Jason L. Speyer

On Modeling and Pulse Phase Estimation of X-Ray Pulsars

Mathematical models are developed to characterize the X-ray pulsar signals, and the pulse phase estimation problem is addressed. The Cramér-Rao lower bound for estimation of the pulse phase is presented. Depending on employing the photon counts or direct use of the measured photon time of arrivals, two different estimation strategies are proposed and analyzed. In the first approach, utilizing the epoch folding procedure, the observed pulsar rate function on the detector is retrieved, and the pulse phase is estimated through a nonlinear least-squares fit of the empirical rate function to the known pulsar rate function. It is shown that this estimator is consistent, but not asymptotically efficient. In the second strategy, a maximum likelihood (ML) estimation problem is formulated using the probability density function of the photon time of arrivals. It is shown that the ML estimator is asymptotically efficient. Computational complexity of the proposed estimators is investigated as well. The analytical results are verified numerically via computer simulations.

Relative Navigation Between Two Spacecraft Using X-ray Pulsars

This paper suggests utilizing X-ray pulsars for relative navigation between two spacecraft in deep space. Mathematical models describing X-ray pulsar signals are presented. The pulse delay estimation problem is formulated, and the Cramér-Rao lower bound (CRLB) for estimation of the pulse delay is given. Two different pulse delay estimators are introduced, and their asymptotic performance is studied. Numerical complexity of each delay estimator, and the effect of absolute velocity errors on its performance is investigated. Using the pulsar measurements, a recursive algorithm is proposed for relative navigation between two spacecraft. The spacecraft acceleration data are provided by the inertial measurement units (IMUs). The pulse delay estimates are used as measurements, and based on models of the spacecraft and IMU dynamics, a Kalman filter is employed to obtain the 3-D relative position and velocity. Furthermore, it is shown that the relative accelerometer biases as well as the differential time between clocks can be estimated. Numerical simulations are also performed to assess the proposed navigation algorithm.

X-Ray Pulsar-Based Relative Navigation using Epoch Folding

How the relative position between two spacecraft can be estimated utilizing signals emitted from X-ray pulsars is explained. The mathematical models of X-ray pulsar signals are developed, and the pulse delay estimation problem is formulated. The Cramér-Rao lower bound (CRLB) for any unbiased estimator of the pulse delay is presented. To retrieve the pulsar photon intensity function, the epoch folding procedure is characterized. Based on epoch folding, two different pulse delay estimators are introduced, and their performance against the CRLB is studied. One is obtained by solving a least squares problem, and the other uses the cross correlation function between the empirical rate function and the true one. The effect of absolute velocity errors on position estimation is also studied. Numerical simulations are performed to verify the theoretical results.

Cauchy estimation for linear scalar systems

An estimation paradigm is presented for scalar discrete linear systems entailing additive process and measurement noises that have Cauchy probability density functions (pdf). For systems with Gaussian noises, the Kalman filter has been the main estimation paradigm. However, many practical system uncertainties that have impulsive character, such as radar glint, are better described by stable non-Gaussian densities, for example, the Cauchy pdf. Although the Cauchy pdf does not have a well defined mean and does have an infinite second moment, the conditional density of a Cauchy random variable, given its linear measurements with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. For a single measurement, simple expressions are obtained for the conditional mean and variance, by deriving closed form expressions for the infinite integrals associated with the minimum variance estimation problem. To alleviate the complexity of the multi-stage estimator, the conditional pdf is represented in a special factored form. A recursion scheme is then developed based on this factored form and closed form integrations, allowing for the propagation of the conditional mean and variance over an arbitrary number of time stages. In simulations, the performance of the newly developed scalar discrete-time Cauchy estimator is significantly superior to a Kalman filter in the presence of Cauchy noise, whereas the Cauchy estimator deteriorates only slightly compared to the Kalman filter in the presence of Gaussian noise. Remarkably, this new recursive Cauchy conditional mean estimator has parameters that are generated by linear difference equations with stochastic coefficients, providing computational efficiency.

Merging of Air Traffic Flows

The problem of merging two or more flows of aircraft in a horizontal plane is considered. The aircraft aim to reach a target flow with the minimal amount of turning, while maintaining a given separation between them. Neither a priori sequencing nor scheduling of the merging aircraft is assumed. The individual trajectories for each aircraft are computed using optimization, and a subset of these optimal trajectories serve as a basis for merging multiple aircraft. A second optimization step produces trajectories for each aircraft to ensure both effective merging into the flow and sufficient separation. This generates the sequencing of the merging aircraft. Simulations of several merging scenarios illustrate the approach and show its results. A statistical study of the influence of the problem parameters on the success of the merging algorithm provides a guideline for the design of the airspace around the merging area. The proposed algorithm can be used for automatic en route and terminal area merging tasks within future more automated air traffic control systems.

Particle Filters With Adaptive Resampling Technique Applied to Relative Positioning Using GPS Carrier-Phase Measurements

Particle filters are widely used when the system is nonlinear and non-Gaussian. In real-time applications, their estimation accuracy and efficiency are significantly affected by the number of particles. For a multivariate state, the appropriate number of particles is estimated adaptively for bounds on the error of the sample mean and variance that are given by the confidence range of a normal distributed probability. The resampling time is determined when the required sample number maintaining the variance accuracy becomes greater than the required sample number maintaining the mean accuracy. The Particle Filter with adaptive resampling is applied to the relative position estimation using GPS carrier-phase measurements. We estimate the probability density function of the position by sampling from the position space with the particle filter and resolve the ambiguity problem without any integer ambiguity search procedures. The adaptive resampling manages the number of position samples for real-time kinematics GPS navigation. The experimental results show the performance of the adaptive resampling technique and the insensitivity of the proposed approach to GPS carrier-phase cycle-slips.

State Estimation for Linear Scalar Dynamic Systems with Additive Cauchy Noises: Characteristic Function Approach

Uncertainties in many practical systems, such as radar glint and sonar noise, have impulsive character and are better described by heavy-tailed non-Gaussian densities, for example, the Cauchy probability density function (pdf). The Cauchy pdf does not have a well defined mean and its second moment is infinite. Nonetheless, the conditional density of a Cauchy random variable, given a scalar linear measurement with an additive Cauchy noise, has a conditional mean and a finite conditional variance. In particular, for scalar discrete linear systems with additive process and measurement noises described by Cauchy pdfs, the unnormalized characteristic function of the conditional pdf is considered. It is expressed as a growing sum of terms that at each measurement update increases by one term, constructed from four new measurement-dependant parameters. The dynamics of these parameters is linear. These parameters are shown to decay, allowing an approximate finite dimensional recursion. From the first two differen...

Empirical pseudo-balanced model reduction and feedback control of weakly nonlinear convection patterns

An empirical model reduction method is performed on nonlinear transient convection patterns near the threshold, using a pseudo-inverse-based projective method called the pseudo-balanced proper orthogonal decomposition (PBPOD). These transient patterns are large-scale amplitude/phase modulations in convection rolls, obtained by prescribing selected spatial input-shape functions. For the nonlinear convection patterns modelled, PBPOD appears to be very effective. Using the nonlinear front example, PBPOD is compared with other existing methods, such as POD and linearized BPOD. The limitations of the methods are discussed. Using a complex prescribed input, complex disturbances are generated and the outputs of the open-loop responses are compared between the original and the reduced-order models. The agreement is good. A feedback-control study is performed using the nonlinear front example. The controller is built by the pseudo-balanced reduced-order model, with a low-order nonlinear estimator. Closed-loop simulations show that the nonlinear travelling fronts can be effectively damped out by the feedback-control actions.

Mode Estimation via Conditionally Linear Filtering: Application to Gyro Failure Monitoring

This work introduces a novel approximate nonlinear filtering paradigm for nonlinear non-Gaussian discrete-time systems. Hinging on the concept of conditional orthogonality of random sequences, the conditionally linear filtering approach is an extension of the classical linear filtering approach. The resulting estimator turns out to be conditionally linear with respect to the most recent measurement, given the passed observations, which is adequate for online filtering. It is shown that the classical Kalman filter and the conditionally Gaussian filter, i.e., the optimal filter for conditionally linearGaussian systems, are special cases of the conditionally linear filter.When applied to the problem of mode estimation in jump-parameter systems with full state information, it provides an approximate nonlinear mode estimator that only requires the first two moments of the process noise, whereas the optimal filter (Wonham filter) requires the complete noise distribution. In the case of noisy measurements in jump-linear systems, the conditionally linear filtering approach lends itself to a meaningful hybrid estimator. The proposed hybrid estimator is successfully applied to the problem of gyro failure monitoring onboard spacecraft using noisy attitude quaternion measurements. Extensive Monte Carlo simulations show that the conditionally linear mode estimator outperforms the linear mode estimator. For Gaussian noises the conditionally linear filter and the optimalWonham filter perform similarly. Moreover, for Weibull-type noises, the conditionally linear filter outperforms a Wonham filter that assumes Gaussian noises.