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NASA NTRS1984

Input-output description of linear systems with multiple time-scales

It is pointed out that the study of systems evolving at multiple time-scales is simplified by studying reduced-order models of these systems valid at specific time-scales. The present investigation is concerned with an extension of results on the time-scale decomposition of autonomous systems to that of input-output systems. The results are employed to study conditions under which positive realness of a transfer function is preserved under singular perturbation. Attention is given to the perturbation theory for linear operators, the multiple time-scale structure of autonomous linear systems, the input-output description of two time-scale linear systems, the positive realness of two time-scale systems, and multiple time-scale linear systems.

Madriz, R. S.↗

NASA NTRS1986

Control design for robust stability in linear regulators: Application to aerospace flight control

Time domain stability robustness analysis and design for linear multivariable uncertain systems with bounded uncertainties is the central theme of the research. After reviewing the recently developed upper bounds on the linear elemental (structured), time varying perturbation of an asymptotically stable linear time invariant regulator, it is shown that it is possible to further improve these bounds by employing state transformations. Then introducing a quantitative measure called the stability robustness index, a state feedback conrol design algorithm is presented for a general linear regulator problem and then specialized to the case of modal systems as well as matched systems. The extension of the algorithm to stochastic systems with Kalman filter as the state estimator is presented. Finally an algorithm for robust dynamic compensator design is presented using Parameter Optimization (PO) procedure. Applications in a aircraft control and flexible structure control are presented along with a comparison with other existing methods.

Yedavalli, R. K.↗

NASA NTRS1986

Generalized energetics for linear and nonlinear symmetric instabilities

The generalized energy conservation integral is applied to linear and nonlinear symmetric instabilities (SIs) and conditional symmetric instabilities (CSIs) in uniform and nonuniform basic states. The generalized energy conservation integral for linear SI in nonuniform basic state is refined to include initial thermal-inertial perturbations. A generalized conservation integral is derived for a nonlinear SI with uniform and nonuniform basic states. The integral reveals that: (1) the stability conditions of nonlinear and linear SI are not the same; (2) when the basic state is symmetrically stable the nonlinear evolution of a symmetric disturbance is energetically bounded by its initial energy; and (3) when the basic state is symmetrically unstable the growth of the symmetric perturbation is approximated by the linear SI theory. Precipitative heating is incorporated into the generalized energy integral for nonlinear CSI; the relation between nonlinear evolution of a CSI circulation and precipitative heating is examined.

Xu, Q.↗

NASA NTRS1987

Piecewise linear approximation for hereditary control problems

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Propst, Georg↗

NASA NTRS1987

A linearized Euler analysis of unsteady flows in turbomachinery

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗

NASA NTRS1986

Evaluation of multi-spectral linear array detectors for spaceborne applications

A NASA laboratory for evaluating linear detector arrays for remote sensing applications from the visible to shortwave infrared is described. Nominal requirements are presented on the critical parameters of signal-to-noise ratio, modulation transfer function, response linearity, dynamic range, and detector-to-detector response uniformity. Results of tests on a custom developed multi-spectral linear array from RCA and two commercial arrays from Texas Instruments and Reticon for the visible and near-infrared are presented. The custom array was clearly superior in signal-to-noise ratio and dynamic range to its off-the-shelf counterparts. Allthe arrays were found to have somewhat less than desirable modulation transfer functions, but excellent response linearity.

Kalshoven, James E., Jr.↗

NASA NTRS1988

A feedback linearization approach to spacecraft control using momentum exchange devices

Recent developments in the area of nonlinear control theory have shown how coordiante changes in the state and input spaces can be used with nonlinear feedback to transform certain nonlinear ordinary differential equations into equivalent linear equations. These feedback linearization techniques are applied to resolve two problems arising in the control of spacecraft equipped with control moment gyroscopes (CMGs). The first application involves the computation of rate commands for the gimbals that rotate the individual gyroscopes to produce commanded torques on the spacecraft. The second application is to the long-term management of stored momentum in the system of control moment gyroscopes using environmental torques acting on the vehicle. An approach to distributing control effort among a group of redundant actuators is described that uses feedback linearization techniques to parameterize sets of controls which influence a specified subsystem in a desired way. The approach is adapted for use in spacecraft control with double-gimballed gyroscopes to produce an algorithm that avoids problematic gimbal configurations by approximating sets of gimbal rates that drive CMG rotors into desirable configurations. The momentum management problem is stated as a trajectory optimization problem with a nonlinear dynamical constraint. Feedback linearization and collocation are used to transform this problem into an unconstrainted nonlinear program. The approach to trajectory optimization is fast and robust. A number of examples are presented showing applications to the proposed NASA space station.

Dzielski, John Edward↗

NASA NTRS1988

A study of the use of linear programming techniques to improve the performance in design optimization problems

This project has two objectives. The first is to determine whether linear programming techniques can improve performance when handling design optimization problems with a large number of design variables and constraints relative to the feasible directions algorithm. The second purpose is to determine whether using the Kreisselmeier-Steinhauser (KS) function to replace the constraints with one constraint will reduce the cost of total optimization. Comparisons are made using solutions obtained with linear and non-linear methods. The results indicate that there is no cost saving using the linear method or in using the KS function to replace constraints.

Young, Katherine C.↗

NASA NTRS1988

Application of linear response theory to experimental data of simultaneous radiation and annealing response of a CMOS device

Results from the application of linear response theory are compared to experimental data from simultaneous radiation and annealing response of a CMOS device. In particular, a method is applied which was developed earlier to determine the characteristic time, t(0), as well as the parameters A and C in the 1n(t) dependence of the linear response function R(t) = -C + A1n(1-t/t(0)). The method is based on a study of the linear response for t being much less than t(0), when R(t) can be expanded in a power series of t: R(t) = R(0) + R'(0)t + 1/2R''(0)t-squared + 1/3R'''(0)t-cubed + ..., where R'(0) and R''(0) are, respectively, the first and second derivatives of R with respect to t. To find the linear response, R(t-t') is substituted in the form of this power series equation into a general equation for the shift of the threshold potential. To test the method, irradiation experiments were conducted on RCA 10(6) rad-hard CMOS IC's. A dose rate of approximately 130 rads/min was used. An IC was irradiated with Co-60 gamma rays for several hours, taking measurements of the threshold potential for one n-channel and one p-channel transistor every ten minutes. For the p-channel transistor, t(0) was found to be approximately 110 min and for the n-channel, t(0) was approximately 70 min. For the p-channel, the theoretical curve deviates from the experimental points only after 70 min; for the n-channel, the deviation takes place after 45 min. Additional findings are discussed and the application of the method to pure annealing is described.

Litovchenko, V.↗

NASA NTRS1989

Equivalent linearization for fatigue life estimates of a nonlinear structure

An analysis is presented of the suitability of the method of equivalent linearization for estimating the fatigue life of a nonlinear structure. Comparisons are made of the fatigue life of a nonlinear plate as predicted using conventional equivalent linearization and three other more accurate methods. The excitation of the plate is assumed to be Gaussian white noise and the plate response is modeled using a single resonant mode. The methods used for comparison consist of numerical simulation, a probabalistic formulation, and a modification of equivalent linearization which avoids the usual assumption that the response process is Gaussian. Remarkably close agreement is obtained between all four methods, even for cases where the response is significantly linear.

Miles, R. N.↗

NASA NTRS1989

Studies with spike initiators - Linearization by noise allows continuous signal modulation in neural networks

It is shown that noise can be an important element in the translation of neuronal generator potentials (summed inputs) to neuronal spike trains (outputs), creating or expanding a range of amplitudes over which the spike rate is proportional to the generator potential amplitude. Noise converts the basically nonlinear operation of a spike initiator into a nearly linear modulation process. This linearization effect of noise is examined in a simple intuitive model of a static threshold and in a more realistic computer simulation of spike initiator based on the Hodgkin-Huxley (HH) model. The results are qualitatively similar; in each case larger noise amplitude results in a larger range of nearly linear modulation. The computer simulation of the HH model with noise shows linear and nonlinear features that were earlier observed in spike data obtained from the VIIIth nerve of the bullfrog. This suggests that these features can be explained in terms of spike initiator properties, and it also suggests that the HH model may be useful for representing basic spike initiator properties in vertebrates.

Yu, Xiaolong↗

NASA NTRS1989

Equivalent linearization for fatigue life estimates of a nonlinear structure

An analysis is presented of the suitability of the method of equivalent linearization for estimating the fatigue life of a nonlinear structure. Comparisons are made of the fatigue life of a nonlinear plate as predicted using conventional equivalent linearization and three other more accurate methods. The excitation of the plate is assumed to be Gaussian white noise and the plate response is modeled using a single resonant mode. The methods used for comparison consist of numerical simulation, a probabilistic formulation, and a modification of equivalent linearization which avoids the usual assumption that the response process is Gaussian. Remarkably close agreement is obtained between all four methods, even for cases where the response is significantly linear.

Miles, R. N.↗

NASA NTRS1989

Calculation of unsteady flows in turbomachinery using the linearized Euler equations

Hall, Kenneth C.↗

NASA NTRS1989

Linear and nonlinear responses to middle latitude surface temperature anomalies

Nonlinear responses to fixed and seasonally varying surface temperature anomalies in a two-level hemispheric time-dependent coupled atmosphere-surface mixed layer model are described. Linear stationary models that are equivalent to the nonlinear time-dependent model are used to analyze these responses. A model linearized around the climatological zonal state of the time dependent model and forced by anomalous surface temperatures does not provide a reasonable estimate for the anomalous reponses, which are considerably underestimated. Better responses are obtained when the anomalous stationary nonlinear eddy fluxes are included in a model linearized around the full climatology. However, this latter model is overly sensitive, and anomalous responses are a small residual balance to the forcing by the surface temperature anomalies and the anomalous transient eddy fluxes. To better understand these linear responses, an eigenanalysis of the climatological state is performed. Seasonal anomalies appear to be dominated by one characteristic pattern near resonance which can be associated with a slowly growing coupled atmosphere-ocean instability.

Roads, John O.↗

NASA NTRS1990

Fan beam generated by a linear-array fed parabolic reflector

The theoretical background and the results of computer simulations and experimental studies for a parabolic reflector fed by a linear array are detailed. The concept of using a parabolic reflector antenna fed by a small linear array to generate fan-beam patterns is validated. Large angle scan along the broad-beam direction of the fan beam can be achieved by offsetting the linear array laterally. It is both empirically and numerically demonstrated that the array feed must be displaced in the reflector's axial direction to an optimum location from the focal plane in order to achieve the best antenna gain performance. As a result, the linear-array-fed parabolic reflector can be used in place of a long planar array in a multifunctional reflector antenna system.

Huang, John↗

NASA NTRS1991

Five degree-of-freedom control of an ultra-precision magnetically-suspended linear bearing

The authors constructed a high precision linear bearing. A 10.7 kg platen measuring 125 mm by 125 mm by 350 mm is suspended and controlled in five degrees of freedom by seven electromagnets. The position of the platen is measured by five capacitive probes which have nanometer resolution. The suspension acts as a linear bearing, allowing linear travel of 50 mm in the sixth degree of freedom. In the laboratory, this bearing system has demonstrated position stability of 5 nm peak-to-peak. This is believed to be the highest position stability yet demonstrated in a magnetic suspension system. Performance at this level confirms that magnetic suspensions can address motion control requirements at the nanometer level. The experimental effort associated with this linear bearing system is described. Major topics are the development of models for the suspension, implementation of control algorithms, and measurement of the actual bearing performance. Suggestions for the future improvement of the bearing system are given.

Trumper, David L.↗

NASA NTRS1991

Feedback linearization for control of air breathing engines

The method of feedback linearization for control of the nonlinear nozzle and compressor components of an air breathing engine is presented. This method overcomes the need for a large number of scheduling variables and operating points to accurately model highly nonlinear plants. Feedback linearization also results in linear closed loop system performance simplifying subsequent control design. Feedback linearization is used for the nonlinear partial engine model and performance is verified through simulation.

Phillips, Stephen↗

NASA NTRS1991

Prediction of unsteady aerodynamic loads in cascades using the linearized Euler equations on deforming grids

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. Using the linearized Euler technique, one decomposes the flow into a mean (or steady) flow plus an unsteady, harmonically varying, small disturbance flow. Linear variable coefficient equations describe the small disturbance behavior of the flow, and are solved using a pseudo-time marching Lax-Wendroff scheme. For the blade motion problem, a harmonically deforming computational rid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition. The paper also presents a new, numerically exact, nonreflecting far-field boundary condition based on an eigenanalysis of the discretized equations. Computed flow solutions demonstrate the computational accuracy and efficiency of the present method. The solution of the linearized Euler equations requires one to two orders of magnitude less computer time than solution of the nonlinear Euler equations using traditional time-accurate time-marching techniques. In addition, the deformable grid significantly improves the accuracy of the solution.

Hall, Kenneth C.↗