Close tracking of equilibrium paths

A method to control a generic system of nonlinear ordinary differential equations between equilibrium states is analyzed. The objective is to ensure that the system's state space trajectory closely tracks an equilibrium path. The control law is obtained via time parameterization of the corresponding equilibrium control path. Conditions which guarantee that the system's state space trajectory closely tracks the equilibrium path are proved using two approaches. One approach uses the mean value theorem, and the other uses the slowly time‐varying systems theory. Importantly, both methods provide relationships between the control rate norm and the tracking error norm. These allow computation of upper bounds on the control rate norm which guarantee a desired upper bound on the tracking error norm. They also enable computation of upper bounds on the tracking error norm for a given upper bound on the control rate norm. Examples illustrate the theoretical results.     

The sharp upper bound on the distance between a parametric patch and its interpolated triangle

In computer aided geometric design （CAGD） and computer graphics, it is a general manipulation to approximate a surface by triangulation mesh. Thus a key problem is to estimate the error of the approximation. So far, many papers have given various estimate bounds of the distance between a parametric patch of a C2 surface and an arbitrary triangle whose vertices are on the patch, but these estimates are all imperfect, some of them have large error, some of them have complicated representation formulae. By using a succinct new method, a sharp upper estimate of the maximum distance between a patch and a triangle is obtained and a strict proof is given. This is very valuable for CAGD.     

Unified analysis of enriched finite elements for modeling cohesive cracks

Aiming to provide a justified theoretical ground based on which different enriched finite elements can be systematically compared, we developed in this work a unified framework for modeling cohesive cracks using the variational multiscale method. The kinematics (i.e. coarse and fine scale displacement and strain fields) and statics (i.e. coarse and fine scale equilibrium equations) are thoroughly investigated in both continuum and discrete settings. With respect to the fine scale kinematics and statics adopted in the embedded finite element method (EFEM) and the extended finite element method (XFEM), we mainly discuss four groups of enriched finite elements with non-uniform discontinuity modes, i.e. the kinematically optimal symmetric EFEM and XFEM, as well as, the kinematically and statically optimal non-symmetric EFEM and XFEM. In all these methods, the enrichment parameters can be regarded either as element-wise local or continuous global variables. The enriched finite elements with material/element/node enrichments are then exemplified in an increasing order of their respective capabilities in representing the fine scale kinematics, and the interrelations between them are also discussed. Owing to this unified framework, we are able to clarify some existing points of view related to EFEM and XFEM. It is found that, if the same fine scale statics is used and the same local/global property is assumed for the involved enrichment parameters, XFEM can be regarded as a kinematically enhanced EFEM since it accounts for a more general fine scale kinematics than that (i.e. relative rigid body motions and self-stretching) considered in EFEM. Finally, several simple, but sufficiently representative, numerical examples are presented to show the significance in appropriately reflecting the fine scale kinematics and statics in an enriched finite element for modeling cohesive cracks.     

Microchoice Bounds and Self Bounding Learning Algorithms

A major topic in machine learning is to determine good upper bounds on the true error rates of learned hypotheses based upon their empirical performance on training data. In this paper, we demonstrate new adaptive bounds designed for learning algorithms that operate by making a sequence of choices. These bounds, which we call Microchoice bounds, are similar to Occam-style bounds and can be used to make learning algorithms self-bounding in the style of Freund (1998). We then show how to combine these bounds with Freund's query-tree approach producing a version of Freund's query-tree structure that can be implemented with much more algorithmic efficiency.     

Bound performance models of heterogeneous parallel processingsystems

Systems of heterogeneous parallel processing are studied such as arising in parallel programs executed on distributed systems. A lower and an upper bound model are suggested to obtain secure lower and upper bounds on the performance of these systems. The bounding models are solved by using a matrix-geometric algorithmic approach. Formal proofs of the bounds are provided along with error bounds on the accuracy of the bounds. These error bounds in turn are reduced to simple computational expressions. Numerical results are included. The results are of interest for application to arbitrary fork-join models with parallel heterogeneous processors and synchronization     

具有不确定噪声的连续时间广义系统确保估计性能的鲁棒Kalman滤波器

讨论了一类具有不确定噪声的连续时间广义随机控制系统的鲁棒Kalman滤波器的设计问题,文中给出了确保估计误差性能指标的不确定噪声协方差矩阵扰动上界,文章研究结果表明,在此界限内采用最坏情况下的最优滤波器实现对状态的估计,它不仅能极小化不确定下的最坏性能,而且也能确保估计误差性能指标达到给定的某个自由度。     

Misclassification Probability Bounds for Multivariate Gaussian Classes

Classification systems based on linear discriminant analysis are employed in a variety of communications applications, in which the classes are most commonly characterized by known Gaussian PDFs. The performance of these classifiers is analyzed in this paper in terms of the conditional probability of misclassification. Easily computed lower and upper bounds on this error probability are presented and shown to provide corresponding bounds on the number of Monte Carlo trials required to obtain a desired level of accuracy. The error probability bounds yield an exact and easily computed expression for the error probability in the case where there are only two classes and a single hyperplane. In the special case where misclassification into a nominated class is independent of all other misclassifications, successively tighter upper and lower bounds can be computed at the expense of successively higher-order products of the individual misclassification probabilities. Finally, bounds are provided on the number of Monte Carlo trials required to improve, with suitably high confidence level, on the confidence interval formed by the error probability bounds.     

A practical approach to disturbance decoupling control

In this paper, a unique dynamic disturbance decoupling control (DDC) strategy, based on the active disturbance rejection control (ADRC) framework, is proposed for square multivariable systems. With the proposed method, it is shown that a largely unknown square multivariable system is readily decoupled by actively estimating and rejecting the effects of both the internal plant dynamics and external disturbances. By requiring as little information on plant model as possible, the intention is to make the new method practical. The stability analysis shows that both the estimation error and the closed-loop tracking error are bounded and the error upper bounds monotonously decrease with the bandwidths. Simulation results obtained on two chemical process problems show good performance in the presence of significant unknown disturbances and unmodeled dynamics.     

A new nonlinear robust disturbance observer

This paper presents a new nonlinear robust disturbance observer by using only the input–output information for minimum phase dynamical systems with arbitrarily relative degrees. The model uncertainties, the nonlinear parts of the system are merged into the disturbance term, and are regarded as a part of the disturbance. The disturbance is assumed bounded. However, the upper and lower bounds are assumed unknown. The new disturbance observer is inspired by the VSS control theory, whereas the upper and lower bounds of the disturbance are adaptively updated. The estimation error of the disturbance can be made as small as necessary by choosing the design parameters. The attraction of the proposed method lies in its robustness to disturbance and easiness to be implemented. Example and simulation results show the effectiveness of the proposed algorithm.     

Interval observer design for continuous-time switched systems under known switching and unknown inputs

ABSTRACT

This paper deals with unknown input estimation for switched linear systems in an unknown but bounded error (UBBE) framework. Based on a known switching signal and under the fulfilment of the relative degree property by all the subsystems, a decoupling method is used to make the state partially affected by the unknown input. Assuming that the disturbances and the measurement noises are unknown but bounded with a priori known bounds, lower and upper bounds of the unmeasured state and unknown input are then computed. A numerical example illustrates the efficiency of the proposed methodology.     

Image noise induced errors in camera positioning

The problem of evaluating worst-case camera positioning error induced by unknown-but-bounded (UBB) image noise for a given object-camera configuration is considered. Specifically, it is shown that upper bounds to the rotation and translation worst-case error for a certain image noise intensity can be obtained through convex optimizations. These upper bounds, contrary to lower bounds provided by standard optimization tools, allow one to design robust visual servo systems.     

Analysis of an extended pressure finite element space for two-phase incompressible flows

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In [6] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.     

Arbitrarily tight upper and lower bounds on the Bayesianprobability of error

This paper presents new upper and lower bounds on the minimum probability of error of Bayesian decision systems for the two-class problem. These bounds can be made arbitrarily close to the exact minimum probability of error, making them tighter than any previously known bounds     

A GES/TS algorithm for the job shop scheduling

The job shop scheduling problem is a difficult combinatorial optimization problem. This paper presents a hybrid algorithm which combines global equilibrium search, path relinking and tabu search to solve the job shop scheduling problem. The proposed algorithm used biased random sampling to have a better covering of the solution space. In addition, a new version of N6 neighborhood is applied in a tabu search framework. In order to evaluate the algorithm, comprehensive tests are applied to it using various standard benchmark sets. Computational results confirm the effectiveness of the algorithm and its high speed. Besides, 19 new upper bounds among the unsolved problems are found.     

Set-theoretic model reference adaptive control with time-varying performance bounds

One of the fundamental problems in model reference adaptive control design is the ability of the controlled system to achieve not only stability but also a user-defined performance in the presence of exogenous disturbances and system uncertainties. To this end, we recently proposed a set-theoretic model reference adaptive control framework, which guarantees the norm of the system error to be less than a user-defined constant performance bound. The contribution of this paper is to generalise the set-theoretic model reference adaptive control framework for enforcing user-defined time-varying performance bounds on the system error, which gives the control designer a flexibility to control the closed-loop system performance as desired on different time intervals (e.g. transient time interval and steady-state time interval). Two adaptive command following control architectures are proposed and their stability and performance properties are rigorously established using system-theoretic methods.     

噪声方差不确定约束系统的滚动时域估计

针对噪声方差不确定的约束系统,讨论了一种鲁棒滚动时域估计(MHE)方法.首先,根据噪声方差不确定模型,找到满足所有不确定性的最小方差上界,在线性矩阵不等式(LMI)框架下求解优化问题,得到近似到达代价的表达形式;然后再融合预测控制的滚动优化原理,把系统的硬约束直接表述在优化问题中,在线优化性能指标,估计出当前时刻系统的状态.仿真时与鲁棒卡尔曼滤波方法进行比较,结果表明了该方法的有效性.     

Enhancing the fundamental limits of sparsity pattern recovery

Detecting the sparsity pattern or support set of a sparse vector from a small number of noisy linear measurements is a challenging problem in compressed sensing. This paper considers the problem of support recovery when statistical side information is available. From the standard linear and noisy measurement model with arbitrary sensing matrix and Gaussian additive noise and by exploiting the side information, a new linear model is derived which benefits from a larger sample size. The common potential benefits of the increase in the number of samples are revealed. The stability guarantees are then analyzed based on the new model. Two decoding schemes are taken for the support recovery task from the new framework, namely, Maximum Likelihood (ML) and Joint-Typicality (JT) decoding. Performance bounds of the support recovery from the new framework are developed and upper bounds are derived on the error probability of these decoders when they are fed with the prior knowledge which is the statistical properties of the new measurement noise. Finally, an extension is provided for when the noise is non-Gaussian. The results show that with the aid of the prior knowledge and using the new framework one can push the performance limits of the sparsity pattern recovery significantly. The approach is supported by extensive simulations including extension of LASSO to the new framework.     

An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices

The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ? ₁-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ? ₁-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.     

Cone-bounded nonlinearities and mean-square bounds—Smoothing and prediction

Mean-square performance bounds are derived for smoothing and prediction problems associated with the broad class of nonlinear dynamic systems which, when modeled by Ito differential equations, contain drift (·dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. Included in this paper are lower bounds on the error covariance attainable by any smoother or any predictor, including the optimum, and upper bounds on the performance of some simple, implementable predictors reminiscent of the designs which are optimal in the linear case. The lower bounds on smoothing and prediction performance are established using measure-transformation techniques to relate a version of the nonlinear problem to its linearization. The upper bound on prediction performance is constructed by a direct analysis of the estimation error. All the bounds hold for correlated system and observation noises. All are rigorously derived and independent of control or control law. In each case, the computational effort is comparable to that for the corresponding optimum linear smoothing or prediction problem. The bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.     

Event-based input and state estimation for linear discrete time-varying systems

In this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.