Presence Of Modeling Errors Research Articles

Accuracy of Some Approximate Gaussian Filters for the Navier--Stokes Equation in the Presence of Model Error

Bayesian state estimation of a dynamical system from a stream of noisy measurements is important in many geophysical and engineering applications where high dimensionality of the state space, spars...

Multidimensional Taylor network optimal control of SISO nonlinear systems for tracking by output feedback

SummaryIt is difficult or even impossible for the existing nonlinear system control methods to realize the real‐time tracking control of nonlinear systems in the presence of modeling errors by output feedback. For that sake, multidimensional Taylor network (MTN) optimal control method is proposed for the real‐time optimal tracking control of the SISO nonlinear constant system with modeling errors as such. On the basis of the original MTN, the control input item is added to the nonlinear dynamic model, which is then termed as the MTN optimal controller (MTNOC). Its initial parameters are trained by the conjugate gradient method and the minimum principle, according to the calculated optimal input and output of the controlled object in open‐loop status. The propagation learning algorithm is adopted to improve the MTN product term weights and eliminate the modeling errors during the real system adjustment process. Simulation results show that the MTNOC promises a high response rate. Despite the errors in the modeling process, MTNOC manages to stabilize the system for tracking the desired output. The experiment of overlaying an additional signal on the input signal proves MTNOC to be of excellent tracking characteristics, capable of inhibiting the disturbance signal to some extent. In short, by realizing the optimal closed‐loop tracking control of the SISO nonlinear constant system by output feedback, MTNOC guarantees its real‐time control accuracy, dynamic performance, robustness, and anti‐disturbance capability.

Sliding mode control of singularly perturbed systems and its application in quad-rotors

ABSTRACTThis paper presents a sliding mode control scheme for tracking control of nonlinear singularly perturbed systems in the presence of model errors and external disturbances. A dual-loop feedback control is developed to provide accurate tracking capability and sufficient robustness to system uncertainties. A sliding mode controller is proposed in the outer-loop feedback design such that the plant states are stabilised for given reference trajectories, while an additional robust controller is designed in the inner loop to increase the adaptability to uncertainties, and reduce the effect of unmodelled high-frequency dynamics on plant dynamics. An appealing feature of the control scheme is the attenuation of chattering. The effectiveness and merits of the new control scheme developed are shown via a verification example of velocity control of a quad-rotor.

Parsimonious cooperative distributed MPC algorithms for offset-free tracking

We propose in this paper novel cooperative distributed MPC algorithms for tracking of piecewise constant setpoints in linear discrete-time systems. The available literature for cooperative tracking requires that each local controller uses the centralized state dynamics while optimizing over its local input sequence. Furthermore, each local controller must consider a centralized target model. The proposed algorithms instead use a suitably augmented local system, which in general has lower dimension compared to the centralized system. The same parsimonious parameterization is exploited to define a target model in which only a subset of the overall steady-state input is the decision variable. Consequently the optimization problems to be solved by each local controller are made simpler. We also present a distributed offset-free MPC algorithm for tracking in the presence of modeling errors and disturbances, and we illustrate the main features and advantages of the proposed methods by means of a multiple evaporator process case study.

Effects of model error on cardiac electrical wave state reconstruction using data assimilation.

Reentrant electrical scroll waves have been shown to underlie many cardiac arrhythmias, but the inability to observe locations away from the heart surfaces and the restriction of observations to only one or two state variables have made understanding arrhythmia mechanisms challenging. Recently, we showed that data assimilation from spatiotemporally sparse surrogate observations could be used to reconstruct a reliable time series of state estimates of reentrant cardiac electrical waves including unobserved variables in one and three spatial dimensions. However, real cardiac tissue is unlikely to be described accurately by mathematical models because of errors in model formulation and parameterization as well as intrinsic but poorly described spatial heterogeneity of electrophysiological properties in the heart. Here, we extend our previous work to assess how model error affects the accuracy of cardiac state estimates achieved using data assimilation with the Local Ensemble Transform Kalman Filter. We focus on one-dimensional states of discordant alternans characterized by significant wavelength oscillations. We demonstrate that data assimilation can provide high-quality estimates under a wide range of model error conditions, ranging from varying one or more parameter values to using an entirely different model to generate the truth state. We illustrate how multiplicative and additive inflation can be used to reduce error in the state estimates. Even when the truth state contains underlying spatial heterogeneity, we show that using a homogeneous model in the data assimilation algorithm can achieve good results. Overall, we find data assimilation to be a robust approach for reconstructing complex cardiac electrical states corresponding to arrhythmias even in the presence of model error.

PD/PID Controller Design for Seismic Control of High-Rise Buildings Using Multi-Objective Optimization: A Comparative Study with LQR Controller

This paper developed multi-objective optimization design of proportional–derivative (PD) and proportional–integral–derivative (PID) controllers for seismic control of high-rise buildings. The case study is an 11-story realistic building equipped with active tuned mass damper (ATMD). Four earthquakes and nine performance indices are taken into account to assess the performance of the controllers. To create a good trade-off between the performance and robustness of the closed-loop structural system, a non-dominated sorting genetic algorithm, NSGA-II, is employed. To evaluate the degree of robustness of the controllers, four structural models with uncertainties in the nominal model of the structure is considered. For comparison purposes, a linear quadratic regulator (LQR) controller is also designed in the numerical simulations. Simulation results show that the proposed PD and PID controllers significantly perform better than the LQR in reduction of structural responses. Also, it is shown that the LQR does not provide a good performance in strong earthquakes. However, PD and PID controllers are able to significantly reduce structural responses. Moreover, it is shown that the PID has a better performance than the PD. The results also show that the proposed controllers are capable of maintaining a desired performance in the presence of modeling errors. They also have several advantages over modern controllers in terms of simplicity and reduction of required sensors and computational resources in tall buildings.

Structure‐independent motion recovery from a monocular image sequence with low fill fraction

SummaryReal‐world image sequences are usually characterized by frequent loss and replacement of tracked feature data, and the key to accurate motion recovery from such image sequences lies in the synthesis of a structure‐independent dynamic filter that enables the replacement of lost point features with newly acquired points in a seamless manner. In this note, a novel nonlinear observer is proposed that relies on filtered estimates of optical flow to accomplish structure‐independent motion recovery from monocular image sequences with a low fill fraction. With a single component of linear velocity assumed known, the proposed scheme relies on the perspective observation of at least five points to yield exponentially convergent estimates of the unknown motion parameters that converge to a uniform, ultimate bound in the presence of model error. The unknown linear and angular velocities are assumed to be generated using an imperfectly known model that incorporates a bounded uncertainty, and optical flow estimation is accomplished using a robust differentiator that is based on the sliding‐mode technique. Numerical results are used to validate and demonstrate superior observer performance compared to a leading alternative design on a real‐world–like image sequence that is characterized by significant measurement noise and high feature turnover rate.

Bayesian inversion of a CRN depth profile to infer Quaternary erosion of the northwestern Campine Plateau (NE Belgium)

Abstract. The rate at which low-lying sandy areas in temperate regions, such as the Campine Plateau (NE Belgium), have been eroding during the Quaternary is a matter of debate. Current knowledge on the average pace of landscape evolution in the Campine area is largely based on geological inferences and modern analogies. We performed a Bayesian inversion of an in situ-produced 10Be concentration depth profile to infer the average long-term erosion rate together with two other parameters: the surface exposure age and the inherited 10Be concentration. Compared to the latest advances in probabilistic inversion of cosmogenic radionuclide (CRN) data, our approach has the following two innovative components: it (1) uses Markov chain Monte Carlo (MCMC) sampling and (2) accounts (under certain assumptions) for the contribution of model errors to posterior uncertainty. To investigate to what extent our approach differs from the state of the art in practice, a comparison against the Bayesian inversion method implemented in the CRONUScalc program is made. Both approaches identify similar maximum a posteriori (MAP) parameter values, but posterior parameter and predictive uncertainty derived using the method taken in CRONUScalc is moderately underestimated. A simple way for producing more consistent uncertainty estimates with the CRONUScalc-like method in the presence of model errors is therefore suggested. Our inferred erosion rate of 39 ± 8. 9 mm kyr−1 (1σ) is relatively large in comparison with landforms that erode under comparable (paleo-)climates elsewhere in the world. We evaluate this value in the light of the erodibility of the substrate and sudden base level lowering during the Middle Pleistocene. A denser sampling scheme of a two-nuclide concentration depth profile would allow for better inferred erosion rate resolution, and including more uncertain parameters in the MCMC inversion.

Accounting for model error in strong‐constraint 4D‐Var data assimilation

The strong‐constraint formulation of four‐dimensional variational data assimilation (4D‐Var) assumes that the model used in the process perfectly describes the true dynamics of the system. However, this assumption often does not hold and the use of an erroneous model in strong‐constraint 4D‐Var can lead to a sub‐optimal estimation of the initial conditions. We show how the presence of model error can be correctly accounted for in strong constraint 4D‐Var by allowing for errors in both the observations and the model when considering the statistics of the innovation vector. We demonstrate that, when these combined model error and observation‐error statistics are used in place of the standard observation error statistics in the strong‐constraint formulation of 4D‐Var, a statistically more accurate estimate of the initial state is obtained.The calculation of the combined model error and observation‐error statistics requires the specification of model error covariances, which in practice are often unknown. We present a method to estimate the combined statistics from innovation data that does not require explicit specification of the model error covariances. Numerical experiments using the linear advection equation and a simple nonlinear coupled model demonstrate the success of the new methods in reducing the error in the estimate of the initial state, even in the case when only the uncorrelated part of the model error is accounted for.

Applications of Polynomial Chaos Expansions in optimization and control of bioreactors based on dynamic metabolic flux balance models

This work proposes model based approaches for on-line or off-line economic optimization of batch reactors in the presence of model error (uncertainty). Polynomial Chaos Expansions are used as an effective and computationally efficient tool to propagate the error in model parameters into the optimizations’ cost functions. The computational efficiency of the proposed uncertainty propagation approach is essential for the implementation of the on-line approach that takes into account feedback corrections. The role of feedback, as applied in the on-line formulation, is proved to be instrumental for reducing conservatism of the optimization results. The proposed approaches can serve to design recipes for maximizing productivity in batch, fed-batch or perfusion operation of bioreactors.

An Analytically Stable Structure and Motion Observer Based on Monocular Vision

This paper presents a novel nonlinear sliding-mode differentiator-based complete-order observer for structure and motion identification with a calibrated monocular camera. In comparison with earlier work that requires prior knowledge of either the Euclidean geometry of the observed object or the linear acceleration of the camera and is restricted to establishing stability and convergence from image-plane measurements of a single tracked feature, the proposed scheme assumes partial velocity state feedback to asymptotically identify the true-scale Euclidean coordinates of numerous observed object features and the unknown motion parameters. The dynamics of the motion parameters are assumed to be described by a model with unknown parameters that incorporates a bounded uncertainty, and a Lyapunov analysis is provided to prove that the observer yields exponentially convergent estimates that converge to a uniform ultimate bound under a generic persistency of excitation condition. Numerical and experimental results are obtained that demonstrate the robust performance of the current scheme in the presence of model error and measurement noise.

Robust dynamic decoupling control for permanent magnet spherical actuators based on extended state observer

This study presents a robust dynamic decoupling control strategy to solve the trajectory tracking problem for the permanent magnet spherical actuator (PMSA). The dynamic model of PMSA obtained by the Lagrange–Euler formalism is obviously a multi-variable non-linear system with strong cross-couplings. Furthermore, uncertainties such as model errors and external disturbances will also affect the precision of the control system. In the active disturbance rejection control (ADRC) framework, the decoupling problem can be reformulated as disturbance rejection by merging the cross channel interference into the lumped disturbance, which consists of internal dynamics and external disturbances. The lumped disturbance is then estimated using extended state observer (ESO) and canceled out in the control law. Herein, the linear active disturbance rejection control is selected for PMSAs, as the tuning process can be greatly simplified by making all the parameters of ESO or controller a function of bandwidth. Simulations and experiments are presented to corroborate the effectiveness and robustness of the proposed strategy, showing that the proposed control algorithm can decouple and linearise the system in the presence of model errors as well as the load and random disturbances. Meanwhile, the modified system has better static and dynamic performances with strong robustness to uncertainties.

A weak-constraint 4DEnsembleVar. Part II: experiments with larger models

ABSTRACTIn recent years, hybrid data-assimilation methods which avoid computation of tangent linear and adjoint models by using ensemble 4-dimensional cross-time covariances have become a popular topic in Numerical Weather Prediction. 4DEnsembleVar is one such method. In spite of its capabilities, its application can sometimes become problematic due to the not-trivial task of localising cross-time covariances. In this work we propose a formulation that helps to alleviate such issues by exploiting the presence of model error, i.e. a weak-constraint 4DEnsembleVar. We compare the weak-constraint 4DEnsembleVar to that of other data-assimilation methods. This is part II of a two-part paper. In part I, we describe the 4DEnsembleVar framework and problems with localised temporal cross-covariances associated with this method are discussed and illustrated on the Korteweg de Vries model. We also introduce our weak-constraint 4DEnsemble-Var formulation and show how it can alleviate—at least partially—the problem of having low-quality time cross-covariances. The second part of this paper deals with experiments on larger and more complicated models, namely the Lorenz 1996 model and a modified shallow-water model with simulated convection, both of them under the presence of model error. We investigate the performance of weak-constraint 4DEnsembleVar against strong-constraint 4DEnsembleVar (both with and without localisation) and other traditional methods (4DVar and the Local Ensemble Transform Kalman Smoother). Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering observation density (in time and space), observation period, ensemble sizes and assimilation window length. In this part we also explain how to perform outer loops in the EnVar methods. We show that their use can be counter-productive if the presence of model error is ignored by the assimilation method. We show that the addition of a weak-constraint generally improves the RMSE of 4DEnVar in cases where model error has time to develop, especially in cases with long assimilation windows and infrequent observations. We have assumed good knowledge of the statistics of this model error.

Effectiveness of Side Force Models for Flow Simulations Downstream of Vortex Generators

Vortex generators are a widely used means of flow control, and predictions of their influence are vital for efficient designs. However, accurate computational fluid dynamics simulations of their effect on the flowfield by means of a body-fitted mesh are computationally expensive. Therefore, the Bender–Anderson–Yagle and jBAY models, which represent the effect of vortex generators on the flow using source terms in the momentum equations, are popular in industry. In this contribution, the ability of the Bender–Anderson–Yagle and jBAY models to provide accurate flowfield results is examined by looking at boundary-layer properties close behind vortex generators. The results are compared with both body-fitted mesh and other source term model Reynolds-averaged Navier–Stokes simulations of three-dimensional incompressible flows over flat-plate and airfoil geometries. The influence of mesh resolution and the domain of application on the accuracy of the models is shown, and the influence of the source term on the generated flowfield is investigated. The results demonstrate the grid dependence of the models and indicate the presence of model errors. Furthermore, it is found that the total applied force has a larger influence on both the intensity and shape of the created vortex than the distribution of the source term over the cells.

Probabilistic damage identification of a designed 9-story building using modal data in the presence of modeling errors

Validity and accuracy of model based identification techniques such as linear finite element (FE) model updating are sensitive to modeling errors. Models used for the design and performance assessment of civil structures often contain large modeling errors for certain frequency ranges of response. In other words, modeling errors have unequal effects on different vibration modes of structures. Therefore, the performance of FE model updating for damage identification is sensitive to the type and the subset of data used and to the residual weight factors. This study proposes a process to mitigate the effects of modeling errors by selecting the optimal subset of modes and the optimal modal residual weights. Multiple model updating classes are defined based on different subsets of modes and different weight factors. Structural damage is then identified using Bayesian model class selection and model averaging techniques over the results of all the considered model updating classes. In addition, a new likelihood function is defined to allow damage identification without the need for calibrating a reference FE model. Performance of the proposed damage identification process and the new likelihood function is evaluated numerically at multiple levels of modeling errors and structural damage on the SAC 9-story steel moment frame. It is shown that the structural damages can be identified with negligible bias when the proposed likelihood and updating process is implemented.

EnKF and 4D-Var data assimilation with chemical transport model BASCOE (version 05.06)

Abstract. We compare two optimized chemical data assimilation systems, one based on the ensemble Kalman filter (EnKF) and the other based on four-dimensional variational (4D-Var) data assimilation, using a comprehensive stratospheric chemistry transport model (CTM). This work is an extension of the Belgian Assimilation System for Chemical ObsErvations (BASCOE), initially designed to work with a 4D-Var data assimilation. A strict comparison of both methods in the case of chemical tracer transport was done in a previous study and indicated that both methods provide essentially similar results. In the present work, we assimilate observations of ozone, HCl, HNO3, H2O and N2O from EOS Aura-MLS data into the BASCOE CTM with a full description of stratospheric chemistry. Two new issues related to the use of the full chemistry model with EnKF are taken into account. One issue is a large number of error variance parameters that need to be optimized. We estimate an observation error variance parameter as a function of pressure level for each observed species using the Desroziers method. For comparison purposes, we apply the same estimate procedure in the 4D-Var data assimilation, where both scale factors of the background and observation error covariance matrices are estimated using the Desroziers method. However, in EnKF the background error covariance is modelled using the full chemistry model and a model error term which is tuned using an adjustable parameter. We found that it is adequate to have the same value of this parameter based on the chemical tracer formulation that is applied for all observed species. This is an indication that the main source of model error in chemical transport model is due to the transport. The second issue in EnKF with comprehensive atmospheric chemistry models is the noise in the cross-covariance between species that occurs when species are weakly chemically related at the same location. These errors need to be filtered out in addition to a localization based on distance. The performance of two data assimilation methods was assessed through an 8-month long assimilation of limb sounding observations from EOS Aura MLS. This paper discusses the differences in results and their relation to stratospheric chemical processes. Generally speaking, EnKF and 4D-Var provide results of comparable quality but differ substantially in the presence of model error or observation biases. If the erroneous chemical modelling is associated with moderately fast chemical processes, but whose lifetimes are longer than the model time step, then EnKF performs better, while 4D-Var develops spurious increments in the chemically related species. If, however, the observation biases are significant, then 4D-Var is more robust and is able to reject erroneous observations while EnKF does not.

Predictive direct power control for three-phase grid-connected converters with online parameter identification

Summary It is well known that predictive direct power control method can be influenced by the presence of modeling errors. Any small variations in model parameters will degrade the performance and stability of the control system. However, this value is difficult to be measured precisely in practical implementation. This paper proposes a predictive direct power control method for three-phase grid-connected converters with online parameter identification technique. Specifically, the proposed method alleviates the parameter mismatch problem by using the online parameter identification technique based on model reference adaptive system estimation theory. The proposed strategy not only identifies the parameter accurately, but also improves dynamic response without any additional sensors. The excellent steady-state and dynamic performance of the proposed method are confirmed through extensive simulations. Copyright © 2016 John Wiley & Sons, Ltd.

Interval Type-2 Fuzzy Sliding Mode Controller Based on Nonlinear Observer for a 3-DOF Helicopter with Uncertainties

In this paper, a robust controller for a three degree of freedom helicopter control is proposed in presence of modeling error and disturbance. In this frame, interval type-2 fuzzy logic control approach (IT2FLC) and sliding mode control (SMC) techniques are used to design a controller named interval type-2 fuzzy sliding mode controller (IT2FSMC). Elevation, pitch and travel angles are considered measurable, whereas unmeasured states are estimated by using a nonlinear observer. The proposed control scheme can not only attenuate the chattering effect of the SMC, but also it reduces the rules number of the fuzzy controller. As for stability, it is exponentially guaranteed by using the Lyapunov criteria. The simulation results show that the IT2FSMC significantly alleviates the chattering effect and provides a good tracking performance, in the presence of modeling error and disturbance. They also illustrate that the IT2FSMC achieves the best tracking performance in comparison with the type-1 fuzzy logic controller, the IT2FLC and the type-1 fuzzy sliding mode controller.

Analysis of Finite-Control-Set Model Predictive Current Control With Model Parameter Mismatch in a Three-Phase Inverter

It is well known that predictive control methods can be affected by the presence of modeling errors. The extent to which finite-control-set model predictive control (FCS-MPC) is influenced by parametric uncertainties is a recurrent concern at the moment of evaluating the viability of this method for power electronics applications. This paper proposes an analytic approach to examine the influence of model parametric uncertainties on the prediction error of FCS-MPC for current control in a three-phase two-level inverter. The analysis shows that the prediction error is not only determined by parametric mismatch but also by the instantaneous values of load current and inverter output voltage. This implies that within each sampling period of the predictive algorithm several conditions of prediction error are generated, as multiple voltage vectors are evaluated. Simulation and experimental results are provided and discussed showing the effects of inaccuracies in the modeling of load resistance and inductance parameters on the performance of FCS-MPC. Even though steady-state performance is noticeably affected with parameter changes, especially when the load inductance is overestimated by the model, its fast transient step response is less affected by parameter changes.

Rejection of unknown periodic disturbances for continuous‐time MIMO systems with dynamic uncertainties

SummaryRejection of unknown periodic disturbances in multi‐channel systems has several industrial applications that include aerospace, consumer electronics, and many other industries. This paper presents a design and analysis of an output‐feedback robust adaptive controller for multi‐input multi‐output continuous‐time systems in the presence of modeling errors and broadband output noise. The trade‐off between robust stability and performance improvement as well as practical design considerations for performance improvements are presented. It is demonstrated that proper shaping of the open‐loop plant singular values as well as over‐parameterizing the controller parametric model can significantly improve performance. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.