Invariant Imbedding Method Research Articles

Trajectory estimation of manoeuvring re-entry vehicles and non-linear filtering

This paper presents a modified Sridhar filter which is applied to the problem of the real-time estimation of the trajectory of a manoeuvring re-entry vehicle (MARV) from its radar observations. The Sridhar filter is based on optimal control concepts, specifically the Pontryagin minimum principle and the method of invariant imbedding. In this approach the unknown forces on the MARV are treated as controls that drive the MARV dynamics to track the noisy observed trajectory. This treatment differs from that used with the extended Kalman filter (EKF), where a new state vector of the unknown forces acting on the MARV considered as Wiener processes is augmented to the MARV state vector. The Sridhar filter does not increase the MARV state vector so its computational time, for the cases studied, is about 20% that of the EKF. The performance, as measured by the point error, of the proposed approach is even better than that of the conventional EKF.

The design of a linear regulator based on an integral-equation representation

This paper presents an alternative on-line algorithm for calculating regulators of linear deterministic dynamical systems which minimize quadratic cost functions employing the invariant-imbedding method. The design scheme used for the optimum linear regulator is based on the integral-equation representation, which enables one to obtain the solution to the corresponding two-point boundary-value problem. The algorithm can be implemented in forward time without memory, unlike the conventional one which uses the Kalman gain function to calculate the feedback gain.

Solution of some problems of continuum mechanics by the method of invariant imbedding

Solution of some problems of continuum mechanics by the method of invariant imbedding

Low-angle tracking radars and non-linear time-delay estimation

A new approach, based on non-linear time-delay estimation, is described for estimating the elevation angle in the combined presence of specular multipath and receiver noise—a situation that is encountered in the low-angle tracking radar environment. Estimation of the elevation angle is accomplished using the time delay between the signals impinging on the antenna array elements. Thus, the accuracy in the elevation angle estimation is dependent on the accuracy of the time-delay estimation algorithm. The formulation of the problem enables one to find the desired estimates independently of the reflection coefficient, which is a significant result. In the proposed approach, a newly derived non-linear filter due to the author (1986) is used. The filter structure is estimated using the Pontryagin minimum principle and the method of invariant imbedding, and is used to find an estimate for the delayed and undelayed signal. Subsequently, the time delay and the elevation angle are estimated.

Target image tracking using a nonlinear filter based on Pontryagin minimum principle

The major contribution of this technical note is the introduction of a new nonlinear filter, based on Pontryagin minimum principle and the method of invariant imbedding, as applied to spatial filtering of FLIR image sequences. This filler is used to improve the signal-to-noise ratio (SNR) of each frame. The target center is then estimated using different approaches. It is shown, through Monte Carlo simulation, that the proposed methods improve the accuracy of the estimate of the target center by 100 pecent on the average, compared to the correlation algorithm.

New Results in Sridhar Filtering Theory

A nonlinear adaptive filter is introduced and applied to the classical problem of detecting a sinusoidal signal, with unknown frequency, in white noise. The filter is basically a new result in what is known as Sridhar filtering theory. In the derivation of the filter, called the “Pontryagin filter”, the Pontryagin minimum principle and the method of invariant imbedding are used. The stability, bias and convergence properties are also studied and presented.

Calculation of neutron albedos for slab with invariant imbedding method.

The albedo data are important for use in solving the radiation streaming problem with albedo techniques, such as albedo Monte Carlo and albedo-Sn methods. This paper describes a method for calculating the energy-angle dependent doubly differential albedos for slab geomery with one-dimensional transport theory, based on the invariant imbedding method as well as the Sn method. Neutron albedo data calculated by the invariant imbedding method, are compared with those calculated by the Sn method and with the experimental data. It is found that the invariant imbedding method can be used to calculate the albedo data for a semi-infinitely thick slab several dozens of times faster than when using the Sn method. The calculated results have excellent agreement with the measured values.

New efficient albedo Monte Carlo method for neutron streaming analyses.

An efficient albedo Monte Carlo method newly developed has been studied by analyzing two types of experiments on neutron streaming. The method is characterized by employing the energy-angle dependent doubly differential albedos for slab, which can be calculated in a short computer time with a one-dimensional transport theory, such as the Sn method and more efficient invariant imbedding method. This paper describes the features of the present albedo Monte Carlo method, including fundamental formulas. In the analyses of the neutron streaming experiments, the calculated results agreed with the measured data within a factor of 2 for a benchmark experiment at the YAYOI reactor and within a factor of 3 for an SNR sodium duct mock-up experiment. It is concluded that the present albedo Monte Carlo method is practical and applicable to the reactor shielding analysis concerning radiation streaming.

Cubic spline and Invariant Imbedding for solving singular two-point boundary value problems

In this paper, the authors present a cubic spline with Invariant Imbedding method for numerically solving linear two-point boundary value problems for certain ordinary differential equations having singular coefficients. These problems arise when reducing partial differential equations to ordinary differential equations by physical symmetry. To remove the singularity, the series solution is used in the neighborhood of the singular point and the boundary condition at a point x = δ away from the singularity is derived. The resulting regular boundary value problem is then efficiently treated by employing cubic splines coupled with Invariant Imbedding method for obtaining numerical solutions. The stability of some recurrence relations involved in the method is investigated. Some numerical experiments have been included and the comparison of the numerical results made with other methods.

Resistance fluctuation in a one-dimensional conductor with static disorder.

The non-self-averaging resistance of a one-dimensional conductor with static disorder is reexamined by the method of invariant imbedding, leading to a Fokker-Planck equation for its probability distribution ${W}_{\ensuremath{\rho}}(\ensuremath{\rho}, l)$, with varying sample length $l$. An exact two-point recursion relation for the moments $〈{\ensuremath{\rho}}^{n}〉$ is given along with a closed-form solution for ${W}_{\ensuremath{\rho}}(\ensuremath{\rho}, l)$ for the case of Gaussian white-noise disorder. The latter confirms $\mathrm{ln}\ensuremath{\rho}$ as the correct scale variable. The treatment admits generalization to the case of $N$ channels and to general disorder.

Discrete invariant imbedding for the numerical solution of boundary-value problems over infinite intervals

A numerical method to solve boundary-value problems posed on infinite intervals is given by reducing the infinite interval to a finite interval which is large, and impossing appropriate asymptotic boundary conditions at the far end. Then the two-point boundary-value problem is solved by using discrete invariant-imbedding method, which is also analyzed for its stability. The theory is illustrated by solving a test example.

Optical bistability in nonlinear media: An exact method of calculation

The method of invariant imbedding is used to obtain exact equations for reflection and transmission coefficients for a plane wave incident on a slab of nonlinear material. These equations are solved for thin slabs of InSb and clear bistability is obtained.

On the non-uniqueness. of non-linear wave solutions in a viscous layer

Solutions of the stationary travelling wave type are considered in draining layers of a viscous fluid. A one-parameter family of waves /1/ is studied that softly branches off into the upper branch of the neutral stability curve of the plane-parallel flow and goes over into a negative soliton (phase velocity c < 3) as the wave number tends to zero. It is shown that this family is not unique: for small values of the parameter δ characterizing the mass flow rate, a second and third family of waves branches off from it with half the period. The critical value δ = δ ∗ is found for which the bifurcation points of the second and third families merge, while for δ > δ ∗ they go into the complex plane; a dependence of the wave number on δ for which the bifurcation occurs is obtained analytically. The properties of the second family of the periodic wave and positive soliton type, for which c > 3 are studied. The solutions are constructed numerically: the periodic solutions are continued in the parameter from the bifurcation points or from the known solutions by using the method of invariant imbedding; the soliton solutions are constructed by joining the linear asymptotic forms as the values of the longitudinal coordinate tend to infinity.

An Approximation of Gamma-Ray Buildup Factors for Two-Layer Shields

An empirical formula for gamma-ray buildup factors in two-layer shields is proposed. The values of the parameters are given for the formula when fitted to the dose buildup factors, calculated by the invariant imbedding method, for normally incident gamma rays penetrating two-layer shields comprised of combinations of water, iron, or lead slabs. The results from the present formula are in excellent agreement with the basic data in the 0.66- to 10-MeV energy range and for total thicknesses up to about20 mfp. The parameters used in the formula change smoothly with the source energy. Therefore, the buildup factor for any arbitrary energy can easily be estimated by interpolation of the parameters with respect to energy. The buildup factors provided by this formula also are in good agreement with those from transport calculations for two-layer shields consisting of water and lead. These factors also agree with those measured by two ionization chambers for /sup 60/Co radiation penetrating two-layer shields comprised of water, iron, or lead slabs.

The design of on-line linear least-squares estimators given covariance specifications via an imbedding method

The method of invariant imbedding is employed to develop on-line algorithms for all types of linear least-squares estimators (filter; fixed-point, fixed-lag, and fixed-interval smoothers; fixed-point, fixed-lead, and fixed-interval predictors) using covariance specifications for nonstationary stochastic signals in the presence of white Gaussian plus colored noise. Two design procedures are given. One is based on the initial-value solution of the function I y ( t) which is the solution to a Fredholm integral equation of the second kind, and the other on that of the function h( t,s) which is the impulse response function obeyed by a generalized Wiener-Hopf integral equation.

On the choice of an appropriate imbedding algorithm for the solution of unstable linear boundary value problems

The method of Extended (or Generalized) Invariant Imbedding solves unstable linear boundary value problems in a stable manner. The method requires the integration of one of four different matrix Riccati differential equations (four different algorithms). In this contribution the questions of existence, stability, boundedness and error propagation of these Riccati equations are discussed in order to help the user to choose the appropriate algorithm.

On the solution of the linear‐quadratic optimal control problem by extended invariant imbedding

AbstractThe solution of the linear‐quadratic output regulator problem by Pontryagin's maximum principle results in an unstable linear boundary value problem. The method of invariant imbedding, which requires the integration of a matrix Riccati differential equation, solves certain boundary value problems in a stable manner. In order to make this well known method applicable, an extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. The resulting algorithm of extended invariant imbedding does not depend on the boundary values. So the matrix Riccati equation has to be integrated only once ‘offline’: The unknown boundary values, the state, and the control are calculated by solution of systems of linear equations (‘online’). So, if the boundary values change, only systems of linear equations have to be solved once more. The algorithm has two variants with usually contrary stability behaviour.

An eigenlength problem arising from a model of patch formation

The patch formation model of Del Grosso, Gerardi and Marchetti[1] is reformulated as an eigenlength problem. The latter is then solved numerically by using recent results of Elder[2, 3], in which the method of invariant imbedding is extended to linear problems having a singularity of the first kind. The results agree very well with those previously obtained[4] for the eigenvalue problem by means of the Frobenius series, Sturmian theory, and a certain monotone iteration.

Anomalous diffusion of liquids in glassy polymers

Anomalous diffusion of liquid penetrants in glassy polymers is formulated as a moving boundry problem. The concept of volume average velocity is utilized in mathematical modelling. The model equation is solved using a method which is a combination of the method of lines and the method of invariant imbedding. The results are shown to be in agreement with the existing experimental observations.

On the solution of linear boundary value problems by extended invariant imbedding

Unstable linear boundary value problems can be solved by the method of Invariant Imbedding in a stable manner. Instead of integration of the system equations this method requires the integration of a matrix Riccati equation, which depends on the boundary values of the problem. The dimension of the Riccati equation is determined by a suitable decoupling of the system equations. Invariant Imbedding now fails, if this decoupling does not correspond with the boundary condition. In addition, the Riccati equation has to be solved once more for each new boundary condition. An extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. This “extended” boundary value is solved by a modified Invariant Imbedding. The resulting “Extended (Dual) Invariant Imbedding” is always applicable and does not depend on the boundary conditions. The corresponding “extended” Riccati equation has to be integrated only once “offline”. If the boundary condition is changed, only systems of linear equations have to be solved “online”.