Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1976"

Algebraic system theory: an analyst's point of view

Abstract: A systematic development of the realization theory of finite dimensional constant linear systems is presented which synthesizes the various currently available approaches. Based on representation theorems for submodules and quotient modules of spaces of polynomial matrices and vectors, this paper combines the abstract algebraic ideas centering around module theory, the use of coprime factorizations of rational transfer functions and state space equations into a unified theory.

Realization of systems with internal passivity and symmetry constraints

Abstract: This expository paper describes the main results and applications of the theory of realization of linear stationary dynamical systems in the case that there are certain internal constraints on the parameters of the state space realization. The internal constraints considered here are those derived from passivity and symmetry requirements. Applications to stability theory, electrical network synthesis, and other areas are outlined.

Reduced integration, function smoothing and non-conformity in finite element analysis (with special reference to thick plates)

Abstract: This paper suggests some reasons for the general success of the reduced integration technique. Furthermore it discusses the particular success of the technique in problems where the true solution arises as a limiting case of the physical statement of a general variational problem in which one of the parameters tends to infinity (penalty type formulation). Such a case arises when thin plates are treated as a limiting case of thick plates in which the shear rigidity tends to infinity. Parabolic isoparametric triangular and quadrilateral elements are used to illustrate the main points.

Observable stored energies of electromagnetic systems

Abstract: The conventional definition of quality factor Q in terms of time-average stored energy is widely assumed to be a measure of the input bandwidth of any ordinary electromagnetic system. But the extent to which this assumption is true has never been established. In the case of all radiating systems, for example, it is known to fail completely. It can be made true quite generally, however, by including only those parts of the total time-average stored energy that give a physically observable contribution to the input bandwidth. Explicit formulas for these newly defined observable stored energies are developed that are valid for any electromagnetic system that is linear, passive, and time-invariant.

Scattering theory and linear least squares estimation—II. Discrete-time problems☆☆☆

Abstract: A certain “star-product” formalism in scattering theory as developed by Redheffer is shown to also be naturally applicable to discrete-time linear least-squares estimation problems. The formalism seems to provide a nice way of handling some of the well-known algebraic complications of the discrete-time case, e.g. the distinctions between time and measurement updates, predicted and filtered estimates, etc. Several other applications of the scattering framework are presented, including doubling formulas for the error covariance, a change of initial conditions formula, equations for a backwards Markov state model and a new derivation of the Chandrasekhar-type equations for the constant parameter case. The differences between the discrete-time and continuous- time cases are noted.

The apparent conflict between estimation and control—a survey of the two-armed bandit problem

Abstract: The two-armed bandit is one of the simplest possible non-deterministic control environments which are not trivial. And yet it is astonishingly difficult to control. For the finite-time problem, dynamic programming methods provide optimal controllers. Optimal control strategies also exist for the infinite-time problem, but their implementation requires infinite storage. Storage can be restricted by allowing only the results of the last r tosses to be recorded: the finite-memory problem—or by considering finite state controllers: the finite-state problem. This paper surveys approaches to the two-armed bandit problem. After introducing the problem and discussing examples of systems where it appears, progress on the three above-mentioned variants is reviewed in turn.

Analysis of bilinear noise models in circuits and devices

Abstract: There are a number of applications in which linear noise models are inappropriate. In the paper, the use of bilinear noise models in circuits and devices is considered. Several physical problems are studied in this framework. These include circuits involving varying parameters (such as variable resistance circuits constructed using field-effect transistors), the effect of switching jitter on sampled data system performance and communication systems involving voltage-controlled oscillators and phase-lock loops. In addition, several types of analytical techniques for stochastic bilinear systems are considered. Specifically, the moment equations of Brockett for bilinear systems driven by white noise are discussed, and closed-form expressions for certain bilinear systems (those that evolve an Abelian or solvable Lie groups) driven by white or colored noise are derived. In addition, an approximate statistical technique involving the use of harmonic expansions is described.

Variational principles for incremental finite element methods

Abstract: Effects of equilibrium imbalance and compatibility mismatch at the beginning of each loading increment are included in the development of variational principles for incremental finite element solutions of small deflection problems in solid continua. Variational principles of the conventional type and modified principles with relaxed continuity requirements along element boundaries are presented. Remarks on applications to elastic–plastic analysis and extensions to large deflection theories of elastostatics are included.

Bond graphs and linear graphs

Abstract: Bond graphs have proven to be useful in modeling a wide variety of physical systems where multiports are needed to represent coupling phenomena. Here we show the relationship between bond graphs and more conventional notations utilizing connection N-ports and system graphs. This provides a rigorous foundation for bond graph manipulations and highlights certain difficulties with sign conventions and casual assignments that arise when junction structure loops are present.

Representation and realization of operational differential equations with time-varying coefficients

Abstract: An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.

Large-scale systems - Stability, complexity, reliability

Abstract: This paper establishes the connective stability concept in the framework of the comparison principle and vector Liapunov functions, as a natural setting for resolving the complexity vs reliability problem in the control of large-scale dynamic systems The central result is the following: a stable complex system when composed as a competitive structure of the interconnected stable subsystems remains stable despite the on-off participation of the subsystems This result is important in that it can be used efficiently to synthesize reliable complex systems by multilevel feedback control

An overview of polynomic system theory

Abstract: s from Papers Appearing in the Proceedings of the IEEE, Special Issue-JanuaT 1976 An Overview of Polynomic System Theory @ W. A. PORTER Department qf ELectrical Engineering, University of Michigan, Ann Arbor, Michigan ABSTRACT: During the past 5 yr multilinear, multipower and polynomic systems have become a most important area of applications and theoretical development. This paper starts with some typical examples and historical remarks. Using this background, it then proceeds to put the recent literature in perspective and to quantify partially some broad problem categories wh,ere intensive work is underway. During the past 5 yr multilinear, multipower and polynomic systems have become a most important area of applications and theoretical development. This paper starts with some typical examples and historical remarks. Using this background, it then proceeds to put the recent literature in perspective and to quantify partially some broad problem categories wh,ere intensive work is underway. Scattering Theory and Linear Least Squares Estimation Part I: Continuous-time Problems by L. LJUNG, T. KAILATH and B. FRIEDLANDER Information Systems Laboratory Department of Electrical Engineering Stanford University, California ABSTRACT : The Riccati equation plays an equally important role in scattering theory as in linear least-squares estimation theory. However, in the scattering literature, a somewhat cllffererrt ~frarnework of treatgnz,ple rlerl’cations of known results as well as to obtain several new results. Examples include the derivation of backwards equations to solve forwards Riccati equations; an analysis of the asymptotic behaviour of the Riccati equation; the derivation of backwards Markovian representations of stochastic processes; and new derivations and new insights into the Chandrasekhar and related Levinson and Cholesky equatioras. The Riccati equation plays an equally important role in scattering theory as in linear least-squares estimation theory. However, in the scattering literature, a somewhat cllffererrt ~frarnework of treatgnz,ple rlerl’cations of known results as well as to obtain several new results. Examples include the derivation of backwards equations to solve forwards Riccati equations; an analysis of the asymptotic behaviour of the Riccati equation; the derivation of backwards Markovian representations of stochastic processes; and new derivations and new insights into the Chandrasekhar and related Levinson and Cholesky equatioras.

Optimum design of finned surfaces

Abstract: The paper describes methods of carrying out the minimum weight design of finned surfaces of various types. For each type of surface (flat, cylindrical, etc.) two cases are considered. First, a method is described by which it is possible to obtain the optimum surface profile of fins required to dissipate a certain amount of heat from the given surface, there being no restriction on the fin height. This analysis is then extended for the case when the fin height is given. The solutions are presented for a number of cases and the results are discussed to illustrate the importance of various parameters in the design.

Minimal realizations and canonical forms for bilinear systems

Abstract: The minimal realization theory for input–output maps that arise from finite- dimensional, continuous time, bilinear systems is discussed. It is shown that an observed bilinear system (i.e. a bilinear system together with an observation functional, but without a fixed initial state) is completely determined by its input–output correspondence, i.e. its multivalued input–output map. A precise formulation and proof of the result are given that continuous canonical forms for bilinear systems do not exist. This is done by adapting to the bilinear case an idea due to Hazewinkel and Kalman for the case of linear systems. In addition, the proof presented here has the advantage of not involving any algebraic geometry, which makes it considerably simpler than the original proof of the linear systems result.

Discretization of stress fields in the finite element method

Abstract: Several aspects of the discretization of stress fields, as opposed to displacement fields, are reviewed. The most classical satisfies rigorous equilibrium, both translational and rotational, in the interior domain of each element and reciprocity of surface tractions at interelement boundaries (strong diffusivity). The difficulties associated with kinematical deformation modes are analysed and resolved by different procedures: the composite element technique; quasi-diffusivity controlled by the dual patch test; discretization of the displacement connectors, or hybridation; discretization of rotational equilibrium. This last and recent approach is discussed in some detail. It involves direct or indirect use of first-order stress functions, whose Co continuity is sufficient for strong diffusivity. One of its advantages is the possibility of curving the boundaries by a geometric isoparametric coordinate transformation. Some numerical convergence tests are presented.

New techniques and results in multidimensional problems

Abstract: The role of elementary decision algebra in the broad area of multidimensional problems is given, with emphasis on algorithms and computational simplification. The encouraging fact that several difficult problems can be solved in a finite number of steps via rational operations opens up unlimited scope for research into the search for more efficient algorithms. It is pointed out via a counterexample that the only existing synthesis procedure for arbitrary passive multiports with prescribed multivariable positive real matrices is incorrect. In the process, several links to the study of positivity preserving linear transformations which map square matrices into square matrices are unravelled.

Finite mode bond graph representation of vehicle-guideway interaction problems

Abstract: Classical modal analysis techniques for the prediction of vehicle-guideway dynamics are developed and interpreted with bond graph representations. Bond graphs are shown to allow easy generalization to multiple span guideways incorporating virtually any dynamic boundary conditions at the support locations. In addition, non-linear vehicle models can be used with any vehicle displacement-time history desired. Straightforward formulation procedures are also provided through the bond graph representation. The analysis procedure is demonstrated for a two-span Bernoulli-Euler Guideway with first-order dynamic boundary conditions. The results for a vehicle traveling at different speeds are shown to compare favorably with current literature.

Foundations of system theory: multidecomposable systems☆

Abstract: The paper discusses multilinear, and more generally multidecomposable, machines. An m-linear machine is shown to be realizable as a network of k-linear machines for k⩽(m-1), linked by certain memoryless m-linear maps. In this way, an m-linear machine can be broken down into linear machines and multilinear memoryless maps.

An algebraic representation of parameter sensitivity in linear time-invariant systems

Abstract: This paper presents a new algebraic representation of trajectory parameter sensitivities for linear time-invariant ordinary differential equation systems. By working from first principles, the parameter sensitivities are obtained from the partial derivatives of the system matrices and state transition matrix. The resulting matrix-operator form allows one to compute the complete set of parameter sensitivities with at most 2nr quadrature integrals where n is the state dimension and r is the control dimension. Additionally, this form provides considerable geometric-insight into the sensitivity system and, in particular, some of the properties related to controllability are discussed. Finally, the results concerning the partial derivatives of the state transition matrix are interesting in their own right, and they allow us to extend some previously reported ( 15 , 16 ) structural properties of the sensitivity system to a more general case.

Algebraic realization theory of bilinear discrete-time input-output maps☆

Abstract: In this paper some new results are presented relating to the abstract realization theory of bilinear input—output maps. Previous results obtained by various authors are also discussed and reconsidered in a unified way. The results obtained are connected with the introduction of the class of realizable series whose structure is strictly related to the possibility of embedding the Nerode state space in a finite dimensional linear space. Updating equations for the state, in the discrete time case, are obtained. The relationships among realizations of bilinear i/o maps and systems described by bilinear equations are also discussed.

On optimal form of arches

Abstract: This paper considers the problem of optimal shaping of arches subjected to general loading. The investigation comprises the search for the optimum centerline form as well as the optimum area distribution along the arch. As objective function and constraints, the appropriate combinations of arch thrust, material volume, total arc length, and enclosed area are considered.

Finite and infinite dimensional bilinear realizations

Abstract: In this paper we discuss the realization of Volterra series by finite and infinite dimensional bilinear systems. We observe that realizing a given Volterra series with various weights leads naturally to a particularly interesting class of bilinear systems with a rich mathematical structure. We are able to use certain results on the shift realization of linear systems to arrive at a suitable analog of shift realizations for bilinear realizations.

Practical stability of discrete-time systems

Abstract: Theorems are stated and proved that provide necessary and sufficient conditions for practical stability of discrete-time systems. The first part of the paper deals with stability and instability with respect to time-varying sets, whereas the second part is devoted to the study of final and semi-final stability. The conditions obtained, which take the form of existence of discrete Lyapunov-like functions, generalize previous results.

On several geometric aspects of nonlinear networks

Abstract: Several qualitative properties of nonlinear networks are considered from a manifold (geometric) point of view. The dynamics of electrical networks are looked at as flows (differential equations) on nontrivial manifolds (nonlinear spaces).

Steady-state invariant control systems under disturbances satisfying differential equations

Abstract: This paper deals with the problem of regulating the output of a linear multivariable system affected by disturbances satisfying a given linear differential equation. Conditions are given for the existence of a linear controller such that the overall control system is asymptotically stable and the regulated output goes asymptotically to zero for any disturbance of the given class. Only the regulated outputs are assumed to be measured. Moreover, procedures are presented for the synthesis of the controller.