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Results for “Optimal Control Problem”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Error Estimates for the Optimal Control of a Parabolic Fractional PDE

In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A Stochastic Gradient Descent Approach for Stochastic Optimal Control

In this work, we introduce a stochastic gradient descent approach to solve the stochastic optimal control problem through stochastic maximum principle. The motivation that drives our method is the gradient of the cost functional in the stochastic optimal control problem is under expectation, and numerical calculation of such an expectation requires fully computation of a system of forward backward stochastic differential equations, which is computationally expensive. By evaluating the expectation with single-sample representation as suggested by the stochastic gradient descent type optimisation, we could save computational efforts in solving FBSDEs and only focus on the optimisation task which aims to determine the optimal control process.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2005

Small Body GN&C Research Report: A Guidance and Control Technique for Small-Body Proximity Operations with Guaranteed Guidance Resolvability and Required Thruster Silent Time

The guidance and control (G&C) algorithms for enabling small-body proximity operations are developed by using a model predictive control approach along with a convexification of the governing dynamics, control constraints, and trajectory/state constraints. The open-loop guidance is solved ahead of time or in a resolvable, real-time manner through the use of PWG (Pseudo Way-point Generation), a technique developed in this research. The PWG scheme ensures required thruster silent times during trajectory maneuvers. The feedback control is implemented to track the PWG trajectories in a manner that guarantees the resolvability for the open-loop problem, enabling the ability to update the G&C in a model-predictive manner. The schemes incorporate gravity models and thruster ring times into discrete dynamics that are solved as a optimal control problem to minimize fuel consumption or thruster energy expenditure. The optimal control problem is cast as an LMI (Linear Matrix Inequality) and then solved through Semi-Definite Programming techniques in a computationally efficient manner that provides convergence and constraint guarantees.

guidance and control (G&C)↗
NASA NTRS1972

An Optimization study on a hybrid computer

The maximum principle is applied to minimum-time optimal-control problems, and an optimization algorithm is presented which can be implemented on a hybrid computer. The state and adjoint equations are set up on ASTRAC 2, a high-speed analog computer capable of 1000 differential equation solutions per second. The optimization algorithm is implemented on a PDP-9, an 18-bit, digital computer. The optimization scheme has global and local search phases and uses a vector optimization criterion. Second and third-order bang-bang control systems are studied as examples.

Gonzalez, R. S.↗
DOE OSTI2020

An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems

The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an Itô type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. In this work, we call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1993

Variational Trajectory Optimization Tool Set: Technical description and user's manual

The algorithms that comprise the Variational Trajectory Optimization Tool Set (VTOTS) package are briefly described. The VTOTS is a software package for solving nonlinear constrained optimal control problems from a wide range of engineering and scientific disciplines. The VTOTS package was specifically designed to minimize the amount of user programming; in fact, for problems that may be expressed in terms of analytical functions, the user needs only to define the problem in terms of symbolic variables. This version of the VTOTS does not support tabular data; thus, problems must be expressed in terms of analytical functions. The VTOTS package consists of two methods for solving nonlinear optimal control problems: a time-domain finite-element algorithm and a multiple shooting algorithm. These two algorithms, under the VTOTS package, may be run independently or jointly. The finite-element algorithm generates approximate solutions, whereas the shooting algorithm provides a more accurate solution to the optimization problem. A user's manual, some examples with results, and a brief description of the individual subroutines are included.

Bless, Robert R.↗
NASA NTRS1993

Development of an hp-version finite element method for computational optimal control

The purpose of this research effort is to develop a means to use, and to ultimately implement, hp-version finite elements in the numerical solution of optimal control problems. The hybrid MACSYMA/FORTRAN code GENCODE was developed which utilized h-version finite elements to successfully approximate solutions to a wide class of optimal control problems. In that code the means for improvement of the solution was the refinement of the time-discretization mesh. With the extension to hp-version finite elements, the degrees of freedom include both nodal values and extra interior values associated with the unknown states, co-states, and controls, the number of which depends on the order of the shape functions in each element.

Hodges, Dewey H.↗
DOE OSTI2022

Autonomous Eco-Driving with Traffic Light and Lead Vehicle Constraints: An Application of Best Constrained Interpolation

Eco-Driving is a critical technology for improving automotive transportation efficiency. It is achieved by modifying the driving trajectory over a particular route to minimize required propulsion energy. Eco-Driving can be approached as an optimal control problem subject to driving constraints such as traffic lights and positions of other vehicles. Best interpolation in a strip is a problem in approximation theory and optimal control. The solution to this problem is a cubic spline. In this research, we demonstrate the connection between Eco-Driving and best interpolation in the strip. By exploiting this connection, we are able to generate optimal Eco-Driving trajectories that could be driven with an autonomous system. Using the simulation tool FASTSim, we can evaluate the trajectories using conventional, hybrid electric, and fully electric vehicle models. Our results quantify the energy efficiency improvements that can be achieved with the proposed approach.

Mahmoud, Yara Hazem↗
DOE OSTI2022

Autonomous Eco-Driving with Traffic Light and Lead Vehicle Constraints: An Application of Best Constrained Interpolation

Eco-Driving is a critical technology for improving automotive transportation efficiency. It is achieved by modifying the driving trajectory over a particular route to minimize required propulsion energy. Eco-Driving can be approached as an optimal control problem subject to driving constraints such as traffic lights and positions of other vehicles. Best interpolation in a strip is a problem in approximation theory and optimal control. The solution to this problem is a cubic spline. In this research, we demonstrate the connection between Eco-Driving and best interpolation in the strip. By exploiting this connection, we are able to generate optimal Eco-Driving trajectories that could be driven with an autonomous system. Using the simulation tool FASTSim, we can evaluate the trajectories using conventional, hybrid electric, and fully electric vehicle models. Our results quantify the energy efficiency improvements that can be achieved with the proposed approach.

Mahmoud, Yara Hazem↗
NASA NTRS1981

Minimum-fuel aerodynamic orbital plane change maneuvers

Several minimum-fuel, aerodynamically controlled, orbital plane change problems are formulated and solved as optimal control problems. A gradient projection algorithm is used to iteratively modify both the control functions, angle of attack and bank angle, and two control parameters to obtain the optimal trajectory. The atmospheric flight profile is combined with two Keplerian (two-body vacuum flight) arcs so that a complete orbit-to-orbit analysis results. The vehicle used in this investigation is the Space Transportation System Shuttle Orbiter. The effects of heat load constraint level and plane change angle are analysed.

Joosten, B. K.↗
DOE OSTI2025

A time-parallel multiple-shooting method for large-scale quantum optimal control

Quantum optimal control plays a crucial role in quantum computing by providing the interface between compiler and hardware. Solving the optimal control problem is particularly challenging for multi-qubit gates, due to the exponential growth in computational complexity with the system's dimensionality and the deterioration of optimization convergence. To ameliorate the computational complexity of time-integration, this paper introduces a multiple-shooting approach in which the time domain is divided into multiple windows and the intermediate states at window boundaries are treated as additional optimization variables. Further, this enables parallel computation of state evolution across time-windows, significantly accelerating objective function and gradient evaluations. Since the initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm, the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits, noting a speedup of 80x for evaluating the gradient in the 4-qubit case, highlighting the method's potential for optimizing control pulses in multi-qubit quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1985

Optimal landing of a helicopter in autorotation

Gliding descent in autorotation is a maneuver used by helicopter pilots in case of engine failure. The landing of a helicopter in autorotation is formulated as a nonlinear optimal control problem. The OH-58A helicopter was used. Helicopter vertical and horizontal velocities, vertical and horizontal displacement, and the rotor angle speed were modeled. An empirical approximation for the induced veloctiy in the vortex-ring state were provided. The cost function of the optimal control problem is a weighted sum of the squared horizontal and vertical components of the helicopter velocity at touchdown. Optimal trajectories are calculated for entry conditions well within the horizontal-vertical restriction curve, with the helicopter initially in hover or forwared flight. The resultant two-point boundary value problem with path equality constraints was successfully solved using the Sequential Gradient Restoration Technique.

Lee, A. Y. N.↗
NASA NTRS1992

Optimal control computer programs

The solution of the optimal control problem, even with low order dynamical systems, can usually strain the analytical ability of most engineers. The understanding of this subject matter, therefore, would be greatly enhanced if a software package existed that could simulate simple generic problems. Surprisingly, despite a great abundance of commercially available control software, few, if any, address the part of optimal control in its most generic form. The purpose of this paper is, therefore, to present a simple computer program that will perform simulations of optimal control problems that arise from the first necessary condition and the Pontryagin's maximum principle.

Kuo, F.↗
NASA NTRS1986

Nearly-grazing optimal trajectories for aeroassisted orbital transfer

In the present treatment of optimal control problems arising in the study of coplanar aeroassisted orbital transfer, the hybrid combination of propulsive parameters in space and aerodynamic maneuvers employing lift modulation in the sensible atmosphere indicates that the optimal energy-viewpoint solution is the grazing trajectory; this trajectory is characterized by favorable values of the peak heating rate and the peak dynamic pressure. Numerical solutions are obtained by means of the sequential gradient restoration algorithm for optimal control problems. It is found that nearly-grazing trajectories yielding the least-square value of the path inclination have desirable characteristics from the standpoints of energy, heating rate, and dynamic pressure.

Miele, A.↗
NASA NTRS1989

Active vibration mitigation of distributed parameter, smart-type structures using Pseudo-Feedback Optimal Control (PFOC)

A new, near-optimal feedback control technique is introduced that is shown to provide excellent vibration attenuation for those distributed parameter systems that are often encountered in the areas of aeroservoelasticity and large space systems. The technique relies on a novel solution methodology for the classical optimal control problem. Specifically, the quadratic regulator control problem for a flexible vibrating structure is first cast in a weak functional form that admits an approximate solution. The necessary conditions (first-order) are then solved via a time finite-element method. The procedure produces a low dimensional, algebraic parameterization of the optimal control problem that provides a rigorous basis for a discrete controller with a first-order like hold output. Simulation has shown that the algorithm can successfully control a wide variety of plant forms including multi-input/multi-output systems and systems exhibiting significant nonlinearities. In order to firmly establish the efficacy of the algorithm, a laboratory control experiment was implemented to provide planar (bending) vibration attenuation of a highly flexible beam (with a first clamped-free mode of approximately 0.5 Hz).

Patten, W. N.↗
NASA NTRS1989

Computational complexities and storage requirements of some Riccati equation solvers

The linear optimal control problem of an nth-order time-invariant dynamic system with a quadratic performance functional is usually solved by the Hamilton-Jacobi approach. This leads to the solution of the differential matrix Riccati equation with a terminal condition. The bulk of the computation for the optimal control problem is related to the solution of this equation. There are various algorithms in the literature for solving the matrix Riccati equation. However, computational complexities and storage requirements as a function of numbers of state variables, control variables, and sensors are not available for all these algorithms. In this work, the computational complexities and storage requirements for some of these algorithms are given. These expressions show the immensity of the computational requirements of the algorithms in solving the Riccati equation for large-order systems such as the control of highly flexible space structures. The expressions are also needed to compute the speedup and efficiency of any implementation of these algorithms on concurrent machines.

Utku, Senol↗
DOE OSTI2025

Neural network approaches for parameterized optimal control

Here, we consider numerical approaches for deterministic, finite-dimensional optimal control problems whose dynamics depend on unknown or uncertain parameters. We seek to amortize the solution over a set of relevant parameters in an offline stage to enable rapid decision-making and be able to react to changes in the parameter in the online stage. To tackle the curse of dimensionality arising when the state and/or parameter are high-dimensional, we represent the policy using neural networks. We compare two training paradigms: First, our model-based approach leverages the dynamics and definition of the objective function to learn the value function of the parameterized optimal control problem and obtain the policy using a feedback form. Second, we use actor-critic reinforcement learning to approximate the policy in a data-driven way. Using an example involving a two-dimensional convection-diffusion equation, which features high-dimensional state and parameter spaces, we investigate the accuracy and efficiency of both training paradigms. While both paradigms lead to a reasonable approximation of the policy, the model-based approach is more accurate and considerably reduces the number of PDE solves.

97 MATHEMATICS AND COMPUTING↗