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NASA NTRS2009

Application of Bounded Linear Stability Analysis Method for Metrics-Driven Adaptive Control

This paper presents the application of Bounded Linear Stability Analysis (BLSA) method for metrics-driven adaptive control. The bounded linear stability analysis method is used for analyzing stability of adaptive control models, without linearizing the adaptive laws. Metrics-driven adaptive control introduces a notion that adaptation should be driven by some stability metrics to achieve robustness. By the application of bounded linear stability analysis method the adaptive gain is adjusted during the adaptation in order to meet certain phase margin requirements. Analysis of metrics-driven adaptive control is evaluated for a second order system that represents a pitch attitude control of a generic transport aircraft. The analysis shows that the system with the metrics-conforming variable adaptive gain becomes more robust to unmodeled dynamics or time delay. The effect of analysis time-window for BLSA is also evaluated in order to meet the stability margin criteria.

Bakhtiari-Nejad, Maryam↗

NASA NTRS2009

Adjustment of Adaptive Gain with Bounded Linear Stability Analysis to Improve Time-Delay Margin for Metrics-Driven Adaptive Control

This paper presents the application of Bounded Linear Stability Analysis (BLSA) method for metrics driven adaptive control. The bounded linear stability analysis method is used for analyzing stability of adaptive control models, without linearizing the adaptive laws. Metrics-driven adaptive control introduces a notion that adaptation should be driven by some stability metrics to achieve robustness. By the application of bounded linear stability analysis method the adaptive gain is adjusted during the adaptation in order to meet certain phase margin requirements. Analysis of metrics-driven adaptive control is evaluated for a linear damaged twin-engine generic transport model of aircraft. The analysis shows that the system with the adjusted adaptive gain becomes more robust to unmodeled dynamics or time delay.

Bakhtiari-Nejad, Maryam↗

NASA NTRS1986

Polynomial elimination theory and non-linear stability analysis for the Euler equations

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Kennon, S. R.↗

NASA NTRS1990

A code for linear stability analysis

A new spectral code, Linear.x, has been written in FORTRAN 77 for the analysis of the linear stability of some basic state. While being generic and unrelated to any particular physical problem, the code provides for various common tasks, including global (eigenvalue spectra), local (single eigenvalues and eigenfunctions), table (one-dimensional and multidimensional tables of eigenvalues), curve (curves in parameter space), and others. A specific problem can be defined as a set of files some of which are included at compile time, while definitions, tasks, and parameters are read during run time. The code is currently used for the stability analysis of compressible flows.

Herbert, Thorwald↗

NASA NTRS1985

Linearization of digital derived rate algorithm for use in linear stability analysis

The digital derived rate (DDR) algorithm is used to calculate the rate of rotation of the Centaur upper-stage rocket. The DDR is highly nonlinear algorithm, and classical linear stability analysis of the spacecraft cannot be performed without linearization. The performance of this rate algorithm is characterized by a gain and phase curve that drop off at the same frequency. This characteristic is desirable for many applications. A linearization technique for the DDR algorithm is investigated. The linearization method is described. Examples of the results of the linearization technique are illustrated, and the effects of linearization are described. A linear digital filter may be used as a substitute for performing classical linear stability analyses, while the DDR itself may be used in time response analysis.

Graham, R. E.↗

NASA NTRS1985

Linearization of digital derived rate algorithm for use in linear stability analysis

Graham, R. E.↗

NASA NTRS2009

Bounded Linear Stability Analysis - A Time Delay Margin Estimation Approach for Adaptive Control

This paper presents a method for estimating time delay margin for model-reference adaptive control of systems with almost linear structured uncertainty. The bounded linear stability analysis method seeks to represent the conventional model-reference adaptive law by a locally bounded linear approximation within a small time window using the comparison lemma. The locally bounded linear approximation of the combined adaptive system is cast in a form of an input-time-delay differential equation over a small time window. The time delay margin of this system represents a local stability measure and is computed analytically by a matrix measure method, which provides a simple analytical technique for estimating an upper bound of time delay margin. Based on simulation results for a scalar model-reference adaptive control system, both the bounded linear stability method and the matrix measure method are seen to provide a reasonably accurate and yet not too conservative time delay margin estimation.

Nguyen, Nhan T.↗

NASA NTRS1992

Linear stability analysis of spherical accretion flows onto compact objects

The steady state structure and linear stability of spherical accretion flows onto compact objects are investigated over a wide range of accretion rates (ARs) for a variety of cooling functions as a function of the ratio of specific heats (gamma). Both radial and nonradial shock perturbations are considered. The flows become more extended as AR decreases. The cooling function has a significant effect on the extended structure of settling solutions with gamma = 5/3. In contrast, the extended structure of settling solutions with gamma = 4/3 is nearly independent of the form of the cooling function. For both gamma = 4.3 and gamma = 5.3, radial shock oscillation in the fundamental and first overtone modes are destabilized in spherically extended accretion envelopes. The application of these results to postsupernova neutron star accretion flows and to luminosity oscillations in AM Her objects are discussed.

Houck, John C.↗

NASA NTRS1973

Linearized stability analysis and design of a flyback dc-dc boost regulator.

Analytic expressions for the small-signal power-stage describing functions of a switched dc-dc boost regulator are derived from an approximate continuous circuit model which is developed by a time-averaging technique. Closed-loop stability is attained through the design of frequency compensation of the loop gain. Open- and closed-loop regulator output impedances are derived from the linearized models for the given configuration. The analysis and design are compared with and confirmed by breadboard measurements.

Wester, G. W.↗

NASA NTRS1995

A parametric study of supersonic laminar flow for swept wings using linear stability analysis

A parametric study to predict the extent of laminar flow on the upper surface of a generic swept-back wing (NACA 64A010 airfoil section) at supersonic speeds was conducted. The results were obtained by using surface pressure predictions from an Euler/Navier-Stokes computational fluid dynamics code coupled with a boundary layer code, which predicts detailed boundary layer profiles, and finally with a linear stability code to determine the extent of laminar flow. The parameters addressed are Reynolds number, angle of attack, and leading-edge wing sweep. The results of this study show that an increase in angle of attack, for specific Reynolds numbers, can actually delay transition. Therefore, higher lift capability, caused by the increased angle of attack, as well as a reduction in viscous drag due to the delay in transition is possible for certain flight conditions.

Cummings, Russell M.↗

NASA NTRS1986

Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

Kaup, D. J.↗

NASA NTRS1997

Structure of the Compressible Reacting Mixing Layer: A Linear Stability Analysis

Previous investigations have demonstrated that a mixing layer in compressible reacting conditions can develop two instability modes in addition to the more common central mode, which exists unaccompanied in incompressible non-reacting flows. These 'outer' modes were shown in the investigations of Planche and Reynolds to have a detrimental effect on mixing and combustion efficiency and therefore their presence has important consequences for applications of supersonic combustion. Results from a parametric study of the compressible reacting mixing layer's regime space using a linear stability technique will be discussed. The purpose is to develop a more complete understanding behind the effects of compressibility, heat release and the density, equivalence and velocity ratios on the amplification rate and structure of each instability mode.

Day, M. J.↗

NASA NTRS1987

Linear stability analysis of three-dimensional compressible boundary layers

A compressible stability analysis computer code is developed. The code uses a matrix finite-difference method for local eigenvale solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used inital-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed that may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.

Malik, Mujeeb R.↗

NASA NTRS2005

Linear Stability Analysis of Gravitational Effects on a Low-Density Gas Jet Injected into a High-Density Medium

The objective of this study was to determine the effects of buoyancy on the absolute instability of low-density gas jets injected into high-density gas mediums. Most of the existing analyses of low-density gas jets injected into a high-density ambient have been carried out neglecting effects of gravity. In order to investigate the influence of gravity on the near-injector development of the flow, a spatio-temporal stability analysis of a low-density round jet injected into a high-density ambient gas was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The Briggs-Bers criterion was combined with the spatio-temporal stability analysis to determine the nature of the absolute instability of the jet whether absolutely or convectively unstable. The roles of the density ratio, Froude number, Schmidt number, and the lateral shift between the density and velocity profiles on the absolute instability of the jet were determined. Comparisons of the results with previous experimental studies show good agreement when the effects of these variables are combined together. Thus, the combination of these variables determines how absolutely unstable the jet will be.

Lawson, Anthony L.↗

NASA NTRS1988

Onset of finger convection in a horizontal porous layer underlying a fluid layer

The problem of the onset of finger convection in a porous layer underlying a fluid layer is considered using linear stability analysis. The linear stability equations for the porous layer are formulated for temperature and salinity gradients existing in both layers. The eigenvalue problem is solved by a shooting method. The solution method and associated computer program are validated by comparison with the results of Sun (1973) for the thermal convection case. Results are also presented for the onset of salt-finger convection.

Chen, F.↗

NASA NTRS2008

Bounded Linear Stability Margin Analysis of Nonlinear Hybrid Adaptive Control

This paper presents a bounded linear stability analysis for a hybrid adaptive control that blends both direct and indirect adaptive control. Stability and convergence of nonlinear adaptive control are analyzed using an approximate linear equivalent system. A stability margin analysis shows that a large adaptive gain can lead to a reduced phase margin. This method can enable metrics-driven adaptive control whereby the adaptive gain is adjusted to meet stability margin requirements.

Nguyen, Nhan T.↗

NASA NTRS1982

Thresholds for the onset of fluid and magnetofluid turbulence

Linear stability calculations conducted in plasma physics are based on the theory of hydrodynamic stability of neutral fluids. The present investigation is concerned with the validity of procedures based on linear stability analysis in six much-studied situations. All have the property that the fluid goes from laminar to turbulent at critical values of some dimensionless number, such as the Reynolds number or the Rayleigh number. The question is, at what threshold values of the dimensionless number do the unstable motions set in, and can these thresholds be predicted by a linear analysis of the stability of the laminar state as the threshold is approached from the stable side. In three cases linear stability analysis clearly seems to fail. These cases include Plane Poiseuille Flow, Plane Couette Flow, and Cylindrical Pipe Flow (Hagen-Poiseuille Flow). Situations in which predictions provided by linear stability analysis are correct are related to Rotating Couette Flow, Thermally-Driven Convection, and Instability of Laminar Boundary Layers.

Montgomery, D.↗

NASA NTRS1984

Microsegregation during directional solidification

During the directional solidification of alloys, solute inhomogeneities transverse to the growth direction arise due to morphological instabilities (leading to cellular or dendritic growth) and/or due to convection in the melt. In the absence of convection, the conditions for the onset of morphological instability are given by the linear stability analysis of Mullins and Sekerka. For ordinary solidification rates, the predictions of linear stability analysis are similar to the constitutional supercooling criterion. However, at very rapid solidification rates, linear stability analysis predicts a vast increase in stabilization in comparison to constitutional supercooling.

Coriell, S. R.↗