Presence Of Modeling Errors Research Articles

Improved Seasonal Precipitation Forecasts for the Asian Monsoon Using 16 Atmosphere–Ocean Coupled Models. Part II: Anomaly

Abstract This is the second part of a paper on the improved seasonal precipitation forecasts for the Asian monsoon using 16 atmosphere–ocean coupled models. This study utilizes a large suite of coupled atmosphere–ocean models; this second part largely addresses the skill of rainfall anomaly forecasts. These include both deterministic and probabilistic skill measures such as the RMS errors, anomaly correlations, equitable threat scores, and the Brier skill score. It was possible to improve the skills of rainfall climatology from the use of a downscaled multimodel superensemble to very high levels, and it is of interest to ask how far this methodology would go toward improving the skills of seasonal rainfall anomaly forecasts. It is possible to go through a sequence of multimodel post processing to improve upon these skills by using a dense rain gauge network over Asia, downscaling forecasts for each member model, and constructing a multimodel superensemble that benefits from the persistence of errors of the member models. This paper addresses the spinup issues of the downscaling and the superensemble results where the number of years of model data needed for training phase, for the downscaling, and for the construction of the superensemble, is addressed. In the context of cross validation, the training phase includes 14 seasons of monsoon data. The forecast phase is only one season; it is this season that was not included in the training phase each time. The relationship between data length and the number of models needed for enhanced skills is another issue that is addressed. Seasonal climate forecasts over the larger monsoon Asia domain and over the regional belts are evaluated. The superensemble forecasts invariably have the highest skill compared to the member models globally and regionally. This is largely due to the presence of large systematic errors in models that carry low seasonal prediction skills. Such models carry persistent signatures of systematic errors, and their errors are recognized by the multimodel superensemble. The probabilistic skills show that the superensemble-based forecasts carry a much higher reliability score compared to the member models. This implies that the superensemble-based forecasts are the most reliable among all the member models. It is possible to examine the performance of models and of the superensemble during periods of heavy monsoon rainfall versus those for deficient monsoon rainfall seasons. One of the conclusions of this study is that given the uncertainties in current modeling for seasonal rainfall forecasts, post processing of multimodel forecasts, using the superensemble methodology, seems to provide the most promising results for the rainfall anomaly forecasts. These results are confirmed by an additional skill metric where the RMS errors and the correlations of forecast skills are evaluated using a normalized precipitation anomaly for the forecasts and the observed estimates.

Periodic control of a reverse osmosis desalination process

This paper addresses the forced periodic operation of a tubular reverse osmosis process for improved performance. The investigation is carried out through simulation of a previously validated model for the RO process. The feed pressure and feed flow rate are transformed into periodic behavior in the form of sinusoidal functions. A nonlinear model predictive control algorithm is utilized to regulate the amplitude and period of the sinusoidal functions that formulate the input signal. The control system managed to generate the cyclic inputs necessary to enhance the closed-loop performance in terms of higher permeate production and lower salt concentration. The proposed control algorithm can attain its objective with and without defining a set point for the controlled outputs. In the latter case, the process is driven to the best achievable performance. Similar successful results in terms of improved production and quality of water were also obtained in the presence of modeling errors and external disturbances.

The filter design from data (FD2) problem: parametric‐statistical approach

SUMMARYA large body of literature exists on the filter design problem, assuming that the system to be filtered is known. However, in most practical situations, the system is not known, but a set of measured data is available. In such situations, a two‐step procedure is typically adopted: a model is identified from this data set, and a filter is designed based on the identified model. In this paper, we consider an alternative approach, which uses the available data not for the identification of a model, but for the direct design of the filter. Such a direct design is investigated within a parametric‐statistical framework for both the cases of linear time‐invariant and nonlinear systems. The noise is assumed to be stochastic, and optimality refers to minimizing the estimation error variance. It is shown that the direct design has superior features with respect to the two‐step design, especially in the presence of modeling errors. Another relevant advantage of the direct design over the two‐step procedure is that minimum variance (Kalman) filters for nonlinear systems are, in general, difficult to derive and/or to implement. On the contrary, the direct approach allows for a very efficient filter design. To demonstrate the effectiveness of the proposed direct design, two examples are presented: the first is related to estimation of the Lorentz chaotic attractor; the second, involving real data, is related to estimation of vehicle yaw rate. Copyright © 2011 John Wiley & Sons, Ltd.

Kalman filtering and smoothing for linear wave equations with model error

Filtering is a widely used methodology for the incorporation of observed data into time-evolving systems. It provides an online approach to state estimation inverse problems when data are acquired sequentially. The Kalman filter plays a central role in many applications because it is exact for linear systems subject to Gaussian noise, and because it forms the basis for many approximate filters which are used in high-dimensional systems. The aim of this paper is to study the effect of model error on the Kalman filter, in the context of linear wave propagation problems. A consistency result is proved when no model error is present, showing recovery of the true signal in the large data limit. This result, however, is not robust: it is also proved that arbitrarily small model error can lead to inconsistent recovery of the signal in the large data limit. If the model error is in the form of a constant shift to the velocity, the filtering and smoothing distributions only recover a partial Fourier expansion, a phenomenon related to aliasing. On the other hand, for a class of wave velocity model errors which are time dependent, it is possible to recover the filtering distribution exactly, but not the smoothing distribution. Numerical results are presented which corroborate the theory, and also propose a computational approach which overcomes the inconsistency in the presence of model error, by relaxing the model.

Adaptive stiffness estimation for compliant robotic manipulation using stochastic disturbance models

To achieve haptic telepresence and proper contact behaviour, the control action of a robotic manipulator must be designed with respect to contact parameters. Unfortunately, it is hard to know these parameters exactly in unknown or partly known environments. In this case, contact instability and poor dynamic accuracy can arise due to the presence of modelling errors in the control design. To overcome these problems, online estimation of the relevant contact parameters can be performed, with corresponding adaptation of control laws. This article presents an algorithm for online stiffness estimation for compliant robotic manipulation based on the extended state-space representation of the system and force signals. No position or velocity measurements are required. The algorithm, supported by theoretical analysis, uses offline data concerning several stiffness mismatch scenarios and, through a least square error analysis, computes an estimate of the stiffness value. Simulation results are presented, with fast and accurate estimation even in the presence of noise, highlighting the merits of the method.

Stable indirect adaptive neural controller for a class of nonlinear system

The paper presents an indirect adaptive neural control scheme for a general high-order nonlinear continuous system. In the proposed scheme a neural controller is constructed based on the single-hidden layer feedforward network (SLFN) for approximating the unknown nonlinearities of dynamic systems. A sliding mode controller is also incorporated to compensate for the modelling errors of SLFN. The parameters of the SLFN are modified using the recently proposed neural algorithm named extreme learning machine (ELM), where the parameters of the hidden nodes are assigned randomly. However different from the original ELM algorithm, the output weights are updated based on the Lyapunov synthesis approach to guarantee the stability of the overall control system, even in the presence of modelling errors which are offset using the sliding mode controller. Finally the proposed adaptive neural controller is applied to control the inverted pendulum system with two different reference trajectories. The simulation results demonstrate that good tracking performance is achieved by the proposed control scheme.

Robust distributed model predictive control: A review and recent developments

AbstractThis study presents a review of distributed model predictive control (DMPC) strategies followed by recent studies conducted by the authors on the robustness of these strategies to model errors and a summary of future challenges in this area. The review identifies three key challenges for the successful application of DMPC: (i) the selection of optimal control structure for DMPC; (ii) the choice of a suitable coordination strategy among the controllers; and (iii) the robustness of DMPC strategies to model errors. Then, the study summarises recent developments related to the robustness of unconstrained and constrained DMPC algorithms. For the unconstrained case, a methodology that is based on the calculation of a performance index is proposed to balance the trade‐off between performance and structure simplicity in the presence of model errors. For the constrained case, a Jacobi iterative‐based method is used to design a robust DMPC algorithm. The proposed techniques are illustrated through case studies involving a high purity binary distillation problem.

EKF-based adaptation of look-up tables with an air mass-flow sensor application

A method for bias compensation and online map adaptation using extended Kalman filters is developed. Key properties of the approach include the methods of handling component aging, varying measurement quality including operating-point-dependent reliability and occasional outliers, and operating-point-dependent model quality. Theoretical results about local and global observability, specifically adapted to the map adaptation problem, are proven. In addition, a method is presented to handle covariance growth of locally unobservable modes, which is inherent in the map adaptation problem. The approach is also applicable to the offline calibration of maps, in which case the only requirement of the data is that the entire operating region of the system is covered, i.e., no special calibration cycles are required. The approach is applied to a truck engine in which an air mass-flow sensor adaptation map is estimated during a European transient cycle. It is demonstrated that the method manages to find a map describing the sensor error in the presence of model errors on a measurement sequence not specifically designed for adaptation. It is also demonstrated that the method integrates well with traditional engineering tools, allowing prior knowledge about specific model errors to be incorporated and handled.

Non-linear robust adaptive excitation controller design in power systems based on a new back-stepping method

This study provides a new method for non-linear robust adaptive excitation controller design for power systems. This method can be developed by using the back-stepping technique, but needs fewer conditions than the conventional methods, which is more desirable in power systems. This method is guaranteed to be robust in the presence of model errors and disturbances, and will adapt to uncertain parameters. Meanwhile, the over-parameterisation problem that usually appears in some adaptive methods is avoided. Simulations on a 4-machine-2-area power system and the East China power system demonstrate the effectiveness of the suggested method.

Regularization techniques for ill-posed inverse problems in data assimilation

Optimal state estimation from given observations of a dynamical system by data assimilation is generally an ill-posed inverse problem. In order to solve the problem, a standard Tikhonov, or L 2, regularization is used, based on certain statistical assumptions on the errors in the data. The regularization term constrains the estimate of the state to remain close to a prior estimate. In the presence of model error, this approach does not capture the initial state of the system accurately, as the initial state estimate is derived by minimizing the average error between the model predictions and the observations over a time window. Here we examine an alternative L 1 regularization technique that has proved valuable in image processing. We show that for examples of flow with sharp fronts and shocks, the L 1 regularization technique performs more accurately than standard L 2 regularization.

On variational data assimilation in continuous time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalization of what is known as weakly constrained four-dimensional variational assimilation (4D-Var) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two-point boundary value problem. For imperfect models, the trade-off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. This trade-off turns out to be trivial if the model is perfect. However, even in this situation, allowing for minute deviations from the perfect model is shown to have positive effects, namely to regularize the problem. The presented formalism is dynamical in character. No statistical assumptions on dynamical or observational noise are imposed. Copyright © 2010 Royal Meteorological Society

Statistical Analysis of the MUSIC Algorithm in the Presence of Modeling Errors, Taking Into Account the Resolution Probability

This paper considers the statistical performance of the MUSIC method under the condition that two closely spaced sources impinging on an array of sensors are effectively resolved, i.e., the spectrum exhibits two peaks in the neighborhood of the true directions-of-arrival (DOA). The MUSIC algorithm is known to have an infinite resolution power in theory. However, in the presence of modeling errors, sources can not be resolved with certainty, even if the array correlation matrix is perfectly known. The focus of this paper is to predict the bias and variance of the DOA estimates taking into account the possible resolution failure of MUSIC. This performance prediction, based on our recent mathematical investigation, is new to the best of our knowledge. A general mathematical framework to derive closed form expressions of the bias and variance versus the model mismatch, conditioned on a general statistical resolution test is proposed. In order to illustrate our mathematical approach, statistical tests with one and two conditions, respectively, are investigated. The accuracy of the performance prediction is illustrated in a simulation study. It is found that the proposed approach outperforms the “classical” technique, which ignores the possible resolution failure of the MUSIC algorithm. Therefore, our results provide better tools for determining the necessary antenna calibration accuracy to achieve some targeted specifications on the estimator performance.

A demonstration of cycled 4D‐Var in the presence of model error

AbstractThe justification for the standard four‐dimensional variational data assimilation (4D‐Var) method used at several major operational centres assumes a perfect forecast model, which is clearly unrealistic. However, the method has been very successful in practice. We investigate the reasons for this using a toy model with fast and slow time‐scales and with non‐random model error. The model error is chosen so that the solution remains predictable on both time‐scales. The fast modes are much less well observed than the slow modes. We show that poorly observed modes can be best forecast by using a regularization matrix in place of the background‐error covariance matrix, and using it to give a much stronger constraint than that implied by the true background error for these modes. The effect is that use can be made of observations over a longer time period. This allows the resulting forecast‐error growth to be reduced to much less than that of random perturbations generated using the analysis‐error covariance matrix and even less than the model error growth given sufficiently accurate observations. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.

Selection of control structure for distributed model predictive control in the presence of model errors

This paper presents a new methodology for selecting control structure in the context of distributed model predictive control. An index was developed to quantify the performance of distributed MPC strategies in the presence of model errors. This index was used for two purposes: to solve the decomposition problem whereby the process is decomposed into parts and to compare distributed MPC strategies with different degrees of coordination. Then, a multi-objective Mixed Integer Nonlinear Programming MINLP formulation is proposed to achieve an optimal trade-off between performance and structure simplicity. Four examples are considered to illustrate the methodology. The simulation results are consistent with the conclusions obtained from the analysis. The proposed methodology can be applied at the design stage to select the best control configuration in the presence of model errors.

Nonlinear Structural Vibration Suppression Using Dynamic Neural Network Observer and Adaptive Fuzzy Sliding Mode Control

Nonlinear vibration control of structures has attracted much attention in recent years. In this paper, the adaptive fuzzy sliding mode (AFSM) control algorithm is adopted to actively control nonlinear structural vibration. Since the AFSM control algorithm needs the full state feedback of the structure, a dynamic neural network (DNN) observer is proposed, considering the nonlinearity of the structure. The neural network weights are adapted on-line, and no off-line learning is required. Furthermore, no exact knowledge of structural nonlinearities is needed. The weight training algorithm is established based on Lyapunov stability theory in the presence of modeling error. A semi-active control algorithm is used for MR damper to track the active control force, which is computed by the AFSM control algorithm based on the DNN observer. Results of the numerical simulation of semi-active control of a 20-story nonlinear building using MR dampers verify the performance of the DNN observer and the effectiveness of the proposed AFSM control.

Broadening the polyethylene molecular weight distribution by controlling the hydrogen concentration and catalyst feed rates

This paper discusses the control of an industrial gas-phase polyethylene reactor to produce a desired molecular weight distribution (MWD) of the polymer. The controller objective is to regulate online the entire molecular weight distribution by either manipulating the hydrogen content inside the reactor or coordinating the feed rates of two different types of catalysts. In this work, the molecular weight distribution is modeled as a function of the reaction kinetics and hydrogen to monomer ratio. Nonlinear model predictive controller (NLMPC) algorithm is used to maintain the desired molecular weight distribution online. The closed-loop simulations indicated the effectiveness of NLMPC to achieve its goal even in the presence of modeling errors. Moreover, the results showed that, altering the hydrogen concentration solely can produce the required polymer quality provided that an efficient mechanism is available to readily alter the hydrogen composition. Alternatively, the desired MWD can also be guaranteed with proper manipulation of the catalyst feed rates while the other process inputs are kept constant.

A unified linear Model Output Statistics scheme for both deterministic and ensemble forecasts

An extension of the classical linear Model Output Statistics (MOS) technique is proposed allowing for the post-processing of ensemble forecasts. In this new approach, the cost function on which the least square parameter estimation is based takes into account the presence of errors in both observations and model observables (referred to as Error-in-Variables MOS, EVMOS), unlike the classical linear MOS cost function whose implicit assumption is the absence of errors in the model observables. It allows for the maintenance of an appropriate variability for the corrected forecasts, even for long lead times and for providing a framework in which both deterministic and probabilistic forecasts can be corrected. The scheme is successfully tested for ensemble correction in the context of an idealized low-order chaotic system, the Lorenz atmospheric model, in the presence of model errors, and compared with a classical technique, known as the non-homogeneous Gaussian regression (NGR) method. The potential use of this approach is also briefly discussed. Copyright © 2009 Royal Meteorological Society

Improvements on Visual Servoing From Spherical Targets Using a Spherical Projection Model

This paper is concerned with improvements to visual feature modeling using a spherical projection model. Three spherical targets are considered: a sphere, a sphere marked with a tangent vector to a point on its surface, and a sphere marked with two points on its surface. A new minimal and decoupled set of visual features is proposed for each target using any central catadioptric camera. Using the proposed set for a sphere, a classical control law is proved to be globally asymptotically stable in the presence of modeling errors and locally asymptotically stable in the presence of calibration errors, considering that perspective and paracatadioptric cameras were used. Simulation and experimental results with perspective, paracatadioptric, and fish-eye cameras validate the proposed theoretical results.

A posteriori parameter choice with an efficient discretization scheme for solving ill-posed problems

Recently, Rajan [M.P. Rajan, An efficient discretization scheme for solving the ill-posed problems, Journal of Mathematical Analysis and Applications 313 (2) (2006) 654–677] considered a discretization scheme for solving ill-posed problems and obtained optimal error estimates under an a priori parameter choice strategy with a particular smoothness assumption on the solution. It is shown that the computational information for solving the system is far less than the traditional projection schemes. Neubauer [A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error, Applied Numerical Mathematics 4 (1988) 507–519] and Engl and Gfrerer [H.W. Engl, H. Gfrerer, A posteriori parameter choice for general regularization methods for solving ill-posed problems, Applied Numerical Mathematics 4 (1988) 395–417] suggested an a posteriori method for choosing the regularization parameter which does not require any knowledge about the exact solution, respectively under finite dimensional as well as general setting. In this paper, we consider an a posteriori parameter choice for finite dimensional approximation which does not require any information about the smoothness of the exact solution and apply to the discretization scheme considered by Rajan. The computational efficiency of the scheme is illustrated through numerical examples.

Optimal Trajectory Regulation for Radar Imaging Guidance

This paper investigates optimal trajectory tracking for air-to-surface missiles using bank-to-turn steering and equipped with Doppler-beam-sharpening seekers. The seeker generates a radar image of the terrain that is continually updated in flight. The guidance problem for this type of sensor is complex, because constraints must be imposed on the flight path of the missile to ensure that the seeker can obtain adequate crossrange resolution for target recognition and aim-point selection. Terminal position, incidence, and angle constraints are also imposed in the guidance problem. An offline optimal trajectory can be obtained through trajectory optimization techniques that have been presented in previous papers. The focus of this work is on following the precomputed optimal trajectories. The method presented uses a linear quadratic regulator to calculate closed-loop in-flight controls. We also investigate the effectiveness of a preview term based on the prediction of the future trajectory from the current state. Preview control is an optimal tracking method that uses future trajectory samples within a preview window to improve the tracking accuracy. A set of preview gains is computed and is used in addition to the regulator gains. Over the last 2 km of flight, an impact-angle guidance law that minimizes the miss distance and impact-angle error is used to achieve the terminal constraints. The guidance scheme is evaluated for tracking offline Doppler-beam-sharpening trajectories in the presence of modeling errors, sensor noise, and disturbances.