Solution of Nonlinear Systems

The problem of solving systems of nonlinear equations has been relatively neglected in the mathematical literature, especially in the textbooks, in comparison to the corresponding linear problem. Moreover, treatments that have an appearance of generality fail to discuss the nature of the solutions and the possible pitfalls of the methods suggested. Probably it is unrealistic to expect that a unified and comprehensive treatment of the subject will evolve, owing to the great variety of situations possible, especially in the applied field where some requirement of human or mechanical efficiency is always present. Therefore we attempt here simply to pose the problem and to describe and partially appraise the methods of solution currently in favor

Observers for discrete-time nonlinear systems

Observer synthesis for discrete-time nonlinear systems with special applications to parameter estimation is analyzed. Two new types of observers are developed. The first new observer is an adaptation of the Friedland continuous-time parameter estimator to discrete-time systems. The second observer is an adaptation of the continuous-time Gauthier observer to discrete-time systems. By adapting these observers to discrete-time continuous-time parameter estimation problems which were formerly intractable become tractable. In addition to the two newly developed observers, two observers already described in the literature are analyzed and deficiencies with respect to noise rejection are demonstrated. improved versions of these observers are proposed and their performance demonstrated. The issues of discrete-time observability, discrete-time system inversion, and optimal probing are also addressed

Restricted maximum likelihood estimation of variance components: computational aspects

Suppose that y is an n x 1 observable random vector, whose distribution is multivariate normal with mean vector X(alpha), where X is;a known n x p matrix of rank p* and (alpha) is a p x 1 vector of unknown parameters. The covariance matrix of y is taken to be;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);where Z(,1), ..., Z(,c) are known matrices and (gamma)(,1), ..., (gamma)(,c), (gamma)(,c+1) are unknown parameters. The parameter space for the vector (gamma) =;((gamma)(,1), ..., (gamma)(,c), (gamma)(,c+1))\u27 is taken to be the set (OMEGA)(,1)* of (gamma)-values for which (gamma)(,c+1) \u3e 0 and the matrix;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);is positive definite (i = 1, ..., c);The problem considered is that of computing a restricted maxi- mum likelihood (REML) estimate of (gamma), that is a point (gamma) at which the log-likelihood function L(,1)((gamma); y) associated with a set of n-p* linearly independent error contrasts attains its supremum over (OMEGA)(,1)*;Aside from certain special cases, (gamma) must be computed numeri- cally, generally by some iterative algorithm. Fourteen algorithms are devised, using the following six approaches: (1) Application of the Newton-Raphson method to the log-likelihood function or, equiva- lently, the application of Newton\u27s root-finding method to the like- lihood equations; (2) Application of Newton\u27s method to linearized likelihood equations, for example (when c = 1) to the equations (xi)(,i)((gamma)) (PAR-DIFF)L(,1)/(PAR-DIFF)(gamma)(,i) = 0 (i = 1, 2), where (xi)(,i)((gamma)) is a function of (gamma) chosen so that, in the special case of balanced data, (xi)(,i)((gamma)) (PAR-DIFF)L(,1)/(PAR-DIFF)(gamma)(,i) (i = 1, 2) are linear in (gamma); (3) Application of the Newton-Raphson method to a concentrated log-likelihood function; (4) The Method of Scor- ing; (5) The EM algorithm; and (6) The Method of Successive Approximations;Efficient procedures for computing the iterates of each algorithm are presented, and some techniques for restricting the iterates to the parameter space are discussed. Also, extensions to the case where;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);for known nonnegative-definite matrices A(,1), ..., A(,c) are devised;Numerical results on the performance of the fourteen iterative algorithms are given for each of four data sets. One very effective algorithm is that obtained by applying Newton\u27s method to a set of linearized, concentrated likelihood equations

A single equivalent representation of a group of induction motors using a reduced order model for power system studies

The research contributes to the identification of dynamic equivalents of large numbers of induction motors for the purpose of load modeling. A model for a group of induction motors for starting as well as running studies is developed. The improvement areas are validity of the model over the whole speed range (zero to full load speed), effects of deep-bar rotor, accuracy, computational requirements and data requirements;The grouping is based on the representation of each individual machine by its equivalent circuit. The proposed model reduces a large number of equivalent circuits into one single equivalency. The new circuit contains variable parameters whose variation depends on the motors to be grouped. The reduced single model also has a reduced single inertia and combined load characteristics. The response of the reduced model and the summation of the responses of all the individual machines were nearly identical for both computer simulation and experimental results;Although the proposed model could be used for any number of machines, considerable simplification was achieved by grouping all induction machines into a small number of categories. These categories are a function of the rating and the run-up times of the individual machines;Voltage dip simulations on a small distribution system were performed and the results are presented. The results from the equivalent reduced model agree well with the results obtained from a detailed analysis of each machine. Also, the reduced load model was compared with a static load model for transient stability studies

Computational analysis of viscoelastic free surface flows

The demand for increasingly small and lightweight products require micro-scale components made of materials which are durable and light. Polymers have therefore become a popular choice since they can be used to produce materials which meet industrial requirements. Many of these polymers are viscoelastic fluids. The reduction in the sizes of components make physical experimentation difficult and costly. Therefore computational tools are being sought to replace old methods of testing. This research has been concerned with the development of a finite volume algorithm for viscoelastic flow which can be readily applied to real world applications. A major part of the research involved the implementation of the Oldroyd-B constitutive equations and associated solution methods, in the 3-D multi-physics software environment PHYSICA+. This provides an unstructured finite volume solution technique for viscoelastic flow. This algorithm is validated using the 4:1 planar contraction and results are reported. The developed viscoelastic algorithm has also been coupled with two interface tracking techniques one of which includes surface tension effects. These techniques are the Scalar Equation Algorithm (SEA) and the Level Set Method (LSM). With both techniques the algorithms are able to take into account flow effects from both fluids (ie. air and polymer) in a two-fluid system. The LSM technique maintains a sharp interface overcoming the smearing of the interface which generally affects interface tracking techniques on Eulerian fixed grids, for example SEA, and enables the curvature of the interface to be calculated accurately to implement surface tension effects. This integrated viscoelastic flow solver and free surface algorithm is then illustrated by predicting two industrial flow processes as used in the electronic packaging industry

Pierre Duhem’s philosophy and history of science

LEITE (Fábio Rodrigo) – STOFFEL (Jean-François), Introduction (pp. 3-6). BARRA (Eduardo Salles de O.) – SANTOS (Ricardo Batista dos), Duhem’s analysis of Newtonian method and the logical priority of physics over metaphysics (pp. 7-19). BORDONI (Stefano), The French roots of Duhem’s early historiography and epistemology (pp. 20-35). CHIAPPIN (José R. N.) – LARANJEIRAS (Cássio Costa), Duhem’s critical analysis of mecha­ni­cism and his defense of a formal conception of theoretical phy­sics (pp. 36-53). GUEGUEN (Marie) – PSILLOS (Stathis), Anti-­scepticism and epistemic humility in Pierre Duhem’s philosophy of science (pp. 54-72). LISTON (Michael), Duhem : images of science, historical continuity, and the first crisis in physics (pp. 73-84). MAIOCCHI (Roberto), Duhem in pre-war Italian philos­ophy : the reasons of an absence (pp. 85-92). HERNÁNDEZ MÁRQUEZ (Víctor Manuel), Was Pierre Duhem an «esprit de finesse» ? (pp. 93-107). NEEDHAM (Paul), Was Duhem justified in not distinguishing between physical and chemical atomism ? (pp. 108-111). OLGUIN (Roberto Estrada), «Bon sens» and «noûs» (pp. 112-126). OLIVEIRA (Amelia J.), Duhem’s legacy for the change in the historiography of science : An analysis based on Kuhn’s writings (pp. 127-139). PRÍNCIPE (João), Poincaré and Duhem : Resonances in their first epistemological reflec­tions (pp. 140-156). MONDRAGON (Damián Islas), Book review of «Pierre Duhem : entre física y metafísica» (pp. 157-159). STOFFEL (Jean-François), Book review of P. Duhem : «La théorie physique : son objet, sa structure» / edit. by S. Roux (pp. 160-162). STOFFEL (Jean-François), Book review of St. Bordoni : «When historiography met epistemology» (pp. 163-165)

The development and use of variables in mathematics and computer science

There are a wide variety of uses of variables in mathematics which we cope with in practice through conventions and tacit assumptions. Experience with computers has made us articulate, criticise and develop these assumptions much more carefully. Historically the term 'variable quantity' was introduced in the context of describing and calculating changing quantities which corresponded to phenomena in the observable world (e.g. the velocity of fluxion of a body moving under the inverse square law). The evolution of the concept has divorced it from these routes of reference and required us to establish the formal apparatus of interpretation and valuation. While the changes considered are highly structured this may be satisfactory, but computing power invites us to cope with change in vastly more complex, unstructured situations such as in simulation of 'real world' processes. We relate this challenge to the distinctive differences in the use of variables in mathematics and practical computing, and we develop a general framework in which all uses of variables can be described in a unified way