Tangent Linearization Research Articles

Development of the tangent linear and adjoint models of the global online chemical transport model MPAS-CO2 v7.3

Abstract. We describe the development of the tangent linear (TL) and adjoint models of the Model for Prediction Across Scales (MPAS)-CO2 transport model, which is a global online chemical transport model developed upon the non-hydrostatic Model for Prediction Across Scales – Atmosphere (MPAS-A). The primary goal is to make the model system a valuable research tool for investigating atmospheric carbon transport and inverse modeling. First, we develop the TL code, encompassing all CO2 transport processes within the MPAS-CO2 forward model. Then, we construct the adjoint model using a combined strategy involving re-calculation and storage of the essential meteorological variables needed for CO2 transport. This strategy allows the adjoint model to undertake a long-period integration with moderate memory demands. To ensure accuracy, the TL and adjoint models undergo vigorous verifications through a series of standard tests. The adjoint model, through backward-in-time integration, calculates the sensitivity of atmospheric CO2 observations to surface CO2 fluxes and the initial atmospheric CO2 mixing ratio. To demonstrate the utility of the newly developed adjoint model, we conduct simulations for two types of atmospheric CO2 observations, namely the tower-based in situ CO2 mixing ratio and satellite-derived column-averaged CO2 mixing ratio (XCO2). A comparison between the sensitivity to surface flux calculated by the MPAS-CO2 adjoint model with its counterpart from CarbonTracker–Lagrange (CT-L) reveals a spatial agreement but notable magnitude differences. These differences, particularly evident for XCO2, might be attributed to the two model systems' differences in the simulation configuration, spatial resolution, and treatment of vertical mixing processes. Moreover, this comparison highlights the substantial loss of information in the atmospheric CO2 observations due to CT-L's spatial domain limitation. Furthermore, the adjoint sensitivity analysis demonstrates that the sensitivities to both surface flux and initial CO2 conditions spread out throughout the entire Northern Hemisphere within a month. MPAS-CO2 forward, TL, and adjoint models stand out for their calculation efficiency and variable-resolution capability, making them competitive in computational cost. In conclusion, the successful development of the MPAS-CO2 TL and adjoint models, and their integration into the MPAS-CO2 system, establish the possibility of using MPAS's unique features in atmospheric CO2 transport sensitivity studies and in inverse modeling with advanced methods such as variational data assimilation.

Flatness‐based sliding mode control for stabilizing a spherical pendulum on a quadrotor

AbstractThis paper presents the problem of transporting a suspended load by a quadrotor. A full model considering the quadrotor and the dynamics of the suspended load, in a three‐dimensional space, is proposed considering as control inputs the torques and the thrust force due to the motors. The solution proposed consists of simple control strategies based on the tangent linearization of the model, which is controllable and therefore flat. A sliding mode control technique is developed for the thrust force and torques associated with the Euler angles in order to track a desired trajectory of the vehicle in a three‐dimensional space with minimum oscillation of the suspended load. The control strategy results to be relatively simple to implement and achieves local stability of the tracking errors, as well as robustness to internal nonlinearities, which are neglected in the linearization process and other external disturbances. The attractiveness of the sliding surfaces and the stability of the tracking errors are formally studied using Lyapunov stability theory. Simulation results are given to show the performance of the proposed control strategy.

Effective time-dependent behavior of three-phase polymer matrix smart composites

This study presents a mathematical framework to predict the effective time-dependent behavior of three-phase smart composites with typical 0-3, 1-3, and 2-2 connectivities. A composite is composed of magnetostrictive and piezoelectric reinforcements that both show nonlinear multiphysics coupling and a polymer matrix that exhibits viscoelastic behavior. For dealing with nonlinear and time-dependent problems, a tangent linearization is employed to linearize the nonlinear constituents and a time-integration algorithm is applied to numerically determine the viscoelastic response of the matrix. A unified constitutive equation can be subsequently formulated to cover various phase constitutive laws following by a simplified unit-cell micromechanics model to set up a composite constitutive relation. The presented formulation is validated by limited experimental data available in literatures. Numerical results show that inclusion of a polymer matrix in a smart composite causes magnetoelectric coupling to be creep due to strain-mediated coupling among these three phases. The established unit-cell micromechanics model can be further integrated into a finite element framework to analyze composite structures, which is a great merit in a variety of practical applications. It will do so by implementing the presented micromechanics method at every integration point.

A three-dimensional variational data assimilation system for aerosol optical properties based on WRF-Chem v4.0: design, development, and application of assimilating Himawari-8 aerosol observations

Abstract. This paper presents a three-dimensional variational (3DVAR) data assimilation (DA) system for aerosol optical properties, including aerosol optical thickness (AOT) retrievals and lidar-based aerosol profiles, developed for the Model for Simulating Aerosol Interactions and Chemistry (MOSAIC) within the Weather Research and Forecasting model coupled to Chemistry (WRF-Chem) model. For computational efficiency, 32 model variables in the MOSAIC_4bin scheme are lumped into 20 aerosol state variables that are representative of mass concentrations in the DA system. To directly assimilate aerosol optical properties, an observation operator based on the Mie scattering theory was employed, which was obtained by simplifying the optical module in WRF-Chem. The tangent linear (TL) and adjoint (AD) operators were then established and passed the TL/AD sensitivity test. The Himawari-8 derived AOT data were assimilated to validate the system and investigate the effects of assimilation on both AOT and PM2.5 simulations. Two comparative experiments were performed with a cycle of 24 h from 23 to 29 November 2018, during which a heavy air pollution event occurred in northern China. The DA performances of the model simulation were evaluated against independent aerosol observations, including the Aerosol Robotic Network (AERONET) AOT and surface PM2.5 measurements. The results show that Himawari-8 AOT assimilation can significantly improve model AOT analyses and forecasts. Generally, the control experiments without assimilation seriously underestimated AOTs compared with observed values and were therefore unable to describe real aerosol pollution. The analysis fields closer to observations improved AOT simulations, indicating that the system successfully assimilated AOT observations into the model. In terms of statistical metrics, assimilating Himawari-8 AOTs only limitedly improved PM2.5 analyses in the inner simulation domain (D02); however, the positive effect can last for over 24 h. Assimilation effectively enlarged the underestimated PM2.5 concentrations to be closer to the real distribution in northern China, which is of great value for studying heavy air pollution events.

Flatness-based Heavy Rope Control: Experimental results with a 6R Industrial Manipulator

Flatness-based control schemes for a heavy rope with freely movable suspension point are compared in simulation and experiment. These schemes are based on a nonlinear distributed parameter model (DPM) of the chain and two different approximations, firstly a rigid body approximation as a physical pendulum, secondly a tangent linearization of the original model. The flat output is given by the free end of the rope for the DPMs and by the center of oscillation for the rigid body approximation. The design is achieved by prescribing a desired path for the flat output and computing the corresponding path of the suspension point. Numerical simulations are performed by means of a finite-difference approximation of the original non-linear DPM. For the experimental studies the chain is suspended from the end effector of a 6R industrial manipulator. The required joint angles of the manipulator are computed by selecting one of the eight solutions of the inverse kinematics of the robot.

Control of single input Hamiltonian systems based on the flatness of their tangent linearization

We explore the incremental flatness based control of single input under-actuated nonlinear Hamiltonian systems exhibiting a controllable tangent linearization model around a given equilibrium point. General properties of controllable linearized Hamiltonian systems are presented, which significantly ease the stabilizing, or output reference trajectory tracking, feedback controller design for the nonlinear system. Controllability of the tangent linear system is equivalent to its flatness. A flat filter controller and a set of nested controllers are presented which are based on incremental position measurements alone. There will be no need for explicit estimations of the conjugate momenta through observers. The proposed controller is constituted by a set of nested, second order linear Flat Filter compensation networks acting on the second order pure integration models, naturally present in the input-to-flat output system structure. For pure integration perturbed systems, Flat Filters have been shown to be equivalent, in general, to reduced order extended state observer based Active Disturbance Rejection controllers. A challenging stabilization problem for an unstable, nonlinear, two spring–mass system carrying an inverted pendulum is considered. Computer simulations illustrate the effectiveness of the proposed controller design in the presence of flat output measurement noise.

Analysis of nonlinear piezomagnetism for magnetostrictive terfenol-D composites

This study presents two micromechanics formulations to investigate the nonlinear piezomagnetism for magnetostrictive Terfenol-D composites subjected to large magnetic driving fields. A simplified unit-cell micromechanics method that hypothesizes the composite microstructure forming a periodic array is developed, and the well-known Mori-Tanaka micromechanics model is reformulated for nonlinear composite problems. This work elucidates the nature of the approximation in applying the unit-cell and Mori-Tanaka micromechanics models to magnetostrictive composites insofar as the concentration tensors are concerned. A first moment secant linearization but conventional tangent linearization is employed for nonlinear magneto-elastic constitutive laws in order to obtain a unified and linearized constitutive equation. Later, a homogenization process on the nonlinear responses of magnetostrictive composites is performed via numerical iterations in order to minimize errors from the linearization process. Different composites comprising of unidirectional Terfenol-D fiber, particle, and lamina reinforcements are studied, respectively. For nonlinear piezomagnetic strain responses simulated in this work, the predictions given by the unit-cell and Mori–Tanaka schemes are in good agreement with the experimental data available in the literature. Numerical results are also presented to illustrate the performance of each micromechanics model for common composite connectivity, i.e., 1–3, 0–3, and 2–2 types. Depending on Terfenol-D geometry, volume fraction, and applied magnetic field, the magnetostrictive composites could exhibit various degrees of nonlinear piezomagnetism. The presented micromechanics models can further be served as composite constitutive equations to analyze smart composite structures such as beams commonly utilized in sensor and actuator applications.

Mean field interaction model accounting for the spatial distribution of inclusions in elastic-viscoplastic composites

A cluster interaction model has been proposed to account for the spatial distribution and morphology of particles when estimating the effective properties of elastic and thermoelastic composites (Molinari and El Mouden, 1996). In the present paper this approach is extended to elastic-viscoplastic composites. To this end the tangent linearization of the non-linear viscoplastic law and the concept of additive interaction equation are used. Although the extension is formulated for the non-linear case, first applications are considered for linear viscoelastic composites, a situation rich enough to evaluate the interest of the cluster interaction approach. Results of the model are compared to numerical homogenization for periodic unit cells with two cubic configurations.

Active disturbance rejection control of nonlinear SISO Lagrangian systems via endogenous injections and exogenous feedback for trajectory tracking

This article deals with a linear classical approach for the robust output reference trajectory tracking control of nonlinear SISO Lagrangian systems with a controllable (flat) tangent linearization around an operating equilibrium point. An endogenous injections and exogenous feedback (EIEF) approach is proposed, which is naturally equivalent to the generalized proportional integral control method and to a robust classical compensation network. It is shown that the EIEF controller is also equivalent, within a frequency domain setting demanding respect for the separation principle, to the reduced order observer based active disturbance rejection control approach. The proposed linear control approach is robust with respect to total disturbances and, thus, it is effective for the linear control of the nonlinear Lagrangian system. An illustrative nonlinear rotary crane Lagrangian system example, which is non-feedback linearizable, is presented along with digital computer simulations.

Active Disturbance Rejection Control for Reference Trajectory Tracking Tasks in the Pendubot System

In this article, an Extended State Observer (ESO) based Active Disturbance Rejection Control (ADRC) scheme is applied to the Pendubot system for a trajectory tracking tasks. The tangent linearization of the system allows to implement the control scheme taking advantage of the differential flatness property, while including a cascade configuration. The proposed method assumes a limited knowledge of the underactuated system where the control input gain and the order of the system are the only needed data. The scheme is experimentally tested leading to accurate tracking results.

On the behavior of ocean analysis and forecast error covariance in the presence of baroclinic instability

The properties of the expected analysis and forecast error covariance matrices are explored using a novel method based on the tangent linearization and adjoint of a 4-dimensional variational (4D-Var) data assimilation system. The method is applied to the mesoscale circulation that develops in the presence of a baroclinically unstable mid-latitude ocean temperature front using a series of paternal twin experiments that employ both strong and weak constraint 4D-Var. Adopting the traditional view of Empirical Orthogonal Functions (EOFs) of a covariance matrix as the semi-major axes of a multi-dimensional hyper-ellipsoid, variations in the volume of the analysis and forecast error hyper-ellipsoids are explored which provides information about the flow of probability through state–space. The complementary variations in the expected total variance of the covariance matrix are also investigated. Two different kinds of behavior are identified that are associated with either the demise or growth of baroclinic instabilities. In both cases, the volume of the hyper-ellipsoid decreases during the 4D-Var analysis cycle. During the subsequent forecasts, the volume of the forecast error hyper-ellipsoid initially continues to collapse under both scenarios. During this time, the hyper-ellipsoid becomes increasingly elongated along some of the semi-major axes as forecast errors grow in preferential directions. Growth in these directions is controlled by the most unstable error modes, and projection of forecast error on to the precursors of these modes has been shown previously to be characterized by upscale energy transfer and non-normal processes. For the case of the growing wave, the forecast error hyper-ellipsoid continues to collapse through to the end of the forecast period. However, for the decaying wave, the hyper-ellipsoid may undergo expansion at longer forecast lead times.

Neural Network Model To Predict Slug Frequency of Low-Viscosity Two-Phase Flow

Summary In this work, the artificial neural network (ANN) is implemented for prediction slug frequency FS for low viscosity μL flow (μL ≤ 1.6 mPa·s) in vertical, horizontal, and inclined pipes. To the authors' knowledge, no ANN model in the literature exists for predicting FS. The input parameters for the suggested ANN are superficial liquid velocity VSL, pipe diameter D, superficial gas velocity VSG, and pipe inclination ϕ. Measured data (450 data points) are gathered from five different resources for developing the ANN model. The ranges of FS, VSL, D, VSG, and ϕ covered by the data set, are (0.03 to 3.167 1/s), (0.05 to 2.073 m/s), (0.01905 to 0.0779 m), (0.133 to 11.84 m/s), and (0 to 90°), respectively. The most popular transfer functions of tangent sigmoid and linear are used in the hidden and output layers, respectively, whereas the Levenberg and Marquardt back propagation algorithm is conducted to train ANN. The experimental data set is divided into 70% for training, 15% for validation, and 15% for testing processes. Due to the absence of a systematic way to find the optimal structure of ANN, an exhaustive search method has been suggested and implemented to find the optimal topology, which is (4-16-1); four neurons in input layer, 16 neurons in the hidden layer, and one neuron in output layer. The proposed ANN predicts correctly the effect of each of the previously mentioned parameters on FS, and it yields a satisfactory prediction and clearly outperforms all the existing models, with a mean square error (MSE) and R2 of 0.0097 and 0.977, respectively.

Cascade Delayed Controller Design for a Class of Underactuated Systems

In this paper, a delayed control strategy for a class of nonlinear underactuated fourth-order systems is developed. The proposal is based on the implementation of the tangent linearization technique, differential flatness, and a study of the σ-stabilization of the characteristic equation of the closed-loop system. The tangent linearization technique allows obtaining a local controllability property for the analyzed class of systems. Also, it can reduce the complexity of the global control design, through the use of a cascade connection of two second-order controllers instead of designing a global controller of the fourth-order system. The stabilizing behavior of the delayed controller design is supported by the σ-stability criterion, which provides the controller parameter selection to reach the maximum exponential decay rate on the system response. To illustrate the efficiency of the theoretical results, the proposal is experimentally assessed in two cases of study: a flexible joint system and a pendubot.

A new nonlinear companding algorithm based on tangent linearization processing for PAPR reduction in OFDM systems

In this paper, a novel nonlinear companding transform (NCT) is proposed to reduce the Peak-to-Average Power Ratio (PAPR) of orthogonal frequency division multiplexing (OFDM) signals. The companding function is designed based on continuously differentiable reshaping of the probability density function (PDF) of signal amplitudes. The original PDF is cut off for PAPR reduction, and lower and medium segments of original PDF are scaled and linearized respectively, for maintaining power and cumulative distribution constraints. The linearized segment is set to be the tangent of the scaled version at the inflexion point, so as to reduce the out-of-band (OOB) radiation as much as possible. Parameters of the proposed scheme are solved under joint constraints of constant power and unity cumulative distribution. A new receiving method is also proposed to improve the bit error rate (BER) performance of OFDM systems. Simulation results indicate the proposed scheme can achieve better OOB radiation and BER performance at same PAPR levels, compared with existing similar companding algorithms.

Flatness based ADRC Control of Lagrangian Systems: A moving crane.

A procedure is described for direct tangent linearization, around a given equilibrium point, of non-linear multivariable Lagrangian systems, in terms of second order variational expansions of the Lagrangian function. When the linearized model is controllable (i.e., it exhibits the flatness property), we present an Active Disturbance Rejection Control (ADRC) scheme, valid for stabilization and flat output reference trajectory tracking tasks designed on the basis of the incremental system. The linear approach requires only generalized incremental position measurements, with no explicit need for incremental velocity observers. The ADRC controller is cast in terms of equivalent classical linear compensation networks. A moving crane example is presented which illustrates, through digital computer simulations, the effectiveness of the proposed control scheme.

Effective behavior of composites with combined kinematic and isotropic hardening based on additive tangent Mori–Tanaka scheme

The goal of the present work is to propose a multi-scale approach for composite materials which accounts for kinematic hardening in the phases. For that purpose, the additive/sequential interaction rule and tangent linearization of viscoplastic response proposed for elastic–viscoplastic material can be extended in a straightforward manner. A two phase composite where each phase is elastic–viscoplastic is considered. The viscoplastic flow is governed by a J2 flow theory with an overstress. To find the overall behavior of the composite, a Mori–Tanaka model is applied. Numerical validation of the proposition is carried out by considering a representative volume element with 30 inclusions. Various configurations have been tested: hard or soft inclusion cases with or without isotropic hardening. It is shown that the quality of the model predictions is not affected by the introduction of the kinematic hardening component in the local constitutive behavior. Namely, in most cases considered in the paper the overall stress–strain response as well as the average stress–strain response per phase is accurately estimated. It has been also verified that the obtained backstress components are consistent with the ones predicted by Finite element calculations with ABAQUS Software.

Active Disturbance Rejection Control of the Inertia Wheel Pendulum through a Tangent Linearization Approach

A flatness based approach is proposed for the linear Active Disturbance Rejection Control (ADRC) stabilization of a nonlinear inertia wheel pendulum (IWP) around its unstable equilibrium point, subject to unmodelled dynamics and disturbances. The approach exploits the cascade structure, provided by the flatness property, of the tangent linearization of the underactuated system which allows designing a high gain linear cascaded Extended State Observer (ESO) of the Generalized Proportional Integral (GPI) type. This class of linear observers is employed to build an Active Disturbance Rejection Control controller with a lower order of complexity regarding other ADRC classic schemes. Experimental results demonstrate the effectiveness and feasibility of the proposed approach, as well as a better behavior with respect to a classic control technique in the presence of disturbances.

Adjoint-Based Observation Impact Estimation with Direct Verification Using Forward Calculation

Abstract The adjoint-based observation impact estimation method has been providing essential information to improve data assimilation systems (DASs) in numerical weather prediction (NWP). This paper has two purposes. The first is to verify the four approximations used in the method: iterative construction of the Kalman gain adjoint operator, the tangent linear (TL) approximation of a forecast model, ignoring cross terms of observation impacts, and approximations for incremental DASs. The second is to add new information to our knowledge of observation impacts. For the verification of the adjoint-based method, we use the TL-based observation impact estimation method that can calculate the same quantity as the adjoint-based method without adjoint calculations. Results of these verifications and observation impact estimations performed on the global NWP system of the Japan Meteorological Agency are as follows. First, observation impacts calculated using the adjoint-based method agree well with those from the TL-based method, with the correlation coefficient between them exceeding 0.97. Second, estimated observation impacts are consistent with previous studies in many aspects. There are also system-dependent properties, such as relatively small impacts from GPS radio occultation data above 13 km. Furthermore, new aspects of observation impacts are found: 1) the probability density function of the observation impact agrees well with the scalar theory when giving the experimental value of the observation and the forecast error standard deviations; 2) later observations in the data assimilation window have larger positive impacts; and 3) impacts of AMSU-A in the Southern Hemisphere are more than double those in the Northern Hemisphere.

An update on the RTTOV fast radiative transfer model (currently at version 12)

Abstract. This paper gives an update of the RTTOV (Radiative Transfer for TOVS) fast radiative transfer model, which is widely used in the satellite retrieval and data assimilation communities. RTTOV is a fast radiative transfer model for simulating top-of-atmosphere radiances from passive visible, infrared and microwave downward-viewing satellite radiometers. In addition to the forward model, it also optionally computes the tangent linear, adjoint and Jacobian matrix providing changes in radiances for profile variable perturbations assuming a linear relationship about a given atmospheric state. This makes it a useful tool for developing physical retrievals from satellite radiances, for direct radiance assimilation in NWP models, for simulating future instruments, and for training or teaching with a graphical user interface. An overview of the RTTOV model is given, highlighting the updates and increased capability of the latest versions, and it gives some examples of its current performance when compared with more accurate line-by-line radiative transfer models and a few selected observations. The improvement over the original version of the model released in 1999 is demonstrated.

Elastic-plastic properties of metal matrix composites: Validation of mean-field approaches

Abstract Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical mean-field estimates of elastic moduli for selected metal matrix composite systems are compared with the results of finite element calculations performed for two simplified unit cells: spherical and cylindrical. Advantages and deficiencies of such numerical verification of analytical homogenization schemes are indicated. Next, predictions of both approaches are compared with available experimental data for two composite systems for tension and compression tests in the elastic-plastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the Mori-Tanaka averaging scheme and the generalized self-consistent scheme lead to reliable predictions when combined with the tangent linearization, while the use of secant moduli results in a too stiff response. It is also found that the mean-field predictions for a small ceramic volume content are very close to the results obtained from the finite-element analysis of a spherical unit cell.