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NASA NTRS2000

Model Checking JAVA Programs Using Java Pathfinder

This paper describes a translator called JAVA PATHFINDER from JAVA to PROMELA, the "programming language" of the SPIN model checker. The purpose is to establish a framework for verification and debugging of JAVA programs based on model checking. This work should be seen in a broader attempt to make formal methods applicable "in the loop" of programming within NASA's areas such as space, aviation, and robotics. Our main goal is to create automated formal methods such that programmers themselves can apply these in their daily work (in the loop) without the need for specialists to manually reformulate a program into a different notation in order to analyze the program. This work is a continuation of an effort to formally verify, using SPIN, a multi-threaded operating system programmed in Lisp for the Deep-Space 1 spacecraft, and of previous work in applying existing model checkers and theorem provers to real applications.

Havelund, Klaus↗

NASA NTRS2002

The Statistical Mechanics of Ideal Homogeneous Turbulence

Plasmas, such as those found in the space environment or in plasma confinement devices, are often modeled as electrically conducting fluids. When fluids and plasmas are energetically stirred, regions of highly nonlinear, chaotic behavior known as turbulence arise. Understanding the fundamental nature of turbulence is a long-standing theoretical challenge. The present work describes a statistical theory concerning a certain class of nonlinear, finite dimensional, dynamical models of turbulence. These models arise when the partial differential equations describing incompressible, ideal (i.e., nondissipative) homogeneous fluid and magnetofluid (i.e., plasma) turbulence are Fourier transformed into a very large set of ordinary differential equations. These equations define a divergenceless flow in a high-dimensional phase space, which allows for the existence of a Liouville theorem, guaranteeing a distribution function based on constants of the motion (integral invariants). The novelty of these particular dynamical systems is that there are integral invariants other than the energy, and that some of these invariants behave like pseudoscalars under two of the discrete symmetry transformations of physics, parity, and charge conjugation. In this work the 'rugged invariants' of ideal homogeneous turbulence are shown to be the only significant scalar and pseudoscalar invariants. The discovery that pseudoscalar invariants cause symmetries of the original equations to be dynamically broken and induce a nonergodic structure on the associated phase space is the primary result presented here. Applicability of this result to dissipative turbulence is also discussed.

Shebalin, John V.↗

NASA NTRS1980

On the existence, uniqueness, and asymptotic normality of a consistent solution of the likelihood equations for nonidentically distributed observations: Applications to missing data problems

A general theorem is given which establishes the existence and uniqueness of a consistent solution of the likelihood equations given a sequence of independent random vectors whose distributions are not identical but have the same parameter set. In addition, it is shown that the consistent solution is a MLE and that it is asymptotically normal and efficient. Two applications are discussed: one in which independent observations of a normal random vector have missing components, and the other in which the parameters in a mixture from an exponential family are estimated using independent homogeneous sample blocks of different sizes.

Peters, C.↗

NASA NTRS2009

Testing First-Order Logic Axioms in AutoCert

AutoCert [2] is a formal verification tool for machine generated code in safety critical domains, such as aerospace control code generated from MathWorks Real-Time Workshop. AutoCert uses Automated Theorem Provers (ATPs) [5] based on First-Order Logic (FOL) to formally verify safety and functional correctness properties of the code. These ATPs try to build proofs based on user provided domain-specific axioms, which can be arbitrary First-Order Formulas (FOFs). These axioms are the most crucial part of the trusted base, since proofs can be submitted to a proof checker removing the need to trust the prover and AutoCert itself plays the part of checking the code generator. However, formulating axioms correctly (i.e. precisely as the user had really intended) is non-trivial in practice. The challenge of axiomatization arise from several dimensions. First, the domain knowledge has its own complexity. AutoCert has been used to verify mathematical requirements on navigation software that carries out various geometric coordinate transformations involving matrices and quaternions. Axiomatic theories for such constructs are complex enough that mistakes are not uncommon. Second, adjusting axioms for ATPs can add even more complexity. The axioms frequently need to be modified in order to have them in a form suitable for use with ATPs. Such modifications tend to obscure the axioms further. Thirdly, speculating validity of the axioms from the output of existing ATPs is very hard since theorem provers typically do not give any examples or counterexamples.

Ahn, Ki Yung↗

NASA NTRS1986

Existence of the time optimal control for robotic manipulators

Using Filipov's Theorem, it is shown that the conditions oif nonfinite escape of trajectories, reachability, and convexity of the dynamics over all admissible controls are needed for the existence of a time optimal solution for the robotic equation. With a lower bound for the finite-escape time established using a Liapunov approach, and an upper bound for the time to reach the target established using the exact linearization idea, a single inequality is found which is closely related to the coriolis and the centrifugal terms, the absence of which implies that the domain of existence of the optimal solution can be made arbitrarily large with a large torque constraint. As the work space is finite, this is essentially a global result in practical situations.

Wen, J.↗

NASA NTRS1975

Almost periodic solutions to difference equations

The theory of Massera and Schaeffer relating the existence of unique almost periodic solutions of an inhomogeneous linear equation to an exponential dichotomy for the homogeneous equation was completely extended to discretizations by a strongly stable difference scheme. In addition it is shown that the almost periodic sequence solution will converge to the differential equation solution. The preceding theory was applied to a class of exponentially stable partial differential equations to which one can apply the Hille-Yoshida theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous equation (which can be approximated by almost periodic sequences) which are the solutions to appropriate discretizations. Two methods of discretizations are discussed: the strongly stable scheme and the Lax-Wendroff scheme.

Bayliss, A.↗

NASA NTRS1997

Quadrature, Interpolation and Observability

Methods of interpolation and quadrature have been used for over 300 years. Improvements in the techniques have been made by many, most notably by Gauss, whose technique applied to polynomials is referred to as Gaussian Quadrature. Stieltjes extended Gauss's method to certain non-polynomial functions as early as 1884. Conditions that guarantee the existence of quadrature formulas for certain collections of functions were studied by Tchebycheff, and his work was extended by others. Today, a class of functions which satisfies these conditions is called a Tchebycheff System. This thesis contains the definition of a Tchebycheff System, along with the theorems, proofs, and definitions necessary to guarantee the existence of quadrature formulas for such systems. Solutions of discretely observable linear control systems are of particular interest, and observability with respect to a given output function is defined. The output function is written as a linear combination of a collection of orthonormal functions. Orthonormal functions are defined, and their properties are discussed. The technique for evaluating the coefficients in the output function involves evaluating the definite integral of functions which can be shown to form a Tchebycheff system. Therefore, quadrature formulas for these integrals exist, and in many cases are known. The technique given is useful in cases where the method of direct calculation is unstable. The condition number of a matrix is defined and shown to be an indication of the the degree to which perturbations in data affect the accuracy of the solution. In special cases, the number of data points required for direct calculation is the same as the number required by the method presented in this thesis. But the method is shown to require more data points in other cases. A lower bound for the number of data points required is given.

Hodges, Lucille McDaniel↗

NASA NTRS1973

Fixed point theorems and dissipative processes.

Operators of the type considered by Hale et al. (1972) are used to show that under certain conditions there is a fixed point in a dissipative map within a Banach space. The conditions required for the existence of this fixed point are discussed in detail. Several fixed point theorems are formulated and proved.

Hale, J. K.↗

NASA NTRS1976

Optimal insensitive-controller synthesis

Proof of two theorems is included in report: local sufficiency condition for existence of insensitive controllers in the case of sufficiently small parameter variations; and necessary condition for optimal controller corresponding to point in boundary of domain of admissible parameter variations to be optimal insensitive controller.

Harvey, C. A.↗

NASA NTRS1980

Towards sub-optimal stochastic control of partially observable stochastic systems

A class of multidimensional stochastic control problems with noisy data and bounded controls encountered in aerospace design is examined. The emphasis is on suboptimal design, the optimality being taken in quadratic mean sense. To that effect the problem is viewed as a stochastic version of the Lurie problem known from nonlinear control theory. The main result is a separation theorem (involving a nonlinear Kalman-like filter) suitable for Lurie-type approximations. The theorem allows for discontinuous characteristics. As a byproduct the existence of strong solutions to a class of non-Lipschitzian stochastic differential equations in dimensions is proven.

Ruzicka, G. J.↗

NASA NTRS1980

Towards sub-optimal stochastic control of partially observable stochastic systems

The paper deals with a class of multidimensional stochastic control problems with noisy data and bounded controls encountered in aerospace design. The emphasis is on suboptimal design, the optimality being taken in quadratic mean sense. To that effect the problem is viewed as a stochastic version of the Lurie problem known from nonlinear control theory. The main result is a separation theorem (involving a nonlinear Kalman-like filter) suitable for Lurie-type approximations. The theorem allows for discontinuous characteristics. As a byproduct the existence of strong solutions to a class of non-Lipschitzian stochastic differential equations in n dimensions is proved.

Ruzicka, G. J.↗

NASA NTRS1991

Magnetic bearings-state of the art

Magnetic bearings have existed for many years, at least in theory. Earnshaw's theorem, formulated in 1842, concerns stability of magnetic suspensions, and states that not all axes of a bearing can be stable without some means of active control. In Beam's widely referenced experiments, a tiny (1/64 in diameter) rotor was rotated to the astonishing speed of 800,000 rps while it was suspended in a magnetic field. Despite a long history, magnetic bearings have only begun to see practical application since about 1980. The development that finally made magnetic bearings practical was solid state electronics, enabling power supplies and controls to be reduced in size to where they are now comparable in volume to the bearings themselves. An attempt is made to document the current (1991) state of the art of magnetic bearings. The referenced papers are large drawn from two conferences publications published in 1988 and 1990 respectively.

Fleming, David P.↗

NASA NTRS1993

Magnetic bearings - State of the art

Fleming, David P.↗

NASA NTRS2003

Applicability of Markets to Global Scheduling in Grids: Critical Examination of General Equilibrium Theory and Market Folklore

Markets are often considered superior to other global scheduling mechanisms for distributed computing systems. This claim is supported by: a casual observation from our every-day life that markets successfully equilibrate supply and demand, and the features of markets which originate in the general equilibrium theory, e.g., efficiency and the lack of necessity of 2 central controller. This paper describes why such beliefs in markets are not warranted. It does so by examining the general equilibrium theory, in terms of scope, abstraction, and interpretation. Not only does the general equilibrium theory fail to provide a satisfactory explanation of actual economies, including a computing-resource economy, it also falls short of supplying theoretical foundations for commonly held views of market desirability. This paper also points out that the argument for the desirability of markets involves circular reasoning and that the desirability can be established only vis-a-vis a scheduling goal. Finally, recasting the conclusion of Arrow's Impossibility Theorem as that for global scheduling, we conclude that there exists no market-based scheduler that is rational (in the sense defined in microeconomic theory), takes into account utility of more than one user, and yet yields a Pareto-optimal outcome for arbitrary user utility functions.

Nakai, Junko↗

NASA NTRS1974

An implicit sampling theorem for bounded bandlimited functions

A rigorous proof of the 'strong bias tone' scheme is embodied in the implicit sampling theorem. The representation of signals that are sample functions of possible nonstationary random processes being of principal interest, the proof could not directly invoke results from classical analysis, which depend on the existence of the Fourier transform of the function under consideration; rather, it is based on Zakai's (1965) theorem on the series expansion of functions, band-limited under a suitably extended definition. A practical circuit that restores an approximate version of the signal from its sine-wave-crossings is presented and possible improvements to it are discussed.

Bar-David, I.↗

NASA NTRS1976

Lagrange duality theory for convex control problems

The Lagrange dual to a control problem is studied. The principal result based on the Hahn-Banach theorem proves that the dual problem has an optimal solution if there exists an interior point for the constraint set. A complementary slackness condition holds, if the primal problem has an optimal solution. A necessary and sufficient condition for the optimality of solutions to the primal and the dual problem is also presented.

Hager, W. W.↗

NASA NTRS2018

The Number of Support Constraints for Overlapping Set Optimization with Nested Admissible Sets Is Equal to One

This paper reports on the formalization of a recent result by Crespo, et al., as found in the references. The formalized result bounds the number of support constraints in a particular type of optimization problem. The problem involves discovering an optimal member of a family of sets that overlaps each member of a constraining collection of sets. The particular case addressed here concerns optimizations in which the family of sets is nested. The primary results were formalized in the interactive theorem prover PVS and support the claim that a single support constraint exists in very general circumstances.

Siratt, John↗

NASA NTRS1979

An algebraic structure of discrete-time biaffine systems

New results on the realization of finite-dimensional, discrete-time, internally biaffine systems are presented in this paper. The external behavior of such systems is described by multiaffine functions and the state space is constructed via Nerode equivalence relations. We prove that the state space is an affine space. An algorithm which amounts to choosing a frame for the affine space is presented. Our algorithm reduces in the linear and bilinear case to a generalization of algorithms existing in the literature. Explicit existence criteria for span-canonical realizations as well as an affine isomorphism theorem are given.

Tarn, T.-J.↗