Efficient a-posteriori error estimation for nonlinear kernel-based reduced systems

In this paper, we consider the topic of model reduction for nonlinear dynamical systems based on kernel expansions. Our approach allows for a full offline/online decomposition and efficient online computation of the reduced model. In particular, we derive an a-posteriori state-space error estimator for the reduction error. A key ingredient is a local Lipschitz constant estimation that enables rigorous a-posteriori error estimation. The computation of the error estimator is realized by solving an auxiliary differential equation during online simulations. Estimation iterations can be performed that allow a balancing between estimation sharpness and computation time. Numerical experiments demonstrate the estimation improvement over different estimator versions and the rigor and effectiveness of the error bounds.     

Robust, reduced-order, nonstrictly proper state estimation via the optimal projection equations with Petersen—Hollot bounds

A state-estimation design problem involving parametric plant uncertainties is considered. An error bound suggested by recent work of Petersen and Hollot is utilized for guaranteeing robust estimation. Necessary conditions which generalize the optimal projection equations for reduced-order state estimation are used to characterize the estimator which minimizes the error bound. The design equations thus effectively serve as sufficient conditions for synthesizing robust estimators. An additional feature is the presence of a static estimation gain in conjunction with the dynamic (Kalman) estimator, i. e., a nonstrictly proper estimator.     

Non-linear observer based on the Euler discretization

In this paper we present an estimator based on the Euler discretization. First, we show that the proposed estimator is an exponential observer in the case when the error between trajectories of continuous time and its associated Euler discretized system can be neglected. Otherwise, we show that the error of estimation is exponentially attracted by some neighbourhood of the origin which depends only on the sampling time. Finally, numerical simulations using a bioreactor model are given in order to highlight the performance of the proposed estimator.     

Sliding-mode estimators for a class of non-linear systems affected by bounded disturbances

The problem of state estimation for a class of non-linear systems with Lipschitz non-linearities is addressed using sliding-mode estimators. Stability conditions have been found to guarantee asymptotic convergence to zero of the estimation error in the absence of noise and non-divergence if the state perturbations and measurement noise are bounded. A method is proposed to find a suitable solution to the algebraic Riccati equation on which the design of the estimator is based. The performance of the resulting sliding-mode filter minimizes an upper bound on the asymptotic estimation error. Based on such an approach, a sliding-mode estimator may be designed so as to outperform the extended Kalman filter in practical applications with models affected by uncertainty and strong, possibly unknown non-linearities, as shown by means of simulations.     

Fully reliable error control for first-order evolutionary problems

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of solution approximation for evolutionary models, i.e., a time-marching technique and a space–time approach. The first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space–time approach. It is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method.The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.     

State estimation for discrete-time neural networks with time-varying delay

This article deals with the problem of delay-dependent state estimation for discrete-time neural networks with time-varying delay. Our objective is to design a state estimator for the neuron states through available output measurements such that the error state system is guaranteed to be globally exponentially stable. Based on the linear matrix inequality approach, a delay-dependent condition is developed for the existence of the desired state estimator via a novel Lyapunov functional. The obtained condition has less conservativeness than the existing ones, which is demonstrated by a numerical example.     

Evaluation of numerical error estimation based on grid refinement studies with the method of the manufactured solutions

This article presents a study on the estimation of the numerical uncertainty based on grid refinement studies with the method of manufactured solutions. The availability of an exact solution and the convergence of the numerical solution to machine accuracy allow the determination of the exact error and of the distinct contributions of the iterative and discretization errors. The study focuses on three different problems of error/uncertainty evaluation (the uncertainty is in this case the error multiplied by a safety factor): the estimation of the iterative error/uncertainty; the influence of the iterative error on the estimation of the discretization error/uncertainty, and the overall numerical error/uncertainty as a combination of the iterative and discretization errors. The results obtained in this study show that it is possible to obtain a reliable iterative error estimator based on a geometric-progression extrapolation of the L_∞ norm of the differences between iterations. In order to obtain a negligible influence of the iterative error on the estimation of the discretization error, the iterative error must be two to three orders of magnitude smaller than the discretization error. If the iterative error is non-negligible it should be added, simply arithmetically, to the discretization error to obtain a reliable estimate of the numerical error; combining by RMS is not conservative.     

Robust filtering under stochastic parametric uncertainties

This paper is concerned with a polynomial approach to robust deconvolution filtering of linear discrete-time systems with random modeling uncertainties. The modeling errors appear in the coefficients of the numerators and denominators of both the input signal and system transfer function models in the form of random variables with zero means and known upper bounds of the covariances. The robust filtering problem is to find an estimator that minimizes the maximum mean square estimation error over the random parameter uncertainties and input and measurement noises. The key to our solution is to quantify the effect of the random parameter uncertainties by introducing two fictitious noises for which a simple way is given to calculate their covariances. The optimal robust estimator is then computed by solving one spectral factorization and one polynomial equation as in the standard optimal estimator design using a polynomial approach. An example of signal detection in mobile communication is given to illustrate the effectiveness of our approach.     

自适应SVD–UKF算法及在穿刺状态估计中的应用

柔性针在实际穿刺过程中会产生不规则形变, 导致柔性针模型存在参数不确定性问题, 影响穿刺精度. 本文针对柔性针穿刺过程存在的不确定性问题以及超声成像等设备存在的量测噪声统计特征不准确性问题, 提出了一种带有噪声估计器的自适应奇异值分解无迹卡尔曼滤波算法. 该算法采用自适应因子实时修正动力学模型误差, 通过奇异值分解抑制系统状态协方差矩阵的负定性, 利用Sage-Husa估计器在线估计噪声的统计特性, 减小了系统状态估计误差. 将新算法应用于带有曲率不定性的柔性针穿刺模型进行计算仿真, 仿真结果显示, 新的算法较现有的UKF算法相比, 估计误差减小了0.28 mm(82.7%), 与AUKF算法相比, 估计误差减小0.06 mm(52%). 因此, 新算法可有效改善滤波性能, 提高穿刺状态的估计精度.     

A robust continuous‐time fixed‐lag smoother for nonlinear uncertain systems

This paper presents a scheme for the design of a robust fixed‐lag smoother for a class of nonlinear uncertain systems. The proposed approach combines a nonlinear robust estimator with a stable fixed‐lag smoother, to improve the estimation error covariance. The robust fixed‐lag smoother is based on the use of integral quadratic constraints and minimax linear quadratic regulator estimation and control theory. The state estimator uses a copy of the system nonlinearity in the estimator and combines an approximate model of the delayed states to produce a smoother signal. Also in this work, a characterization of the delay approximation error is presented, and the corresponding integral quadratic constraint is included in the design, which gives a guaranteed bound on the performance cost function. In order to see the effectiveness of the method, it is applied to a quantum optical phase estimation problem. Results show a significant improvement in the error covariance of the estimator when compared with a robust nonlinear filter. Copyright © 2015 John Wiley & Sons, Ltd.     

Moments and root-mean-square error of the Bayesian MMSE estimator of classification error in the Gaussian model

The most important aspect of any classifier is its error rate, because this quantifies its predictive capacity. Thus, the accuracy of error estimation is critical. Error estimation is problematic in small-sample classifier design because the error must be estimated using the same data from which the classifier has been designed. Use of prior knowledge, in the form of a prior distribution on an uncertainty class of feature-label distributions to which the true, but unknown, feature-distribution belongs, can facilitate accurate error estimation (in the mean-square sense) in circumstances where accurate completely model-free error estimation is impossible. This paper provides analytic asymptotically exact finite-sample approximations for various performance metrics of the resulting Bayesian Minimum Mean-Square-Error (MMSE) error estimator in the case of linear discriminant analysis (LDA) in the multivariate Gaussian model. These performance metrics include the first, second, and cross moments of the Bayesian MMSE error estimator with the true error of LDA, and therefore, the root-mean-square (RMS) error of the estimator. We lay down the theoretical groundwork for Kolmogorov double-asymptotics in a Bayesian setting, which enables us to derive asymptotic expressions of the desired performance metrics. From these we produce analytic finite-sample approximations and demonstrate their accuracy via numerical examples. Various examples illustrate the behavior of these approximations and their use in determining the necessary sample size to achieve a desired RMS. The Supplementary Material contains derivations for some equations and added figures.     

A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

Humanoid state estimation using a moving horizon estimator

In this research, a new state estimator based on moving horizon estimation theory is suggested for the humanoid robot state estimation. So far, there are almost no studies on the moving horizon estimator (MHE)-based humanoid state estimator. Instead, a large number of humanoid state estimators based on the Kalman filter (KF) have been proposed. However, such estimators cannot guarantee optimality when the system model is nonlinear or when there is a non-Gaussian modeling error. In addition, with KF, it is difficult to incorporate inequality constraints. Since a humanoid is a complex system, its mathematical model is normally nonlinear, and is limited in its ability to characterize the system accurately. Therefore, KF-based humanoid state estimation has unavoidable limitations. To overcome these limitations, we propose a new approach to humanoid state estimation by using a MHE. It can accommodate not only nonlinear systems and constraints, but also it can partially cope with non-Gaussian modeling error. The proposed estimator framework facilitates the use of a simple model, even in the presence of a large modeling error. In addition, it can estimate the humanoid state more accurately than a KF-based estimator. The performance of the proposed approach was verified experimentally.     

Two approaches to partial-nodes-based state estimation for delayed complex networks with intermittent measurement transmissions

This paper is concerned with the state estimation problem for delayed complex dynamic networks with non-identical local dynamical systems. The state estimation is conducted based on constrained information of the measurement outputs. Specifically, the network outputs are available only from a portion of network nodes, and such outputs are transmitted from the network nodes to the estimator in an intermittent way. By utilizing the Halanay inequality method as well as the average dwell-time approach, two sets of sufficient conditions are established that ensure the error dynamics of the state estimation to converge to zero exponentially, and explicit expressions of the estimator gains are further characterized. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed approaches.     

Kullback–Leibler average,consensus on probability densities,and distributed state estimation with guaranteed stability

This paper addresses distributed state estimation over a sensor network wherein each node–equipped with processing, communication and sensing capabilities–repeatedly fuses local information with information from the neighbors. Estimation is cast in a Bayesian framework and an information-theoretic approach to data fusion is adopted by formulating a consensus problem on the Kullback–Leibler average of the local probability density functions (PDFs) to be fused. Exploiting such a consensus on local posterior PDFs, a novel distributed state estimator is derived. It is shown that, for a linear system, the proposed estimator guarantees stability, i.e. mean-square boundedness of the state estimation error in all network nodes, under the minimal requirements of network connectivity and system observability, and for any number of consensus steps. Finally, simulation experiments demonstrate the validity of the proposed approach.     

Minimax state estimation for linear discrete-time differential-algebraic equations

This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator’s equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example.     

复杂网络的事件驱动故障估计

针对一类非线性耦合的复杂网络系统，提出了一种基于复杂网络估计器的近似最优故障估计方法.首先将复杂网络的状态与故障进行增广，然后对增广后的状态和故障进行了联合状态估计.为了处理多信号传输可能发生的数据冲突，采用了事件驱动的方法使复杂网络的输出传输至远程估计器.通过递推矩阵方程方法给出了估计误差协方差矩阵的上界，并通过设计估计器参数使得该上界在迹的意义下最小.最后，通过仿真例子验证了所提联合估计方案的可行性和有效性.     

A posteriori error estimators for stabilized P1 nonconforming approximation of the Stokes problem

In this paper we derive and analyze some a posteriori error estimators for the stabilized P1 nonconforming approximation of the Stokes problem involving the strain tensor. This will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions. The error estimator for the nonconforming error is obtained in the standard way, and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method. A crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated Neumann data for local Stokes problems. It turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local Stokes problems. We also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error. Numerical results are provided to illustrate the performance of our error estimators.     

On a posteriori error estimates for the linear triangular finite element

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.     

Discretization error estimator for transcient dynamic simulations

In this paper, we demonstrate that the concept of error in constitutive relation provides an answer to the problem of error estimation in transient dynamic analysis.The construction of our error measure is based on a reformulation of the transient dynamic problem. From the solution to the discretized model, we build a set of fields, which satisfy the kinematic constraints, the initial conditions and the equilibrium equation exactly. The quality of this numerical solution depends on the extent to which the constitutive relations are satisfied.Our error estimator can be used with explicit as well as implicit time integration schemes. Here, it is first calculated on a simple single-degree-of-freedom linear dynamic problem. Its satisfactory behavior is demonstrated by different tests. Moreover, it is compared with several other indicators from the literature.Next, we explain how this error measure can be applied to problems involving both time and space. Then, preliminary one-dimensional test results for a bar fixed at one end are presented and discussed.Finally, we introduce a new error indicator which turns out to be an indicator of the error on the time integration for the initial reference problem. This indicator enables us to extract from the global error estimation the main contribution, which is relative to the time integration scheme chosen. Then, this quantity is calculated in order to evaluate the error due to the lumped mass assumption for problems solved by the explicit central difference method.