Polynomial Description Research Articles

Residual Generator Computation for Fault Detection of a General Aviation Aircraft

This paper addresses the problem of the detection and isolation of actuator faults on a general aviation aircraft, characterised by a nonlinear model, in the presence of wind gust disturbances. In particular, this work investigates the design of residual generators in order to realise complete diagnosis schemes when additive faults are present. The use of a canonical input-output polynomial description for the linearised model of the aircraft allows to compute in a straightforward way minimal order residual generators. These tools lead to dynamic filters that can guarantee both disturbance signal decoupling and robustness properties with respect to linearisation errors. The results obtained in the simulation of the faulty behaviour of a Piper PA30 are finally reported.

TCP-aware power and rate adaptation in DS∕CDMA networks

A novel joint power and rate adaptation scheme for DS/CDMA networks is proposed using information from the TCP state machine. The impetus behind this cross-layer optimisation approach is to satisfy the varying transmission rates of TCP flows which depend on the congestion window and round-trip time. Initially, a new family of objective functions is defined that encompasses TCP based information. Then by solving the proposed bi-objective optimisation problem using the ɛ-constraint method the Pareto front is calculated, which gives insight into the trade-off between the desired transmission rate and the corresponding transmitted power. Using a polynomial description of the Pareto front, the solution with the minimum L2-norm from a defined utopia point is selected as the optimum trade-off between power consumption and required rate transmission, even though different heuristics can be used. Although the Pareto frontier has been calculated using derivative based optimisation techniques, prominent paradigms of knowledge-based techniques can increase the computational efficiency. Previously proposed schemes allocate resources based on lower-layer information, such as channel conditions and link-layer buffer occupancy. The novelty and prerogative of this cross-layer optimisation approach is that critical transport layer information is integrated into the design of a power and rate adaptation algorithm for DS/CDMA networks.

High-order free vibration of sandwich panels with a flexible core

Free vibration analysis of sandwich panels with a flexible core based on the high-order sandwich panel theory approach is presented. The mathematical formulation uses the Hamilton principle and includes derivation of the governing equations along with the appropriate boundary conditions. The formulation embodies a rigorous approach for the free vibration analysis of sandwich plates with a general construction, having high-order effects owing to the non-linear patterns of the in-plane and the vertical displacements of the core through its height. As such, it improves on the available classical and high-order theories. The formulation uses the classical thin plate theory for the face sheets and a three-dimensional elasticity theory or equivalent one for the core. The analyses are valid for any type of loading scheme, localized as well as distributed, and distinguish between loads applied at the upper or the lower face. It can also deal with any type of boundary conditions that may be different at the upper and the lower face sheets at the same edge. The effects of the rotary inertia of the various constituents of the sandwich construction are included. Two types of computational models are considered. The first model uses the vertical shear stresses in the core in addition to the displacements of the upper and the lower face sheets as its unknowns. The second model assumes a polynomial description of the displacement fields in the core that is based on the displacement fields of the first model. In this case the unknowns are the coefficients of these polynomials in addition to the displacements of the various face sheets. The two computational models have been validated numerically through a very good comparison with the well known classical and high-order plate theories. The numerical study consists of free vibration eigenmodes of two typical simply-supported panels, including higher modes that cannot be detected by other high-order computational models, and a parametric study that compares the results of the various computational models and the first-order shear deformable results.

A POLYNOMIAL DESCRIPTION OF THE RIJNDAEL ADVANCED ENCRYPTION STANDARD

The paper gives a polynomial description of the Rijndael Advanced Encryption Standard recently adopted by the National Institute of Standards and Technology. Special attention is given to the structure of the S-Box.

Polynomial approach to the control of SISO periodic systems subject to input constraint

The problem of stabilization of single-input single-output discrete-time periodic systems subject to input constraint is discussed in this paper. Precisely, the initial state of the system is assumed to belong to a given polyhedron and the scalar input is constrained to be bounded. It is shown how this problem can be posed in terms of the polynomial description of the system. The solution can be found by using a linear programming optimization algorithm.

An accurate lock-free finite element for buckling analysis of laminated plates under combined in-plane loads

The objective of this paper is to investigate the buckling behavior of moderately thick laminated plates subjected to various in-plane edge loads. For this purpose, an accurate four noded rectangular plate element based on coupled displacement field is developed. Moment-shear and in-plane equilibrium equations are employed to derive the polynomial displacement field of the element. The resulting element has six degrees of freedom per node namely three translations, two bending rotations and a twist and allows higher-order polynomial description for the field variables. The element is not only lock-free but also yields excellent convergence characteristics. A series of numerical examples are solved to demonstrate the efficacy of the proposed element. The influence of lay-up sequence, side-to-thickness ratios, boundary conditions on the prediction capability of the element is studied in detail.

Thermodynamics of mixing and ordering in the diopside–jadeite system: II. A polynomial fit to the CVM results

AbstractIn Part I (Vinograd, 2002) the relationship between temperature, composition and free energy of the diopside–jadeite solid solution was described using a CVM model. Here the same relationship is described using a combination of Redlich-Kister (RK) polynomials and Gaussian functions. The use of the Gaussians permits an accurate description of the free energy dip centred at Jd50, which is caused by the long-range ordering, while the effect of the short-range ordering can be easily modelled using the RK polynomial The simplified RK-Gaussian description is extended to the three-component diopside–jadeite–hedenbergite solid solution. The polynomial description permits an easy incorporation of the derived activity-composition relations into phase-equilibrium computation software.

Pareto analysis in multiobjective optimization using the collinearity theorem and scaling method

This paper presents a method to predict the relative objective weighting scheme necessary to cause arbitrary members of a Pareto solution set to become optimal. First, a polynomial description of the Pareto set is constructed utilizing simulation and high performance computing. Then, using geometric relationships between the member of the Pareto set in question, the location of the utopia point and the polynomial coefficients, the weighting of the performance metrics which causes a particular member of the Pareto set to become optimal is determined. The use of this technique, termed the scaling method, is examined via using a sample problem from the field of vehicle dynamics optimization. The scaling method is based on the collinearity theorem which is also presented in the paper.

Numerically solving physiological models based on a polynomial approach.

Much research effort has been directed in different physiological contexts towards describing realistic behaviors with differential equations. One observes obviously that more state-variables give the model more accuracy. Unfortunately, the computational cost involved is higher. A new algorithm is presented for simulating a model described by a system of differential equations in which efficiency may not be altered by its size. In order to do this, the method is based on a polynomial description of the state-variables' evolution and on a computation distributed control. Evaluations and results performed with classical models like Fitzhugh Nagumo or Hodgkin Huxley, allow validation of the method and exhibits its potential to decrease the computational costs.

Optimal filtered predictive control – a delta operator approach

Using the framework of the discrete delta operator a new generalized predictive controller is developed. Nominal stability is assured by an end-point state weighting. Starting with a controller description in state space form, the nominal stability is proven and the minimal end-point weighting matrix is derived. The new predictive control method is analyzed using the state-space techniques and a polynomial description suitable for real-time implementation is given. An optimal filtering capability is involved by the Kalman design. A final simulation example of a flexible mechanical system underlines the improved stochastic disturbance rejection.

Computing a Hurwitz factorization of a polynomial

A polynomial is called a Hurwitz polynomial (sometimes, when the coefficients are real, a stable polynomial) if all its roots have real part strictly less than zero. In this paper we present a numerical method for computing the coefficients of the Hurwitz factor f( z) of a polynomial p( z). It is based on a polynomial description of the classical LR algorithm for solving the matrix eigenvalue problem. Similarly with the matrix iteration, it turns out that the proposed scheme has a global linear convergence and, moreover, the convergence rate can be improved by considering the technique of shifting. Our numerical experiments, performed with several test polynomials, indicate that the algorithm has good stability properties since the computed approximation errors are generally in accordance with the estimated condition numbers of the desired factors.

Analysis of acceleration strategies for restarted minimal residual methods

We provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual (MR) Krylov subspace methods due to restarting. We evaluate the popular practice of using nearly invariant subspaces to either augment Krylov subspaces or to construct preconditioners which invert on these subspaces. In the case where these spaces are exactly invariant, the augmentation approach is shown to be superior. We further show how a strategy recently introduced by de Sturler for truncating the approximation space of an MR method can be interpreted as a controlled loosening of the condition for global MR approximation based on the canonical angles between subspaces. For the special case of Krylov subspace methods, we give a concise derivation of the role of Ritz and harmonic Ritz values and vectors in the polynomial description of Krylov spaces as well as of the use of the implicitly updated Arnoldi method for manipulating Krylov spaces.

Polynomial Approach to the control of SISO Periodic Systems Subject to Input Constraint

The problem of stabilization of single-input single-output discrete-time periodic systems subject to input constraint is discussed in this paper. Precisely, the initial state of the system is assumed to belong to a given polyhedron and the scalar input is constrained to be bounded. It is shown how this problem can be posed in terms of the polynomial description of the system. The solution can be found by using a linear programming optimization algorithm.

Three-dimensional finite element analysis of the human temporomandibular joint disc

A three-dimensional finite element model of the articular disc of the human temporomandibular joint has been developed. The geometry of the articular cartilage and articular disc surfaces in the joint was measured using a magnetic tracking device. First, polynomial functions were fitted through the coordinates of these scattered measurements. Next, the polynomial description was transformed into a triangulated description to allow application of an automatic mesher. Finally, a finite element mesh of the articular disc was created by filling the geometry with tetrahedral elements. The articulating surfaces of the mandible and skull were modeled by quadrilateral patches. The finite element mesh and the patches were combined to create a three-dimensional model in which unrestricted sliding of the disc between the articulating surfaces was allowed. Simulation of statical joint loading at the closed jaw position predicted that the stress and strain distributions were located primarily in the intermediate zone of the articular disc with the highest values in the lateral part. Furthermore, it was predicted that considerable deformations occurred for relatively small joint loads and that relatively large variations in the direction of joint loading had little influence on the distribution of the deformations.

Wavefront fitting using Gaussian functions

Often the polynomial description of a wavefront shape is inaccurate because sharp local deformations are difficult to represent. In this case an analytical representation in terms of Gaussians may give better results. We have made a study of properties of Gaussians in fitting wavefronts. We analyzed an specific array of Gaussian functions and show their easy Fourier transformation. In Fourier space some criteria are proposed for setting the Gaussian width, the separation between these functions and making an estimation of the wavefront fitting error. Two simulated wavefronts are fitted and a comparison with a Zernike polynomial is made. It is well known that when fitting using Zernike polynomials one needs to find the optimal number of terms beyond which the errors in the approximation become larger. We show that using Gaussians the accuracy of the fit increases with the number of terms. We demonstrate some interesting properties, such as their facility to fit local deformations and fast parameter determination.

Construction of 2-designs

This paper presents a construction of 2-(2 n −1, n,λ) designs and gives the polynomial description of this class of designs. The construction applies to the cases of all integer n≥2.

Polynomial description of linear block codes and its applications to soft-input, soft-output decoding

This paper reviews the representation of binary linear block codes by multi-indeterminate “descriptive polynomials” and extends it to versions permuted with respect to the usual systematic form. The main decoding rules (word-by-word and symbol-by-symbol) can then be expressed in terms of descriptive polynomials, the computation of which is easily implemented using trellises. Particular emphasis is given to the symbol-by-symbol decoding rule where the decoder input and output are both real numbers, assuming a logarithmic measure of the likelihood ratio is employed. The same form of input and output enables successive soft decodings of concatenated codes and iterated decoding. Keeping only the largest monomial of descriptive polynomials widely simplifies decoding and, moreover, results in the same decoded word as the word-by-word decoding rule although the decisions on the symbols are still expressed as real numbers. These tools are applied to convolutional codes (for a finite message length). Decoding a non-recursive non-systematic code by means of the trellis as defined here is shown to be of same complexity as using the trellis conventionally associated with the Viterbi algorithm. On the contrary, decoding a recursive systematic code, of better potential performance, is of greater and prohibitive complexity. Keeping reasonable complexity in this case can however use iteration. Two distinct reasons for iteration of decoding are identified, which both apply to turbo coding. Cet article reprend la rep'esentation de codes en blocs lineaires par des « polynomes descripteurs » a plusieurs indeterminees et la generalise a des versions permutees par rapport a la forme systematique usuelle. Les principales regies de decodage (par mot et par symbole) peuvent etre exprimees en fonction de polynomes descripteurs que l’on peut aisement calculer au moyen de treillis. On insiste surtout sur la regie de decodage symbole par symbole, ou l’entree comme la sortie du decodeur sont des nombres reels, en supposant qu’une mesure logarithmique des rapports de vraisemblance est employee. La meme forme a l’entree et a la sortie permet des decodages ponderes successifs et le decodage itere. On simplifie largement le decodage en ne conservant que le plus grand monome des polynomes descripteurs et, de plus, le resultat obtenu est le meme que celui de la regie de decodage mot par mot bien que les decisions sur les symboles soient toujours exprimees par des nombres reels. Ces outils sont appliques aux codes convolutifs (pour un message de longueur finie). On montre que la complexite du decodage d’un code non recursif et non systematique a l’aide du treillis dejini ici est la meme qu’en employant le treillis habituellement associe a l’algorithme de Viterbi. Au contraire, le decodage d’un code recursif systematique, de meilleures performances potentielles, est d’une complexite plus grande et prohibitive. On peut cependant conserver une complexite raisonnable, dans ce cas, en faisant usage de l’iteration. Deux raisons distinctes de l’iteration du decodage sont identifiees, qui toutes deux s’appliquent aux turbo codes.

On the use of patient data for the definition of reference intervals in clinical chemistry

To improve the definition of reference intervals using patient data in clinical chemistry we studied the use of single results as a selection criterion (i.e. those results judged by the clinician not to require confirmation or which are considered unimportant for clinical screening purposes). Using this criterion 95% intervals were defined for S-albumin, S-cholesterol, S-creatinine, B-glucose, B-haemoglobin and S-urea similar to the corresponding reference intervals determined for healthy adults. Data collected from patients in the age interval 45-95 years demonstrated a significant decrease in levels of S-albumin and B-haemoglobin with age, an increase in S-creatinine, B-glucose and S-urea levels, and a biphasic variation in level of S-cholesterol. It is suggested that these changes are not caused by disease but result from the ageing process. The use of selected patient data for characterization of these changes enables a linear or polynomial description to be made for the age-related changes in the reference intervals.

Cooperative kinetic processes described by polynomial expressions

The time evolution of a certain natural property of a complex kinetic process is usually investigated by linearization techniques, which lead to approximate results valid in the vicinity of equilibrium. In this article we present a polynomial or multiorder description, which involves cooperative processes of any order and is valid less close to equilibrium. As the kinetics evolves in time the higher order or faster cooperative processes begin to vanish in stepwise fashion, and the whole kinetics tends towards the first order. The polynomial representation has therefore a number of terms which depend on the nature of the problem as well as on the relative importance of the polynomial coefficients, that encompass a number of constant or almost constant parameters. On the other hand, the polynomial roots correspond to unstable and stable equilibrium points, which can be established a particular procedure shown in the following treatment.

Bicriteria and restricted 2-Facility Weber Problems

In this paper we look at two interesting extensions to the classical 2-Facility Weber Problem in ℝ d : At first problems are investigated where we do not allow the optimal locations to be in a specific region. Efficient algorithms for this Global Optimization problem are presented as well as new structural results. Secondly we consider 2-Facility Weber Problems with two decision makers where each decision maker can choose his own preferences for the location problem. We give an efficient algorithm for determining all pareto locations for this multicriteria problem as well as a polynomial description of the set of all pareto locations (in ℝ 2d ). All the results presented in this paper are based on a discretization of the original continuous problem using geometrical and combinatorial arguments. The time complexity of all the presented algorithms isO(dM logM), whereM is the number of existing facilities.