Study of non-linear optimization techniques
Nonlinear optimization techniques in dynamic programming and solution of ordinary nonlinear differential equations by Runge-Kutta method
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Nonlinear optimization techniques in dynamic programming and solution of ordinary nonlinear differential equations by Runge-Kutta method
Finite dimensional approximations of optimal nonlinear filters for class of large-noise, nonlinear, continuous-time problems
Design study for optimal nonlinear receiver demodulator
Recent work with NASA's Jet Propulsion Laboratory has allowed for external access to five of JPL's real-world requirements models, anonymized to conceal proprietary information, but retaining their computational nature. Experimentation with these models, reported herein, demonstrates a dramatic speedup in the computations performed on them. These models have a well defined goal: select mitigations that retire risks which, in turn, increases the number of attainable requirements. Such a non-linear optimization is a well-studied problem. However identification of not only (a) the optimal solution(s) but also (b) the key factors leading to them is less well studied. Our technique, called KEYS, shows a rapid way of simultaneously identifying the solutions and their key factors. KEYS improves on prior work by several orders of magnitude. Prior experiments with simulated annealing or treatment learning took tens of minutes to hours to terminate. KEYS runs much faster than that; e.g for one model, KEYS ran 13,000 times faster than treatment learning (40 minutes versus 0.18 seconds). Processing these JPL models is a non-linear optimization problem: the fewest mitigations must be selected while achieving the most requirements. Non-linear optimization is a well studied problem. With this paper, we challenge other members of the PROMISE community to improve on our results with other techniques.
Telban and Cardullo have developed and successfully implemented the non-linear optimal motion cueing algorithm at the Visual Motion Simulator (VMS) at the NASA Langley Research Center in 2005. The latest version of the non-linear algorithm performed filtering of motion cues in all degrees-of-freedom except for pitch and roll. This manuscript describes the development and implementation of the non-linear optimal motion cueing algorithm for the pitch and roll degrees of freedom. Presented results indicate improved cues in the specified channels as compared to the original design. To further advance motion cueing in general, this manuscript describes modifications to the existing algorithm, which allow for filtering at the location of the pilot's head as opposed to the centroid of the motion platform. The rational for such modification to the cueing algorithms is that the location of the pilot's vestibular system must be taken into account as opposed to the off-set of the centroid of the cockpit relative to the center of rotation alone. Results provided in this report suggest improved performance of the motion cueing algorithm.
Signal and noise problem as solution to nonlinear stochastic differential equations
A new technique which converts a constrained optimization problem to an unconstrained one where conflicting figures of merit may be simultaneously considered has been combined with a complex mission analysis system. The method is compared with existing single and multiobjective optimization methods. A primary benefit from this new method for multiobjective optimization is the elimination of separate optimizations for each objective, which is required by some optimization methods. A typical wide body transport aircraft is used for the comparative studies.
Several parallel-vector computational improvements to the unconstrained optimization procedure are described which speed up the structural analysis-synthesis process. A fast parallel-vector Choleski-based equation solver, pvsolve, is incorporated into the well-known SAP-4 general-purpose finite-element code. The new code, denoted PV-SAP, is tested for static structural analysis. Initial results on a four processor CRAY 2 show that using pvsolve reduces the equation solution time by a factor of 14-16 over the original SAP-4 code. In addition, parallel-vector procedures for the Golden Block Search technique and the BFGS method are developed and tested for nonlinear unconstrained optimization. A parallel version of an iterative solver and the pvsolve direct solver are incorporated into the BFGS method. Preliminary results on nonlinear unconstrained optimization test problems, using pvsolve in the analysis, show excellent parallel-vector performance indicating that these parallel-vector algorithms can be used in a new generation of finite-element based structural design/analysis-synthesis codes.
There are indications that persistent contrails can lead to adverse climate change, although the complete effect on climate forcing is still uncertain. A flight trajectory optimization algorithm with fuel and contrails models, which develops alternative flight paths, provides policy makers the necessary data to make tradeoffs between persistent contrails mitigation and aircraft fuel consumption. This study develops an algorithm that calculates wind-optimal trajectories for cruising aircraft while avoiding the regions of airspace prone to persistent contrails formation. The optimal trajectories are developed by solving a non-linear optimal control problem with path constraints. The regions of airspace favorable to persistent contrails formation are modeled as penalty areas that aircraft should avoid and are adjustable. The tradeoff between persistent contrails formation and additional fuel consumption is investigated, with and without altitude optimization, for 12 city-pairs in the continental United States. Without altitude optimization, the reduction in contrail travel times is gradual with increase in total fuel consumption. When altitude is optimized, a two percent increase in total fuel consumption can reduce the total travel times through contrail regions by more than six times. Allowing further increase in fuel consumption does not seem to result in proportionate decrease in contrail travel times.
Aircraft flying over the airspace of different countries are subject to over-flight charges. These charges vary from country to country. Airspace charges, while necessary to support the communication, navigation and surveillance services, may lead to aircraft flying routes longer than wind-optimal routes and produce additional carbon dioxide and other gaseous emissions. This paper develops an optimal route between city pairs by modifying the cost function to include an airspace cost whenever an aircraft flies through a controlled airspace without landing or departing from that airspace. It is assumed that the aircraft will fly the trajectory at a constant cruise altitude and constant speed. The computationally efficient optimal trajectory is derived by solving a non-linear optimal control problem. The operational strategies investigated in this study for minimizing aircraft fuel burn and emissions include flying fuel-optimal routes and flying cost-optimal routes that may completely or partially reduce airspace charges en route. The results in this paper use traffic data for transatlantic flights during July 2012. The mean daily savings in over-flight charges, fuel cost and total operation cost during the period are 17.6 percent, 1.6 percent, and 2.4 percent respectively, along the cost- optimal trajectories. The transatlantic flights can potentially save $600,000 in fuel cost plus $360,000 in over-flight charges daily by flying the cost-optimal trajectories. In addition, the aircraft emissions can be potentially reduced by 2,070 metric tons each day. The airport pairs and airspace regions that have the highest potential impacts due to airspace charges are identified for possible reduction of fuel burn and aircraft emissions for the transatlantic flights. The results in the paper show that the impact of the variation in fuel price on the optimal routes is to reduce the difference between wind-optimal and cost-optimal routes as the fuel price increases. The additional fuel consumption is quantified using the 30 percent variation in fuel prices during March 2014 to March 2015.
The applicability and usefulness of several classical and other methods for solving the two-point boundary-value problem which arises in non-linear singularly perturbed optimal control are assessed. Specific algorithms of the Picard, Newton and averaging types are formally developed for this class of problem. The computational requirements associated with each algorithm are analysed and compared with the computational requirement of the method of matched asymptotic expansions. Approximate solutions to a linear and a non-linear problem are obtained by each method and compared.
We address problem of on-line scheduling of the aircraft descent and approach trajectory. We formulate a general multiphase optimal control problem for optimization of the descent trajectory and review available methods of its solution. We develop a fast algorithm for solution of this problem using two key components: (i) fast inference of the dynamical and control variables of the descending trajectory from the low dimensional flight profile data and (ii) efficient local search for the resulting reduced dimensionality non-linear optimization problem. We compare the performance of the proposed algorithm with numerical solution obtained using optimal control toolbox General Pseudospectral Optimal Control Software. We present results of the solution of the scheduling problem for aircraft descent using novel fast algorithm and discuss its future applications.
Two narrow-band radar systems are developed for high resolution target range estimation in inhomogeneous media. They are reformulations of two presently existing systems such that high resolution target range estimates may be achieved despite the use of narrow bandwidth radar pulses. A double sideband suppressed carrier radar technique originally derived in 1962, and later abandoned due to its inability to accurately measure target range in the presence of an interfering reflection, is rederived to incorporate the presence of an interfering reflection. The new derivation shows that the interfering reflection causes a period perturbation in the measured phase response. A high resolution spectral estimation technique is used to extract the period of this perturbation leading to accurate target range estimates independent of the signal-to-interference ratio. A non-linear optimal signal processing algorithm is derived for a frequency-stepped continuous wave radar system. The resolution enhancement offered by optimal signal processing of the data over the conventional Fourier Transform technique is clearly demonstrated using measured radar data. A method for modeling plane wave propagation in inhomogeneous media based on transmission line theory is derived and studied. Several simulation results including measurement of non-uniform electron plasma densities that develop near the heat tiles of a space re-entry vehicle are presented which verify the validity of the model.
To cope with air traffic growth and congested airports, two solutions are apparent on the supply side: 1) use larger aircraft in the hub and spoke system; or 2) develop new routes through secondary airports. An enlarged route system through secondary airports may increase the proportion of route monopolies in the air transport market.The monopoly optimal non linear pricing policy is well known in the case of one dimension (one instrument, one characteristic) but not in the case of several dimensions. This paper explores the robustness of the one dimensional screening model with respect to increasing the number of instruments and the number of characteristics. The objective of this paper is then to link and fill the gap in both literatures. One of the merits of the screening model has been to show that a great varieD" of economic questions (non linear pricing, product line choice, auction design, income taxation, regulation...) could be handled within the same framework.VCe study a case of non linear pricing (2 instruments (2 routes on which the airline pro_ddes customers with services), 2 characteristics (demand of services on these routes) and two values per characteristic (low and high demand of services on these routes)) and we show that none of the conclusions of the one dimensional analysis remain valid. In particular, upward incentive compatibility constraint may be binding at the optimum. As a consequence, they may be distortion at the top of the distribution. In addition to this, we show that the optimal solution often requires a kind of form of bundling, we explain explicitly distortions and show that it is sometimes optimal for the monopolist to only produce one good (instead of two) or to exclude some buyers from the market. Actually, this means that the monopolist cannot fully apply his monopoly power and is better off selling both goods independently.We then define all the possible solutions in the case of a quadratic cost function for a uniform distribution of agent types and explain the implications for airlines in terms of service differentiation.
This paper presents a summary of a series of recent analytical studies conducted to investigate one-engine-inoperative (OEI) optimal control strategies and the associated optimal trajectories for a twin engine helicopter in Category-A terminal-area operations. These studies also examine the associated heliport size requirements and the maximum gross weight capability of the helicopter. Using an eight states, two controls, augmented point-mass model representative of the study helicopter, continued takeoff (CTO), rejected takeoff (RTO), balked landing (BL), and continued landing (CL) are investigated for both vertical-takeoff-and-landing (VTOL) and short-takeoff-and-landing (STOL) terminal-area operations. The formulation of the non-linear optimal control problems with considerations for realistic constraints, solution methods for the two-point boundary-value problem, a new real-time generation method for the optimal OEI trajectories, and the main results of this series of trajector optimization studies are presented. In particular, a new balanced-weight concept for determining the takeoff decision point for VTOL Category-A operations is proposed, extending the balanced-field length concept used for STOL operations.
Analysis of interplanetary trajectories is a crucial area for both manned and unmanned missions of the Space Exploration Initiative. A deep space maneuver (DSM) can improve a trajectory in much the same way as a planetary swingby. However, instead of using a gravitational field to alter the trajectory, the on-board propulsion system of the spacecraft is used when the vehicle is not near a planet. The purpose is to develop an algorithm to determine where and when to use deep space maneuvers to reduce the cost of a trajectory. The approach taken to solve this problem uses primer vector theory in combination with a non-linear optimizing program to minimize Delta(V). A set of necessary conditions on the primer vector is shown to indicate whether a deep space maneuver will be beneficial. Deep space maneuvers are applied to a round trip mission to Mars to determine their effect on the launch opportunities. Other studies which were performed include cycler trajectories and Mars mission abort scenarios. It was found that the software developed was able to locate quickly DSM's which lower the total Delta(V) on these trajectories.
In this paper, we describe an initial optimization study of a Variable-Camber Continuous Trailing-Edge Flap (VCCTEF) system. The VCCTEF provides a light-weight control system for aircraft with long flexible wings, providing efficient high-lift capability for takeoff and landing, and greater efficiency with reduced drag at cruising flight by considering the effects of aeroelastic wing deformations in the control law. The VCCTEF system is comprised of a large number of distributed and individually-actuatable control surfaces that are constrained in movement relative to neighboring surfaces, and are non-trivially coupled through structural aeroelastic dynamics. Minimzation of drag results in a constrained, coupled, non-linear optimization over a high-dimension search space. In this paper, we describe the modeling, analysis, and optimization of the VCCTEF system control inputs for minimum drag in cruise. The purpose of this initial study is to quantify the expected benefits of the system concept. The scope of this analysis is limited to consideration of a rigid wing without structural flexibility in a steady-state cruise condition at various fuel weights. For analysis, we developed an optimization engine that couples geometric synthesis with vortex-lattice analysis to automate the optimization procedure. In this paper, we present and describe the VCCTEF system concept, optimization approach and tools, run-time performance, and results of the optimization at 20%, 50%, and 80% fuel load. This initial limited-scope study finds the VCCTEF system can potentially gain nearly 10% reduction in cruise drag, provides greater drag savings at lower operating weight, and efficiency is negatively impacted by the severity of relative constraints between control surfaces.
Accelerated gradient method for numerical solution of parameter optimization problems with nonlinear constraints