Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1982"

Laguerre Functions in Signal Analysis and Parameter Identification

Abstract: For the approximation of real functions in L 2 (0, ∞) that are frequently encountered in signal analysis and parameter identification, analytical and computer studies suggest the use of Laguerre functions. Such functions can form at least locally optimal or near-optimal sets. The results are shown for continuous systems to be encouragingly flat, indicating low sensitivity to the position of the Laguerre multiple pole. Relationships to linear time-invariant discrete systems are given, using discrete Laguerre functions.

Thermodynamic Bond Graphs and the Problem of Thermal Inertance

Abstract: It is shown that an isolated thermal inertance does not obey the second law of thermodynamics. Consequently, such an element should not be used in physical systems theory. To eliminate the structural gap in the thermal domain of current physical systems theory, a new framework is introduced using Bond Graph concepts. These Thermodynamic Bond Graphs are the result of synthesis of methods used in thermodynamics and in mechanics.

Walsh series analysis of multi-delay systems

Abstract: A matrix, called the “delay operational matrix”, is constructed from the Walsh matrix. This matrix, together with some matrices obtained from the delay operational matrix after performing right-shift operations, is used to solve multi-delay linear dynamic systems. A simple example is given to compare the actual solution and the solution obtained by the techniques of this paper.

A Relative Efficiency study of Some Popular Detectors

Abstract: The relative efficiency between two detectors is a measure of the amount of data one detector requires, relative to the other detector, to attain a prescribed level of performance. In this paper the behavior of the relative efficiencies of two pairs of popular detection systems is investigated, and it is shown that the commonly employed asymptotic value of the relative efficiency can sometimes be a poor indicator of finite-sample-size detection performance even for some very large sample sizes.

Solution of linear state-space equations and parameter estimation in feedback systems using laguerre polynomial expansion

Abstract: This paper analyzes the application of Laguerre polynomial expansion to linear systems. It can be applied to the solution of linear state equations by using an algebraic matrix to determine the coefficients of the Laguerre expansion. It also can be applied to system identification by using the expansion to determine the coefficients in the transfer function. Examples are given to demonstrate the accuracy of finite order expansion by Laguerre polynomials.

Bilinear, dynamic single-ports and bond graphs of economic systems

Abstract: It is shown that reinvestment and inventory effects can be described in 0431 0368 bond-graph terms. The associated differential equations in price and order-flow 0431 0368 V variables are bilinear forms. These effects are the fundamental inertia and compliance 0431 0368 V 3 of economic bond graph theory. Properties of the components and example market 0431 0368 V 3 graphs are discussed.

Optimal circularly symmetric quantizers

Abstract: Polar coordinates quantization of the bivariate Gaussian and other cir- cularly symmetric sources has already been investigated. The schemes quantize the polar coordinates representation of the random variables independently in an attempt to reduce the mean square error below that of an analogous rectangular coordinates quantizer. It has been shown for the Gaussian case that the polar quantizer outperforms the rectangular quantizer when the number of levels N is large, while for small N, the rectangular form is often better than the polar form. This paper is an investigation of the optimality of polar quantizers with the subsequent development of optimal circularly symmetric quantizers (labelled Dirichlet polar quantizers).

Feedback characteristic polynomial controller design of 3-D systems in state-space

Abstract: For a general state-space model of three-dimensional (3-D) systems the characteristic polynomial (eigenvalue) control problem via state and output feedback is considered. A frequency domain approach is employed which in the scalar input case leads to a set of necessary and sufficient conditions. The multi-input problem is treated by assuming that the state or output feedback gain matrix is expressed as the dyadic product ⊙F = ⊙ ⊙f T of a column vector ⊙β and a row vector ⊙f T . This assumption leads to an equivalent scalar input problem β which is directly solved by using the scalar input results. Concerning the dynamic feedback compensator design problem, the important particular case of proportional plus integral plus derivative (PID) control is considered and treated by essentially the same algorithm, which leads to a linear algebraic system in the unknown parameters, along with some constraint equations upon the closed-loop characteristic polynomial sought.

Disturbance Decoupled Observers Having Unrestricted Spectrum

Abstract: The problem of observing the state of a linear finite dimensional system in the presence of unknown disturbance noise is considered. Necessary and sufficient conditions are developed for the existence of a minimal order observer capable of producing an asymptotic estimate of the state of the given system. The approach used leads to a computationally attractive procedure for obtaining an observer having specified eigenvalues.

On the Input—Output Decoupling of 2-D Systems by State Feedback

Abstract: The problem of designing a linear state feedback controller for single-input single-output decoupling of linear multivariable 2-D (two-dimensional) systems is discussed. A method is presented for the determination of the decoupling controller matrices which, when applied to the open-loop system, yield a closed-loop system whose transfer function matrix is diagonal and nonsingular. Necessary and sufficient conditions are established for the state feedback decoupling problem to have a solution. Two examples are included to illustrate the proposed decoupling method.

Truncation error for the generalized Bessel type sampling series

Abstract: Numerous upper bounds for the truncation error of the Shannon sampling expansion associated with the Fourier transforms are available in the literature. In this paper we derive an upper bound for the truncation error of the generalized sampling expansion associated with the Hankel (Bessel) transform. A clear example of the application of such series is in the analysis of optical systems with circular symmetry. A lower bound for the Bessel function Jm(jḿ,N+iy), which agrees with the known asymptotic one and is necessary to this analysis, is derived. The method, which employs complex contour integration, can be applied to other generalized sampling expansions.

Formal numerical characteristics of closed constrained incremental successive substitution algorithms

Abstract: This paper establishes the formal numerical characteristics of closed constrained incremental successive substitution algorithms. The overall formalism gives detailed consideration to such items as the determination of: safety zones wherein the algorithm possesses inherently stable convergence characteristics; existence, uniqueness and convergence properties; formal limitations on the class of functions to which the scheme applies and lastly, establishes self-adaptive attributes. To simplify the presentation, the operators employed are cast in the functional format associated with nonlinear heat conduction.

Spatial effects on the stability of a food-limited moth population

Abstract: Having obtained anomalous results in an attempt to continue the simulation study of moth-wasp interaction in Auslander, Oster and Huffaker (J. Franklin Inst. 297 , 345-375), attention was centered on the role of spatial heterogeneity in an environment with the moth (Anagasta kuhniella) alone. Because of the probable presence of chaotic components in the population behavior (random appearing behavior that is actually caused by deterministic influences), a statistically-based parameter sensitivity and parameter identification method was used. By defining a binary performance criterion that measured the ability of a model with a specific set of parameters to maintain a stable population, the importance of spatial heterogeneity was confirmed. In addition, the use of Monte-Carlo type simulation studies, combined with a binary performance criterion, was demonstrated to be effective for parameter identification and/or parameter sensitivity determination of at least some systems with chaotic or nearly chaotic behavior.

Reduction of Transfer Functions from the Stability-Equation Method and Complex Curve Fitting

Abstract: In this paper, the reduction method uses the concepts of stability-equation and important poles to find the denominator of the reduced model. Then the numerator of the reduced model is found by complex curve fitting. This method tends to simultaneously guarantee a stable reduced model from a stable system and obtain a satisfactory result, since it considers the distribution of important poles. Examples are presented to illustrate this advantage.

Dynamic forces in multiport mechanics: Direct and Lagrangian formulations

Abstract: Holonomic systems can be represented by a bond graph in which inertial variables are related to generalized variables using a multiport displacement-modulated transformer structure. It is shown that the part of the generalized forces due to inertial elements can be found either by direct calculation or by the operations of Lagrange's equations. Although no gyrators are needed for the representation, gyrational coupling may exist. Alternative representations may involve gyrators explicitly and it is possible to construct kinetic co-energy functions for these gyrators for use in Lagrange's equations.

The reconstruction of bordered- diagonal and Jacobi matrices from spectral data

Abstract: Simple, new, direct methods are derived for constructing real, symmetric, bordered-diagonal and tridiagonal matrices from their eigenvalues and the eigenvalues of any one of their principal submatrices. A direct method is also presented for constructing, from its eigenvalues, a real tridiagonal matrix which is symmetric about both its main and secondary diagonals. The techniques described make use of special properties of positive, real, odd, rational functions which occur in electric circuit theory. Examples are given which demonstrate the various methods.

Near insensitivity of linear feedback systems

Abstract: In this paper, the notion of near insensitivity with respect to disturbance and parameter variations is introduced as a performance measure for linear feedback systems and conditions for near insensitivity are derived. It is shown that near insensitivity is attainable with high gain feedback provided that certain geometric conditions relating the system matrices and the parameter variation matrices are satisfied.

Eddy Current Effects on the Asynchronous Performance of the Hysteresis Machine

Abstract: An analysis of the asynchronous behavior of the hysteresis machine is undertaken based on a “rectified” model of its structure. The hysteresis shell and the yoke of the rotor are assumed as being manufactured out of materials with finite conductivities. The field analysis undertaken via Maxwell's equations and Poynting's theorem shows the influence of the induced eddy currents on the machine performance at nonvanishing slips. The characteristics of the machine are computed as functions of slip, revealing three particularly interesting cases.

Comments on ‘Simplification of Large Linear Systems using A Two-Step Iterative Method’: A Two-Step Projection Algorithm

Abstract: This brief communication establishes a two-step iterative algorithm based on the orthogonal projection for reducing order of the high-order system transfer function or state variable equations. A two-step iterative algorithm which has been developed by the authors ( 1 ) consists of the residue and pole (or eigenvalue) optimization with respect to the objective function. Here, the optimum residues in the first step can be determined by using the reciprocal basis in the projection theorem. The reciprocal basis allows one to avoid performing the Grammian inversion. Selecting the new basis, the optimum poles in the second step can be also applied for the orthogonal projection. Although the resulting reduced-order models derived from this geometrical point of view are consistent with models of a two-step iterative algorithm, the algorithm is thus a computationally much simpler way to derive the formula.

Multi-terminal representations and diakoptics

Abstract: Multi-terminal representations constitute a central concept in the analysis of a large system through its subsystems and are based on linear graph-theoretic methods. The subject of diakoptics also has an identical aim, but the origins of its development are quite different. The purpose of this paper is first to establish links between these two parallel sets of concepts. Having achieved this, the advantages of the approach based on multi-terminal representations are emphasized.

On the definition of likelihood in abstract spaces

Abstract: The aim of this work is to give a rigorous definition of likelihood without any reference to the peculiarities of Euclidean spaces, and is thus applicable to a larger class of problems with a more complex result space. Here we intend to offer the simplest possible discussion of the ideas on which likelihood methods are based, with some remarks about their links to some classical measure-theoretical concepts, such as Radon—Nikodym derivatives. Since the definition of likelihood relies on the topological structure of the result space, it is necessary to point out the connections that it has with the measure-theoretical one, mostly caused by the fact that singletons, i.e. the actual observable results, are usually measure-zero sets.

Three Dimensional Electro-magnetic Field Diffusion in Multilayer Cylindrical Structures

Abstract: The problem of the magnetic field diffusion through cylindrical structures is considered. An analytic solution is given for the problem of diffusion, through any number of resistive coaxial layers, of the three components of a quite general input vector magnetic field assigned on a boundary cylindrical surface. The limitations on the class of input functions (or more generally distributions) are the existence of the double space Fourier transform and the time Laplace transform; this makes the solution suitable for studying the transient-state behaviour in cylindrical structures of finite or infinite length.

Doubly-terminated two-variable lossless ladder networks

Abstract: In this paper necessary and sufficient conditions are derived for a two-variable positive real function to be the driving-point impedance of certain classes of doubly-terminated lossless ladder networks. Specifically, two classes of networks are studied: (a) the class of networks in which the lossless structure is a cascade of p 1 - and p 2 -variable two-ports, each two-port having its transmission zeros at the origin and/or at infinity; (b) the class of networks in which the lossless structure is a lowpass or highpass ladder network with series arms having p 1 - and p 2 -type elements in series and shunt arms having the p 1 - and p 2 -type elements in parallel. It is indicated that via suitable transformations of the variables, conditions for many other types of ladder structures can be derived.

Discrete analogues of Green's identities through the graph—theoretic field model

Abstract: In this paper, we derive discrete versions of Green's identities (which appear in the study of potential field theory) as direct consequences of applying Tellegen's Theorem to the Graph—Theoretic Field Model (GTFM) of a field. The procedure herein is in marked contrast to the existing procedures where Green's Identities are derived from the Divergence Theorem by using some strictly mathematical operations. In particular, Green's third identity, which is the starting point formulation for the Boundary Integral Method, is singled out for special attention in terms of its discrete counterpart in the Graph—Theoretic Field Model. The first discrete identity is used to establish certain properties of solutions for the GTFM and a limiting process is applied to the three discrete identities to derive the traditional vector-calculus forms of Green's identities.

Overdamped and Underdamped Linear Dynamical Systems

Abstract: Conditions are given for underdamped or overdamped linear dynamical systems in terms of loop matrix parameters, (√(C)R√(C)g,g)2 ≷ 4(√(C)L√(C)g,g) for all [boxV]g[boxV] = 1, g ∈ H. These criteria are looked upon as natural generalizations of the elementary one-loop RLC series scalar criteria (R/2L)2 ≷ 1/LC, when written in the more suggestive form: (√(C)R√(C))2 ≷ 4√(C)L√(C). A simplified test for determining dynamical systems with all complex natural modes or all real modes are presented with additional comments.

The Bivariate Distribution of a Median Smoothed Markov Chain

Abstract: Median based smoothing algorithms have received considerable attention in the last few years. Their properties make them sometimes superior to linear smoothers. In this paper we develop an expression for the bivariate distribution of a median- smoothed Markov chain and we illustrate one application of it by comparing the power spectra of the input and the output of a median smoother when the input is binary valued.

Principal partition of graphs and connectivity games

Abstract: A two-person game related to the connectivity among all vertices of a graph is defined. Necessary and sufficient conditions for the short, cut and neutral games are given in terms of the principal partition of a graph. Winning strategies are described.

Data flow analysis of program nets

Abstract: This paper deals with the data flow analysis of program nets by means of firing sequences, and presents some basic properties on maximal firing numbers and terminating conditions. Specifically, it is shown that a node in a program net is firable if a certain subgraph containing the node does not exist. An algorithm for the determination of the maximal firing number is also presented.

Analysis of trajectory errors in integrating ordinary differential equations

Abstract: A method of analyzing and interpreting trajectory errors in the numerical solution of ordinary differential equations by digital computers is discussed Truncation in integrating a set of differential equations leads to errors in the trajectory of the solution An explanation is given for the use of diagrams in the complex plane to evaluate errors in the trajectory, with a discussion of the properties of a number of frequently used integration formulas via the diagrams The diagrams portray the characteristics of an integration method in more detail than do the absolutely stable regions presented by Dahlquist Based on the diagrams, guidelines are listed as to how to choose a proper integration formula for the given set of differential equations A method is presented to check whether or not the numerical solution is satisfactory