Nonlinear Inverse Model Research Articles

Identification of nonlinear system model and inverse model based on conditional invertible neural network

In applications such as adaptive inverse control, internal model control, and active noise control, the identification accuracy of the system model and the inverse model directly affects the performance. However, it is not easy to identify inverse models for nonlinear systems. Moreover, existing methods require two identification calculations to obtain the system model and the inverse model. Therefore, an identification method of nonlinear system model and inverse model based on conditional invertible neural network (cINN) is proposed. The invertible structure of cINN enables simultaneous approximation of complex nonlinear functions and simultaneous acquisition of the corresponding inverse functions. Consequently, both the nonlinear system model and the inverse model can be identified concurrently through cINN. Moreover, the identification performance of the cINN-based method is validated and applied to disturbance cancellation in a classical nonlinear simulation system. Finally, the inverse model of the actuator is identified by cINN, and the inverse model is applied to the nonlinear compensation of the actuator.

Flight control design for a hypersonic waverider configuration: A non-linear model following control approach

AbstractDLR is currently investigating the potential of hypersonic flight systems in the context of different mission scenarios. One configuration type of higher interest for civil and military purposes is the hypersonic glide vehicle (HGV). Such HGVs operate over broad flight envelopes and pose complex flight dynamic characteristics. This article presents DLR’s generic hypersonic glide vehicle 2 (GHVG-2) and proposes an integrated non-linear flight control architecture that is based on the idea of the non-linear dynamic inversion and non-linear model following control methodologies. The proposed control scheme is designed to sufficiently and robustly handle the system dynamics of the over-actuated flight vehicle. The approach is first discussed, and the performance of the suggested control laws is later investigated via simulations of a high-fidelity non-linear flight dynamic model in the nominal case and under the presence of parametric uncertainties. The presented results demonstrate that the proposed NMFC approach provides benefits for the robust control of the regarded hypersonic system.

Investigation of goose breeding in Turkiye by linear and nonlinear regression models

In this study, the change in the number of geese breeding in Turkiye over the years was examined by linear and non-linear regression models. Among linear and non-linear regression models, linear, quadratic, cubic, logarithmic, and inverse regression models were used. R2 and MSE values were taken as criteria for comparing the models. As a result of the study, the cubic regression model with the highest R2 value and the lowest MSE value was found to be the best fitting model for the number of geese. According to the cubic regression model, the number of geese in Turkiye in 2023 and 2024 was estimated to be 1849304 and 2107588, respectively.

Geobody estimation by Bhattacharyya method utilizing nonlinear inverse modeling of magnetic data in Baba-Ali iron deposit, NW Iran

One of the essential geophysical concerns is the estimation of the physical and geometrical parameters of the reserve (geobody), which is done by exploiting the nonlinear inverse modeling of magnetic data. The present study includes preparing and modeling magnetic data to suggest the Baba Ali Iron ore deposit's drilling locations in NW Iran. The area is covered with 1000 points of geomagnetic reading with an almost 5 × 10 m2 regularly spaces grid trending WE. The areal and depth extent of the iron ore geobody was unknown. The Bhattacharyya method by MATLAB software coding was employed to underestimate the misfit function and re-construct potential field data, providing the most suitable fitting with measured magnetic data. In this order, the residual calculated anomaly exhibited an excellent two-dimensional conformation with forward modeling. Furthermore, 3D modeling correctly reconstructs the anomalies' productive resources' properties. After preparing full magnetic maps, the magnetic lenses distinguished in four anomalies of surface depths, 20, 50, and deeper than 50 m for this zone. This magnetite lens for the first zone was estimated based on analytical signal filters applied on the entire magnetic map so that the lens's depth is trivial and almost zero. Due to specific gravity calculated as 4.77 t/m3, initial storage capacity is suggested to be about 95,400 tons of magnetite, pyrite, and hematite minerals at most in an area about 6 Km2. Finally, to complete the preliminary explorations of the specified area, exploratory drilling is suggested for three points by inverse modeling. Regarding this study as the first try in magnetic reconnaissance step of Iron mineral exploration in the study area, there is no geological constraints available based on drilling evidences. However, the model is well satisfies the surface anomalies considering residual magnetic property.

Three stages in the variation of the depth of hypoxia in the California Current System 2003–2020 by satellite estimation

The depth of hypoxia (DOH) is the shallowest depth at which the waters become hypoxic (oxygen concentration < 60 μmol kg−1), is a crucial indicator of the formation and expansion of oxygen minimum zones (OMZs). In this study, a nonlinear polynomial regression inversion model was developed to estimate the DOH in the California Current System (CCS), based on the dissolved oxygen profile detected by the Biogeochemical-Argo (BGC-Argo) float and remote sensing data. Satellite-derived net community production was used in the algorithm development, to denote the combined effect of phytoplankton photosynthesis and O2 consumption. Our model performs well, with a coefficient of determination of 0.82 and a root mean square error of 37.69 m (n = 80) from November 2012 to August 2016. Then, it was used to reconstruct the variation in satellite-derived DOH in the CCS from 2003 to 2020, and three stages of the DOH variation trend were identified. From 2003 to 2013, the DOH showed a significant shallowing trend due to the intense subsurface O2 consumption caused by strong phytoplankton production in the CCS coastal region. The trend was interrupted by two successive strong climate oscillation events from 2014 to 2016, which led to a significant deepening of the DOH and a slowing, or even reversal, of the variations in other environmental parameters. After 2017, the effects of climate oscillation events gradually disappeared, and the shallowing pattern in the DOH recovered slightly. However, by 2020, the DOH had not returned to the pre-2014 shallowing characteristic, which would lead to continuing complex ecosystem responses in the context of global warming. Based on the satellite inversion model of DOH in the CCS, we provide a new insight on the high-resolution spatiotemporal OMZ variations during an 18-year period in the CCS, which will aid in the evaluation and prediction of local ecosystems variation.

Estimation of Depth, Physical Parameters, and Location Using Nonlinear Inverse Modeling of Magnetic Data by Bhattacharya Method in a Field of NW Iran

Estimation of Depth, Physical Parameters, and Location Using Nonlinear Inverse Modeling of Magnetic Data by Bhattacharya Method in a Field of NW Iran

Extreme partial least-squares

We propose a new approach, called Extreme-PLS, for dimension reduction in conditional extreme values settings. The objective is to find linear combinations of covariates that best explain the extreme values of the response variable in a non-linear inverse regression model. The asymptotic normality of the Extreme-PLS estimator is established in the single-index framework and under mild assumptions. The performance of the method is assessed on simulated data. A statistical analysis of French farm income data, considering extreme cereal yields, is provided as an illustration.

On the Development and Performance Evaluation of Improved Radial Basis Function Neural Networks

This article deals with the development of four modified radial basis function neural network (RBFNN) models. The corresponding learning algorithms associated with the updating of internal parameters of the models are derived. The conventional inputs are used in the first and second modified RBFNN models (models 3 and 4) whereas exponential nonlinear inputs are used in the fifth and sixth RBFNN models to provide additional nonlinearity for achieving a better solution of nonlinear classification, and direct and inverse modeling problems. To assess and compare the performance potentiality of the proposed four new RBFNN models, one classification problem, one direct modeling problem, and one inverse modeling problem are solved through computer simulation-based experiments. For comparison and to assign the performance rank of each of the four modified RBFNN models, two conventional and commonly used RBFNN models (models 1 and 2) are also simulated. To access the performance of different models during the training phase of Examples 1 and 2, the root mean-square error (RMSE) value, mean absolute deviation (MAD), and the number of iterations required to achieve convergence are obtained. For the third example, only the first two performance measures are found. During the testing or validation phase, the output responses of the different models of Example 2 are compared with the desired response analysis. For Example 3, the bit-error rate (BER) plots are compared. The observation of all the results demonstrates consistent ranks of all models in the case of all three examples. It is, in general, found that the ranks of the models 1–6 are 6, 4, 3, 2, 5, and 1, respectively. In essence, in terms of all performance measures, model M-6 with an exponential version of inputs with weights on both layers occupies the first position whereas model M-4 with conventional inputs as the second position.

Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors

Bayesian inference and uncertainty quantification in a general class of nonlinear inverse regression models is considered. Analytic conditions on the regression model {G(θ):θ∈Θ} and on Gaussian process priors for θ are provided such that semiparametrically efficient inference is possible for a large class of linear functionals of θ. A general Bernstein–von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence, posterior-based credible sets are valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in nonlinear tomography problems: an elliptic inverse problem for a Schrödinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces.

Intelligent Speed Regulation Control in Switched Reluctance Motor of Electric Vehicle Based on Neural Network Parameter Identification

This paper studies the intelligent speed regulation control of switched reluctance motor of electric vehicle based on neural network parameter identification. Starting with the analysis of the performance of switched reluctance motor, the nonlinear flux linkage characteristic inversion model and torque characteristic model of switched reluctance motor are established based on BP neural network. This paper studies and improves the fast self configuration algorithm of BP neural network. Finally, the nonlinear simulation model of switched reluctance motor is established under Matlab/Simulink. The model can be used for further control research. In this paper, the integrated control method of instantaneous torque control based on torque observation and three-step commutation control is studied, and the simulation analysis is carried out. The results show that this method can effectively reduce the torque ripple of switched reluctance motor and improve the performance of its drive system.

A METHOD OF WATER DEPTH INVERSION IN COASTAL AREA CONSIDERING TEMPERATURE INFORMATION

Abstract. The remote sensing method for water depth inversion is fast, flexible, and low in cost, which has become an important means of method for water depth detection. This paper takes the coastal area where is around Gulangyu Island as the research area. Based on the spectral reflectance, sea surface temperature (SST) and measured water depth data, a nonlinear inversion model of water depth is established by using BP neural network. Combined with the tide data, the water depth and underwater topography in coastal area is obtained. The average relative error is 0.27. The root mean square error is 1.92. The results show that the participation of sea surface temperature in the model construction can improve the inversion error of offshore water depth to a certain extent, and can help improve the accuracy of the model.

A nonlinear inverse model for airborne wind energy system analysis, control, and design optimization

AbstractThis paper describes a nonlinear model inversion for performance analysis and system optimization for airborne wind energy (AWE) systems. Airborne wind energy systems are lighter and potentially lower in cost than comparable conventional wind turbines due to using a tether and bridle rather than blades and a tower which must support high bending and compressive loads. They are also easier to install due to using a tension anchor rather than a foundation. There are many AWE systems in various states of R&amp;D, but no commercial AWE wind farms currently exist. Because AWE systems are novel and significantly more complex to design, analyze, and test than conventional wind turbines, better analysis tools are important for the technology to mature.AWE system design involves many interdependent design trade‐offs, which are difficult to analyze with traditional tools; using simulations to iterate through parameters for design optimization is undesirable. A nonlinear model inversion based on appropriate simplifying assumptions is better suited to this task. The inverse model uses a trajectory input then calculates attitudes, forces, and control values required to follow that trajectory as well as power produced. Applicable constraints ensure that the trajectory is realizable. A validation is presented, using the output of the inverse model as a feed‐forward controller for a high fidelity simulation. The analysis shows good agreement with the simulation, despite simplifications such as a straight rigid tether, constant lift and drag coefficients, and a simplified rotor model.

Prediction of baking quality using machine learning based intelligent models

A domestic cookie baking process was modeled using nonlinear forward and inverse models to predict surface temperature, moisture content and browning index that describe the baking quality of the end product. The baking processes were carried out at different oven temperatures (160, 180, 200 °C) and the changes in surface temperature, moisture content and browning index were determined to construct the identification models namely nonlinear polynomial models (PLN) and nonlinear artificial-neural network (ANN) model. The parameters of the artificial models were optimized using least-squares estimation and Levenberg-Marquardt optimization, respectively. The predicted baking characteristics in both forward and inverse phases were in good agreement with the measured ones even for the browning index which was difficult to model because of the its nonminimum-phase dynamics. The application results indicated that the developed intelligient models were very accurate, having low root mean-squared errors, the ANN model approximated the desired values better than the PLN models for all the state variables. Thus, the designed ANN models are applicable for the automatized industrial and domestic oven designs of the future.

The Effect of Meteorological Factors, Seasonal Factors and Air Pollutions on the Formation of Particulate Matter

To investigate the effect of meteorological factors, seasonal factors and Air Pollutions on the formation of particulate matter (PM2.5) in Jinan, China. Nonlinear dynamic inversion model was established to analyze the effect of meteorological factors, seasonal factors and CO, PM10, SO2, NO2, O3 on PM2.5 formations in different seasons. Temperature has a great influence on PM2.5 concentration variation. Precipitation exacerbate the formation of PM2.5 in Winter. Wind speed make a little contribution to PM2.5 formation in Jinan during different season. The formation of PM2.5 was influenced by confounding factors.

SIMULATION OF A SALT DOME USING 2D LINEAR AND NONLINEAR INVERSE MODELING OF RESIDUAL GRAVITY FIELD DATA

In gravity field inversion we usually dealing with underdetermined problems and for obtaining realistic solutions can introduce a depth-weighting function to the inversion algorithm. We employ a linear inversion method for determining the underground density distribution of the gravity causative mass. The validation and accuracyof method is tested on two synthetic gravity anomaly from different models, while the data are noise- free and corrupted with noise. In this paper, We also invert the 2D gravity anomaly produced by a salt dome from the northwest of Iran. The salt domes in the region under investigation are a rich source of Potash. The inverted structure demonstrate on average a depth to top and bottom of 27 m and 65 m, respectively. For comparison, we also have simulated the salt dome using the nonlinear inverse modeling. The results are mostly similar.

Automated Estimation of Coastal Bathymetry from High Resolution Multi-Spectral Satellite Images

Coastal bathymetry is the most essential tool for marine planning, monitoring and management, modelling, nautical navigation and scientific studies of marine environments. The techniques have been developed over the last decade to derive bathymetry using remote sensing technology with efficiently and low costly. Log linear bathymetric inversion model and non-linear bathymetric inversion model provides two empirical approaches for deriving bathymetry from multispectral satellite imagery, which have been refined and widely applied. This paper compares these two approaches by means of a geographical error analysis for the site Kankesanturai using WorldView-2 satellite imagery. In order to calibrate both models; Single Beam Echo Sounding (SBES) data in this study area were used as reference points. Corrections for atmospheric and sun-glint effects are applied prior to the water depth algorithm. The algorithm was tuned and both models were calibrated by performing the necessary algorithm with available single beam echo sounding data in the study area. The coefficients of standard R2 is estimated as 0.846 for log-linear and 0.692 for non-linear model. Log linear model performs better than the non-linear model. The model residuals were mapped and the spatial auto-correlation was calculated based on the bathymetric estimation model. A spatial error model was constructed to generate more reliable estimates of bathymetry by calculating the spatial autocorrelation of model error and integrating this into an improved regression model. Finally, the spatial error model improved the bathymetric estimates of R2 up to 0.854 for log-linear and 0.704 non-linear model respectively. The Root Mean Square Error (RMSE) was calculated for the different depth ranges and also for all reference points. The overall accuracy for the log linear and the non-linear inversion model after the geographical error analysis is estimated as ±1.532 m and ±2.089 m for this study. The spatial error model improved bathymetric estimates than those derived from a conventional log-linear and non-linear technique although these methods perform very similar estimates overall.

Automated Estimation of Coastal Bathymetry from High Resolution Multi-Spectral Satellite Images

Coastal bathymetry is the most essential tool for marine planning, monitoring and management, modelling, nautical navigation and scientific studies of marine environments. The techniques have been developed over the last decade to derive bathymetry using remote sensing technology with efficiently and low costly. Log linear bathymetric inversion model and non-linear bathymetric inversion model provides two empirical approaches for deriving bathymetry from multispectral satellite imagery, which have been refined and widely applied. This paper compares these two approaches by means of a geographical error analysis for the site Kankesanturai using WorldView-2 satellite imagery. In order to calibrate both models; Single Beam Echo Sounding (SBES) data in this study area were used as reference points. Corrections for atmospheric and sun-glint effects are applied prior to the water depth algorithm. The algorithm was tuned and both models were calibrated by performing the necessary algorithm with available single beam echo sounding data in the study area. The coefficients of standard R2 is estimated as 0.846 for log-linear and 0.692 for non-linear model. Log linear model performs better than the non-linear model. The model residuals were mapped and the spatial auto-correlation was calculated based on the bathymetric estimation model. A spatial error model was constructed to generate more reliable estimates of bathymetry by calculating the spatial autocorrelation of model error and integrating this into an improved regression model. Finally, the spatial error model improved the bathymetric estimates of R2 up to 0.854 for log-linear and 0.704 non-linear model respectively. The Root Mean Square Error (RMSE) was calculated for the different depth ranges and also for all reference points. The overall accuracy for the log linear and the non-linear inversion model after the geographical error analysis is estimated as ±1.532 m and ±2.089 m for this study. The spatial error model improved bathymetric estimates than those derived from a conventional log-linear and non-linear technique although these methods perform very similar estimates overall.

Evaluating the methods and influencing factors of satellite-derived estimates of NOX emissions at regional scale: A case study for Yangtze River Delta, China

The top-down estimation of NOX emissions and their influencing factors were evaluated based on the “synthetic” and real satellite observation methods at different spatial scales in eastern China. Using the “synthetic” NO2 vertical column densities (VCD) simulated from a hypothetical “true” emission inventory, the top-down estimates of NOX emissions for the Yangtze River Delta (YRD) region at 9 km resolution and the Southern Jiangsu City Cluster (SJC) at 3 km resolution were obtained using various inverse modeling approaches and the a priori emissions for January and July 2012. The normalized mean biases (NMBs) between the top-down and the hypothetical “true” emissions for all the cases were smaller than 6%, which indicates that both linear and nonlinear approaches could effectively constrain the total amount of emissions, with limited influence from spatial resolution, a priori emissions, and seasons. Larger differences for most cases were found for the normalized mean errors (NMEs), implying that the inverse modeling approach and other influencing factors played a more important role on the spatial distribution of the top-down estimates. Two NO2 VCD products from real satellite observation (Dutch OMI NO2 data product v2 (DOMINO v2) and Peking University OMI NO2 data product v2 (POMINO v2)) were then applied to emissions constraints. The NMEs between the top-down estimates derived from the two products were calculated at 182% and 99% for January and July, respectively, indicating the great importance of satellite observation in constraining emissions. With the nonlinear inverse modeling approach, the top-down estimates of NOX emissions based on POMINO v2 were 25%–60% smaller than the national bottom-up inventory for the four seasons in the YRD, which indicates overestimation by the bottom-up method due to the insufficient consideration of recent air pollution control policy. At the 9 km resolution, the simulated NO2 concentrations with air quality modeling based on the top-down estimates were much closer to available ground observation than the bottom-up ones for all seasons, which suggests improved emissions estimation from the inverse model at regional scales.

Multidisciplinary modeling and simulation framework for launch vehicle system dynamics and control

Future concepts and key technologies for reusable launch vehicles are currently investigated by the DLR project AKIRA, focusing on vertical takeoff and horizontal landing (VTHL), as well as horizontal takeoff and horizontal landing (HTHL) concepts. Dedicated developments of multidisciplinary frameworks for launch vehicle modeling and preliminary design optimization have been presented in the relevant literature. These activities are often performed by several independent and discipline-specific tools; such an approach can only account for limited interactions of the involved disciplines with the overall system dynamics.Therefore, the objective of this paper is to focus on a multidisciplinary launch vehicle dynamics modeling, guidance, and control framework to support reusable launch vehicle design activities at DLR while taking into account the highly interconnected disciplines involved and changing environmental conditions. The modeling framework is based on the object-oriented, multidisciplinary, and equation-based modeling language MODELICA. Dedicated 3-DOF and 6-DOF model implementations, covering the kinematics and dynamics formulation, environmental effects, aerodynamics, and propulsion models are presented.Within this framework, a method to obtain a direct connection between 3-DOF and 6-DOF models is shown. This is done by considering results from the trajectory optimization package ‘trajOpt’ in combination with nonlinear 6-DOF inverse models obtained automatically by MODELICA. Angular rates and resulting moments can be retrieved by this intermediate 6-DOF modeling approach for subsequent controllability studies. We discuss some of these benefits in terms of nonlinear flight control simulations for an HTHL reusable launch vehicle concept.

Decentralised adaptive synchronisation of a class of discrete-time and nonlinearly parametrised coupled multi-agent systems

ABSTRACT In this paper, the adaptive decentralised control problem of the multi-agent system is considered. Each agent is described by a discrete-time nonlinear inverse model with the unknown parameter. The output of each agent has an influence from the history outputs of its neighbour agents. To deal with uncertainty, a projection-type parameter estimation algorithm is adopted to identify parameter. According to the certainty equivalence principle, the local control law of each agent is designed only using the history outputs of its own and neighbour agents. Under the parameter update law and the local dual control, it is shown that the output of each agent tracks the average value of history outputs from its neighbour agents, and the closed-loop system eventually achieves strong synchronisation. Finally, two simulations for an example are provided to illustrated the validity of the results.