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NASA NTRS1980

Realization theory and quadratic optimal controllers for systems defined over Banach and Frechet algebras

It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.

Byrnes, C. I.↗

NASA NTRS1982

Moving frames and prolongation algebras

Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

Estabrook, F. B.↗

NASA NTRS1983

Symmetric linear systems - An application of algebraic systems theory

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Hazewinkel, M.↗

NASA NTRS1983

Three-dimensional algebraic grid generation

One of the major approaches to numerical grid generation is the explicit algebraic expression of a physical grid as a function of a uniform grid in a rectangular computational coordinate system. The algebraic methods are based on mathematical interpolation, and the primary advantages are speed and directness. The relation between interpolation and grid generation is described. For three-dimensional grid generation, transfinite interpolation using the coordinate control processes developed in the multisurface method and two-boundary technique are advocated. Grid singularities encountered in three dimensions are discussed, and the exploration of multiple overlapping grids is proposed. Some aspects of interactive algebraic grid computation in three dimensions are discussed.

Smith, R. E.↗

NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗

NASA NTRS1989

Real gas flow computations using algebraic reduction

It is shown that algebraic reduction can be extensively used in the computation of source terms for chemical species in chemically reacting flow fields. Cases of computation from chemical equilibrium and frozen chemistry are considered. For the case of chemical equilibrium the implementation of algebraic reduction method is shown to produce satisfactory results with computational efficiency better than comparable methods. For the case of finite rate chemistry, algebraic reduction of the source term expression for the species is used to obtain a directly integrable expression.

Nagaraj, N.↗

NASA NTRS1990

Applications of algebraic grid generation

Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.

Eiseman, Peter R.↗

NASA NTRS1990

Extensions of algebraic image operators: An approach to model-based vision

Researchers extend their previous research on a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, researchers devised an innovative, efficient edge detection scheme. An accurate method for deriving gradient component information from this edge detector is presented. Based upon this new edge detection system researchers developed a robust method for linear feature extraction by combining the techniques of a Hough transform and a line follower. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The algebraic operators are global operations which are easily reconfigured to operate on any size or shape region. This provides a natural platform from which to pursue dynamic scene analysis. A method for optimizing the linear feature extractor which capitalizes on the spatially reconfiguration nature of the edge detector/gradient component operator is discussed.

Lerner, Bao-Ting↗

NASA NTRS1991

On the application of algebraic turbulence models to high Mach number flows

In this paper the implementation of embedded-function algorithms for conventional algebraic turbulence models is described; using this methodology it is shown that the entire wall layer may be removed from the calculation algorithm (as well as near-wall mixing-length models) with no degradation in the accuracy of predicted results for skin friction. Various difficulties that are associated with the application of conventional algebraic models in high-speed flows are discussed. Alternative algebraic models are proposed which are cast in terms of the Howarth-Dorodnitsyn variable and which lead to a simple 'law of the wall' for both velocity and total enthalpy.

Degani, A. T.↗

NASA NTRS1991

An algebra of discrete event processes

This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.

Heymann, Michael↗

NASA NTRS1995

Algebraic theory of molecules

An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.

Iachello, Franco↗

NASA NTRS1997

Predicting NonInertial Effects with Algebraic Stress Models which Account for Dissipation Rate Anisotropies

Three types of turbulence models which account for rotational effects in noninertial frames of reference are evaluated for the case of incompressible, fully developed rotating turbulent channel flow. The different types of models are a Coriolis-modified eddy-viscosity model, a realizable algebraic stress model, and an algebraic stress model which accounts for dissipation rate anisotropies. A direct numerical simulation of a rotating channel flow is used for the turbulent model validation. This simulation differs from previous studies in that significantly higher rotation numbers are investigated. Flows at these higher rotation numbers are characterized by a relaminarization on the cyclonic or suction side of the channel, and a linear velocity profile on the anticyclonic or pressure side of the channel. The predictive performance of the three types of models are examined in detail, and formulation deficiencies are identified which cause poor predictive performance for some of the models. Criteria are identified which allow for accurate prediction of such flows by algebraic stress models and their corresponding Reynolds stress formulations.

Jongen, T.↗

NASA NTRS1999

Priority in Process Algebras

This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.

Cleaveland, Rance↗

NASA NTRS2011

Process Algebra Approach for Action Recognition in the Maritime Domain

The maritime environment poses a number of challenges for autonomous operation of surface boats. Among these challenges are the highly dynamic nature of the environment, the onboard sensing and reasoning requirements for obeying the navigational rules of the road, and the need for robust day/night hazard detection and avoidance. Development of full mission level autonomy entails addressing these challenges, coupled with inference of the tactical and strategic intent of possibly adversarial vehicles in the surrounding environment. This paper introduces PACIFIC (Process Algebra Capture of Intent From Information Content), an onboard system based on formal process algebras that is capable of extracting actions/activities from sensory inputs and reasoning within a mission context to ensure proper responses. PACIFIC is part of the Behavior Engine in CARACaS (Cognitive Architecture for Robotic Agent Command and Sensing), a system that is currently running on a number of U.S. Navy unmanned surface and underwater vehicles. Results from a series of experimental studies that demonstrate the effectiveness of the system are also presented.

process algebras↗

NASA NTRSDate not supplied

Towards Formalization of Advanced Linear Algebra with Applications to Dynamical Systems using PVS

Linear Algebra is essential for numerous aerospace problems of interest. Formal reasoning about hybrid systems that contain variables modeled by differential equations rely on concepts from Linear Algebra such as eigenvalues, matrix decompositions, and matrix valued functions. For example, the long-term dynamics of a system of differential equations depend on the stability/instability of its equilibrium points, which often reduces to an eigenvalue problem. This talk will embark on a quest to formalize theorems and results about eigenvalues and eigenvectors using PVS. We shall start our journey with 2 x 2 complex matrices, where we will apply our PVS code to a simple example of a dynamical system. Since it can be difficult or impossible to give simple expressions of eigenvalues for larger matrices (i.e. 5 x 5 or higher), we then move towards specifying the power method for verified computation of eigenvalue approximations in PVS. This effort requires development of multivariate complex arithmetic. At the end of the day, having such additions to the PVS NASA libraries will help move towards the use of formal methods to verify concepts of control theory and system level verification.

Linear Algebra↗

NASA NTRS1973

Algebraic criteria for positive realness relative to the unit circle.

A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.

Siljak, D. D.↗

NASA NTRS1974

Algebraically growing waves in ducts with sheared mean flow

Analysis of the behavior of standing and traveling acoustic waves in a smooth duct with a fluid flow having a sheared mean velocity profile, when the waves grow algebraically as they travel along the duct axis. It is shown that standing waves growing algebraically with the axial distance cannot exist in a smooth duct when the duct wall have a finite resistance. The existence of traveling waves subject to the same law of growth is also dismissed under realistic flow conditions.

Nayfeh, A. H.↗

NASA NTRS1977

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Djaferis, T. E.↗