ВЛИЯНИЕ ВИДА СРЕДНЕЙ ВЗВЕШЕННОЙ ОЦЕНКИ НА ЗАВИСИМОСТЬ КОМПЛЕКСНОГО ПОКАЗАТЕЛЯ КАЧЕСТВА ОТ ПАРАМЕТРОВ ОБЪЕКТА

Objects quality is usually assessed by a complex indicator. It includes single quality indicators with their significance factors. The convolution of the corresponding dependencies represents average weighted quantities: arithmetic, geometric, harmonic, quadratic, etc. At the same time, the influence of the convolution type on the level of the complex quality index, the stability of the calculation results and, the reliability of the quality comparison among a number of similar objects is unknown in advance. Therefore, the aim of the study was to assess the influence of the average weighted type on the level and stability of the calculating results of the complex quality index in different objects compressing.For typical private objects compared the values of the complex quality index calculated according to the formulas of various average weighted estimates. Significance of the corresponding unit quality indicators, incompleteness of the object description and control factors influence on the object took into account.The results of the research were got using the method of virtual experiment planning. They showed that the influencing parameters changes, the calculated levels and stability of the complex quality index essentially depend on the type of convolution. It was shown that under the priori uncertainty of the necessary convolution for the best representative choosing of the corresponding class of objects, the arithmetic average weighted estimate is the best for using.The obtained data can serve as a basis for an informed choice of the type of average weighted in the quality assessment of various objects and decision-making on rational levels of controlled factors.Качество объектов обычно оценивают соответствующим комплексным показателем. Он формируется единичными показателями качества с их коэффициентами значимости. Свертку соответствующих зависимостей представляют средними взвешенными величинами: арифметической, геометрической, гармонической, квадратической и др. При этом заранее неизвестным является влияние вида свертки на уровень комплексного показателя качества, стабильность результатов расчета и достоверность сопоставления качества сходных объектов. Поэтому целью исследования являлась оценка влияния вида среднего взвешенного на уровень и стабильность результатов расчета комплексного показателя качества при сравнении различных объектов.Для типичных частных объектов сопоставили рассчитанные значения комплексного показателя качества с использованием формул указанных средних взвешенных оценок. При этом учли значимости соответствующих единичных показателей качества, показатель неполноты описания объекта и влияние на объект управляющих факторов.Результаты расчетов получили с применением планирования виртуального эксперимента. Они показали, что уровень и стабильность комплексного показателя качества существенно зависят от вида свертки. В результате выявили, что при выборе лучшего представителя из соответствующего класса объектов целесообразно пользоваться средней арифметической взвешенной оценкой.Полученные данные могут служить основой при выборе вида среднего взвешенного при оценке качества разнообразных объектов и принятии решений о рациональных уровнях управляемых факторов

Operator means of probability measures and generalized Karcher equations

In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a large part of Kubo-Ando theory to arbitrary many variables, in fact, to the case of probability measures with bounded support on the cone of positive definite operators. This framework characterizes each operator mean extrinsically as unique solutions of generalized Karcher equations which are obtained by exchanging the matrix logarithm function in the Karcher equation to arbitrary operator monotone functions over the positive real half-line. If the underlying Hilbert space is finite dimensional, then these generalized Karcher equations are Riemannian gradients of convex combinations of strictly geodesically convex log-determinant divergence functions, hence these new means are the global minimizers of them, in analogue to the case of the Karcher mean as pointed out. Our framework is based on fundamental contraction results with respect to the Thompson metric, which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above.Comment: arXiv admin note: text overlap with arXiv:1208.560

A survey and comparison of contemporary algorithms for computing the matrix geometric mean

In this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the Karcher mean. These new algorithms are constructed using the standard definition of the Riemannian Hessian. The survey includes the ALM list of desired properties for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed algorithms. We conclude that currently first-order algorithms are best suited for this optimization problem as the size and/or number of the matrices increase. Copyright © 2012, Kent State University

Coping with Algebraic Constraints in Power Networks

In the intuitive modelling of the power network, the generators and the loads are located at different subset of nodes. This corresponds to the so-called structure preserving model which is naturally expressed in terms of differential algebraic equations (DAE). The algebraic constraints in the structure preserving model are associated with the load dynamics. Motivated by the fact the presence of the algebraic constraints hinders the analysis and control of power networks, several aggregated models are reported in the literature where each bus of the grid is associated with certain load and generation. The advantage of these aggregated models is mainly due to the fact that they are described by ordinary differential equations (ODE) which facilitates the analysis of the network. However, the explicit relationship between the aggregated model and the original structure preserved model is often missing, which restricts the validity and applicability of the results. Aiming at simplified ODE description of the model together with respecting the heterogenous structure of the power network has endorsed the use of Kron reduced models; see e.g. [2]. In the Kron reduction method, the variables which are exclusive to the algebraic constraints are solved in terms of the rest of the variables. This results in a reduced graph, the (loopy) Laplaican matrix of which is the Schur complement of the (loopy) Laplacian matrix of the original graph. By construction, the Kron reduction technique restricts the class of the applicable load dynamics to linear loads. The algebraic constraints can also be solved in the case of frequency dependent loads where the active power drawn by each load consists of a constant term and a frequencydependent term [1],[3]. However, in the popular class of constant power loads, the algebraic constraints are “proper”, meaning that they are not explicitly solvable. In this talk, first we revisit the Kron reduction method for the linear case, where the Schur complement of the Laplacian matrix (which is again a Laplacian) naturally appears in the network dynamics. It turns out that the usual decomposition of the reduced Laplacian matrix leads to a state space realization which contains merely partial information of the original power network, and the frequency behavior of the loads is not visible. As a remedy for this problem, we introduce a new matrix, namely the projected pseudo incidence matrix, which yields a novel decomposition of the reduced Laplacian. Then, we derive reduced order models capturing the behavior of the original structure preserved model. Next, we turn our attention to the nonlinear case where the algebraic constraints are not readily solvable. Again by the use of the projected pseudo incidence matrix, we propose explicit reduced models expressed in terms of ordinary differential equations. We identify the loads embedded in the proposed reduced network by unveiling the conserved quantity of the system