Trajectory Control and Optimization for Responsive Spacecraft

The concept of responsive space has been gaining interest, and growing to include systems that can be re-tasked to complete multiple missions within their lifetime. The purpose of this study is to develop an algorithm that produces a maneuver trajectory that will cause a spacecraft to arrive at a particular location within its orbit earlier than expected. The time difference, delta t, is used as a metric to quantify the effects of the maneuver. Two separate algorithms are developed. The first algorithm is an optimal control method and is developed through Optimal Control Theory. The second algorithm is a feedback control method and is developed through Lyapunov Theory. It is shown that the two algorithms produce equivalent results for the maneuvers discussed. In-plane maneuver results are analyzed analytically, and an algebraic expression for delta t is derived. Examples are provided of how the analytic expression can be used for mission planning purposes. The feedback control algorithm is then further developed to demonstrate the simplicity of implementing additional capabilities. Finally, a set of simulations is analyzed to show that in order to maximize the amount of delta t achieved, a spacecraft must be allowed as much lead time as possible, and begin thrusting as early as possible

Trajectory Control and Optimization for Responsive Spacecraft

The concept of responsive space has been gaining interest, and growing to include systems that can be re-tasked to complete multiple missions within their lifetime. The purpose of this study is to develop an algorithm that produces a maneuver trajectory that will cause a spacecraft to arrive at a particular location within its orbit earlier than expected. The time difference, delta t, is used as a metric to quantify the effects of the maneuver. Two separate algorithms are developed. The first algorithm is an optimal control method and is developed through Optimal Control Theory. The second algorithm is a feedback control method and is developed through Lyapunov Theory. It is shown that the two algorithms produce equivalent results for the maneuvers discussed. In-plane maneuver results are analyzed analytically, and an algebraic expression for delta t is derived. Examples are provided of how the analytic expression can be used for mission planning purposes. The feedback control algorithm is then further developed to demonstrate the simplicity of implementing additional capabilities. Finally, a set of simulations is analyzed to show that in order to maximize the amount of delta t achieved, a spacecraft must be allowed as much lead time as possible, and begin thrusting as early as possible

Optimal Collision Avoidance Trajectories for Unmanned/Remotely Piloted Aircraft

The post-911 environment has punctuated the force-multiplying capabilities that Remotely Piloted Aircraft (RPA) provides combatant commanders at all echelons on the battlefield. Not only have unmanned aircraft systems made near-revolutionary impacts on the battlefield, their utility and proliferation in law enforcement, homeland security, humanitarian operations, and commercial applications have likewise increased at a rapid rate. As such, under the Federal Aviation Administration (FAA) Modernization and Reform Act of 2012, the United States Congress tasked the FAA to provide for the safe integration of civil unmanned aircraft systems into the national airspace system (NAS) as soon as practicable, but not later than September 30, 2015. However, a necessary entrance criterion to operate RPAs in the NAS is the ability to Sense and Avoid (SAA) both cooperative and noncooperative air traffic to attain a target level of safety as a traditional manned aircraft platform. The goal of this research effort is twofold: First, develop techniques for calculating optimal avoidance trajectories, and second, develop techniques for estimating an intruder aircraft\u27s trajectory in a stochastic environment. This dissertation describes the optimal control problem associated with SAA and uses a direct orthogonal collocation method to solve this problem and then analyzes these results for different collision avoidance scenarios

A numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics

In this thesis, we develop accurate and efficient numerical methods for solving partial differential equation (PDE) constrained optimization problems arising from multiscale particle dynamics, with the aim of producing a desired time-dependent state at the minimal cost. A PDE-constrained optimization problem seeks to move one or more state variables towards a desired state under the influence of one or more control variables, and a set of constraints that are described by PDEs governing the behaviour of the variables. In particular, we consider problems constrained by one-dimensional and two-dimensional advection-diffusion problems with a non-local integral term, such as the associated mean-field limit Fokker-Planck equation of the noisy Hegselmann-Krause opinion dynamics model. We include additional bound constraints on the control variable for the opinion dynamics problem. Lastly, we consider constraints described by a two-dimensional robot swarming model made up of a system of advection-diffusion equations with additional linear and integral terms. We derive continuous Lagrangian first-order optimality conditions for these problems and solve the resulting systems numerically for the optimized state and control variables. Each of these problems, combined with Dirichlet, no-flux, or periodic boundary conditions, present unique challenges that require versatility of the numerical methods devised. Our numerical framework is based on a novel combination of four main components: (i) a discretization scheme, in both space and time, with the choice of pseudospectral or fi nite difference methods; (ii) a forward problem solver that is implemented via a differential-algebraic equation solver; (iii) an optimization problem solver that is a choice between a fi xed-point solver, with or without Armijo-Wolfe line search conditions, a Newton-Krylov algorithm, or a multiple shooting scheme, and; (iv) a primal-dual active set strategy to tackle additional bound constraints on the control variable. Pseudospectral methods efficiently produce highly accurate solutions by exploiting smoothness in the solutions, and are designed to perform very well with dense, small matrix systems. For a number of problems, we take advantage of the exponential convergence of pseudospectral methods by discretising in this way not only in space, but also in time. The alternative fi nite difference method performs comparatively well when non-smooth bound constraints are added to the optimization problem. A differential{algebraic equation solver works out the discretized PDE on the interior of the domain, and applies the boundary conditions as algebraic equations. This ensures generalizability of the numerical method, as one does not need to explicitly adapt the numerical method for different boundary conditions, only to specify different algebraic constraints that correspond to the boundary conditions. A general fixed-point or sweeping method solves the system of equations iteratively, and does not require the analytic computation of the Jacobian. We improve the computational speed of the fi xed-point solver by including an adaptive Armijo-Wolfe type line search algorithm for fixed-point problems. This combination is applicable to problems with additional bound constraints as well as to other systems for which the regularity of the solution is not sufficient to be exploited by the spectral-in-space-and-time nature of the Newton-Krylov approach. The recently devised Newton-Krylov scheme is a higher-order, more efficient optimization solver which efficiently describes the PDEs and the associated Jacobian on the discrete level, as well as solving the resulting Newton system efficiently via a bespoke preconditioner. However, it requires the computation of the Jacobian, and could potentially be more challenging to adapt to more general problems. Multiple shooting solves an initial-value problem on sections of the time interval and imposes matching conditions to form a solution on the whole interval. The primal-dual active set strategy is used for solving our non-linear and non-local optimization problems obtained from opinion dynamics problems, with pointwise non-equality constraints. This thesis provides a numerical framework that is versatile and generalizable for solving complex PDE-constrained optimization problems from multiscale particle dynamic

Computational Techniques for Optimal Control of Quantum System

The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time—dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite—dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite—dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2— versus H1—based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. vi A matrix—free cascadic BFGS algorithm is introduced in the L2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov—Newton scheme applied to a norm—preserving discrete model of a dipole quantum control problem. The use of second—order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first— and second—order optimality conditions. Criteria for the discretization of the non—convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust line- search procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high—energy state transitions. Control of bound—to—bound and bound—to—continuum transitions in open quantum systems and vector field control of two—dimensional systems is presented. An efficient space—time spectral discretization of the time—dependent Schrödinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined

Hybrid Solution of Stochastic Optimal Control Problems using Gauss Pseudospectral Method and Generalized Polynomial Chaos Algorithms

Two numerical methods, Gauss Pseudospectral Method and Generalized Polynomial Chaos Algorithm, were combined to form a hybrid algorithm for solving nonlinear optimal control and optimal path planning problems with uncertain parameters. The algorithm was applied to two concept demonstration problems: a nonlinear optimal control problem with multiplicative uncertain elements and a mission planning problem sponsored by USSTRATCOM. The mission planning scenario was constructed to find the path that minimizes the probability of being killed by lethal threats whose locations are uncertain to statistically quantify the effects those uncertainties have on the flight path solution, and to use the statistical properties to estimate the probability that the vehicle will be killed during mission execution. The results demonstrated that the method is able to effectively characterize how the optimal solution changes with uncertainty and that the results can be presented in a form that can be used by mission planners and aircrews to assess risks associated with a mission profile

Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization

Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, brewing, and printing. In real-world applications it is often relevant to not only model a process of interest, but to also optimize it in order to achieve a desired outcome with minimal resources, such as time, money, or energy. Mathematically, the dynamics of interacting particle systems can be described using Dynamic Density Functional Theory (DDFT). The resulting models are nonlinear, nonlocal partial differential equations (PDEs) that include convolution integral terms. Such terms also enter the naturally arising no-flux boundary conditions. Due to the nonlocal, nonlinear nature of such problems they are challenging both to analyse and solve numerically. In order to optimize processes that are modelled by PDEs, one can apply tools from PDE-constrained optimization. The aim here is to drive a quantity of interest towards a target state by varying a control variable. This is constrained by a PDE describing the process of interest, in which the control enters as a model parameter. Such problems can be tackled by deriving and solving the (first-order) optimality system, which couples the PDE model with a second PDE and an algebraic equation. Solving such a system numerically is challenging, since large matrices arise in its discretization, for which efficient solution strategies have to be found. Most work in PDE-constrained optimization addresses problems in which the control is applied linearly, and which are constrained by local, often linear PDEs, since introducing nonlinearity significantly increases the complexity in both the analysis and numerical solution of the optimization problem. However, in order to optimize real-world processes described by nonlinear, nonlocal DDFT models, one has to develop an optimal control framework for such models. The aim is to drive the particles to some desired distribution by applying control either linearly, through a particle source, or bilinearly, though an advective field. The optimization process is constrained by the DDFT model that describes how the particles move under the influence of advection, diffusion, external forces, and particle–particle interactions. In order to tackle this, the (first-order) optimality system is derived, which, since it involves nonlinear (integro-)PDEs that are coupled nonlocally in space and time, is significantly harder than in the standard case. Novel numerical methods are developed, effectively combining pseudospectral methods and iterative solvers, to efficiently and accurately solve such a system. In a next step this framework is extended so that it can capture and optimize industrially relevant processes, such as brewing and nano-filtration. In order to do so, extensions to both the DDFT model and the numerical method are made. Firstly, since industrial processes often involve tubes, funnels, channels, or tanks of various shapes, the PDE model itself, as well as the optimization problem, need to be solved on complicated domains. This is achieved by developing a novel spectral element approach that is compatible with both the PDE solver and the optimal control framework. Secondly, many industrial processes, such as nano-filtration, involve more than one type of particle. Therefore, the DDFT model is extended to describe multiple particle species. Finally, depending on the application of interest, additional physical effects need to be included in the model. In this thesis, to model sedimentation processes in brewing, the model is modified to capture volume exclusion effects