General Nonlinear Model Research Articles

On the ensemble Kalman inversion under inequality constraints

Abstract The ensemble Kalman inversion (EKI), a recently introduced optimisation method for solving inverse problems, is widely employed for the efficient and derivative-free estimation of unknown parameters. Specifically in cases involving ill-posed inverse problems and high-dimensional parameter spaces, the scheme has shown promising success. However, in its general form, the EKI does not take constraints into account, which are essential and often stem from physical limitations or specific requirements. Based on a log-barrier approach, we suggest adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. Finally, we showcase our results through two examples involving partial differential equations.

CT reconstruction using diffusion posterior sampling conditioned on a nonlinear measurement model.

Recently, diffusion posterior sampling (DPS), where score-based diffusion priors are combined with likelihood models, has been used to produce high-quality computed tomography (CT) images given low-quality measurements. This technique permits one-time, unsupervised training of a CT prior, which can then be incorporated with an arbitrary data model. However, current methods rely on a linear model of X-ray CT physics to reconstruct. Although it is common to linearize the transmission tomography reconstruction problem, this is an approximation to the true and inherently nonlinear forward model. We propose a DPS method that integrates a general nonlinear measurement model. We implement a traditional unconditional diffusion model by training a prior score function estimator and apply Bayes' rule to combine this prior with a measurement likelihood score function derived from the nonlinear physical model to arrive at a posterior score function that can be used to sample the reverse-time diffusion process. We develop computational enhancements for the approach and evaluate the reconstruction approach in several simulation studies. The proposed nonlinear DPS provides improved performance over traditional reconstruction methods and DPS with a linear model. Moreover, as compared with a conditionally trained deep learning approach, the nonlinear DPS approach shows a better ability to provide high-quality images for different acquisition protocols. This plug-and-play method allows the incorporation of a diffusion-based prior with a general nonlinear CT measurement model. This permits the application of the approach to different systems, protocols, etc., without the need for any additional training.

Enhancing DC distribution network efficiency through optimal power coordination in lithium-ion batteries: A sparse nonlinear optimization approach

This work deals with the problem regarding the optimal operation of lithium-ion batteries to improve the economic, technical, and environmental indices of standalone and grid-connected direct current (DC) distribution networks.To this effect, a general nonlinear programming model has been developed, which considers three objective functions: (i) the daily energy purchasing costs at the terminals of the substation, considering the maintenance costs of renewable generation (photovoltaic) and energy storage technologies (batteries); (ii) the daily energy losses associated with energy transport; and (iii) the CO2 emissions per day of operation, associated with conventional generators. This is done by integrating all the operations and technical constraints of the DC network and the devices it comprises. The sparse nonlinear optimization (SNOPT) solution method is employed, comparing its results against those of three recently reported metaheuristic optimization methodologies: the continuous genetic algorithm and the parallel versions of the particle swarm optimizer and the vortex search algorithm. The performance of the proposed methodology is validated on standalone and grid-connected networks, considering the technical and economic conditions of two regions of Colombia (rural and urban territories) and evaluating its effectiveness in terms of solution quality, standard deviation, and processing times. The results show that the SNOPT tool of the General Algebraic Modeling System (GAMS) achieves the global optimum in all test scenarios, exhibiting a standard deviation of zero and requiring less than three seconds on average to generate a solution for an entire day of operation.

High-Order Extended Kalman Filter for State Estimation of Nonlinear Systems

In general, the extended Kalman filter (EKF) has a wide range of applications, aiming to minimize symmetric loss function (mean square error) and improve the accuracy and efficiency of state estimation. As the nonlinear model complexity increases, rounding errors gradually amplify, leading to performance degradation. After multiple iterations, divergence may occur. The traditional extended Kalman filter cannot accurately estimate the nonlinear model, and these errors still have an impact on the accuracy. To improve the filtering performance of the extended Kalman filter (EKF), this paper proposes a new extended Kalman filter (REKF) method that utilizes the statistical properties of the rounding error to enhance the estimation accuracy. After establishing the state model and measurement model, the residual term is used to replace the higher-order term in the Taylor expansion, and the least squares method is applied to identify the residual term step by step. Then, the iterative process of updating the extended Kalman filter is carried out. Within the Kalman filter framework, a higher-order rounding error-based extended Kalman filter (REKF) is designed for the joint estimation of rounding error and random variables, and the solution method for the rounding error is considered for the multilevel approximation of the original function. Through numerical simulations on a general nonlinear model, the higher-order rounding error-based extended Kalman filter (REKF) achieves better estimation results than the extended Kalman filter (EKF) and improves the filtering accuracy by utilizing the higher-order rounding error information, which also proves the effectiveness of the proposed method.

Nonlinear model predictive control—Cross-coupling control with deep neural network feedforward for multi-hydraulic system synchronization control

This paper studies a multi-hydraulic system (MHS) synchronization control algorithm. Firstly, a general nonlinear asymmetric MHS state space entirety model is established and subsequently the model form is simplified by nonlinear feedback linearization. Secondly, an entirety model-type solution is proposed, integrating a nonlinear model predictive control (NMPC) algorithm with a cross-coupling control (CCC) algorithm. Furthermore, a novel disturbance compensator based on the system's inverse model is introduced to effectively handle disturbances, encompassing unmodeled errors and noise. The proposed innovative controller, known as nonlinear model predictive control—cross-coupling control with deep neural network feedforward (NMPC-CCC-DNNF), is designed to minimize synchronization errors and counteract the impact of disturbances. The stability of the control system is rigorously demonstrated. Finally, simulation results underscore the efficacy of the NMPC-CCC-DNNF controller, showcasing a remarkable 60.8% reduction in synchronization root mean square error (RMSE) compared to other controllers, reaching up to 91.1% in various simulations. These results affirm the superior control performance achieved by the NMPC-CCC-DNNF controller.

Adomian Decomposition Method With Inverse Differential Operator and Orthogonal Polynomials for Nonlinear Models

A proficient Adomian decomposition method is proposed amidst the presence of inverse differential operator and orthogonal polynomials for solving nonlinear differential models. The method is indeed a reformation of the standard Adomian method thereby improving the rapidity of the solution's convergence rate. A generalized recurrent scheme for a general nonlinear model was derived and further utilized to solve certain nonlinear test models. Lastly, numerical results are reported in comparative tables, demonstrating absolute error differences between the exact and approximate solutions with regards to various employed orthogonal polynomials.

Group Variable Selection via Group Sparse Neural Network

Group variable selection is an important issue in high-dimensional data modeling and most of existing methods consider only the linear model. Therefore, a new method based on the deep neural network (DNN), an increasingly popular nonlinear method in both statistics and deep learning communities, is proposed. The method is applicable to general nonlinear models, including the linear model as a special case. Specifically, a group sparse neural network (GSNN) is designed, where the definition of nonlinear group high-level features (NGHFs) is generalized to the network structure. A two-stage group sparse (TGS) algorithm is employed to induce group variables selection by performing group structure selection on the network. GSNN is promising for complex nonlinear systems with interactions and correlated predictors, overcoming the shortcomings of linear or marginal variable selection methods. Theoretical results on convergence and group-level selection consistency are also given. Simulations results and real data analysis demonstrate the superiority of our method.

A flexible planning approach for integrated lot sizing and rework planning with random proportion of defective products

We consider a stochastic capacitated lot sizing problem with in-line rework. Both defective and defect-free products can be the result of an imperfect production process in real-world production systems. However, the proportion of defective items in a production lot is subject to uncertainty. For economic and environmental reasons, the possibility to rework defective products appears reasonable since these products usually have considerable value. In this paper, a nonlinear model formulation for integrated lot sizing and rework planning is proposed. We use a sample average approach to approximate the generic nonlinear model formulation. To cope with the randomness of the proportion of defective products, we apply a flexible planning approach that allows the adjustment of production and rework quantities. We conduct extensive numerical investigations to evaluate the performance of the proposed planning approach.

Analysis of the Strength of Different Minerals-Modified MPC Based on Mathematical Models

The study discussed the effects of different mineral incorporations and the curing time on the strength of modified magnesium phosphate cement (MPC) mortars through mechanical tests, mathematical model analysis and microstructure characterization. Fly ash (FA), silica fume (SF), and metakaolin (MK), which exhibit excellent durability and bonding properties, were used to modify the MPC. A quantitative relationship was established between the strength of modified MPC mortars and the mineral incorporation and curing time. First, the strength of each mineral-modified MPC mortar cured in air with different mineral incorporations and curing durations was evaluated. The strengths of MPC mortars containing 10% fly ash, 15% silica fume, and 10% metakaolin—which perform best in their incorporations—were compared to analyze the function of the three minerals. To establish the relationship between strength and mineral incorporation and curing time, three mathematical models, linear model, general nonlinear model, and data distribution shape nonlinear model (DDSNM), are commonly used for material property analysis based on statistics. DDSNM best describes the trend of strength change among the three models and the error is small for three minerals. Based on DDSNM, the influence of various minerals on the strength of MPC mortar was quantitatively evaluated by calculating the variable partial derivatives, and verified by scanning electron microscopy and X-ray diffraction. MK performs the best in improving the flexural strength performance of MPC, while SF performs the best in the compressive strength. FA-MPC has low sensitivity to dosage fluctuations and is easy to prepare.

Adaptive neural network sliding mode controller design for load following of nuclear power plant

The power control and load following of nuclear reactors have always been a research topic of concern in the field of nuclear power. In order to improve the control system performance and load following performance, this paper proposes a sliding mode control method based on adaptive RBF neural network (RBFNN) for pressurized water reactor (PWR) power control and load following. Firstly, based on the operation mechanism and energy transport process of PWR, the PWR power model is established based on the point-reactor equation. Then, based on Lyapunov stability theory, an ideal sliding mode controller is designed for a general nonlinear second-order model. Secondly, considering that some state variables in the actual PWR model cannot be directly measured, which leads to the problem that the actual control law cannot be directly applied, RBFNN is used as the controller, and its output is used to replace the actual control law and approximate the ideal control law. Adaptive algorithm is adopted to improve the approximation effect of RBFNN. Finally, the simulation results show that the proposed control method can achieve the PWR load following task well under the set working conditions, and has good robustness to disturbances. Compared with the traditional sliding mode control (TSMC) method, this method can effectively eliminate chattering phenomenon and greatly improve the control performance of the sliding mode controller.

Three-Axes Mems Calibration Using Kalman Filter and Delaunay Triangulation Algorithm

MEMS-IMUs are widely used in research, industry, and commerce. A proper calibration technique must reduce their innate errors. In this study, a turntable-based IMU calibration approach was presented. Parameters such as the bias, lever arm, and scale factor, in addition to misalignment, are included in the general nonlinear model of the IMU output. Accelerometer error parameters were estimated using the transformed unscented Kalman filter (TUKF) with triangulation algorithm is suggested for calibrating inertial measurement unit (MPU6050) three-axes accelerometer. In contrast to the present methods, the suggested method uses the gravitational signal as a constant reference and necessitates no external equipment. The technique requires that the sensor be positioned in a rough orientation and that basic rotations be adopted. This technology also offers a quicker and easier calibration. Comparing the experimental findings with other works, Allan deviation shows significant improvements for the bias instability, where a bias instability of (0.116 μg) is achieved at temperatures between (−15°C) and (80°C).

Remaining useful life prediction using nonlinear multi-phase Wiener process and variational Bayesian approach

Due to the changeable internal mechanisms or external working conditions, the degradation trend of products usually presents multiple phase characteristics. However, most existing multi-phase degradation methods treat each phase as linear and rely on artificial designation to directly specify the forms of drift models, which may result in an inaccurate description of degradation characteristics for complex equipment. To this end, we formulate a general nonlinear multi-phase degradation model with three-source variability based on the Wiener process. Meanwhile, a stage division method is developed to automatically determine the degradation phase number, change-point locations, and the forms of drift models. Then, we obtain the expressions of remaining useful life (RUL) by considering the uncertainty of change-point degradation observations. Especially, we derive the approximate analytical solution of RUL based on the linear model. Furthermore, to fully consider the unit-to-unit heterogeneity and utilize the degradation observations of the in-service unit and prior information simultaneously, we propose a parameter estimation method based on variational Bayesian approach, which adaptively updates all parameters as random variables. Finally, two numerical examples and three practical examples are provided to verify the effectiveness of the proposed method.

Nonlinear modeling and analysis of spatial pipeline with arbitrary shapes supported by clamps considering displacement-dependent characteristic

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes.

General Nonlinear Function-on-Function Regression via Functional Universal Approximation

Various linear or nonlinear function-on-function (FOF) regression models have been proposed to study the relationship between functional variables, where certain forms are assumed for the relationship. However, because functional variables take values in infinite-dimensional spaces, the relationships between them can be much more complicated than those between scalar variables. The forms in existing FOF models may not be enough to cover a wide variety of relationships between functional variables, and hence the applicability of these models can be limited. We consider a general nonlinear FOF regression model without any specific assumption on the model form. To fit the model, inspired by the universal approximation theorem for the neural networks with “arbitrary width,” we develop a functional universal approximation theorem which asserts that a wide range of general maps between functional variables can be approximated with arbitrary accuracy by members in our proposed family of maps. This family is “fully” functional in that the complexity of the maps within the family is completely determined by the smoothness of the component functions in the map. With this functional universal approximation theorem, we develop a novel method to fit the general nonlinear FOF regression model, which includes all existing FOF models as special cases. The complexity of the fitted model is controlled by smoothness regularization, without the necessity to choose the number of hidden neurons. Supplemental materials for code and additional information are available online.

A Novel Reconfigurable Nonlinear Cascaded MZM Mixer, Amplitude Shift Key Modulator (ASK), Frequency Hopping and Phase Shifter

A novel reconfigurable Microwave Photonics (MWP) mixer is presented in this paper, which can be configured to work as a frequency hopper, ASK modulator and phase shifter. This mixer is based on a cascaded Mach–Zehnder Modulator (MZM) structure. A general nonlinear analytical model for the structure is presented. This model is platform free, which means it can be applied to several MWP and integrated MWP platforms. Based on the analytical model, the performance of the structure and output results, such as the optical and electrical spectrum of the structure, are derived in mathematical closed-form expressions. The results of the presented analytical model are compared and evaluated with the results obtained from the simulation to prove the correctness of the analytical model. The presented structure has a simple form, which can be fabricated at a low cost. In addition, according to the obtained results from the analytical model, there is no need to change the arrangement of the structure to operate in any of the mentioned configurations, and the desired function is achievable only by changing the bias voltage of the modulators at the desired frequency.

Aspects of Program Engagement in an Online Physical Activity Intervention and Baseline Predictors of Engagement.

Participant engagement in an online physical activity (PA) intervention is described and baseline factors related to engagement are identified. Longitudinal Study Within Randomized Controlled Trial. Online/Internet. Primary care patients (21-70years). ActiveGOALS was a 3-month, self-directed online PA intervention (15 total lessons, remote coaching support, and a body-worn step-counter). Engagement was measured across six outcomes related to lesson completion (total number and time to complete), coach contact, and behavior tracking (PA, sedentary). Self-reported baseline factors were examined from seven domains (confidence, environment, health, health care, demographic, lifestyle, and quality of life). General linear and nonlinear mixed models were used to examine relationships between baseline factors and engagement outcomes within and across all domains. Seventy-nine participants were included in the sample (77.2% female; 74.7% white non-Hispanic). Program engagement was high (58.2% completed all lessons; PA was tracked ≥3 times/week for 11.3 ± 4.0weeks on average). Average time between completed lessons (days) was longer than expected and participants only contacted their coach about 1 of every 3weeks. Individual predictors related to health, health care, demographics, lifestyle, and quality of life were significantly related to engagement. Examining multiple aspects of engagement and a large number of potential predictors of engagement is likely needed to determine facilitators and barriers for high engagement in multi-faceted online intervention programs.

A General Approach for the Modelling of Negative Feedback Physiological Control Systems.

Mathematical models can improve the understanding of physiological systems behaviour, which is a fundamental topic in the bioengineering field. Having a reliable model enables researchers to carry out in silico experiments, which require less time and resources compared to their in vivo and in vitro counterparts. This work's objective is to capture the characteristics that a nonlinear dynamical mathematical model should exhibit, in order to describe physiological control systems at different scales. The similarities among various negative feedback physiological systems have been investigated and a unique general framework to describe them has been proposed. Within such a framework, both the existence and stability of equilibrium points are investigated. The model here introduced is based on a closed-loop topology, on which the homeostatic process is based. Finally, to validate the model, three paradigmatic examples of physiological control systems are illustrated and discussed: the ultrasensitivity mechanism for achieving homeostasis in biomolecular circuits, the blood glucose regulation, and the neuromuscular reflex arc (also referred to as muscle stretch reflex). The results show that, by a suitable choice of the modelling functions, the dynamic evolution of the systems under study can be described through the proposed general nonlinear model. Furthermore, the analysis of the equilibrium points and dynamics of the above-mentioned systems are consistent with the literature.

Observer‐based sliding mode H∞$H_\infty$ control for offshore structures with nonlinear energy sink mechanisms

AbstractIt is significant to reduce the wave‐induced vibration and ensure the safety of the offshore structure. This paper deals with the observer‐based sliding mode vibration control problem of offshore structures with nonlinear energy sink (NES) mechanisms. By considering the coupled nonlinear characteristics of unmodelled dynamics, system uncertainty, and external wave force, a more general nonlinear system model of the offshore structure equipped with an NES mechanism is presented. Then, an observer‐based sliding mode control scheme for the nonlinear structure‐NES system is developed. The existence condition and the design algorithm of the observer‐based sliding mode controller are provided. Simulation results show that the observer‐based sliding mode controller is capable of reducing the wave‐induced vibration amplitude of offshore structures effectively. Compared with the NES‐based passive control, the observer‐based sliding mode controller can further reduce the response amplitudes of the offshore structure remarkably. In addition, the designed observer‐based sliding mode controller is robust the variations of structure parameters and the external wave loads, thereby presenting potential advantages of the active damping control of the offshore structure.

Synthesis of Output-Feedback Controllers for Mixed Traffic Systems in Presence of Disturbances and Uncertainties

In this paper, we study mixed traffic systems that move along a single-lane ring-road or open-road. The traffic flow forms a platoon, which includes a number of heterogeneous human-driven vehicles (HDVs) together with only one connected and automated vehicle (CAV) that receives information from a subset of neighbors. The dynamics of HDVs are assumed to follow a general continuous-time nonlinear car-following model, which is a function of their velocity, spacing, and the relative velocity. The acceleration of the single CAV is also directly controlled by a dynamical output-feedback controller. The ultimate goal of this work is to present a robust control strategy that can smoothen the traffic flow in the presence of undesired disturbances (e.g. abrupt deceleration) and parametric uncertainties. A prerequisite for synthesizing a dynamical output controller is the stabilizability and detectability of the underlying system. Accordingly, a theoretical analysis is presented first to prove the stabilizability and detectability of the mixed traffic flow system. Then, two <inline-formula> <tex-math notation="LaTeX">$H_\infty$</tex-math> </inline-formula> control strategies, with and without considering uncertainties in the system dynamics, are designed. The efficiency of the two control methods is subsequently illustrated through numerical simulations, and various experimental results are presented to demonstrate the effectiveness of the proposed controller to mitigate disturbance amplification and achieve platoon stability.

Parameter Estimation of Linear and Nonlinear Systems Connected in Parallel

Presently, the problem of parameter estimation is addressed for a more general nonlinear model. In this respect, the proposed nonlinear system is composed of the parallel connection of linear and nonlinear blocks. The interconnected structure of linear and nonlinear blocks systems leads to a highly nonlinear problem involving several unknown parameters and inaccessible internal signals. In this parameter estimation method, the linear and nonlinear block parameters are estimated in one stage by using simple a sine signal or multi-cosine wave.