Nonlinear Stochastic Dynamical Systems Research Articles

A novel method for response probability density of nonlinear stochastic dynamic systems

AbstractThis paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

Reliability control of nonlinear stochastic dynamical system based on discrete data

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

Extensive numerical simulations on competitive growth between the Edwards–Wilkinson and Kardar–Parisi–Zhang universality classes

Extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the model of random deposition with surface relaxation (RDSR), the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), etching, and restricted solid on solid (RSOS). The scaling exponents are obtained in both the (1 + 1)- and (2 + 1)-dimensional competitive growth with the nonlinear growth probability p and the linear proportion 1 – p. Our results show that, when p changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growths of cooperation and competition.

Nonlinear Optimal Control for Stochastic Dynamical Systems

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

Multiple adaptive fuzzy Nussbaum-type functions design for stochastic nonlinear systems with fixed-time performance

Nussbaum design has been successfully exploited to deal with uncertain systems with a single unknown control coefficient. However, it is challenging for the traditional Nussbaum method to efficiently handle the feedback control design with multiple unknown time-varying control coefficients because the multiple Nussbaum-type gains might mutually cancel out, which makes the Lyapunov-based stability analysis infeasible. To tackle this issue, a novel type of Nussbaum function is designed for the tracking problem of a class of stochastic strict-feedback nonlinear systems. In the presented Nussbaum design, the multiple unknown control coefficients are separately compensated, where the effect of mutual cancellation no longer exists and the Lyapunov stability analysis can be directly applied. Moreover, the proposed Nussbaum design is combined with the fixed-time stability theory, which ensures the settling time is bounded and independent of the initial condition. Within the adaptive fuzzy backstepping framework, it is shown that all signals in the closed-loop system remain bounded through the theoretical discussions. The simulation experiment is conducted to verify the effectiveness of the presented design for the continuous-time stochastic nonlinear dynamical system.

DeepBayes—An estimator for parameter estimation in stochastic nonlinear dynamical models

Stochastic nonlinear dynamical systems are ubiquitous in modern, real-world applications. Yet, estimating the unknown parameters of stochastic, nonlinear dynamical models remains a challenging problem. The majority of existing methods employ maximum likelihood or Bayesian estimation. However, these methods suffer from some limitations, most notably the substantial computational time for inference coupled with limited flexibility in application. In this work, we propose DeepBayes estimators that leverage the power of deep recurrent neural networks. The method consists of first training a recurrent neural network to minimize the mean-squared estimation error over a set of synthetically generated data using models drawn from the model set of interest. The a priori trained estimator can then be used directly for inference by evaluating the network with the estimation data. The deep recurrent neural network architectures can be trained offline and ensure significant time savings during inference. We experiment with two popular recurrent neural networks — long short term memory network (LSTM) and gated recurrent unit (GRU). We demonstrate the applicability of our proposed method on different example models and perform detailed comparisons with state-of-the-art approaches. We also provide a study on a real-world nonlinear benchmark problem. The experimental evaluations show that the proposed approach is asymptotically as good as the Bayes estimator.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Learning Control Policies for Stochastic Systems with Reach-Avoid Guarantees

We study the problem of learning controllers for discrete-time non-linear stochastic dynamical systems with formal reach-avoid guarantees. This work presents the first method for providing formal reach-avoid guarantees, which combine and generalize stability and safety guarantees, with a tolerable probability threshold p in [0,1] over the infinite time horizon. Our method leverages advances in machine learning literature and it represents formal certificates as neural networks. In particular, we learn a certificate in the form of a reach-avoid supermartingale (RASM), a novel notion that we introduce in this work. Our RASMs provide reachability and avoidance guarantees by imposing constraints on what can be viewed as a stochastic extension of level sets of Lyapunov functions for deterministic systems. Our approach solves several important problems -- it can be used to learn a control policy from scratch, to verify a reach-avoid specification for a fixed control policy, or to fine-tune a pre-trained policy if it does not satisfy the reach-avoid specification. We validate our approach on 3 stochastic non-linear reinforcement learning tasks.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

A Computational Technique for Nonlinear Nonlocal Stochastic Dynamical Systems with Variable Order Fractional Brownian Noise

A Computational Technique for Nonlinear Nonlocal Stochastic Dynamical Systems with Variable Order Fractional Brownian Noise

Grid-Based Bayesian Filters With Functional Decomposition of Transient Density

The paper deals with the state estimation of nonlinear stochastic dynamic systems with special attention to grid-based Bayesian filters such as the point-mass filter (PMF) and the marginal particle filter (mPF). In the paper, a novel functional decomposition of the transient density describing the system dynamics is proposed. The decomposition approximates the transient density in a closed region. It is based on a non-negative matrix/tensor factorization and separates the density into functions of the future and current states. Such decomposition facilitates a thrifty calculation of the convolution involving the density, which is a performance bottleneck of the standard PMF/mPF implementations. The estimate quality and computational costs can be efficiently controlled by choosing an appropriate decomposition rank. The performance of the PMF with the transient density decomposition is illustrated in a terrain-aided navigation scenario and a problem involving a univariate non-stationary growth model.

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.

Prequasiideal of the type weighted binomial matrices in the Nakano sequence space of soft functions with some applications

Consider the space of weighted binomial matrices in the Nakano sequence space of soft functions. We have offered some geometric and topological structures of the multiplication operator acting on this space and its associated operator ideal. The existence of a fixed point of the Kannan contraction operator in this prequasioperator ideal is confirmed. Finally, we discuss many applications of solutions to nonlinear stochastic dynamical matrix systems and illustrative examples of our findings.

A sparse Bayesian framework for discovering interpretable nonlinear stochastic dynamical systems with Gaussian white noise

Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equation discovery are mostly applicable to deterministic systems and require both input and state measurements. We here propose a novel data-driven framework for discovering nonlinear stochastic dynamical systems with Gaussian white noise. The proposed framework blends concepts of stochastic calculus, sparse learning algorithms, and Bayesian statistics to learn the governing physics from data. In particular, we combine sparsity–promoting spike and slab prior, Bayes law, and the Kramers–Moyal formula to identify stochastic differential equations from data. The proposed framework is highly efficient and works with sparse, noisy, and incomplete output measurements. The efficacy and robustness of the proposed approach are illustrated in several numerical examples involving both complete and partial state measurements. The results obtained indicate the potential of the proposed approach in discovering nonlinear stochastic dynamical systems subjected to Gaussian white noise excitation.

Nonlinear stochastic seismic dynamic response analysis of submerged floating tunnel subjected to non-stationary ground motion

Owing to the randomness of seismic excitations, the submerged floating tunnel (SFT), which is a new structure subjected to the combined action of self-weight, buoyancy, and anchorage under seismic excitation, is a nonlinear stochastic dynamic system and the evaluation of its dynamic performance is an issue that must be addressed. In this study, the nonlinear dynamic differential equation of an SFT subjected to seismic dynamic action was systematically derived by considering the wave–structure dynamic coupling interaction, and its numerical solution was obtained based on the Newmark-β integration algorithm. Furthermore, based on the theoretical framework of nonlinear stochastic dynamics and the stochastic seismic model reflecting the characteristics of the site-specific response spectrum, the probability density function (PDF) of the nonlinear seismic dynamic response of the SFT was analyzed, and its seismic dynamic performance was evaluated based on dynamic reliability. Thus, it was found that the randomness of ground motion propagates, fluctuates, evolves, and dissipates in the SFT, and finally reaches the nonlinear seismic dynamic response of the SFT. Additionally, the nonlinear dynamic response of the underwater suspension tunnel is highly sensitive to earthquake excitations. The proposed evaluation framework is an effective tool for evaluating the seismic performance of underwater tunnels.

First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems by a fractional moments-based mixture distribution approach

First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed ‘fractional moments-based mixture distribution’, to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moment and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem.

Decision-Making on the Solution of a Stochastic Nonlinear Dynamical System of Kannan-Type in New Sequence Space of Soft Functions

In this paper, we construct and investigate the space of weighted Gamma matrix of order r in Nakano sequence space of soft functions. The idealization of the mappings has been achieved through the use of extended s -soft functions and this sequence space of soft functions. This new space’s topological and geometric properties, the multiplication mappings that stand in on it, and the mappings’ ideal that correspond to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.

A feature mapping strategy of metamodelling for nonlinear stochastic dynamical systems with low to high-dimensional input uncertainties

This paper deals with the issue on metamodelling (a.k.a. surrogate modelling) of nonlinear stochastic dynamical systems, which are often with multiple input uncertainties Θ∈Rn, viz., the dimension n may range from low to high (e.g., n≥10). In this paper, to circumvent the problem of “curse of dimensionality” of high-dimensional input uncertainties, the feature spaces of outputs and inputs are firstly extracted from the original output and input spaces, and thus a feature mapping strategy is proposed. To form the feature output space, the nonlinear autoregressive with exogenous inputs (NARX) and the proper orthogonal decomposition (POD) are adopted, while the feature input space is detected by the active subspace method (ASM). It is found that the dimension of feature input (output) space may be much less than the one of original input (output) space, thus the applicability of many metamodelling methods can be naturally enhanced. On the constructed input–output feature space, the procedure of metamodelling is completed by the polynomial chaos expansion (PCE) combined with Kriging, which can capture global behaviours as well as local characteristics of the computational model. Two techniques are introduced to accelerate the proposed feature mapping strategy, consisting of the GF-discrepancy minimization algorithm for the design of experiments (DoEs), and the manifold optimization technique for the parameter identification of ASM. Four benchmarks, including a mathematical function (n=2), a dynamical quarter car model (n=10), a Bouc–Wen nonlinear oscillator subjected to earthquake ground motions (n=30), and the first sub-system (as a black box) of the NASA UQ Challenge 2019 (n=100), are studied to demonstrate the accuracy and efficiency of the proposed method. Some problems to be further studied are also outlined.

Globally-evolving-based generalized density evolution equation for nonlinear systems involving randomness from both system parameters and excitations

It has long been one of the main challenges in science and engineering to capture the probabilistic response of high-dimensional nonlinear stochastic dynamic systems involving double randomness, i.e. randomness in both system parameters and excitations. For this purpose, a globally-evolving-based generalized density evolution equation (GE-GDEE) is established. Generally, for a multi-dimensional nonlinear system involving double randomness, if one single physical quantity as a response of the system is of interest, a GE-GDEE, as a two-dimensional partial differential equation (PDE) governing the probability density function (PDF), can be derived. The effective drift coefficients, which represent the physically driving force function in the GE-GDEE, can be determined based on the data from some representative deterministic dynamic analyses of the underlying physical system. A new estimator for effective drift coefficients is developed based on the vine copulas. Once the effective drift coefficients are determined, the GE-GDEE can be solved to capture the probability distributions of the quantities of interest. Several numerical examples, including linear and nonlinear multi-degree-of-freedom (MDOF) systems subjected to white noise or non-stationary earthquake ground motions, are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Universal MATLAB‐based square‐root solutions in the family of continuous‐discrete Gaussian filters for state estimation in nonlinear stochastic dynamic systems

AbstractThis article concerns with the issue of square‐rooting in continuous‐discrete Gaussian filters intended for state estimation in nonlinear stochastic dynamic systems of continuous‐discrete sort. These cover all methods devised within the quadrature, cubature, and unscented Kalman filtering approaches as well as those that can be constructed in the future. Based on the universal moment differential equations developed by Särkkä and Sarmavuori in 2013, we advance further that study and design two square‐root solutions grounded on MATLAB ODE solvers in the mentioned Gaussian filtering framework. The main problem addressed is a potential negativity of some weights utilized in calculations of the predicted and filtering means and covariances, which precludes from orthogonal square‐rooting schemes to be applied. In practice, such square‐root implementations are often requested because of their exceptional numerical robustness to round‐off and other disturbances. These also preserve the symmetry and positivity of the covariances computed, automatically. Here, we employ the recently‐devised ‐orthogonal square‐rooting technique for designing our universal MATLAB‐based square‐root solutions in the realm of quadrature, cubature, and unscented Kalman filters, which are easily adjusted to any particular method by using its weights and deterministically selected samples exploited in calculations of the sampled means and covariances. Such a ‐orthogonal square‐rooting approach is grounded on hyperbolic factorizations. It leads to two novel algorithms covering any continuous‐discrete Gaussian filter of the quadrature, cubature, or unscented Kalman‐like kind. Practical performances of our square‐root solutions are validated, assessed, and compared within two simulated ill‐conditioned scenarios in aeronautical and chemical engineering.