Scaling and redundancy for ill-conditioned two input, two output plants

It is well known in the multivariable control literature that ill-conditioned plants may pose significant difficulties in control design. In this paper, we explore two causes of ill-conditioning. The first arises when one input has much smaller gain than does the other. The second occurs when the inputs are almost redundant, in the sense that they affect the plant outputs in approximately the same manner. To quantify these two sources of ill-conditioning, we define input and output redundancy angles, and relate these angles to the size of the plant inverse and condition number. We also investigate the dependence of the redundancy angles upon scaling, and show that certain plants have nearly redundant inputs regardless of the choice of units.     

Numerical results for linear Fredholm integral equations of the first kind over surfaces in three dimensions

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method – the Tikhonov method – to deal with this ill-conditioning and to obtain accurate numerical results.     

A recursive state estimator in the presence of state inequality constraints

This paper proposes an optimal recursive estimator to estimate the states of a stochastic discrete time linear dynamic system when the states of the system are constrained with inequality constraints. The case when the constraints are strictly satisfied is treated independently from the case when some of the constraints are violated. For the first case, the well known Kalman filter estimator is used. In the second case, an algorithm which uses a series of successive orthogonalizations on the measurement subspaces is employed to obtain the optimal estimate. It is shown that the proposed estimator has several attractive properties such that it is an unbiased estimator. More importantly, compared to other estimator found in the literature, the proposed estimator needs less computational efforts, is numerically more stable and it leads to a smaller variance. To show the effectiveness of the proposed estimator, several simulation results are presented and discussed.     

N4SID and MOESP algorithms to highlight the ill-conditioning into subspace identification

In this paper,an analysis for ill conditioning problem in subspace identifcation method is provided.The subspace identifcation technique presents a satisfactory robustness in the parameter estimation of process model which performs control.As a frst step,the main geometric and mathematical tools used in subspace identifcation are briefly presented.In the second step,the problem of analyzing ill-conditioning matrices in the subspace identifcation method is considered.To illustrate this situation,a simulation study of an example is introduced to show the ill-conditioning in subspace identifcation.Algorithms numerical subspace state space system identifcation(N4SID)and multivariable output error state space model identifcation(MOESP)are considered to study,the parameters estimation while using the induction motor model,in simulation(Matlab environment).Finally,we show the inadequacy of the oblique projection and validate the efectiveness of the orthogonal projection approach which is needed in ill-conditioning;a real application dealing with induction motor parameters estimation has been experimented.The obtained results proved that the algorithm based on orthogonal projection MOESP,overcomes the situation of ill-conditioning in the Hankel s block,and thereby improving the estimation of parameters.     

On the ill-conditioning of subspace identification with inputs

There is experimental evidence that the performance of standard subspace algorithms from the literature (e.g. the N4SID method) may be surprisingly poor in certain experimental conditions. This happens typically when the past signals (past inputs and outputs) and future input spaces are nearly parallel. In this paper we argue that the poor behavior may be attributed to a form of ill-conditioning of the underlying multiple regression problem, which may occur for nearly parallel regressors. An elementary error analysis of the subspace identification problem, shows that there are two main possible causes of ill-conditioning. The first has to do with near collinearity of the state and future input subspaces. The second has to do with the dynamical structure of the input signal and may roughly be attributed to “lack of excitation”. Stochastic realization theory constitutes a natural setting for analyzing subspace identification methods. In this setting, we undertake a comparative study of three widely used subspace methods (N4SID, Robust N4SID and PO-MOESP). The last two methods are proven to be essentially equivalent and the relative accuracy, regarding the estimation of the (A,C) parameters, is shown to be the same.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

Stochastic proportionate flowshop scheduling with setups

This paper addresses the problem of scheduling n jobs on a proportionate two-machine flowshop where the machines are subject to random breakdowns and setup times are considered separate from processing times. The considered performance measure is makespan. Sequences that minimize makespan with probability 1 are obtained when the first or the second machine is subject to random breakdowns without making any assumptions about downtime distributions or counting processes. It is assumed that the processing and setup times on one machine dominate the corresponding times on the other machine. In the case that processing and setup times on the first and second machines are proportionate, it is shown that the longest processing time (LPT) rule gives an optimal solution when only the first machine is subject to breakdowns, while the shortest processing time (SPT) rule yields an optimal solution when only the second machine suffers breakdowns.     

Application of the Monte Carlo Method for Simulation of Pattern Formation by Ion-Beam Sputtering of Amorphous Bodies

The formation of ordered structures by ion-beam sputtering of a surface of an amorphous body in the case when a strong nonlinearity has a significant effect on the morphology of the irradiated surface is studied. Three modifications of the Monte Carlo method are used for the numerical simulation of the process, and the first of them is a kind of imitational simulation. It is shown that the direct (imitating) statistical simulation of the ion bombardment of the surface of the target, which best matches the considered physical process and is widely used in other papers, has a serious disadvantage. In the case of imitational simulation, random fluctuations of the depth of the sputtering roughen the target surface to such an extent that none of the modes provided by the continuous model can be observed. It especially concerns the modes set after the long ion bombardment of the surface of the target. However, solutions of the continuous model can be investigated numerically by means of other modifications of the Monte Carlo method with decreased dispersion. Two of these modifications are developed in this paper. Applying these methods, under certain conditions, an ordered structure composed of hexagonally symmetrical hollows is obtained on the surface of the target after the target’s long-term exposure to irradiation by a normal ion beam.     

Convolutional Neural Network Structure Transformations for Complexity Reduction and Speed Improvement

Two methods of convolution-complexity reduction, and therefore acceleration of convolutional neural network processing, are introduced. Convolutional neural networks (CNNs) are widely used in computer vision problems. In the first method, we propose to change the structure of the convolutional layer of the neural network into a separable one, which is more computationally simple. It is shown experimentally that the proposed structure makes it possible to achieve up to a 5.6-fold increase in the operating speed of the convolutional layer for 11 × 11-sized convolutional filters without loss in recognition accuracy. The second method uses 1 × 1 fusing convolutions to increase the number of convolution outputs along with decreasing the number of filters. It decreases the computational complexity of convolution and provides an experimental processing speed increase of 11% in the case of large convolutional filters. It is shown that both proposed methods preserve accuracy when tested with the recognition of Russian letters, CIFAR-10, and MNIST images.     

Simultaneous structural analysis and design based on augmented Lagrangian duality

Solution procedures in structural optimization are commonly based on a nested approach where approximations of the analysis and design problems are solved alternately in an iterative scheme. In this paper, we study a simultaneous approach based on an integrated formulation of the analysis and design problems. An advantage of the simultaneous approach, when compared to the nested one, is that the dependence between the analysis and design variables is imposed explicitly. In the nested approach, this dependence is implicitly determined through the solution of the analysis problem. Earlier simultaneous approaches mostly utilize various penalty function reformulations. In this paper, we make use of two augmented Lagrangian schemes, which avoid the numerical ill-conditioning inherent in penalty reformulations. These schemes give rise to Lagrangian subproblems with somewhat different properties, and two efficient techniques are adapted for their solution. The first is a projected Newton method, and the second is a simplicial decomposition scheme. Computational results for bar-truss structures show that the proposed schemes are viable approaches for solving the integrated formulation, and that they are promising for future developments.     

A method for optimal limit design of structures with alternative loads

Three problems of structural optimization are formulated and solved either by a direct or an iterative method. The first problem, the most general considered here, concerns “cost-optimization” of perfectly plastic structures for alternative loads. The second and third ones are more specific problems with linear objective functions, quadratic yield conditions and either a single or several load conditions. The linear-quadratic case can be solved directly when only one load condition is considered. This problem is shown to be related to elastic analysis. For alternative loads an iterative method based on this analysis is proposed. It shows strong convergence and forms the basis of an iterative solution of the general problem.     

Subspace identification by data orthogonalization and model decoupling

It has been observed that identification of state-space models with inputs may lead to unreliable results in certain experimental conditions even when the input signal excites well within the bandwidth of the system. This may be due to ill-conditioning of the identification problem, which occurs when the state space and the future input space are nearly parallel.We have in particular shown in the companion papers (Automatica 40(4) (2004) 575; Automatica 40(4) (2004) 677) that, under these circumstances, subspace methods operating on input-output data may be ill-conditioned, quite independently of the particular algorithm which is used. In this paper, we indicate that the cause of ill-conditioning can sometimes be cured by using orthogonalized data and by recasting the model into a certain natural block-decoupled form consisting of a “deterministic” and a “stochastic” subsystem. The natural subspace algorithm for the identification of the deterministic subsystem is then a weighted version of the PI-MOESP method of Verhaegen and Dewilde (Int. J. Control 56 (1993) 1187-1211). The analysis shows that, under certain conditions, methods based on the block-decoupled parametrization and orthogonal decomposition of the input-output data, perform better than traditional joint-model-based methods in the circumstance of nearly parallel regressors.     

有理Bézier三角曲面片低阶导矢界的估计

基于Bézier三角曲面的de Casteljau算法,同时运用一些恒等式和基本不等式,给出了两类有理Bézier三角曲面片低阶导矢的上界.第一类上界是用控制顶点凸包直径表示的,在一阶偏导的情况下,它是对已有上界的改进;在二阶偏导情况下,当最大权因子与最小权因子比值大于2时,它也是对已有上界的改进.第二类上界是用相邻控制顶点间距离的最大值来表示的.     

An improved approach for determining the optimal orientation of orthotropic material

Several approaches for determining the optimal orientation in the topological optimization process are discussed in this paper. It is shown that the use of a strain based method may give an even worse result than the case in which the discrete orientational variables are fixed in the optimization process, because strong couplings exist among the orientational variables when the strain field is used. The stress field is less sensitive than the strain field with respect to the variations of the orientational variables. Therefore, the coupling between the orientational variables is relatively weak when the stress field is used. This explains why the stress based approaches (e.g. Suzuki and Kikuchi 1991; Díaz and Bendsøe 1992) are more efficient than the strain based method. This kind of approach will be generalized to deal with more general optimization problems. It will be shown that the new approach can not only obtain the best result for the problem as compared to the other methods, but also has the generality for various optimization problems. Several examples will be presented to support the above statements.     

Path Coordinates in a 3D Path Following Problem

Two approaches to 3D path coordinates used when solving a path following problem for aerial vehicles are proposed. The first approach lies in reducing the problem to a two-dimensional case by projection. The second approach is based on introducing an adaptive frame at the target point. The choice of the adaptive frame determines the complexity of the algorithm of the control synthesis. It is shown that the parallel transport frame, or the Bishop frame, is most convenient.     

The impact of collinearity involving the intercept term on the numerical accuracy of regression

It is well known that multiple linear regression models with ill-conditioning can produce coefficient estimates with degraded numerical accuracy. This study examines the numerical accuracy of regression algorithms in the presence a particular type of ill-conditioning, that arising from collinear relationships that involve the intercept term and the independent variables. A benchmark data set is used to produce ill-conditioned data by introducing near linear relationships among the independent variables and the intercept term. The experiments reported here demonstrate that centering does not prevent a loss in numerical accuracy for this particular type of ill-conditioning. In addition, the ability of commonly used diagnostic checks to detect these problems is studied. As an example of the problems that arise from ignoring the relationships studied here we demonstrate that the regression procedures in two widely used statistical packages, SAS and SPSS-X, fail to detect this type of ill-conditioning and report highly inaccurate coefficient estimates.     

Procedural coordination and offshored software tasks: Lessons from two case studies

This paper presents two case studies of offshored software tasks. The success of such tasks is critically dependent on managing an inherent interdependence between onshore and offshore teams. In one case study, both teams belong to the vendor organization, while in the other they are affiliated respectively with client and vendor. It is shown that that interdependence is best addressed through procedural coordination, which entails two complementary strategies. The first consists of carefully specifying and partitioning tasks, and the second of implementing integration mechanisms to bridge communication gaps. Despite contextual differences, the two case studies offer common lessons.     

THE EQUAL PROBABILITY TEST AND OPTIMAL ALLOCATION OF SUBJECTS TO THE GROUPS WHEN COMPARING A NEW DRUG AND A PLACEBO

It is shown that the Fisher-Irwin “usual” (nonrandomized) conditional test for differences between proportions, using independent binomial samples, is very conservative (in the sense that the actual significance level attributable to an outcome is often one-fourth to one-half of the anticipated value) and hence is not appropriate for comparing a new drug and a placebo when the response is quantal and the sample sizes are small. Instead of this test and choice of the significance level (which is, in practice, made by an arbitrary conventional choice), we suggest the equal probability test representing a randomized analog of the “usual” conditional test and determining the critical region so that the errors of the first and second kind are equal. An optimal allocation of subjects to the groups for testing the difference between the probabilities of a response of the two groups is investigated when we are limited to the total number of subjects available for the drug trial. We choose an allocation of subjects to the groups that minimizes the upper bound of the unconditional error of the first kind and hence is greatly preferred over equal allocation of subjects to the groups. The above procedure can be also used for determination of the minimal total number of subjects (for comparative trials under optimal allocation of subjects to the groups) required for detecting the difference between the probabilities of a response of the two groups with the specified upper bound of type 1 error. Numerical examples illustrating the use of the proposed procedures are included.     

Recursive identification of overparametrized systems

A recursive identification algorithm based on extended least squares is proposed to deal with the contingency of overparametrization. The algorithm is relatively simple compared to those involving online order determination, being based on adaptively introducing suitable excitation into the algorithm to avoid ill-conditioning. In the case of extended-least-squares-based adaptive estimation, then the regressors are appropriately stochastically perturbed. The algorithm is shown to converge to a uniquely defined signal model with any pole-zero cancellations at the origin     

The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D

The inverse scattering problem for sound-soft obstacles is considered for both smooth and piecewise smooth surfaces in 3D. The nonlinear and ill-posed integral equation of the first kind is solved by the nonlinear Landweber method. It is an iterative regularization scheme to obtain approximations for the unknown boundary of the obstacle. It is stable with respect to noise and essentially no extra work is required to incorporate several incident waves. So far, it has only been applied to the two dimensional case. Two different integral equations are presented to obtain far-field data. Furthermore, the domain derivative and its adjoint are characterized. The integral equations of the second kind are approximated by a boundary element collocation method. The two-grid method is used to solve the large and dense linear systems. Numerical examples are illustrated to show that both smooth and piecewise smooth obstacles can be reconstructed with this method, where the latter case has not yet been reported.