Some consequences on the planar three-index transportation problem

In this study, some algebraic characterizations of the coefficient matrix A of the planar three-index transportation problem are derived and the equivalent formulation of this problem is obtained using the Kronecker product. It is shown that eigenvectors of the matrix G ⁺ G are characterized in terms of eigenvectors of the matrix A ⁺ A , where G ⁺ is the Moore–Penrose inverse of the coefficient matrix G of the equivalent problem.     

DYNAMICS AND CONTROL OF INTEGRATED PRODUCTION-INVENTORY-SHIPPING SYSTEMS

Abstract

In this paper, multivariable system theory is used to synthesize piecewise-constant control policies for production-inventory systems which incorporate unfilled-order backlogs. The state variables of the systems are the actual production rate, the inventory level discrepancy, and the unfilled-order backlog respectively. The control inputs are the desired production rate and the desired shipping rate, and the disturbance input is the demand for finished parts. It is shown that the original multi-input system can be decoupled into a set of single-input sub-systems in the modal domain by using a suitably chosen generalised modal matrix. The required policies for the original system are then readily obtained by considering each of the decoupled sub-systems separately. The behaviour of the controlled production-inventory system is illustrated by the presentation of the results of computer simulation studies showing typical transient and steady state response characteristics.     

Design of input shapers using modal cost for multi-mode systems

This paper presents two techniques for the determination of modal weights for the design of robust input shapers for multi-mode systems. The weights for the first technique are derived based on the scaling factor for the potential energy in each mode, such that the modal displacement is the same. The second technique exploits the modal cost analysis which uses the angle between the right eigenvectors and the row vectors of the output matrix to determine the contribution of each mode to the output. Two examples are used to illustrate the improvement in performance of the modal weighted minimax shapers compared to the traditional minimax shaper which equally weights all the modes.     

New Results on Modal Participation Factors: Revealing a Previously Unknown Dichotomy

This paper presents a new fundamental approach to modal participation analysis of linear time-invariant systems, leading to new insights and new formulas for modal participation factors. Modal participation factors were introduced over a quarter century ago as a way of measuring the relative participation of modes in states, and of states in modes, for linear time-invariant systems. Participation factors have proved their usefulness in the field of electric power systems and in other applications. However, in the current understanding, it is routinely taken for granted that the measure of participation of modes in states is identical to that for participation of states in modes. Here, a new analysis using averaging over an uncertain set of system initial conditions yields the conclusion that these quantities (participation of modes in states and participation of states in modes) should not be viewed as interchangeable. In fact, it is proposed that a new definition and calculation replace the existing ones for state in mode participation factors, while the previously existing participation factors definition and formula should be retained but viewed only in the sense of mode in state participation factors. Several examples are used to illustrate the issues addressed and the results obtained.      

Application of matrix perturbation theory in robust control of large-scale systems

The aim of this paper is to develop new results for robust centralized and decentralized control of large-scale systems using matrix perturbation theory. A new robustness measure is proposed which is more appropriate for large-scale systems than the existing measures. The use of this new index allows for an evaluation of system eigenvalue sensitivity to perturbations without calculating the eigenvectors and the condition number of the modal matrix. In addition, a novel robust decentralized controller design method is proposed to assign the closed-loop eigenvalues of the large-scale system in a desired region. This method is based on eigenstructure assignment of isolated subsystems. The assignment of overall closed-loop poles in the desired region is guaranteed provided that certain sufficient conditions for the isolated subsystems are satisfied. Simulation results are given to show the efficiency of the new methods.     

Modal synthesis method of lateral vibration analysis for rotor-bearing system

Rigid coupling and flexible connection made up of elastic coupling units are widely applied to rotor-bearing system with multi-branched shafting system. This paper proposes a modal synthesis method of lateral vibration analysis for such kind of rotor-bearing system. When the proposed approach is developed, the elastic coupling unit is defined as “flexible substructure” which is treated individually and the other parts are partitioned into some substructures which are analyzed by finite element method. The lower-frequency normal modes of the substructures are retained; whereas the higher-frequency normal modes are neglected by a frequency truncation criterion and the residual flexibility of those omitted modes is considered. The lower-frequency normal modes and residual flexibility are considered to be the assumed modes of Rayleigh–Ritz analysis of whole structure. The approach to the treatment of higher-frequency modes leads to a great reduction in the calculation time and a significant improvement in the efficiency of modal synthesis. Examples show that the proposed approach is very effective for the vibration analysis of rotor-bearing system with multi-branched shafting system and the results derived from this approach are in a good agreement with those from transfer matrix method.     

A probabilistic spectral framework for grouping and segmentation

This paper presents an iterative spectral framework for pairwise clustering and perceptual grouping. Our model is expressed in terms of two sets of parameters. Firstly, there are cluster memberships which represent the affinity of objects to clusters. Secondly, there is a matrix of link weights for pairs of tokens. We adopt a model in which these two sets of variables are governed by a Bernoulli model. We show how the likelihood function resulting from this model may be maximised with respect to both the elements of link-weight matrix and the cluster membership variables. We establish the link between the maximisation of the log-likelihood function and the eigenvectors of the link-weight matrix. This leads us to an algorithm in which we iteratively update the link-weight matrix by repeatedly refining its modal structure. Each iteration of the algorithm is a three-step process. First, we compute a link-weight matrix for each cluster by taking the outer-product of the vectors of current cluster-membership indicators for that cluster. Second, we extract the leading eigenvector from each modal link-weight matrix. Third, we compute a revised link weight matrix by taking the sum of the outer products of the leading eigenvectors of the modal link-weight matrices.     

On the effects of redundant control inputs

This paper considers effects of redundant control inputs on system performance. A necessary and sufficient condition for strict decrease of the quadratic performance index with input extension is introduced. For the non-strict decrease cases, the initial state set is determined by using Hamilton matrix eigenvectors and generalized eigenvectors. The $H_2$ optimal control problem is similarly studied on the basis of state feedback and dynamic output feedback controllers. Both linear matrix inequality (LMI) and Riccati equation based methods are established. The guaranteed cost problem is also solved for uncertain systems with input extension. An iterative optimization algorithm is presented, for choosing the input matrix columns for a better performance index. A satellite launch vehicle (SLV) system is used to illustrate the results.     

Systolic networks for iterative methods for linear equations with cyclic reduction II

This paper presents systolic networks for the application of cyclic reduction to iterative methods for the solution of a linear system of equations A x=b where A is a p-cyclic matrix derived from multi-colouring ordered difference schemes on a regular mesh.     

三维实体转子系统热-结构耦合响应分析

以某型航空发动机高压转子系统为研究对象,基于不均匀分布稳态温度场,建立了某高压转子系统三维实体单元有限元模型以及稳态温度场下转子系统热-结构耦合振动方程,利用热-结构-动力学耦合理论,采用间接耦合法,通过稳态温度场分析和静力分析生成热应力,然后进行预应力模态分析,最后利用模态叠加法进行不平衡量和热弯曲耦合响应分析,实现热-结构-动力学耦合计算.通过稳态温度场对典型级盘稳态响应影响的分析以及不平衡量与热弯曲耦合稳态响应分析,发现耦合响应对转子系统各级盘的振动响应有较大影响.     

Parametric eigenvalue-eigenvector assignment in reduced-order observer design

The design of reduced-order state observers for linear multivariable systems is considered. The procedure used is based on the assignment of prespecified sets of eigenvalues and associated eigenvectors to the observer model, whereby the state estimation error is effectively governed. It is shown, in particular, that the restrictions on the assignable eigenvectors are removed when the number of system outputs equals or exceeds half the number of state variables. Moreover, with the outputs in excess of half the state variables, some design parameters remain unaffected by the assignment of both sets of eigenvalues and eigenvectors. A certain part of these additional parameters can be adjusted to alleviate the impact of misalignment between the system and the observer initial states. The remaining part contributes to the identification of the observer matrices—other than the dynamical coefficient matrix. The one-to-one correspondence between the observability of the system and the eigenvalue assignability of the observer in the framework of this approach is demonstrated.     

The Self—Determinism of Teleogenic Systems

Abstract

The existence of uncontrollable state variables is shown to be necessary for a system to be teleogenic, capable of generating its own goals. Since human beings and other living organisms are manifestly teleogenic, the behavioristic paradigm is fundamentally flawed. A triadic theory of human development, in which heredity and environment are joined by a self-determined component consisting of the evolving uncontrollable state variables, is outlined. The behavior of a simple teleoid under two different modes of development is described.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.     

Dynamic coherency identification of generators using self-organising feature maps

An important step in constructing dynamic equivalents of large power systems is the coherency identification and grouping of generators. Self-organising feature maps can do this task, if the attribute vectors, which characterise the features of the generator dynamics inside the network are well chosen. It is shown in the paper that the principal components of the correlation matrix of the simulated time responses of the generators after faults are especially suitable to form the attribute vectors. The results are compared with the use of right eigenvectors and participation factors of the linearised system matrix as attribute vectors.     

考虑推进剂晃动的火箭液固耦合分析的比拟算法

对于弹性容器与不可压无黏液体之间的线性耦合问题,已有缩聚对称形式的液固耦合系统有限元方程.利用比拟算法获得液固耦合系统的系统矩阵,将问题转化为通用有限元程序可以解决的问题.以包含贮箱的火箭模型为例,求解火箭的模态特性,其中包括由液体晃动所引起的火箭振动模态.结果表明此类模态与重力加速度有关,频率随重力加速度的增大而增大.     

陀螺系统微振动模态摄动分析与灵敏度计算

在状态空间下,将线性陀螺系统微振动问题导向哈密顿体系,可以得到一组加权共轭辛正交关系和模态展开定理．利用这种特点构造了陀螺系统模态摄动计算式与灵敏度计算式,从而解决了拉格朗日体系下陀螺系统模态摄动分析与灵敏度计算的困难,算例显示了文中计算方法的有效性．     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

A Right-Preconditioning Process for the Formal–Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets

Aright-preconditioning process for linear interval systems has been presented by Neumaier in 1987. It allows the construction of an outer estimate of the united solution set of a square linear interval system in the form of a parallelepiped. The denomination “right-preconditioning” is used to describe the preconditioning processes which involve the matrix product AC in contrast to the (usual) left-preconditioning processes which involve the matrix product AC, where A and C are respectively the interval matrix of the studied linear interval system and the preconditioning matrix.The present paper presents a new right-preconditioning process similar to the one presented by Neumaier in 1987 but in the more general context of the inner and outer estimations of linear AEsolution sets. Following the spirit of the formal-algebraic approach to AE-solution sets estimation, summarized by Shary in 2002, the new right-preconditioning process is presented in the form of two new auxiliary interval equations. Then, the resolution of these auxiliary interval equations leads to inner and outer estimates of AE-solution sets in the form of parallelepipeds. This right-preconditioning process has two advantages: on one hand, the parallelepipeds estimates are often more precise than the interval vectors estimates computed by Shary. On the other hand, in many situations, it simplifies the formal algebraic approach to inner estimation of AE-solution sets. Therefore, some AE-solution sets which were almost impossible to inner estimate with interval vectors, become simple to inner estimate using parallelepipeds. These benefits are supported by theoretical results and by some experimentations on academic examples of linear interval systems.