Nonlinear Measurement Equations Research Articles

Tracking of a Maneuvering Target with Unknown Course and Speed Using Extended Kalman Filter

In this paper, an effective tracking method for a maneuvering target moving at unknown variable speed on the 2-dimensional plane is proposed. First, dynamics are derived for the target which moves at variable speed by introducing the first-order dynamics for the target's jerk motion as a modified Singer model. Next, in order to get highly accurate tracking, the kinematic constraints are incorporated into measurement equation as pseudomeasurements. Based on the extended Kalman filter algorithm, the tracking filter is derived from the linear dynamics and the augmented nonlinear measurement equation. The efficacy of this method is shown by simulations using real data.

Mixed coordinate tracking of generalized maneuvering targets using acceleration and jerk models

In a recent paper the authors introduced an all-Cartesian formulation of a jerk model for tracking highly maneuvering targets. Here a more complex but realistic case is considered, where target motion modeling and tracking are carried out in the 3-D Cartesian frame using measurements obtained in a spherical system. The transformation of the measurements into the Cartesian system results in nonlinear measurement equations. We solve the problem using an extended Kalman filter (EKF) approach, and also treat the earlier acceleration model similarly for comparison of results.

Genetic Algorithm Filter for Non-Linear State Space Estimation.

A new method for non-linear state space estimation using genetic algorithm is developed. This method is named the Genetic algorithm filter (GA filter). State vactors are coded with genes and their survival rates are decided by a probability density functions. A crowd of genes represents the characteristic of a current state of a system. GA are general techniques for doing numerical optimization and are based on stochastic simulation. Their strength is that they can be applied to problems where an analytical approach is difficult or impossible. GA filter can handle non-linear system equations and measurement equations as well as non-Gaussian and non-additive noise. Implementing the GA requires less analytical work than implementing the extended Kalman filter, but the GA filters are more computer intensive. We will demonstrate effectiveness of the resulting new filters with simple examples.

Conditional densities for continuous-time nonlinear hybrid systems with applications to fault detection

Continuous-time nonlinear stochastic differential state and measurement equations, all of which have coefficients capable of abrupt changes at a random time, are considered; finite-state jump Markov chains are used to model the changes. Conditional probability densities, which are essential in obtaining filtered estimates for these hybrid systems, are then derived. They are governed by a coupled system of stochastic partial differential equations. When the Q matrix of the Markov chain is either lower or upper diagonal, it is shown that the system of conditional density equations is finite-dimensional computable. These findings are then applied to a fault detection problem to compute state estimates that include the failure time.

Comments on "Discrete-time observability and estimability analysis for bearings-only target motion analysis"

Discusses the extensive mathematical analysis carried out by the authors of the original paper [see ibid., vol. 33, no. 1, p. 178-201, 1997] and submits the following points. The authors used pseudo measurements for recasting the observability problem into a linear framework. They treated the bearings-only passive target tracking system as a deterministic system. It is already established that for deterministic systems, the pseudo measurements are linear functions of the states of the system, though the coefficient matrix is a nonlinear function of the original measurements, By using the pseudo measurements in a linear observer, global stability can be shown. However, if the pseudo measurement observer, for which the analysis is mostly carried out by the authors, is used in a noisy environment as a pseudo measurement filter (PMF), biased estimates are arrived at. Hence, though the approach of authors is quite direct and provides insights about the algebraic structure of the BOT problem, as pseudo measurements are used throughout the analysis is not of much use to the TMA community, as the nonlinear measurement equation along with measurement noise are required to be considered in the BOT problem to obtain unbiased results.

Nonlinear filters based on taylor series expansions∗

We have been dealing with the linear transition and measurement equations. The equations derived from economic theories, however, are nonlinear in general. Unless the distributions are normal and the measurement and transition equations are linear, we cannot derive the explicit expression for the filtering algorithm. Therefore, some approximation is necessary for estimation. In Chapter 3, the nonlinear filtering algorithms are derived by approximating the nonlinear measurement and transition equations, based on the Taylor series expansion.

Passive Doppler-bearing tracking using a pseudo-linear estimator

In this paper an explicit pseudo-linear estimator for Doppler-bearing tracking is proposed. It overcomes the problems with the bias of earlier pseudo-linear estimators and with the nonlinear frequency measurement equation by using another representation than the Cartesian one and by using the logarithm of the frequency, respectively. It is fast, inherently stable, and easy to implement. The Doppler shift is, for a nonmoving own-ship, determined by the target velocities, while the bearing rate is determined by the same velocities divided by the range. The special representation in this paper uses this difference in behavior to give fast and bias free estimation of the range. Instead of iterating a weighted feast squares problem using bearing and frequency measurements simultaneously, the true bearings and the course are estimated in a bearings only step followed by a frequency only step, which estimates range and frequency. The range estimate then gives the speed estimate. Modifications for scenarios with multiple emitted frequency lines and/or for frequency lines that disappear during parts of a scenario are shown.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the optimal state estimation of a class of discrete-time nonlinear systems

The optimal linear estimator is given for a class of discretetime nonlinear stochastic systems with nonlinear noisy measurement equations. First, the one-step predictor is considered and then the limiting behavior of the estimator is examined.

Fundamental properties and performance of conventional bearings-only target motion analysis

This paper considers the problem of estimating the position and velocity of an object from noise corrupted bearing measurements obtained by a single moving observation platform. The process is inherently nonlinear and exhibits unusual observability properties that are geometry-dependent. A maximum likelihood estimate (MLE) of the target motion analysis solution is developed and its performance analyzed. A comparison is drawn between the MLE and two previously reported methods, a nonlinear modified-instrumental variable estimate (MIV) and the pseudo-linear estimate (PLE). Both the MIV and PLE are shown to derive from approximations to the nonlinear measurement equation and therefore share some common properties with the MLE. The limits on performance that can be expected from processing bearing data are detailed. Specifically, for long range-to-baseline geometries, approximate expressions for the Cramer-Rao bound are derived. Extension of the results to the practical filters approximately predicts numerically observed behavior. For less restrictive geometries, bounds are presented. Incorporation of a target speed constraint on the MLE results in a transition to a lower dimensional problem as noise level and range increases. Monte Carlo experimental results are presented and the improvements realized by the MLE techniques are evident.

Estimation Of Blood Vessel Boundaries In X-Ray Images

A new approach to blood vessel boundary estimation is presented in this paper. By appropriately modeling the blood vessel as a dynamically evolving state vector, and by taking into account the Poisson statistics of the x-ray imaging noise, we arrive at a state-space system with a nonlinear measurement equation which includes non-Gaussian, nonadditive noise. Maximum a posteriori (MAP) smoothing equations are derived for the state vector describing the vessel, and the optimally smoothed state vector is found by a dynamic programming search. This method performs especially well in images with low signal-to-noise ratio and low sampling rate. The performance of the proposed method is demonstrated by the boundary estimates obtained by applying the algorithm to a simulated vessel and measurement data as well as to real vessel phantom measurement data at various SNRs.

On-Line Tracking of Six Orbital Elements of a Thrust Maneuvering Space Vehicle

Abstract : The problem of on-line tracking of the state variables of a thrust maneuvering space vehicle in three dimensional space described by non-linear discrete dynamical and measurement equations is studied for several different estimation schemes. A second order filtering relation is derived for a discrete non-linear plant which adds bias correction terms to the quasilinear filter. The rapid estimation scheme for a nonlinear system is employed both to provide the starting algorithm and whenever the state jump is detected. The digital computer evaluation indicates that the estimates of the states by the combination of a second order filter with the rapid estimation scheme is superior to the results obtained from quasilinear filtering techniques for the application of tracking the thrust maneuvering space vehicle. (Author)

Can atomic processes be described by non-linear wave mechanics in space-time?

Generalizing from a classical application of the paradigm of Elementary Measurement discussed elsewhere (Leiter, 1969), we consider a non-linear, spinor wave-mechanical field theory of Elementary Measurement. In this theory, charged particles are represented by complex spinorc-number fields interacting through their associated electromagnetic fields in space-time. The paradigm of Elementary Measurement implies that the particle fields, and their associatedc-number electromagnetic fields, are interdependent degrees of freedom in an action principle associated with the measurement interaction, and are not elementary in themselves. Making the action stationary with respect to the interacting field degrees of freedom gives the equations of motion of the measurement. The application of this model theory to atomic hydrogen yields the result that the inherent ‘limit cycle solutions’ (LCS) of the non-linear measurement equations correspond to the quantum levels of conventional relativistic Dirac quantum mechanics of hydrogen, in the approximation that the nucleus has infinite mass. Superpositions of these Dirac-LCS solutions have the property of collapsing (reduction of the wave packet) into one of the LCS in the superposition,in a characteristic time which is identical to the ‘lifetime’ of the associated atomic levels as calculated from conventional quantum mechanics. Hence, in thisc-number electromagnetic theory,both spontaneous and induced transitions can be accounted for. ‘Photons’, in this theory, are not elementary particles, but instead are associated with the secondary dynamics related to the inherent nonlinear structure in the elementary measurement equations of motion. The ‘hidden variable’ characteristics of this measurement theory (as seen from the point of view of ordinary quantum mechanics), in describing a universe made up of such hydrogen atoms, is discussed. Within this context, a consistent derivation of the Planck blackbody radiation formula is given, in which the associated electromagnetic fields arec-numbers and arenot second quantized. Finally, a generalization of this prototype model theory, to a more consistent form which can account for the presence of ‘vacuum interaction processes’ and negative energy states, is suggested.