Showing papers on "Realizability published in 1980"

A shift operator approach to bilinear system theory

Abstract: Using a transform representation, we present a bilinear realization theory for a Volterra series input–output map. The approach involves the definition of appropriate shift operators on linear spaces associated with the transforms of the kernals in the Volterra series. This approach yields in a very simple manner a theory of minimality and connections with the concepts of span reachability and observability. It also leads to a characterization of finite dimensional realizability in terms of rationality properties of the transforms.

Realizability inequalities and closed moment equations

Abstract: where the y , are real variables and P, is a polynomial expression of degree R . The moments are { y , ( t ) } , {y,(l)y,(t’)}, { v , ( t ) y , ( t ’ ) y l ( r” ) } , . . . , where { 1 denotes an average taken over an ensemble of realizations. The realizability inequalities are the conditions that all realizations (except for a possible zero-measure set) enter the ensemble with non-negative probability. This work was motivated by certain problems in the theory of turbulence described by the incompressible Navier-Stokes equation, which can be written in the form (1) as

Synthesis of symmetrical dual mode in-line prototype networks

Abstract: An analytic synthesis technique is developed for symmetrical dual mode in-line prototype networks up to and including degree 12 networks. Due to the symmetrical realization, the even mode coupling matrix completely defines the network and one particular form representing the double cross-coupled array, may be obtained directly from the transfer function. Commencing with this matrix, rotational transformation are used to transform the even mode matrix into the form required for a dual mode in line structure. Analytical solutions are obtained for degrees up to and including 12 and the necessary realizability conditions are determined for each case. Explicit formulas for the angles of rotation are given and for degrees up to and including 10, a complete set of explicit formulas are provided for the final coupling matrix. To assist in the design of the in-line structure, tables of element values for filters having 20 dB and 26 dB return loss and possessing the minimum number of transmission zeros at infinity are tabulated.

Distortion-Rate Theory and Filtering

Abstract: The constraint of realizability is introduced and with it the Distortion Rate Function is redefined. For the Orenstein Uhlenbeck process, the constrained Distortion Rate function is shown to satisfy a dynamic programming equation. Further, a filtering problem is shown to be equivalent to determination of the constrained distortion rate function.

Synthesis of Lumped-Distribut ed Networks: Lossless Cascades

Abstract: A study of the composition of the polynomials in the expression for the input impedances to a general (not necessarily lossless) mixed lumped-distributed cascade shows that a simpler and more efficient set of realizability conditions based on a generalization of the Kinariwala [lo] expansion process rather than on a Richards’ theorem synthesis is possible for this network configuration. These specifications are given entirely in terms of determinants formed from the coefficients of the input-impedance expression of the cascades. First the composition of the polynomials in the inputimpedance expression for the specified cascades is studied in some detail. The results of this study are then used to show that sufficient information is contained in basically one determinant formed from the polynomials to permit the PR matrix specification of the lumped-parameter networks in the cascade. Working only with the polynomials restricts this problem to the single-variable domain. Sufficiency is proved by first showing that any function satisfying the specified form for Z,, and for which none of n polynomial determinants is identically zero has a cascade representation without necessarily satisfying a specific cascade condition. Subsequently, it is shown that all lumped coupling networks in the cascade are lossless pr two-ports. Examples are given and a detailed comparison for the lossless case is made between the two-variable formulation in the literature and the single-variable formulation of this paper, in which we demonstrate their equivalence, and also show the more complicated or obscure nature of the two-variable formulation and the less lengthy computational effort with a simpler synthesis algorithm for the one-variable formulation. We carry out the analysis in terms of general (lossy or lossless) cascades to show that the results of this paper also apply to lossy cascades. Extension to the lossy case will be discussed in a forthcoming paper.

Synthesis of lumped-distributed networks: Lossless cascades

Abstract: This paper establishes single-variable realizability conditions and simple synthesis procedures for the driving-point impedance of lossless lumped-distributed cascades. The cascades considered consist of commensurate, uniform, lossless transmission lines interconnected by lossless lumped two-ports and terminated by either lumped passive or resistive loads. The reaizability conditions are stated and the analysis is carried out entirely in terms of a single variable, which allows the extension of the results to cascades with lossy lumped networks In addition, a detailed comparison between the realizability conditions in this paper and those in the literature using a two-variable formulation is given with the two sets shown equivalent. However, the computational effort required to check for realizability and to carry out the synthesis procedures is simpler and less obscure for the methods of this paper.

Elastic realizability of force and displacement systems in structures

Abstract: This paper discusses the constraints imposed upon equilibrium and compatibility solutions in structures through the use of constitutive equations. It is shown, for example, that \"realizable\" force systems are bounded by statically determinate force systems for the case of trusses. The analysis used depends heavily upon the concept of \"basic solutions\" (statically determinate substructures) for linear systems which appears most commonly in the theory of linear programming. Introduction. As used in this paper, the term \"physically realizable\" or simply \"realizable\" applies to force and displacement solutions which satisfy all the equations of structures. It is the intention here to study the properties of these solutions with respect to solutions of the equilibrium or compatibility equations which may or may not also satisfy the constitutive equations. The idea, of course, is that if equilibrium or compatibility systems are to be \"realized\" or built they must also satisfy constitutive equations. While questions such as realizability have a certain academic interest, the motivation here is somewhat deeper and lies in structural optimization. Having tried unsuccessfully to take structural optimization problems head on, the next step is to attempt to simplify matters. In some cases [1] a convenient simplification is just to neglect the constitutive equations and work with, for example, objective functions and constraints written in terms of forces. When the constitutive equations are dropped from the set of constraints it can happen—notably in cases of multiple loading conditions—that the resulting force system cannot be realized (built). Rather than a practical solution, these results must be regarded as bounds. The question then to be addressed here is the effect of constitutive equations on force and compatibility solutions. It will be shown that the answer lies within the segment of the theory of linear systems which deals with the question of non-negative solutions of linear equations and some concepts of convex analysis. In spite of the fact that this type of analysis is basic to the study of linear programming, much of the supporting material is not easily available. There is, however, an excellent summary of this material in the first chapter of Gale's book on economic models [2] which is highly recommended to the reader. In an earlier effort [3] the authors examined some aspects of the realizability problem for a simple example while here it is hoped to develop a general theory of realizability. In this earlier case it was in fact true that realizable forces were bounded by statically determinate force systems, although this fact was not noted. It is proposed to show here that * Received November 16, 1978. This work has been supported by the National Science Foundation. 412 SPILLERS and LEFCOCHILOS in general realizable solutions are bounded by statically determinate force systems. Other properties of realizable solutions will also be developed. The work presented here proceeds in the following manner. First of all a notation is introduced. Then the case of a single redundant is considered. Finally the general case of an arbitrary number of redundants is developed from the case of the single redundant. The entire paper relies heavily on the concept of a \"basic\" solution of a linear system which of course corresponds to a statically determinate substructure of a given structure. When specific examples are considered they will be trusses, but it may be added that the notation used applies to any type of structure including continuous systems. (The case of the truss, which has a diagonal primitive stiffness matrix, must, however, be extended to the general case in which the primitive stiffness matrix is partitioned-diagonal.) Notation. It is proposed to present the node and mesh methods of structural analysis in the following form (see [5] or any good book on matrix structural analysis): The Node Method: NF = P—node equilibrium, F = KA—constitutive equation, (1) A = N8—member/joint displacement equation, The Mesh Method: CA = 0—compatibility equations, A = K 'F—constitutive equation, (2) F = F°+CF„—member/mesh force equation. In these equations F, A—member force and displacement, P, 8—node force and displacement, K—primitive stiffness matrix, F°—any equilibrium force system (N F° = P), Fm—mesh force matrix, N—generalized incidence matrix, C—generalized branch-mesh matrix. Ordinarily the node and mesh methods are solved as (.NKN)8 = P or (CK'C)F„= -CK'F° (3) in which 8 and Fm are to be computed given the other matrices. In this paper the interest lies then in determining the ranges of 5 and Fm as K varies in some arbitrary manner while remaining positive definite (as required by the particular class of structure under study). For the case of trusses the primitive stiffness matrix is particularly simple: it is just a diagonal matrix with non-negative diagonal terms. For this case the terms in the system matrices in Eq. (3) are linear in either the elements of AT or AT\"1 and these equations can be rewritten in the form ELASTIC REALIZABILITY 413 Dk= p and = 0 (4) where k and k~l are simply column matrices whose elements are the diagonal terms of K and K~' are simply column matrices whose elements are the diagonal terms of K and K ' respectively. The elements of the matrices D and .^are the linear in the displacements S and forces Fm . From Eq. (4) the readability problem is reduced to finding all values of S and Fm for which these equations have semi-positive solutions k and k'\\ (In Gale's terminology, x is semi-positive if x > 0 but x ¥= 0.) There are basic differences in the forces and displacement formulations as they appear in Eqs. (3-4). From one point of view the displacement formulation deals with the solutions of a non-homogeneous system while the force formulation deals with a homogeneous system; from another point of view the displacements S vary inversely along a ray in K-space while the forces are constant along such a ray. Finally it should be noted that the form of Eq. (4) is reminiscent of work on linear inequalities. But the fact that the coefficients of D and <^are linear in the displacements S and the forces Fm adds a degree of difficulty not common in this area. Structures with a single redundant. In this section it will be shown that realizable force systems are bounded by statically determinant force systems for structures which are statically indeterminant to the first degree. Displacement realizability will also be discussed. In this case it is convenient to start with the node equilibrium equation, NF=P. (5) If n is the number of nodal degrees of freedom and b is the number of branch forces (b is the number of bars in the case of the truss), a single redundant implies that b = n + 1. It is furthermore assumed that the structure geometrically stable, which implies that the rank of the matrix N is n. Fig. 1 shows a plane truss which will be useful in discussing this case of a single redundant. The loading itself is of a certain interest since it corresponds to a situation in which the sign of a bar force can be changed by changing the values of the member stiffnesses. For example, as k} -* 0 (bar 3 is removed from the structure) bar 2 goes into tension, while as k, —» 0 it goes into compression. Some of the appropriate matrices are also indicated in this figure. At this point it is convenient to invoke the following theorem. ([5, theorem 2.9]): Theorem 1. Exactly one of the following alternatives holds. Either the equation Ax = 0 has a semi-positive solution or the inequality Ay > 0 has a solution. When Theorem 1 is applied to the system J*k~l = 0 for the case of a single redundant, it simply states that either (a) the system is realizable or (b) all the terms in the row matrix & must have the same sign. The regions of realizability are therefore defined by points at which the terms in ^change sign (pass through zero). Since a term in ^becoming zero corresponds to the formation of a statically determinate substructure, the region of real414 SPILLERS and LEFCOCHILOS

Realizability of multilinear input/output maps

Abstract: This paper examines the realization theory of multilinear input/output maps. It is shown that the existence of a transfer function representation with separable denominator does not guarantee finite state realizability as in the linear and bilinear cases. Conditions for quasi-reachability of a class of multilinear realizations are demonstrated using linear methods, and a form of stability condition obtained for an associated class of multilinear transfer functions. Conditions for observability, which may also be derived from the theory of polynomial response maps, are outlined.

On the realizability of magnetohydroelectric interaction

Abstract: A variant of the initially proposed magnetohydroelectric theory that may bring the theory within the realm of experimental verifiability and practical applicability is suggested.

Realizability conditions on short‐circuit conductance matrix by means of grounded active resistive networks embedding grounded VCVSs

Abstract: The proof is given of a new theorem concerning the minimum number of ideal grounded voltage-controlled voltage sources (VCVSs) necessary and sufficient to realize an arbitrary real square matrix as the short-circuit conductance matrix of a grounded transformerless active resistive multiport network embedding grounded VCVSs.

Kleene's Readability and “Divides” Notions for Formalized Intuitionistic Mathematics

Abstract: S.C. Kleene's two major continuing research interests have been recursive function theory and the foundations of intuitionistic mathematics. They come together in his various notions of recursive realizability, which in turn give rise to his “divides” notions. We review the original definitions and results, and indicate a few further developments.

Semantical Models for Intuitionistic Logics

Abstract: It is ironic that intuitionism, whose origins are rooted in the concept of “proofs”, should produce so many (apparently) different kinds of models: Kripke models, Beth models, topological models, realizability, Swart models, and so on. Furthermore there appears to be a general view that most of the modellings are equivalent, although occasionally it is observed that they are not! In this talk we consider the concept of an abstract semantics for a logic L which we believe satisfies the minimum requirements in order to be called a “truth-value semantics” for L. We then discuss possible notions of equivalence between different semantics for L and in particular we catalogue just about all the truth-value semantics for intuitionistic logic and some of its extensions. We conclude with a Beth-like modelling for the extension CD (constant domains) of intuitionistic logic.

On the realizability of state equations with RC 3-terminal network

Abstract: Necessary and sufficient conditions are derived for a minimal realization system \dot{x} = A_x + b_u , y = cx + \du to have a transfer function which is realizable as a voltage transfer function of a transformerless RC 3-terminal network

On the generation and optimization of equivalent ladder networks

Abstract: The impedance scaling process and Norton's transformation are formulated in a uniform manner as a Cauer-Howitt transformation with diagonal matrices. For passive two-element-kind ladder networks the realizability conditions are reduced to linear inequality constraints. An algorithm is given to enumerate all vertices of the realizability polyhedrons which represent all equivalent networks with a minimum number of elements. In addition a cost function meeting practical requirements is stated and optimized, taking into account various orders of network complexity.