High-dimensional Nonlinear Systems Research Articles

Time-delay-locked dynamic mode decomposition for extracting flow patterns of compressor in experiment

Dynamic mode decomposition (DMD) is widely used in extracting the main features of high-dimensional nonlinear systems in computational fluid dynamics. However, it cannot be directly applied to real compressors due to the limitations of data acquisition over the entire flow field. In order to overcome this limitation, our paper proposes a time-delay-locked DMD algorithm. It combines the idea of phase locking, Hankel time-delay method, and DMD method. Based on it, the flow patterns are successfully extracted and the unsteady flow phenomena including rotating stall and rotating instability are explained as modal waves. Multi-peak spectrum nearly 1/2 BPF observed under off-design RI conditions is explained. In addition, multi-scale of rotating stall is analyzed and shock wave is captured.

Ensemble Riemannian data assimilation: towards large-scale dynamical systems

Abstract. This paper presents the results of the ensemble Riemannian data assimilation for relatively high-dimensional nonlinear dynamical systems, focusing on the chaotic Lorenz-96 model and a two-layer quasi-geostrophic (QG) model of atmospheric circulation. The analysis state in this approach is inferred from a joint distribution that optimally couples the background probability distribution and the likelihood function, enabling formal treatment of systematic biases without any Gaussian assumptions. Despite the risk of the curse of dimensionality in the computation of the coupling distribution, comparisons with the classic implementation of the particle filter and the stochastic ensemble Kalman filter demonstrate that, with the same ensemble size, the presented methodology could improve the predictability of dynamical systems. In particular, under systematic errors, the root mean squared error of the analysis state can be reduced by 20 % (30 %) in the Lorenz-96 (QG) model.

A parametric and feasibility study for data sampling of the dynamic mode decomposition: range, resolution, and universal convergence states

Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The dynamic mode decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. It aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject—the turbulent free-shear flow over a prism—for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent.

Credit Valuation Adjustment with Replacement Closeout: Theory and Algorithms

The replacement closeout convention has drawn more and more attention since the 2008 financial crisis. Compared with the conventional risk-free closeout, the replacement closeout convention incorporates the creditworthiness of the counterparty and thus providing a more accurate estimate of the Mark-to-market value of a financial claim. In contrast to the risk-free closeout, the replacement closeout renders a nonlinear valuation system, which constitutes the major difficulty in the valuation of the counterparty credit risk. In this paper, we show how to address the nonlinearity attributed to the replacement closeout in the theoretical and computational analysis. In the theoretical part, we prove the unique solvability of the nonlinear valuation system and study the impact of the replacement closeout on the credit valuation adjustment. In the computational part, we propose a neural network-based algorithm for solving the (high dimensional) nonlinear valuation system and effectively alleviating the curse of dimensionality. We numerically compare the computational cost for the valuations with risk-free and replacement closeouts. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm for the valuation of the replacement closeout.

Safe Reinforcement Learning Algorithm and Its Application in Intelligent Control for CPS

PDF HTML XML Export Cite reminder Safe Reinforcement Learning Algorithm and Its Application in Intelligent Control for CPS DOI: 10.21655/ijsi.1673-7288.00284 Author: Affiliation: Clc Number: Fund Project: Article | Figures | Metrics | Reference | Related | Cited by | Materials | Comments Abstract:The design of a safe controller for a Cyber-Physical System (CPS) is a hot research topic. The existing safety controller design based on formal methods has problems such as excessive reliance on models and poor scalability. Intelligent control based on Deep Reinforcement Learning (DRL) can handle high-dimensional nonlinear complex systems and uncertain systems and is becoming a very promising CPS control technology, but it lacks safety guarantees. This study addresses the safety issues of Reinforcement Learning (RL) control by analyzing a typical case of an industrial oil pump control system and carries out research on a Safe Reinforcement Learning (SRL) algorithm and intelligent control application. First, the SRL issue of the industrial oil pump control system is formalized, and a simulation environment of the oil pump is built. Then, by designing the structure and activation function of the output layer, an oil pump controller in the form of a neural network is constructed to satisfy the linear inequality constraints of the on-off operations of the oil pump. Finally, in order to better balance the safety and optimality control objectives, a new SRL algorithm is designed based on the Augmented Lagrange Multiplier (ALM) method. A comparative experiment on the industrial oil pump shows that the controller synthesized by the proposed algorithm surpasses existing similar algorithms both in safety and optimality. During the evaluation, the neural network controllers synthesized in this study pass rigorous formal verification with a probability of 90%. Meanwhile, compared with the theoretically optimal controller, neural network controllers achieve an optimal objective value loss of 2%. The proposed method is expected to be applied in more scenarios, and the case study scheme may provide a reference for other researchers in the field of safe intelligent control and formal verification. Reference Related Cited by

Neural Network Optimal Feedback Control With Guaranteed Local Stability

Recent research shows that supervised learning can be an effective tool for designing near-optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of neural network controllers is still not well understood. In particular, some neural networks with high test accuracy can fail to even locally stabilize the dynamic system. To address this challenge we propose several novel neural network architectures, which we show guarantee local asymptotic stability while retaining the approximation capacity to learn the optimal feedback policy semi-globally. The proposed architectures are compared against standard neural network feedback controllers through numerical simulations of two high-dimensional nonlinear optimal control problems: stabilization of an unstable Burgers-type partial differential equation, and altitude and course tracking for an unmanned aerial vehicle. The simulations demonstrate that standard neural networks can fail to stabilize the dynamics even when trained well, while the proposed architectures are always at least locally stabilizing and can achieve near-optimal performance.

Analysis of Internal Resonance of a 3DOF Dynamic System Reduced from the Tower-Cable-Beam Structure

To study the complex mechanism of the high-dimensional nonlinear cable systems, a 3 degree-of-freedom model reduced from the tower-cable-beam structure is proposed and investigated in this paper. Based on the D’Alembert Principle, the dynamic equations of in-plane and out-of-plane vibration are established and simulated by the 4th-order Runge-Kutta method. The results exhibit the phenomenon of coupling internal resonance under the systematical conditions revealed by the analytical analysis on the dynamic equations. The smaller mass ratio of the cable-beam would lead to a greater vibration intensity while the tensile stiffness and initial force of the cable have no significant effect. The in-plane and out-plane cable vibrations are independent, and the internal resonance would not be excited by the harmonic excitation in the cable axis. Additionally, applying damping on any component of the system is verified to be an effective approach to vibration reduction. Compared with ordinary cables, cables with less-weight and high-strength materials would be exited to less vibration intensity under the same external excitation.

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, an efficient 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 66 to 4,417x speed-up with 1 to 5% relative errors.

Greedy sampling and approximation for realizing feedback control for high dimensional nonlinear systems

The problem of closed-loop control for high dimensional nonlinear systems is considered. Three different types of kernel interpolation surrogates for the optimal feedback policy are compared. The data for this comes from open-loop controlled solutions. Here, additional information about the system originating from the Pontryagin Maximum Principle is exploited. The starting position for the open-loop control is adaptively selected by a geometric greedy selection criterion, and a so-called vectorial kernel orthogonal greedy algorithm is performed to set up the surrogate. With this procedure we can overcome the curse of dimensionality and still receive a very precise, robust and real-time capable feedback control.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

QRnet: Optimal Regulator Design With LQR-Augmented Neural Networks

In this paper we propose a new computational method for designing optimal regulators for high-dimensional nonlinear systems. The proposed approach leverages physics-informed machine learning to solve high-dimensional Hamilton-Jacobi-Bellman equations arising in optimal feedback control. Concretely, we augment linear quadratic regulators with neural networks to handle nonlinearities. We train the augmented models on data generated without discretizing the state space, enabling application to high-dimensional problems. We use the proposed method to design a candidate optimal regulator for an unstable Burgers' equation, and through this example, demonstrate improved robustness and accuracy compared to existing neural network formulations.

Probing the phase space of coupled oscillators with Koopman analysis.

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.

On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system.

In this paper, infinite homoclinic orbits existing in the Lorenz system are analytically presented. Such homoclinic orbits are induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. A period-doubling bifurcation tree of period-1, period-2, and period-4 motions are generated as an example. The homoclinic orbits relative to unstable period-1, period-2, and period-4 motions are determined. Illustrations of homoclinic orbits and periodic orbits are given. This study presents how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.

Penalized ensemble Kalman filters for high dimensional non-linear systems.

The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of models, updated with data, to track the time evolution of a usually non-linear system. It does so by using an empirical approximation to the well-known Kalman filter. However, its performance can suffer when the ensemble size is smaller than the state space, as is often necessary for computationally burdensome models. This scenario means that the empirical estimate of the state covariance is not full rank and possibly quite noisy. To solve this problem in this high dimensional regime, we propose a computationally fast and easy to implement algorithm called the penalized ensemble Kalman filter (PEnKF). Under certain conditions, it can be theoretically proven that the PEnKF will be accurate (the estimation error will converge to zero) despite having fewer ensemble members than state dimensions. Further, as contrasted to localization methods, the proposed approach learns the covariance structure associated with the dynamical system. These theoretical results are supported with simulations of several non-linear and high dimensional systems.

A New Approach for Nonlinear Transformation of Means and Covariances in Direct Statistical Analysis of Nonlinear Systems

Covariance analysis describing function technique is a conventional method to solve the performance analysis of the nonlinear missile guidance system. Aiming at the faultiness of covariance analysis describing function technique and its improved method, this paper proposes a new method to solve the mean and covariance transfer issue by using the high-degree spherical-radial cubature rule. In terms of simulation models, the nonlinear missile guidance system contains the saturation link to limit the lateral acceleration command and the initial heading error is introduced. In addition, the zero-mean target evasive maneuver with exponentially-decaying autocorrelation function is chosen as the input of the missile guidance system. Furthermore, compared with the existing methods, the proposed method has higher computational efficiency, accuracy and better applicability in high-dimensional nonlinear systems by theoretical derivation and simulation results from the mean and covariance values of relative lateral separation. Finally, the presented analysis method can be used as a general method for the design and statistical analysis of multidimensional nonlinear systems.

Symmetry classification and exact solutions of (3 + 1)-dimensional fractional nonlinear incompressible non-hydrostatic coupled Boussinesq equations

The symmetry classifications of two fractional higher dimensional nonlinear systems, namely, (3 + 1)-dimensional incompressible non-hydrostatic Boussinesq equations and (3 + 1)-dimensional Boussinesq equations with viscosity, are discussed. Both the fractional Boussinesq equations are considered to have derivatives with respect to all variables of fractional type, and some exact solutions are reported along with graphical illustrations.

Adaptive Deep Learning for High-Dimensional Hamilton--Jacobi--Bellman Equations

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially-trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body, and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems.

First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based generalized density evolution equation

The reliability analysis of high-dimensional stochastic dynamical systems subjected to random excitations has long been one of the major challenges in civil and various engineering fields. Despite great efforts, no satisfactory method with high efficiency and accuracy has been available as yet for high-dimensional systems even when they are linear systems, not to mention generic nonlinear systems. In the present paper, a novel method by imposing appropriate absorbing boundary condition on the newly developed ensemble-evolving-based generalized density evolution equation (EV-GDEE) combined with a feasible numerical method is proposed to capture the time-variant first-passage reliability of high-dimensional systems enforced by additive white noise excitation. In the proposed method, the equivalent drift coefficients in EV-GDEE can be estimated by analytical expression or captured by some representative deterministic dynamic analyses. Further, imposing the absorbing boundary condition and then solving the EV-GDEE, a one-or two-dimensional partial differential equation (PDE), yield the remaining probability density of the response of interest. Consequently, by integrating the remaining probability density, the numerical solution of time-variant first-passage reliability can be obtained. Several numerical examples are illustrated to verify the efficiency and accuracy of the proposed method. Compared to the Monte Carlo simulation, the proposed method is of much higher efficiency. Problems to be further studied are finally discussed.

Approximate Simulation for Template-Based Whole-Body Control

Reduced-order template models are widely used to control high degree-of-freedom legged robots, but existing methods for template-based whole-body control rely heavily on heuristics and often suffer from robustness issues. In this letter, we propose a template-based whole-body control method grounded in the formal framework of approximate simulation. Our central contribution is to demonstrate how the Hamiltonian structure of rigid-body dynamics can be exploited to establish approximate simulation for a high-dimensional nonlinear system. The resulting controller is passive, more robust to push disturbances, uneven terrain, and modeling errors than standard QP-based methods, and naturally enables high center of mass walking. Our theoretical results are supported by simulation experiments with a 30 degree-of-freedom Valkyrie humanoid model.

Balanced Reduced-Order Models for Iterative Nonlinear Control of Large-Scale Systems

We propose a new framework to design controllers for high-dimensional nonlinear systems. The control is designed through the iterative linear quadratic regulator (ILQR), an algorithm that computes control by iteratively applying the linear quadratic regulator on the local linearization of the system at each time step. Since ILQR is computationally expensive, we propose to first construct reduced-order models (ROMs) of the high-dimensional nonlinear system. We derive nonlinear ROMs via projection, where the basis is computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT). Numerical experiments are performed on a semi-discretized nonlinear Burgers equation. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT, with BT-ROM based controllers outperforming LQG-BT slightly for very low-dimensional systems.