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At least 19 records

Complex symmetric matrices with strongly stable iterates

Complex-valued symmetric matrices are studied. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. A sharp criterion is provided for identifying those symmetric matrices whose spectral norm is not exceeding one: such strongly stable matrices are usually sought in connection with convergent difference approximations to partial differential equations. As an example, the derived criterion is applied to conclude the strong stability of a Lax-Wendroff scheme.

Tadmor, E.↗

Stability bounds for the control of large space structures

Balas (1977) has discussed the stability problem of reduced-order regulators and estimators in terms of control and observation 'spillover'. The term 'control spillover' was used to define that part of the feedback control which excites the uncontrolled (or residual) modes, and 'observation spillover' was used to define that part of the measurement which is contaminated by residual modes. In this paper, two sufficient conditions are derived via Lyapunov methods for asymptotic stability of large space structures using a class of reduced-order controllers. These conditions give allowable bounds on the spectral norms of control and observation spillover terms. The sufficient condition given by a specified inequality equation appears to be less conservative, and should be useful as a design tool for the control of large space structures.

Joshi, S. M.↗

Robustness of multiloop linear feedback systems

This paper presents a new approach to the frequency-domain analysis of multiloop linear feedback systems. The properties of the return difference equation are examined using the concepts of singular values, singular vectors and the spectral norm of a matrix. A number of new tools for multiloop systems are developed which are analogous to those for scalar Nyquist and Bode analysis. These provide a generalization of the scalar frequency-domain notions such as gain, bandwidth, stability margins and M-circles, and provide considerable insight into system robustness.

Doyle, J. C.↗

New insights in the numerical reliability properties of existing Kalman filter implementations

The convergence properties of Kalman filter algorithms are investigated analytically. A theoretical error analysis is performed on four types of algorithms, as defined by Anderson and Moore (1979): (1) a conventional Kalman filter, (2) a square-root (SR) covariance filter, (3) the Chandrasekhar SR filter, and (4) an SR information filter. The derivations are given in detail, and numerical results for the flight-path reconstruction problem studied by Verhaegen (1987) are presented in tables and graphs. It is shown that error propagation in algorithms (1) and (2) is sensitive to the condition number of the innovation-signal covariance matrix and the spectral norm of the filter state-transition matrix, whereas other parameters are dominant in (3) and (4). Filter (2) is found to be the most reliable for the class of problems studied.

Verhaegen, M. H.↗

A linear programming approach to characterizing norm bounded uncertainty from experimental data

The linear programming spectral overbounding and factorization (LPSOF) algorithm, an algorithm for finding a minimum phase transfer function of specified order whose magnitude tightly overbounds a specified nonparametric function of frequency, is introduced. This method has direct application to transforming nonparametric uncertainty bounds (available from system identification experiments) into parametric representations required for modern robust control design software (i.e., a minimum-phase transfer function multiplied by a norm-bounded perturbation).

Scheid, R. E.↗

NASA/SPoRT's GOES-R Activities in Support of Product Development, Management, and Training

The NASA Short-term Prediction Research and Transition (SPoRT) Center supports many activities within the GOES-R Proving Grounds (PG). These include the development of imagery from existing instrumentation as a proxy to future Advanced Baseline Imager (ABI) capabilities on GOES-R. The Moderate Resolution Imaging Spectroradiometer (MODIS) and the Visible/Infrared Imager/Radiometer Suite (VIIRS) instruments are used to provide a glimpse of the multi-spectral capabilities that will become the norm as the number of channels and data rate dramatically increase with GOES-R. The NOAA/NWS has plans to provide operational users with all ABI channels at the highest resolution. Data fusion of individual channels into composite red, green, and blue imagery products will assist the end user with this future wave of information. While increasing the efficiency in the operational use of ABI channels, these composites provide only qualitative information. Within the GOES-R PG, SPoRT and other partners are exploring ways to include quantitative information as part of the composite imagery. However, limitations in local hardware processing and/or data bandwidth for users of the GOES-R data stream are challenges to overcome. This presentation will discuss the creation of these composite images as well as possible solutions to address these processing challenges. In a similar manner the Geostationary Lightning Mapper (GLM) to be launched on GOES-R presents several data management challenges. The GLM is a pioneering instrument to quantify total lightning from a geostationary platform. The expected data frequency from the GLM is to be at a sub-minute interval. Users of such a data set may have little experience in handling such a rapid update of information. To assist users, SPoRT is working with the NWS to develop tools within the user fs decision support system to allow tracking and analysis of total lightning from a storm-based perspective. This presentation will discuss the challenges and progress of this tool development work. With new data and products comes the need for user Training. Within the GOES-R PG SPoRT is supporting the demonstration of these future products by providing various training materials to end users. A summary of training provided to operational users will be discussed.

Fuell, Kevin K.↗

Robust control of systems with real parameter uncertainty and unmodelled dynamics

During this research period we have made significant progress in the four proposed areas: (1) design of robust controllers via H infinity optimization; (2) design of robust controllers via mixed H2/H infinity optimization; (3) M-delta structure and robust stability analysis for structured uncertainties; and (4) a study on controllability and observability of perturbed plant. It is well known now that the two-Riccati-equation solution to the H infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H infinity controllers if the optimal H infinity norm or gamma, an upper bound of a suboptimal H infinity norm, is given. In this research, we discovered some useful properties of these H infinity Riccati solutions. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma in the domain of interest. Based on these properties, quadratically convergent algorithms are developed to compute the optimal H infinity norm. We also set up a detailed procedure for applying the H infinity theory to robust control systems design. The desire to design controllers with H infinity robustness but H(exp 2) performance has recently resulted in mixed H(exp 2) and H infinity control problem formulation. The mixed H(exp 2)/H infinity problem have drawn the attention of many investigators. However, solution is only available for special cases of this problem. We formulated a relatively realistic control problem with H(exp 2) performance index and H infinity robustness constraint into a more general mixed H(exp 2)/H infinity problem. No optimal solution yet is available for this more general mixed H(exp 2)/H infinity problem. Although the optimal solution for this mixed H(exp 2)/H infinity control has not yet been found, we proposed a design approach which can be used through proper choice of the available design parameters to influence both robustness and performance. For a large class of linear time-invariant systems with real parametric perturbations, the coefficient vector of the characteristic polynomial is a multilinear function of the real parameter vector. Based on this multilinear mapping relationship together with the recent developments for polytopic polynomials and parameter domain partition technique, we proposed an iterative algorithm for coupling the real structured singular value.

Chang, Bor-Chin↗

On convexity of H-infinity Riccati solutions and its applications

The celebrated two-Riccati-equation solution to a standard H-infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H-infinity controllers if the optimal H-infinity norm or gamma, an upper bound of a suboptimal H-infinity norm, is given. In this note, some properties of these H-infinity Riccati solutions are revealed. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma on the domain of interest. Based on these properties, a quadratically convergent algorithm is developed to compute the optimal H-infinity norm.

Li, X. P.↗

Stability analysis of spectral methods for hyperbolic initial-boundary value systems

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations.

Gottlieb, D.↗

An adaptive pseudo-spectral method for reaction diffusion problems

The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.

Bayliss, A.↗

An adaptive pseudo-spectral method for reaction diffusion problems

The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.

Bayliss, A.↗

Environmental processes and spectral reflectance characteristics associated with soil erosion in desert fringe regions

Two Thematic Mapper (TM) scenes were acquired. A scene was acquired for the Bahariya, Egypt field area, and one was acquired covering the Okavango Delta site. Investigations at the northwest Botswana study sites have concentrated upon a system of large linear (alab) dunes possessing an average wavelength of 2 kilometers and an east-west orientation. These dunes exist to the north and west of the Okavango Swamp, the pseudodeltaic end-sink of the internal Okavango-Cubango-Cuito drainage network. One archival scene and two TM acquisitions are on order, but at present no TM data were acquired for the Tombouctou/Azaouad Dunes, Mali. The three areas taken together comprise an environmental series ranging from hyperarid to semi-arid, with desertization processes operational or incipient in each. The long range goal is to predict normal seasonal variations, so that aperiodic spectral changes resulting from soil erosion, vegetation damage, and associated surface processes would be distinguishable as departures from the norm.

Jacobberger, P. A.↗

The Observed Spectral Evolution of Solar Flare Hard X-Ray Emission

The spectral evolution of flare emission may be characterized using color-color diagrams (CCDs), a technique which has been widely employed by the astrophysics community, but not yet by the solar flare community. CCDs are constructed for a sample of flares observed simultaneously by CGRO/BATSE and Yohkoh/HXT. It is found that flare spectral evolution follows one of only a few patterns, which generally evolve differently than the soft-hard-soft pattern put forth as the norm in previous work. The implications for the nature of flare energy release and acceleration/propagation models are discussed

Newton, Elizabeth↗

The Waypoint Planning Tool: Real Time Flight Planning for Airborne Science

NASA Earth science research utilizes both spaceborne and airborne real time observations in the planning and operations of its field campaigns. The coordination of air and space components is critical to achieve the goals and objectives and ensure the success of an experiment. Spaceborne imagery provides regular and continual coverage of the Earth and it is a significant component in all NASA field experiments. Real time visible and infrared geostationary images from GOES satellites and multi-spectral data from the many elements of the NASA suite of instruments aboard the TRMM, Terra, Aqua, Aura, and other NASA satellites have become norm. Similarly, the NASA Airborne Science Program draws upon a rich pool of instrumented aircraft. The NASA McDonnell Douglas DC-8, Lockheed P3 Orion, DeHavilland Twin Otter, King Air B200, Gulfstream-III are all staples of a NASA's well-stocked, versatile hangar. A key component in many field campaigns is coordinating the aircraft with satellite overpasses, other airplanes and the constantly evolving, dynamic weather conditions. Given the variables involved, developing a good flight plan that meets the objectives of the field experiment can be a challenging and time consuming task. Planning a research aircraft mission within the context of meeting the science objectives is complex task because it is much more than flying from point A to B. Flight plans typically consist of flying a series of transects or involve dynamic path changes when "chasing" a hurricane or forest fire. These aircraft flight plans are typically designed by the mission scientists then verified and implemented by the navigator or pilot. Flight planning can be an arduous task requiring frequent sanity checks by the flight crew. This requires real time situational awareness of the weather conditions that affect the aircraft track. Scientists at the University of Alabama-Huntsville and the NASA Marshall Space Flight Center developed the Waypoint Planning Tool, an interactive software tool, that enables scientists to develop their own flight plans (also known as waypoints) with point-and-click mouse capabilities on a digital map draped with real time satellite imagery. The Waypoint Planning Tool has further advanced to include satellite orbit predictions and seamlessly interfaces with the Real Time Mission Monitor which tracks the aircraft s position when the planes are flying. This presentation will describe the capabilities and features of the Waypoint Planning Tool highlighting the real time aspect, interactive nature and the resultant benefits to the airborne science community.

He, Yubin↗

Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

Parsani, Matteo↗

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Gottlieb, David↗

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Gottlieb, David↗

A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

Bey, Kim S.↗