An algorithm for the solution of dynamic linear programs

The algorithm's objective is to efficiently solve Dynamic Linear Programs (DLP) by taking advantage of their special staircase structure. This algorithm constitutes a stepping stone to an improved algorithm for solving Dynamic Quadratic Programs, which, in turn, would make the nonlinear programming method of Successive Quadratic Programs more practical for solving trajectory optimization problems. The ultimate goal is to being trajectory optimization solution speeds into the realm of real-time control. The algorithm exploits the staircase nature of the large constraint matrix of the equality-constrained DLPs encountered when solving inequality-constrained DLPs by an active set approach. A numerically-stable, staircase QL factorization of the staircase constraint matrix is carried out starting from its last rows and columns. The resulting recursion is like the time-varying Riccati equation from multi-stage LQR theory. The resulting factorization increases the efficiency of all of the typical LP solution operations over that of a dense matrix LP code. At the same time numerical stability is ensured. The algorithm also takes advantage of dynamic programming ideas about the cost-to-go by relaxing active pseudo constraints in a backwards sweeping process. This further decreases the cost per update of the LP rank-1 updating procedure, although it may result in more changes of the active set that if pseudo constraints were relaxed in a non-stagewise fashion. The usual stability of closed-loop Linear/Quadratic optimally-controlled systems, if it carries over to strictly linear cost functions, implies that the saving due to reduced factor update effort may outweigh the cost of an increased number of updates. An aerospace example is presented in which a ground-to-ground rocket's distance is maximized. This example demonstrates the applicability of this class of algorithms to aerospace guidance. It also sheds light on the efficacy of the proposed pseudo constraint relaxation scheme

Development of control strategies for safe microburst penetration: A progress report

A single-engine, propeller-driven, general-aviation model was incorporated into the nonlinear simulation and into the linear analysis of root loci and frequency response. Full-scale wind tunnel data provided its aerodynamic model, and the thrust model included the airspeed dependent effects of power and propeller efficiency. Also, the parameters of the Jet Transport model were changed to correspond more closely to the Boeing 727. In order to study their effects on steady-state repsonse to vertical wind inputs, altitude and total specific energy (air-relative and inertial) feedback capabilities were added to the nonlinear and linear models. Multiloop system design goals were defined. Attempts were made to develop controllers which achieved these goals

Optimization of aircraft trajectories through severe microbursts

A method of defining performance envelopes for aircraft microburst penetration is being developed. A trajectory is computed for a given aircraft/control law configuration and given microburst parameters (either a downdraft or a head/tailwind type microburst). The maximum deviation from the nominal altitude is recorded for that trajectory. Then the microburst parameters are varied, and the process is repeated. Thus a three dimensional plot of maximum altitude deviation versus microburst range scale and intensity is generated. Finally, a certain maximum altitude deviation, say 50 feet, is defined as the safe penetration limit. Then the 50 foot level contour becomes the performance limit for safe operation as a function of microburst intensity and range. Control inputs from deterministic trajectory optimization are used in the above described calculations to define the maximal performance limit. These limits provide targets for the designer of practical control laws

Optimal aircraft performance during microburst encounter

The effects of microburst characteristics on the optimal penetration performance of jet transport and general aviation aircraft are presented. The purpose is to determine the best possible performance that can be achieved in a broad range of microbursts. A secondary goal is to illustrate good strategies for dealing with a range of microbursts during takeoff and landing. Over 1100 optimal trajectories were computed for two aircraft types flying through idealized microbursts using a Successive Quadratic Programs trajectory optimization algorithm. Contours of safety metrics are plotted as functions of the length scales, magnitudes, and locations of horizontal wind shears and vertical downdrafts. These performance contours show three length-scale regimes for optimal microburst penetration. At short length scales, hazards usually associated with gustiness predominate (e.g., high normal load factor, rotational upset). At intermediate length scales, a degraded ability to maintain flight path and/or vertical velocity poses the most serious threat. At very long microburst length scales, excessive touchdown velocities may result. The ability to transit a microburst successfully also varies strongly with microburst location. The results show that both aircraft types could penetrate some very severe microbursts if optimal control histories were followed. Nevertheless, these control strategies assume perfect prior knowledge of the wind, and practical limits to successful encounter with real-time control capabilities would be lower. The optimally controlled jet transport can successfully penetrate higher intensity microbursts than can the general aviation aircraft

A Technique for Determining the Carrier Phase Differences between Independent GPS Receivers during Scintillation

A method for recovering the carrier phase differences between pairs of independent GPS receivers has been developed and demonstrated in truth-model simulations. This effort is in support of a project that intends to image the disturbed ionosphere with diffraction tomography techniques using GPS measurements from large arrays of receivers. Carrier phase differential GPS techniques, common in surveying and relative navigation, are employed to determine the phase relationships between the receivers in the imaging array. Strategies for estimating the absolute carrier phase disturbances at each receiver are discussed. Simulation results demonstrate that the system can rapidly detect the onset of scintillation, identify one non-scintillating reference signal, and recover the carrier phase differences accurate to 0.1 cycles.Aerospace Engineering and Engineering Mechanic

Real-time trajectory optimization on parallel processors

A parallel algorithm has been developed for rapidly solving trajectory optimization problems. The goal of the work has been to develop an algorithm that is suitable to do real-time, on-line optimal guidance through repeated solution of a trajectory optimization problem. The algorithm has been developed on an INTEL iPSC/860 message passing parallel processor. It uses a zero-order-hold discretization of a continuous-time problem and solves the resulting nonlinear programming problem using a custom-designed augmented Lagrangian nonlinear programming algorithm. The algorithm achieves parallelism of function, derivative, and search direction calculations through the principle of domain decomposition applied along the time axis. It has been encoded and tested on 3 example problems, the Goddard problem, the acceleration-limited, planar minimum-time to the origin problem, and a National Aerospace Plane minimum-fuel ascent guidance problem. Execution times as fast as 118 sec of wall clock time have been achieved for a 128-stage Goddard problem solved on 32 processors. A 32-stage minimum-time problem has been solved in 151 sec on 32 processors. A 32-stage National Aerospace Plane problem required 2 hours when solved on 32 processors. A speed-up factor of 7.2 has been achieved by using 32-nodes instead of 1-node to solve a 64-stage Goddard problem

Safe microburst penetration techniques: A deterministic, nonlinear, optimal control approach

A relatively large amount of computer time was used for the calculation of a optimal trajectory, but it is subject to reduction with moderate effort. The Deterministic, Nonlinear, Optimal Control algorithm yielded excellent aircraft performance in trajectory tracking for the given microburst. It did so by varying the angle of attack to counteract the lift effects of microburst induced airspeed variations. Throttle saturation and aerodynamic stall limits were not a problem for the case considered, proving that the aircraft's performance capabilities were not violated by the given wind field. All closed loop control laws previously considered performed very poorly in comparison, and therefore do not come near to taking full advantage of aircraft performance

GPS Carrier Tracking Loop Performance in the presence of Ionospheric Scintillations

The performance of several GPS carrier tracking loops is evaluated using wideband GPS data recorded during strong ionospheric scintillations. The aim of this study is to determine the loop structures and parameters that enable good phase tracking during the power fades and phase dynamics induced by scintillations. Constant-bandwidth and variable-bandwidth loops are studied using theoretical models, simulation, and tests with actual GPS signals. Constant-bandwidth loops with loop bandwidths near 15 Hz are shown to lose phase lock during scintillations. Use of the decision-directed discriminator reduces the carrier lock threshold by ∼1 dB relative to the arctangent and conventional Costas discriminators. A proposed variablebandwidth loop based on a Kalman filter reduces the carrier lock threshold by more than 7 dB compared to a 15-Hz constant-bandwidth loop. The Kalman filter-based strategy employs a soft-decision discriminator, explicitly models the effects of receiver clock noise, and optimally adapts the loop bandwidth to the carrier-to-noise ratio. In extensive simulation and in tests using actual wideband GPS data, the Kalman filter PLL demonstrates improved cycle slip immunity relative to constant bandwidth PLLs.Aerospace Engineering and Engineering Mechanic

Three-axis attitude determination via Kalman filtering of magnetometer data

A three-axis Magnetometer/Kalman Filter attitude determination system for a spacecraft in low-altitude Earth orbit is developed, analyzed, and simulation tested. The motivation for developing this system is to achieve light weight and low cost for an attitude determination system. The extended Kalman filter estimates the attitude, attitude rates, and constant disturbance torques. Accuracy near that of the International Geomagnetic Reference Field model is achieved. Covariance computation and simulation testing demonstrate the filter's accuracy. One test case, a gravity-gradient stabilized spacecraft with a pitch momentum wheel and a magnetically-anchored damper, is a real satellite on which this attitude determination system will be used. The application to a nadir pointing satellite and the estimation of disturbance torques represent the significant extensions contributed by this paper. Beyond its usefulness purely for attitude determination, this system could be used as part of a low-cost three-axis attitude stabilization system

Analysis of Ionospheric Scintillations using Wideband GPS L1 C/A Signal Data

A non-real-time GPS receiver has been developed and tested for use in scintillation analysis. The receiver consists of a digital storage receiver and non-real-time software acquisition and tracking algorithms. The goal of this work is to shed light on the behavior of strongly scintillating signals: signals which cause conventional GPS receivers to lose carrier lock. The receiver collects wideband GPS L1 digital data sampled at 5.7 MHz using an RF front-end and stores it on disk for post-processing. It processes the data off-line to determine carrier signal amplitude and phase variations during scintillations. The main processing algorithms are traditional code delay and carrier frequency acquisition algorithms and special signal processing algorithms that effectively function as a delay-locked loop and phase-locked loop. The tracking algorithms use non-causal smoothing techniques in order to optimally reconstruct the phase and amplitude variations of a scintillating signal. These techniques are robust against the deep power fades and strong phase fluctuations characteristic of scintillating signals. To test the receiver, scintillation data were collected in Cauchoeira Paulista, Brazil, from December 4 to 6, 2003. The data set spans several hours and includes times when one or more satellite signals are scintillating. The smoothing algorithm has been used to determine the carrier amplitude and phase time histories of the scintillating signals along with the distortion of the pseudorandom noise (PRN) code’s autocorrelation function. These quantities provide a characterization of scintillation that can be used to study the physics of scintillations or to provide off-line test cases to evaluate a tracking algorithm’s ability to maintain signal lock during scintillations.Aerospace Engineering and Engineering Mechanic