncertain large-scale time-delay interconnected non-linear systems, a new method is presented whereby a class of continuous memoryless decentralised local adaptive robust state feedback controllers with a rather simpler structure is proposed. It is also shown that the solutions of uncertain large-scale time-delay interconnected systems can be guaranteed as uniformly exponentially convergent towards a ball that can be as small as desired. Finally, a numerical example is given to demonstrate the validity of the results.

The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems

AbstractThe probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.

Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

AbstractIn this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.

Chance constrained optimization of process systems under uncertainty: I. Strict monotonicity

An approach for chance constrained programming of large-scale nonlinear dynamic systems is presented. The stochastic property of the uncertainties is explicitly considered in the problem formulation in which some input and state constraints are to be complied with predefined probability levels. The method considers a nonlinear relation between the uncertain input and the constrained variables. It also involves efficient algorithms so as to compute the probabilities and, simultaneously, the gradients through integration by collocation in finite elements. The formulation of single or joint probability limits incorporates the issue of feasibility and the contemplation of trade-off between robustness and profitability regarding the objective function values. The approach is relevant to all cases when uncertainty can be described by any kind of joint correlated multivariate distribution function. Thus, chance constrained programming is a promising technique in solving optimization problems under uncertainty in system engineering. The potential and the efficiency of the presented systematic methodology, which assumes a strict monotonic relationship between the uncertain input and the uncertain constrained output, are illustrated with application to a reactive batch distillation processes under uncertainty.

Fault Diagnostics and Security of Critical Infrastructure Systems

Modern societies have reached a point where everyday life relies heavily on the reliable operation and intelligent management of critical infrastructures, such as electric power systems, telecommunication networks, water distribution networks, and transportation systems. Monitoring and controlling such systems is becoming increasingly more challenging as their size, complexity and interactions are steadily growing. Moreover, these critical infrastructures are susceptible to natural disasters, frequent failures, as well as malicious attacks. There is a need to develop a common system-theoretic framework for modelling the behavior of critical infrastructure systems and for designing algorithms for intelligent monitoring, control and security of such systems. The goal of this presentation is to motivate the key role of fault diagnostics and fault tolerant control methodologies in securing the reliable operation of critical infrastructure systems, and to provide a framework for achieving early detection, isolation and accommodation of faults in large-scale nonlinear dynamic systems. Various ideas based on distributed fault diagnostics and adaptive approximation techniques with learning algorithms will be presented, and directions for future research will be discussed.