An earth pole-sitter using hybrid propulsion

In this paper we investigate optimal pole-sitter orbits using hybrid solar sail and solar electric propulsion (SEP). A pole-sitter is a spacecraft that is constantly above one of the Earth's poles, by means of a continuous thrust. Optimal orbits, that minimize propellant mass consumption, are found both through a shape-based approach, and solving an optimal control problem, using a direct method based on pseudo-spectral techniques. Both the pure SEP case and the hybrid case are investigated and compared. It is found that the hybrid spacecraft allows consistent savings on propellant mass fraction. Finally, is it shown that for sufficiently long missions (more than 8 years), a hybrid spacecraft, based on mid-term technology, enables a consistent reduction in the launch mass for a given payload, with respect to a pure SEP spacecraft

On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems

We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. In contrast to the existing results, the optimal controller addressed in this paper is allowed to be discontinuous. This generalization requires a substantial modification to the existing convergence analysis in terms of both the framework as well as the notion of convergence around points of discontinuity. Although the nonlinear system is assumed to be feedback linearizable, the optimal control does not necessarily linearize the dynamics. Such problems frequently arise in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. We prove that a sequence of solutions obtained using the Legendre pseudospectral method converges to the optimal solution of the continuous-time problem under mild conditions

Advances in Pseudospectral Methods for Optimal Control

The article of record as published may be located at http://dx.doi.org/10.2514/6.2008-7309AIAA Guidance, Navigation and Control Conference and Exhibit ; Paper no. AIAA-2008-7309, Honolulu, Hawaii, 2008Recently, the Legendre Pseudospectral (PS) method migrated from theory to fight application onboard the International Space Station for performing a finite-horizon, zero- propellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for infinite-horizon optimal control problems recently, the Legendre Pseudospectral (PS) method migrated from theory to flight application onboard the International Space Station for performing a finite-horizon, zero-propellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for in finite-horizon optimal control problems. Motivated by these technicalities, the concept of primal-dual weighted interpolation, introduced earlier by the authors, is used to articulate a united theory for all PS methods for optimal control. This theory illuminates the previously hidden fact of the unit weight function implicit in the Legendre PS method based on Legendre-Gauss-Lobatto points. The united framework also reveals why this Legendre PS method is the most appropriate method for solving finite-horizon optimal control problems with arbitrary boundary conditions. This conclusion is borne by a proper definition of orthogonality needed to generate convergent approximations in Hilbert spaces. Special boundary conditions are needed to ensure the convergence of the Legendre PS method based on the Legendre-Gauss-Radau (LGR) and the Legendre-Gauss (LG) points. These facts are illustrated by simple examples and counter examples which reveal when and why PS methods based on LGR and LG points fail. A new kind of consistency in the primal-dual weight functions allows us to generate dual maps (such as Hamiltonians, adjoins etc) without resorting to solving difficult two-point boundary-value problems. These concepts are encapsulated in a united Convector Mapping Theorem.We gratefully acknowledge partial funding for this research provided to one of the authors (Ross) by the Air Force Office of Scientific Research under AFOSR Grant F1ATA0-60-6-2G002.Approved for public release; distribution is unlimited

Algorithms of data generation for deep learning and feedback design: A survey

Recent research reveals that deep learning is an effective way of solving high dimensional Hamilton–Jacobi–Bellman equations. The resulting feedback control law in the form of a neural network is computationally efficient for real-time applications of optimal control. A critical part of this design method is to generate data for training the neural network and validating its accuracy. In this paper, we provide a survey of existing algorithms that can be used to generate data. All the algorithms surveyed in this paper are causality-free, i.e., the solution at a point is computed without using the value of the function at any other points. An illustrative example is given for the optimal feedback design using supervised learning in which the data is generated using causality-free algorithms

Lärares uppfattningar om skriftliga omdömen : En intervjustudie om bedömningens didaktiska betydelse

Studiens syfte är att undersöka lärares uppfattningar om skriftliga omdömens utvecklande funktion för deras arbete samt hur undervisningen påverkas av arbetet med bedömning. Avsikten är vidare att undersöka om de praxisnära uppfattningarna stämmer överens med Skolverkets intention med skriftliga omdömen som verktyg för att utveckla undervisningen, samt om uppfattningar hos lärare skiljer sig beroende på vilken ålder de undervisar i. Vår utgångspunkt för studien är didaktiska konsekvenser av bedömning, och i synnerhet formativ bedömning då denna bedömningsform påtalas av Skolverket i samband med skriftliga omdömen. För att uppnå syftet med studien har vi valt att intervjua sex lärare verksamma på låg- och mellanstadiet och undersökt vilken betydelse arbetet med skriftliga omdömen har för didaktiskt handlande i form av planering och utvärdering.Resultatet av vår undersökning visar att skriftliga omdömen har betydelse för det didaktiska handlandet, samt att vissa skillnader kan urskiljas i lärarnas uppfattningar beroende på vilket åldersstadium de är verksamma inom. Resultatet visar även tydligt att lärarnas uppfattningar har förändrats sedan införandet av skriftliga omdömen och vår slutsats är att de befinner sig i en process med att utveckla bedömning som ett didaktiskt redskap. Nyckelord: didaktik, lärare, intervju, bedömning, skriftliga omdöme