Showing papers on "Linearization published in 1980"

Goodness of Link Tests for Generalized Linear Models

Abstract: SUMMARY Data analytic procedures are proposed to examine the adequacy of the hypothesized link used in fitting a generalized linear model. Through model expansion and linearization, tests and estimation techniques are provided. These procedures, along with the release of GLIM3, enable the user to examine routinely and objectively the fit of an hypothesized model. Examples are presented to illustrate the testing and fitting procedure.

The mixed discrete‐continuous inverse problem: Application to the simultaneous determination of earthquake hypocenters and velocity structure

Abstract: Some inverse problems are characterized by a model consisting of a piecewise continuous function and a set of discrete parameters. For linear problems of this general type, which we call mixed, we show that when the number of data d is greater than the number of parameters p, it is always possible to construct a set of a least d - p equations that are independent of the values of the discrete part of the model. These equations, which we call the annulled data set, can be used to estimate the continuous part of the model. The discrete part of the model can be estimated from a second set of p equations that relate the discrete and continuous parts of the model. The linearization of the nonlinear travel time functional that enter in the hypocenter location problem leads to a mixed inverse problem. The splitting procedure is natural to this problem if the hypocenters are estimated initially by conventional nonlinear least squares by using travel times calculated from some initial estimate of the velocity model. The annulled data are a set of linear combinations of the residuals that are unbiased by that initial location, and as a result, they can be used directly to estimate a perturbation to the velocity model by a Backus-Gilbert procedure. This makes an iterative algorithm possible that consists of a conventional hypocenter location followed by estimating a perturbation of the velocity model from the annulled data set. The uniqueness of the final velocity model is assessed via the linear resolution analysis of Backus and Gilbert (1968, 1970). We also construct a set of Frechet derivatives that relate perturbations of each hypocenter component to perturbations of the velocity model. These kernels are used to assess the possible error of the hypocenters due to inadequate knowledge of the velocity structure by an application of the generalized prediction approach of Backus (1970a). Good results are obtained when the procedure is applied to a simple synthetic data set.

Real‐time forecasting with a conceptual hydrologic model: 1. Analysis of uncertainty

Abstract: The optimal control of watershed systems requires accurate real-time short-term forecasts of river flows. For the first time, this paper formulates a large, nonlinear conceptual model (the National Weather Service catchment model) in a mode amenable to analysis of uncertainty and the utilization of real-time information (measurements, forecasts, guesses) to update system states and improve streamflow predictions. The proposed methodology is based on the state space formulation of the equations describing the hydrologic model and the assumption of sources of uncertainty in the data and in the model structure. The first two moments of random variables are estimated in a computationally efficient way using on-line linear estimation techniques. Linearization of functional relationships is performed with the uncommon but powerful multiple-input describing function technique for the most strongly nonlinear responses and Taylor expansion for the rest. The linear feedback rule developed is based on the Kalman filter.

Studies on Power System Load Flow Equivalencing

Abstract: This paper reviews the principles and main methods for load-flow equivalencing, with due emphasis given to on-line applications. Unreduced Load Flow, Ward, REI, Linearization, and Identification methods are covered, introducing a number of new ideas and versions. A critical analysis of the properties of the different methods is made, to assist in choosing the most suitable ones for practical application.

Dynamical Systems and Evolution Equations: Theory and Applications

Abstract: I. Evolution Equations on ?n.- 1. The Space ?n.- 2. Evolution Equations on ?n.- 3. Finite Escape Times.- 4. Processes on ?sun X ?.- 5. Linear Dynamical Systems.- 6. The Simplest Class of Processes.- 7. Stability of a Particular Motion: Processes.- 8. Stability of Equilibrium: Dynamical Systems.- 9. Stability of Equilibrium: Linear Dynamical Systems.- 10. Quadratic Liapunov Functions and Linearization.- 11. The Invariance Principle and Asymptotic Behavior.- 12. Comments and Extensions.- References.- II. Preliminaries for Abstract Evolution Equations.- 1. Abstract Spaces.- 2. Functions.- 3. Linear Functions.- 4. Differentiation of Functions.- 5. Abstract Evolution Equations.- References.- III. Abstract Dynamical Systems and Evolution Equations.- 1. Dynamical Systems and C0-Semigroups.- 2. Linear Dynamical Systems.- 3. Generation of Linear Dynamical Systems.- 4. Choosing the State Space in Applications.- 5. Generation of Nonlinear Dynamical Systems.- 6. Comments and Extensions.- References.- IV. Some Topological Dynamics.- 1. Liapunov Functions and Positive Invariance.- 2. Computation of V.- 3. Stability and Liapunov's Direct Method.- 4. Positive Limit Sets and the Invariance Principle.- 5. Orbital Precompactness and Use of the Invariance Principle.- 6. Comments and Extensions.- References.- V. Applications and Special Topics.- 1. A Feedback Control Problem.- 2. The Thermoelastic Stability Problem.- 3. The Viscoelastic Stability Problem.- 4. A Fission Reactor Stability Problem.- 5. A Supersonic Panel-Flutter Problem.- 6. Discrete Dynamical Systems.- 7. Finite-Dimensional Approximation.- References.

Equivalent linearization for systems subjected to non-stationary random excitation

Abstract: The method of eauivalent linearization is applied to the general problem of the response of non-linear discrete systems to non-stationary random excitation. Conditions for minimum equation difference are determined which do not depend explicitly on lime but only on the instantaneous statistics of the response process. Using the equivalent linear parameters, a deterministic non-linear ordinary differential equation for the covariance matrix is derived. An example is given of a damped Duffing oscillator subjected to modulated white noise.

Linear Theory Methods for Pipe Network Analysis

Abstract: A previously unpublished numerical method for the steady-state solution of the flow distribution in a pipe network is presented. It relies on an iterative linearization procedure to solve for junction heads, and is highly suited for use on small computers. Its relationship to other solution methods is explicitly developed. This new method is extremely simple to formulate, and requires minimum data preorganization for computation.

Electroacoustic equations for one‐domain ferroelectric bodies

Abstract: Based on a fully Galilean electrodynamics of dielectric deformable bodies, a completely deductive derivation of the coupled, dynamical, electromagnetic, electroacoustic, and energetic equations and constitutive equations for elastic, pyroelectric, ferroelectric dielectrics accommodating viscosity and dielectric‐relaxation phenomena is presented. Of necessity the theory is first constructed in a fully nonlinear framework even though applications to signal processing are envisaged. Then a linearization is performed by Lagrangian variation about an initial ferroelectric state in which there exist no strains, but there are present both stresses and local electric field as well as an initial electric polarization. The latter provokes a breaking of the ideal symmetry of the material, so that finally linearized coupled electroacoustic and heat‐propagation equations are deduced on which can be based the study of coupled acoustic‐soft optic modes in ferroelectrics of the BaTiO3‐type. No thermodynamical restriction...

Implicit computations of unsteady transonic flow governed by the full-potential equation in conservation form

Abstract: An alternating-direction implicit algorithm is presented for solving the conservative, full-potential equation for unsteady, transonic flow. A new development is the time-linearization of the density function. This linearization reduces the solution process from one of solving a system of two equations at each mesh point to one of solving a single equation. Two sample cases are computed. First, a one-dimensional traveling shock wave is computed and compared with the analytic solution. Second, a two-dimensional case is computed of a flow field that results from a thickening and subsequently thinning airfoil. The resulting flow field, which includes a traveling shock wave, is compared to the flow field obtained from the low-frequency, small-disturbance, transonic equation.

Analysis of Boundary Value Problems on Infinite Intervals

Abstract: In this paper boundary value problems of ordinary differential equations on infinite intervals are analysed. There is a theory for problems of this kind which requires the fundamental matrix of the system of differential equations to have certain decay properties near infinity. The aim of this paper is to establish a theory which holds under weaker assumptions. The analysis for linear problems is done by determining the fundamental matrix of the system of differential equations asymptotically. For inhomogeneous problems a suitable particular solution having a “nice” asymptotic behaviour is chosen and so global existence and uniqueness theorems are established in the linear case. The asymptotic behaviour of this solution follows immediately. Nonlinear problems are treated by using perturbation techniques meaning linearization near infinity and by using the methods for the linear case. Moreover, some problems from fluid dynamics and thermodynamics are dealt with and they illustrate the power of the asymptot...

Propagation of High-Frequency Electromagnetic Waves Through a Magnetized Plasma in Curved Space-Time. I

Abstract: This is the first of two papers on the propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. We first show that the nonlinear system of equations governing the plasma and the electromagnetic field in a given, external gravitational field has locally a unique solution for any initial data set obeying the appropriate constraints, and that this system is linearization stable at any of its solu­tions. Next we prove that the linearized perturbations of a ‘background’ solution are characterized by a third-order (not strictly) hyperbolic, constraint-free system of three partial differential equations for three unknown functions of the four space-time coordinates. We generalize the algorithm for obtaining oscillatory asymptotic solutions of linear systems of partial differential equations of arbitrary order, depending polynomially on a small parameter such that it applies to the previously established perturbation equation when the latter is rewritten in terms of dimensionless variables and a small scale ratio. For hyperbolic systems we then state a sufficient condition in order that asymptotic solutions of finite order, constructed as usual by means of a Hamiltonian system of ordinary differential equations for the characteristic strips and a system of transport equations determining the propagation of the amplitudes along the rays, indeed approximate solutions of the system. The pro­cedure is a special case of a two-scale method, suitable for describing the propagation of locally approximately plane, monochromatic waves through a dispersive, inhomogeneous medium. In the second part we shall apply the general method to the perturbation equation referred to above.

Flutter analysis of an airplane with multiple structural nonlinearities in the control system

Abstract: Experience has shown that the flutter prediction process for airplanes can be greatly affected by strong concentrated nonlinearities which may be localized in the linking elements of the control mechanism, in the pivot joints of variable-sweep-wing systems, and in the connecting points between wing and pylon-mounted external stores. The principle of equivalent linearization offers an efficent possibility for solving the related nonlinear flutter equations in the frequency domain as a complement to the well-known time domain procedures. Taking as an example an airplane with nonlinear control characteristics, it is demonstrated how the equivalent linearization approach can be extended to rather complicated systems with multiple sets of strongly interacting, concentrated nonlinearities.

Servovalve flow linearization circuit

Abstract: A linearization circuit for use with servovalves for controlling reciprocating actuators. The circuit will maintain a substantially linear flow as a function of spool position from the servovalve regardless of the differential pressure across the servovalve orifices, for example the differential pressures which result from changing direction of movement of the controlled actuator under a high load. A differential pressure signal from the actuator, which correlates to differential pressure on the servovalve orifices is processed to provide a compensation signal combined with the normal control signal to achieve the desired output.

Slurry Flow in Pipe Networks

Abstract: The basic equations for steady-state flow of a solid-liquid slurry in a network of pipes are developed. The equations are expressed in terms of unknown flow rates and solids concentrations for each pipe in the network. Two sets of simultaneous algebraic equations are obtained. These are nonlinear and a direct solution of the equation sets is not possible. A procedure for solving the equations based on linearization of nonlinear terms is descirbed. This procedure requires several trials and converges rapidly. A computer program was developed based on the proposed method and an example system is analyzed using this program and the results are presented.

Rounding Error Analysis of Elementary Numerical Algorithms

Abstract: The paper presents a new rounding error analysis of product and summation algorithms, Horner’s scheme, evaluations of finite continued fractions, computations of determinants of tridiagonal systems, of determinants of second order and a ’fast’ complex multiplication. The error analysis uses the linearization method and new condition numbers constituting optimal bounds in appraisals of the possible errors. The new error estimates are tested by numerous numerical examples.

Computer control algorithms for a tubular ammonia reactor

Abstract: The specific objective of this paper is to develop direct digital control strategies for an ammonia reactor using the quadratic regulator theory and compare the performance of the resultant control system with that under conventional PID regulators. The controller design studies are based on a ninth-order state-space model obtained from the exact nonlinear distributed model using linearization and lumping approximations. The evaluation of these controllers, with reference to their disturbance rejection capabilities and transient response characteristics, is carried out using hybrid computer simulation. The liner state-space model is simulated on the analog processor of the AD511 hybrid computer and the control algorithms are implemented on the digital computer.

Partitioning identification algorithms

Abstract: In this paper, batch processing partitioning parameter identification agorithms are obtained using the "partitioning" approach to estimation. The algorithms, herein denoted the GPIA's, are applicable to linear as well as nonlinear systems and are derived by a natural applicalton of the generalized partitioned algorithms (GPA's) of Lainiotis; namely, by selecting a natural partitioning of the augmented state vector (the system state and unknown parameters); by linearization of the model equations; and then by using, in an iterative fashion, the GPA algorithms for the augmented state. The relationships between the GPIA's and maximum-likelihood identification methods, which employ gradient based numerical techniques to obtain a solution, are also established. An example of the application of the GPIA to aircraft parameter identification from actual flight test data is presented, as well as a direct comparison with the results obtaining using an iterated extended Kalman filter algorithm.