Two new weak constraint qualifications and applications

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.Facultad de Ciencias ExactasDepartamento de Matemátic

TWO NEW WEAK CONSTRAINT QUALIFICATIONS AND APPLICATIONS

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.RONEX-Optimization (PRONEX-CNPq/FAPERJ) [E-26/171.510/2006-APQ1]Fapesp [2006/53768-0, 2009/09414-7, 2010/19720-5]CNPq [300900/2009-0, 303030/2007-0, 305740/2010-5, 474138/2008-9

On-board Trajectory Computation for Mars Atmospheric Entry Based on Parametric Sensitivity Analysis of Optimal Control Problems

This thesis develops a precision guidance algorithm for the entry of a small capsule into the atmosphere of Mars. The entry problem is treated as nonlinear optimal control problem and the thesis focuses on developing a suboptimal feedback law. Therefore parametric sensitivity analysis of optimal control problems is combined with dynamic programming. This approach enables a real-time capable, locally suboptimal feedback scheme. The optimal control problem is initially considered in open loop fashion. To synthesize the feedback law, the optimal control problem is embedded into a family of neighboring problems, which are described by a parameter vector. The optimal solution for a nominal set of parameters is determined using direct optimization methods. In addition the directional derivatives (sensitivities) of the optimal solution with respect to the parameters are computed. Knowledge of the nominal solution and the sensitivities allows, under certain conditions, to apply Taylor series expansion to approximate the optimal solution for disturbed parameters almost instantly. Additional correction steps can be applied to improve the optimality of the solution and to eliminate errors in the constraints. To transfer this strategy to the closed loop system, the computation of the sensitivities is performed with respect to different initial conditions. Determining the perturbation direction and interpolating between sensitivities of neighboring initial conditions allows the approximation of the extremal field in a neighborhood of the nominal trajectory. This constitutes a locally suboptimal feedback law. The proposed strategy is applied to the atmospheric entry problem. The developed algorithm is part of the main control loop, i.e. optimal controls and trajectories are computed at a fixed rate, taking into account the current state and parameters. This approach is combined with a trajectory tracking controller based on the aerodynamic drag. The performance and the strengthsa and weaknesses of this two degree of freedom guidance system are analyzed using Monte Carlo simulation. Finally the real-time capability of the proposed algorithm is demonstrated in a flight representative processor-in-the-loop environment