Strict anisotropic norm bounded real lemma in terms of matrix inequalities

This paper is aimed at extending the H _∞ Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in the terms of information theory using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H _∞ norm. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. These convex constraints can be approximated by two linear matrix inequalities.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems

This paper presents a robust a posteriori residual error estimator for diffusion-convection-reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in R^d, d=2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results.     

线性模型中的一类随机约束$s$--$K$估计

In this paper, we propose a stochastic restricted s–K estimator in the linear model with additional stochastic linear restrictions by combining the ordinary mixed estimator(OME) with the s–K estimator. It is shown that the proposed estimator is superior to the OME and the s–K estimator under the mean squared error matrix criterion under some conditions. Finally, a numerical example and a Monte Carlo simulation study are given to verify the theoretical results.     

On designing H∞ fault estimator for switched nonlinear systems of neutral type

This paper deals with the problem of fault estimation for a class of switched nonlinear systems of neutral type. The nonlinearities are assumed to satisfy global Lipschitz conditions and appear in both the state and measured output equations. By employing a switched observer-based fault estimator, the problem is formulated as an H_∞ filtering problem. Sufficient delay-dependent existence conditions of the H_∞ fault estimator (H_∞-FE) are given in terms of certain matrix inequalities based on the average dwell time approach. In addition, by using cone complementarity algorithm, the solutions to the observer gain matrices are obtained by solving a set of linear matrix inequalities (LMIs). A numerical example is provided to demonstrate the effectiveness of the proposed approach.     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Information-Based Complexity and Nonparametric Worst-Case System Identification

In this paper we review recent results on nonparametric approaches to identification of linear dynamic systems, under nonprobabilistic assumptions on measurement uncertainties. Two main categories of problems are considered in the paper: H_∞ and l₁ settings. The H_∞ setting assumes that the true system is linear time-invariant and the available information is represented by samples of the frequency response of the system, corrupted by an l_∞-norm bounded noise. The aim is to estimate a proper, stable finite-dimensional model. The estimation error is quantified according to an H_∞ norm, measuring the "distance" of the estimated model from the worst-case system in the class of allowable systems, for the worst-case realization of the measurement error. In the l₁ setting, the aim is to identify the samples of the impulse response of an unknown linear time-invariant system. The available information is given by input/output measurements corrupted by l_∞-bounded noise and the estimation error is measured according to an l₁ norm, for the worst case with respect to allowable systems and noise. In this paper, the main results available in the literature for both settings are reviewed, with particular attention to (a) evaluation of the diameter of information under various experimental conditions, (b) convergence to zero of the diameter of information (i.e., existence of robustly convergent identification procedures), and (c) computation of optimal and almost-optimal algorithms. Some results are also reported for the l_∞ setting, similar to the l₁ setting, with the exception of the estimation error, which is measured by an l_∞ norm.     

Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems

This paper studies the robust and resilient finite-time H_∞ control problem for uncertain discrete-time nonlinear systems with Markovian jump parameters. With the help of linear matrix inequalities and stochastic analysis techniques, the criteria concerning stochastic finite-time boundedness and stochastic H_∞ finite-time boundedness are initially established for the nonlinear stochastic model. We then turn to stochastic finite-time controller analysis and design to guarantee that the stochastic model is stochastically H_∞ finite-time bounded by employing matrix decomposition method. Applying resilient control schemes, the resilient and robust finite-time controllers are further designed to ensure stochastic H_∞ finite-time boundedness of the derived stochastic nonlinear systems. Moreover, the results concerning stochastic finite-time stability and stochastic finite-time boundedness are addressed. All derived criteria are expressed in terms of linear matrix inequalities, which can be solved by utilizing the available convex optimal method. Finally, the validity of obtained methods is illustrated by numerical examples.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

The general linear model and the generalized singular value decomposition

A new derivation is given for the generalized singular value decomposition of two matrices X and F having the same number of rows. It is shown how this decomposition reveals the structure of the general Gauss-Markov linear model (y, Xβ, σ²FF′), and exhibits the structure and solution of the generalized linear least squares problem used to provide the best linear unbiased estimator for the model. The decomposition is used to prove optimality of the estimator and to reveal the structure of the covariance matrix of the error of the estimator.     

Robust Mean-Squared Error Estimation of Multiple Signals in Linear Systems Affected by Model and Noise Uncertainties

This paper is a continuation of the work in [11] and [2] on the problem of estimating by a linear estimator, N unobservable input vectors, undergoing the same linear transformation, from noise-corrupted observable output vectors. Whereas in the aforementioned papers, only the matrix representing the linear transformation was assumed uncertain, here we are concerned with the case in which the second order statistics of the noise vectors (i.e., their covariance matrices) are also subjected to uncertainty. We seek a robust mean-squared error estimator immuned against both sources of uncertainty. We show that the optimal robust mean-squared error estimator has a special form represented by an elementary block circulant matrix, and moreover when the uncertainty sets are ellipsoidal-like, the problem of finding the optimal estimator matrix can be reduced to solving an explicit semidefinite programming problem, whose size is independent of N. The research was partially supported by BSF grant #2002038     

Robust finite-time H∞ control of singular stochastic systems via static output feedback

This paper addresses the problem of robust finite-time stabilization of singular stochastic systems via static output feedback. Firstly, sufficient conditions of singular stochastic finite-time boundedness on static output feedback are obtained for the family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Then the results are extended to singular stochastic H_∞ finite-time boundedness for the class of singular stochastic systems. Designed algorithm for static output feedback controller is provided to guarantee that the underlying closed-loop singular stochastic system is singular stochastic H_∞ finite-time boundedness in terms of strict linear matrix equalities with a fixed parameter. Finally, an illustrative example is presented to show the validity of the developed methodology.     

变系数部分线性模型的加权随机约束s-K估计

研究随机约束条件下半参数变系数部分线性模型的参数估计问题,当回归模型线性部分变量存在多重共线性时,基于Profile最小二乘方法、s-K估计和加权混合估计构造参数向量的加权随机约束s-K估计,并在均方误差矩阵准则下给出新估计量优于s-K估计和加权混合估计的充要条件,最后通过蒙特卡洛数值模拟验证所提出估计量的有限样本性质.     

State estimation for Markovian jumping genetic regulatory networks with random delays

In this paper, the state estimation problem is investigated for stochastic genetic regulatory networks (GRNs) with random delays and Markovian jumping parameters. The delay considered is assumed to be satisfying a certain stochastic characteristic. Meantime, the delays of GRNs are described by a binary switching sequence satisfying a conditional probability distribution. The aim of this paper is to design a state estimator to estimate the true states of the considered GRNs through the available output measurements. By using Lyapunov functional and some stochastic analysis techniques, the stability criteria of the estimation error systems are obtained in the form of linear matrix inequalities under which the estimation error dynamics is globally asymptotically stable. Then, the explicit expression of the desired estimator is shown. Finally, a numerical example is presented to show the effectiveness of the proposed results.     

Multiplicative stochastic systems: Optimization and analysis

We consider the H ₂/H _∞-optimal control problem for a dynamical system defined by a linear stochastic Itô equation whose drift and diffusion coefficients linearly depend on the state vector, the control signal, and the external disturbance. The optimization is carried out under the a priori requirement of maximum possible damping of the harmful influence of external disturbances on the system operation. We present theorems on the solvability of matrix Riccati differential equations to which the original optimization problem is reduced.     

On the optimal design of the output transformation for discrete-time linear systems

The optimization of the output matrix for a discrete-time, single-output, linear stochastic system is approached from two different points of view. Firstly, we investigate the problem of minimizing the steady-state filter error variance with respect to a time-invariant output matrix subject to a norm constraint. Secondly, we propose a filter algorithm in which the output matrix at timek is chosen so as to maximize the difference at timek+1 between the variance of the prediction error and that of the a posteriori error. For this filter, boundedness of the covariance and asymptotic stability are investigated. Several numerical experiments are reported: they give information about the limiting behavior of the sequence of output matrices generated by the algorithm and the corresponding error covariance. They also enable us to make a comparison with the results obtained by solving the former problem.This work was supported by the Italian Ministry of Education (MPI 40%), Rome, Italy.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Gain-scheduled fault detection on stochastic nonlinear systems with partially known transition jump rates

In this paper, the problem of continuous gain-scheduled fault detection (FD) is studied for a class of stochastic nonlinear systems which possesses partially known jump rates. Initially, by using gradient linearization approach, the nonlinear stochastic system is described by a series of linear jump models at some selected working points. Subsequently, observer-based residual generator is constructed for each jump linear system. Then, a new observer-design method is proposed for each re-constructed system to design H_∞ observers that minimize the influences of the disturbances, and to formulate a new performance index that increase the sensitivity to faults. Finally, continuous gain-scheduled approach is employed to design continuous FD observers on the whole nonlinear stochastic system. Simulation example is given to show the effectiveness and potential of the developed techniques.