Nonlinear Process Model Research Articles

A novel performance prediction method for harmonic reducers based on the trend-enhanced Fisher’s discriminant ratio-Wiener process

Abstract Harmonic reducers, as core components of industrial robots, play a critical role in maintaining robot health. Performance degradation assessment (PDA) and remaining useful life (RUL) prediction are essential for ensuring the operational reliability of such robots. To address the multistage characteristics and stochastic uncertainties in the performance degradation of harmonic reducers, a performance prediction method based on Fisher’s discriminant ratio and Wiener process is proposed. Firstly, the health indicator (HI) construction is defined as an optimization problem based on Fisher’s discriminant ratio and trend monotonicity constraints. By leveraging the sum of the multiplication of frequency amplitudes and optimized weights, precise segmentation of the multistage degradation process over the entire lifecycle is achieved. Notably, the optimized weights can automatically identify the resonance frequency bands caused by damage. Subsequently, a nonlinear Wiener process model with a drift coefficient in the form of a power function is established. The probability density function (PDF) expression for RUL is derived based on the concept of first hit time (FHT). Additionally, a logarithmic likelihood function (Log-LF) for the unknown parameters in the degradation model is constructed. The experimental results indicate that the proposed method surpasses the other two Wiener process models in terms of root mean square error (RMSE), mean absolute percentage error (MAPE), and cumulative relative accuracy (CRA). This provides robust support for preventive maintenance decision-making for harmonic reducers.

UTILIZING GAUSSIAN PROCESS REGRESSION FOR NONLINEAR MAGNETIC SEPARATION PROCESS IDENTIFICATION

This paper presents a novel approach utilizing Gaussian Process Regression (GPR) to identify dynamic models with nonlinear parameters in magnetic separation processes. It aims to address the complex and dynamic nature of these processes by employing advanced modeling methods. The effectiveness of GPR is demonstrated through its application to simulated signals representing real iron ore separation processes, highlighting its potential to enhance existing models and optimize processes. Conducted within the MATLAB, this research lays the groundwork for further advancement and practical implementation. The utilization of GPR in magnetic separation offers innovative modeling of nonlinear dynamic processes, promising improved efficiency and precision in industrial applications.

Machine learning-based input-augmented Koopman modeling and predictive control of nonlinear processes

Koopman-based modeling and model predictive control have been a promising alternative for optimal control of nonlinear processes. Good Koopman modeling performance significantly depends on an appropriate nonlinear mapping from the original state-space to a lifted state space. In this work, we propose an input-augmented Koopman modeling and model predictive control approach. Both the states and the known inputs are lifted using two deep neural networks (DNNs), and a Koopman model with nonlinearity in inputs is trained within the higher-dimensional state space. A Koopman-based model predictive control problem is formulated. To bypass non-convex optimization induced by the nonlinearity in the Koopman model, we further present an iterative implementation algorithm, which approximates the optimal control input via solving a convex optimization problem iteratively. The proposed method is applied to a chemical process and a biological water treatment process via simulations. The efficacy and advantages of the proposed modeling and control approach are demonstrated.

Recursive identification of kernel-based hammerstein model for online energy efficiency estimation of large-scale chemical plants

Accurate energy efficiency prediction is crucial for decision-making in large-scale chemical production, especially considering energy management and environmental concerns. The complexity of chemical processes, characterized by strong nonlinearities, dynamics, and uncertainties, challenges traditional prediction methods. This paper introduces KHSM (Kernel-based Hammerstein Subspace Model), which models nonlinear static and linear dynamic processes in series by embedding kernel function-based nonlinear mapping into subspace modeling for energy efficiency estimation. KHSM identifies nonlinear process models without explicit nonlinear mapping functions, constructing dynamic models that encompass both energy input and product output variables. An adaptive KHSM variant dynamically updates the energy efficiency model, improving accuracy by iteratively selecting and excluding samples. Comparative analysis shows KHSM outperforms existing methods, achieving a coefficient of determination (R2) closest to 1 and reducing the root-mean-square error (RMSE) by up to an order of magnitude, with the smallest mean and standard deviation. Additionally, modeling efficiency increases by an order of magnitude. The developed model enables ongoing energy efficiency estimation and proactive identification of efficiency trends. Validation through mathematical and real-world ethylene process cases demonstrates KHSM's efficacy and applicability. These estimates facilitate informed adjustments to energy management strategies and optimization of production approaches in energy-intensive chemical processes.

Creating Fuzzy Models from Limited Data

The design of experiments is a methodological approach in which measurement experiments are carefully planned to obtain highly informative data. This paper addresses the challenge of constructing mathematical models for complex nonlinear processes when the available measurement data have low information content. This problem often arises when data are collected without the guidance of an experimental modeling expert. We examine two practical examples to illustrate this issue: a textile wastewater decolorization process and atmospheric corrosion of structural metal materials. In both cases, the measured data were insufficient to construct highly accurate models. It is, therefore, necessary to make a trade-off between model complexity and accuracy by adapting modeling techniques to work effectively with the limited data available. The main aim of the paper is, therefore, to focus on simple but effective techniques that allow as much information as possible to be extracted from low-quality measurements and to maximize the usefulness of the model for its intended purpose.

Adaptive soft sensor modeling of chemical processes based on an improved just‐in‐time learning and random mapping partial least squares

AbstractThe just‐in‐time learning‐based partial least squares (JIT‐PLS) has been extensively applied to adaptive soft sensor modeling of complex nonlinear processes. However, it still has the problems of unreasonable relevant samples selection and unsatisfactory local modeling. Aiming at these problems, this paper proposes an improved just‐in‐time learning‐based random mapping partial least squares (IJIT‐RMPLS), including an improved relevant samples selection strategy and a random mapping PLS (RMPLS) model. On the one hand, considering the different correlation degrees between input variables and output variable, this method applies mutual information to evaluate the importance of each input variable and designs a variable‐weighted Euclidean distance to select relevant samples for local modeling. On the other hand, in order to prompt the prediction precision of local soft sensor models, this method combines the idea of nonlinear random mapping in extreme learning machines with PLS and builds a RMPLS with multiple activation functions. Applications on a numerical example and a real chemical process show that the proposed IJIT‐RMPLS has smaller prediction error compared with traditional JIT‐PLS.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Optimization-based multi-source transfer learning for modeling of nonlinear processes

This work develops an optimization-based multi-source transfer learning scheme for modeling of nonlinear chemical processes. Specifically, a transfer learning neural network model for a target process with limited data is developed using the pre-trained model obtained with multiple source processes. Since the performance of transfer learning models depends on the quality of the pre-trained models, we propose a novel Bayesian optimization problem to optimize the selection of multi-source data for the pre-trained models by first deriving a generalization error bound for multi-source domain adaptation using Y-discrepancy distance. Subsequently, the optimization problem is formulated using the theoretical error bound to select the optimal set of multiple sources, which can provide a good initial guess of the weight parameters for transfer learning model. Finally, a simulation study of a chemical reactor process in Aspen Plus Dynamics is conducted to illustrate the effectiveness of the optimization-based multi-source transfer learning scheme.

Automatic initial value generation procedure for nonlinear process models by interval arithmetic based cutting and splitting

To solve large nonlinear equation systems, we present a hybrid method based on interval arithmetic and real-valued root-finding. The proposed approach includes two new interval arithmetic based methods, the so-called cutting and a special kind of bisection of consistent variable spaces. Applied to three examples from chemical engineering, it is shown that the approach is now able to find all system solutions within a variable space as well as only one process-relevant solution in much less time thanks to the built-in root-finding step. For the latter, a conventional Newton method as well as IPOPT have been examined. The hybrid approach no longer needs a well-estimated initial point to converge to a solution, only rough initial variable bounds are required.

A neural network based model for multi-dimensional non-linear Hawkes processes

This paper introduces the Neural Network for Non-linear Hawkes processes (NNNH), a non-parametric method based on neural networks to fit non-linear Hawkes processes. Our method is suitable for analysing large datasets in which events exhibit both mutually-exciting and inhibitive patterns. The NNNH approach models the individual kernels and the base intensity of the non-linear Hawkes process using feed forward neural networks and jointly calibrates the parameters of the networks by maximizing the log-likelihood function. We utilize Stochastic Gradient Descent to search for the optimal parameters and propose an unbiased estimator for the gradient, as well as an efficient computation method. We demonstrate the flexibility and accuracy of our method through numerical experiments on both simulated and real-world data, and compare it with state-of-the-art methods. Our results highlight the effectiveness of the NNNH method in accurately capturing the complexities of non-linear Hawkes processes.

Maximum Principle in Autonomous Multi-Object Safe Trajectory Optimization

The following article presents the task of optimizing the control of an autonomous object within a group of other passing objects using Pontryagin’s bounded maximum principle. The basis of this principle is a multidimensional nonlinear model of the control process, with state constraints reflecting the motion of passing objects. The analytical synthesis of optimal multi-object control became the basis for the algorithm for determining the optimal and safe object trajectory. Simulation tests of the algorithm on the example of real navigation situations with various numbers of objects illustrate their safe trajectories in changing environmental conditions. The optimal object trajectory obtained using Pontryagin’s maximum principle was compared with the trajectory calculated using the Bellman dynamic programming method. The analysis of the research allowed for the formulation of valuable conclusions and a plan for further research in the field of autonomous vehicle control optimization. The maximum principle algorithm allows one to take into account a larger number of objects whose data are derived from ARPA anti-collision radar systems.

NIRK-based mixed-type accurate continuous–discrete Gaussian filters with deterministically sampled expectation and covariance for state estimation in continuous-time stochastic process models with discrete measurements

This paper advances further the idea of overall continuous–discrete Gaussian filtering with deterministically sampled expectation and covariance towards efficient mixed-type accurate Kalman-like schemes based on self-regulated Ordinary Differential Equation (ODE) solvers with automatic local and global error controls intended for treating continuous-time nonlinear stochastic process models with discrete measurements. The universal state estimation algorithms of such sort designed recently by Kulikov and Kulikova (2022) possess three potential limitations, which might affect their successful applications in practice. First, these deal with complicated Moment Differential Equations (MDEs) and, hence, can be rather time-consuming in estimating stochastic process models of large size, that compromises their efficiency for online computations. Second, the aforementioned filters do not employ any kind of global error regulation and, hence, can be insufficiently accurate in solving difficult MDEs arisen. Third, these are hardly effective in treating stiff continuous–discrete stochastic systems. Below, all the listed issues are addressed and solved via utilization of the much simpler but more robust MDEs arisen in the usual continuous–discrete Extended Kalman Filter (EKF) within the mixed-type accurate filtering approach initiated by Kulikova and Kulikov (2015), which is rooted in the variable-stepsize Nested Implicit Runge–Kutta (NIRK) methods. For the sake of stability of our novel methods to the numerical integration and round-off errors, we devise square-root versions of the mixed-type Gaussian filters, which are based either on hyperbolic QR factorizations or on hyperbolic Singular Value Decompositions (SVDs). Performances of these new mixed-type square-root Gaussian filters are assessed and compared to their original (non-mixed-type) counterparts as well as to their non-square-root versions in a simulated non-stiff ill-conditioned radar tracking scenario and on a stiff stochastic Oregonator reaction.

Mathematical modeling of non-linear reaction-diffusion process in autocatalytic reaction: Akbari-Ganji method

A theoretical model of the electroreduction of halogen oxoanions via autocatalytic redox mediation by halide anions is discussed. This autocatalytic process depends upon a system reaction-diffusion equation including a nonlinear term for homogeneous reactions. This study solves these equations using the robust and user-friendly analytical technique, the Akbari-Ganji method. The current and the redox component concentration are presented in terms of the rate constant. The influence of variables on current is examined using sensitivity analysis. The most important factor affecting current is the ratio between the bulk concentrations of halogen molecules and non-electroactive halogen oxoanions. The theoretically predicted results are compared with the numerical results. These approximate results enhance our understanding of system behavior.

Robust reduced-order machine learning modeling of high-dimensional nonlinear processes using noisy data

Autoencoder-based reduced-order machine learning models have been developed for modeling and predictive control of nonlinear chemical processes with high dimensionality such as discretization of reaction–diffusion processes. However, in the presence of data noise, autoencoders may over-fit the training data and subsequently learn an inaccurate low-dimensional representation of the process variables. This leads to an inaccurate prediction model when the models are integrated with model predictive control (MPC). To address this issue, this work develops a novel machine-learning-based reduced-order modeling method by integrating SpectralDense layers into autoencoders and incorporating them with recurrent neural networks. We demonstrate that the new architecture of autoencoders using SpectralDense layers is more robust against over-fitting than conventional autoencoders in the presence of data noise, which improves the prediction accuracy in MPC. A diffusion–reaction process simulation example is used to demonstrate that the robust autoencoders outperform those using conventional layers for reduced-order modeling in predictive control.

Modeling and Control of Non-Linear CSTH Process using Hybrid Optimized Technique

Modeling and Control of Non-Linear CSTH Process using Hybrid Optimized Technique

Adaptive Multioutput Gradient RBF Tracker for Nonlinear and Nonstationary Regression.

Multioutput regression of nonlinear and nonstationary data is largely understudied in both machine learning and control communities. This article develops an adaptive multioutput gradient radial basis function (MGRBF) tracker for online modeling of multioutput nonlinear and nonstationary processes. Specifically, a compact MGRBF network is first constructed with a new two-step training procedure to produce excellent predictive capacity. To improve its tracking ability in fast time-varying scenarios, an adaptive MGRBF (AMGRBF) tracker is proposed, which updates the MGRBF network structure online by replacing the worst performing node with a new node that automatically encodes the newly emerging system state and acts as a perfect local multioutput predictor for the current system state. Extensive experimental results confirm that the proposed AMGRBF tracker significantly outperforms existing state-of-the-art online multioutput regression methods as well as deep-learning-based models, in terms of adaptive modeling accuracy and online computational complexity.

Reduced-order Koopman modeling and predictive control of nonlinear processes

In this paper, we propose an efficient data-driven predictive control approach for general nonlinear processes based on a reduced-order Koopman operator. A Kalman-based sparse identification of nonlinear dynamics method is employed to select lifting functions for Koopman identification. The selected lifting functions are used to project the original nonlinear state–space into a higher-dimensional linear function space, in which Koopman-based linear models can be constructed for the underlying nonlinear process. To curb the significant increase in the dimensionality of the resulting full-order Koopman models caused by the use of lifting functions, we propose a reduced-order Koopman modeling approach based on proper orthogonal decomposition. A computationally efficient linear robust predictive control scheme is established based on the reduced-order Koopman model. A case study on a benchmark chemical process is conducted to illustrate the effectiveness of the proposed method. Comprehensive comparisons are conducted to demonstrate the advantage of the proposed method.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

A Deep Learning Feature Fusion Based Health Index Construction Method for Prognostics Using Multiobjective Optimization

Degradation modeling and prognostics serve as the basis for system health management. Recently, various sensors provide plentiful monitoring data that can reflect the system status. A multitude of feature fusion techniques based on multisensor data have been proposed to generate a composite health index (HI) for prognostics, which can represent the underlying degradation mechanism. Most existing methods have used linear fusion models and neglected the practical requirements for HI construction, which are insufficient to reveal the nonlinear relations among features and difficult to obtain accurate HIs for complicated systems. This study proposes a novel feature fusion-based HI construction method with deep learning and multiobjective optimization. Multiple degradation features are fused by a deep neutral network (DNN). Several desired properties that the HIs should have for prognostics are adopted to formulate the objective functions of DNN training. To balance the spatial complexity and performance of the fusion model, a multiobjective optimization model is generated for training the DNN. Then, a generalized nonlinear Wiener process model is used to predict the remaining useful life with the resulted HIs. Finally, two cases are analyzed to illustrate the effectiveness and robustness of the proposed method.

Deep Cascade Gradient RBF Networks With Output-Relevant Feature Extraction and Adaptation for Nonlinear and Nonstationary Processes.

The main challenge for industrial predictive models is how to effectively deal with big data from high-dimensional processes with nonstationary characteristics. Although deep networks, such as the stacked autoencoder (SAE), can learn useful features from massive data with multilevel architecture, it is difficult to adapt them online to track fast time-varying process dynamics. To integrate feature learning and online adaptation, this article proposes a deep cascade gradient radial basis function (GRBF) network for online modeling and prediction of nonlinear and nonstationary processes. The proposed deep learning method consists of three modules. First, a preliminary prediction result is generated by a GRBF weak predictor, which is further combined with raw input data for feature extraction. By incorporating the prior weak prediction information, deep output-relevant features are extracted using a SAE. Online prediction is finally produced upon the extracted features with a GRBF predictor, whose weights and structure are updated online to capture fast time-varying process characteristics. Three real-world industrial case studies demonstrate that the proposed deep cascade GRBF network outperforms existing state-of-the-art online modeling approaches as well as deep networks, in terms of both online prediction accuracy and computational complexity.