Extended Trust-Tech Methodology For Nonlinear Optimization: Analyses, Methods And Applications

Many theoretical and practical problems can be formulated as a global optimization problem. Traditional local optimization methods can only attain a local optimal solution and be entrapped in the local optimal solution; while existing global optimization algorithms usually sparsely approximates the global optimal solution in a stochastic manner. In contrast, the transformation under stability-retaining equilibrium characterization (TRUST-TECH) methodology prevails over existing algorithms due to its capability of locating multiple, if not all, local optimal solutions to the optimization problem deterministically and systematically in a tier-by-tier manner. The TRUST-TECH methodology was developed to solve unconstrained and constrained nonlinear optimization problems. This work extends the TRUST-TECH methodology by incorporating new analytical results, developing new solution methods and solving new problems in practical applications. This work first provides analytical results regarding the invariance of partial stability region in quasi-gradient systems. Our motivation is to resolve numerical difficulties arising in implementations of trajectory based methods, including TRUST-TECH. Improved algorithms were developed to resolve these issues by altering the original problem to speed-up movement of the trajectory. However, such operations can lead the trajectory converge to a different solution, which could be undesired under specific situations. This work attempts to answer the question regarding invariant convergence for a special class of numerical operations whose dynamical behaviours can be characterized by a quasi-gradient dynamical system. To this end, we study relationship between a gradient dynamical system and its associated quasi-gradient system and reveal the invariance of partial stability region in the quasi-gradient system. These analytical results lead to methods for checking invariant convergence of the trajectory starting from a given point in the quasi-gradient system and the algorithm to maintain invariant convergence. This work also develops new solution methods to enhance TRUST-TECH's capability of solving constrained nonlinear optimization problems and applies them to solve practical problems arising in different applications. Specifically, TRUST-TECH based methods are first developed for feasibility computation and restoration and are applied to power system applications, including power flow computation and feasibility restoration for infeasible optimal power flow problems. Indeed, a unified framework based on TRUST-TECH is introduced for analysing feasibility and infeasibility for nonlinear problems. Secondly, the TRUST-TECH based interior point method (TT-IPM) and the reduced projected gradient method are developed to better tackle constrained nonlinear optimization problems. As application, the TT-IPM method is used to solve mixed-integer nonlinear programs (MINLPs). Finally, this work develops the ensemble of optimal, input-pruned neural networks using TRUST-TECH (ELITE) method for constructing high-quality neural network ensembles and applies ELITE to build a short-term load forecaster named ELITE-STLF with promising performance. Possible extensions of the TRUST-TECH methodology to a much broader range of optimization models, including multi-objective optimization and variational optimization, are suggested for future research efforts

Sequential quadratic programming solutions to related aircraft trajectory optimization problems

Aircraft performance optimization continues to play an important role in the aerospace sciences. The studies undertaken in this dissertation explore the performance of high-speed aircraft with regard to missile evasion, minimum-time-to-climb, minimum-time-to-turn, and the unorthodox approach of obtaining a robust optimality-based control law for real-time aircraft control. The dissertation includes four papers presented or accepted for presentation at major conferences and presently in various stages of review for publication in scholarly journals;The similarity in each paper, in addition to the focus on optimal aircraft trajectories, is that an existing nonlinear programming method, sequential quadratic programming (SQP), is used to treat each trajectory optimization problem. This approach is suitable since the emphasis is on applications and problem solving, and the method is accurate and computationally inexpensive. Also, the flexibility of SQP allows for performance index, mathematical model, and constraint changes with relatively little reprogramming. This enables a wide range of trajectory optimization problems to be formulated and studied;In the study of the aircraft missile-evasion problem in horizontal planar flight, unlike earlier investigations, the full original equations of motion are used. Also, no linearization about a nominal pursuit triangle is done. The velocity ratio, that is, the velocity of the aircraft to the velocity of the missile for the duration of the confrontation, becomes a major factor in deciding optimal evasive strategies. Evasion against a surface-to-air missile involves a large nonlinear optimal control problem of dynamic order of at least thirteen. Inward , outward , pull-up, dive, and inverted pull-down evasive maneuvers are investigated. The results show that the missile enters the hit region of the aircraft for constrained vertical plane flight, but not for constrained horizontal flight. The optimal throttle setting for constrained horizontal plane flight is of bang-bang type;For the minimum-time turn problems, having free final velocity provides the biggest impact on turn times, which can be reduced by as much as fifty percent. For a wide range of final energies studied in the three-dimensional turns, it was found that the aircraft tends to initially lose altitude in the optimal turn even though the nominal control from which the optimization process started corresponds to an initial climbing turn. This tendency of favoring kinetic energy over potential energy had not been featured in earlier papers;Finally, the investigation of optimality-based control laws for real-time aircraft control is a significant departure from the usual open-loop solutions to trajectory optimization problems. It was found that the robustness of the optimal control obtained from the optimality-condition is not guaranteed, but by introducing a certain correction term, it can be enhanced significantly. It appears that this technique of enhancing robustness has not been used until now

Gaussian Process Model Predictive Control of An Unmanned Quadrotor

The Model Predictive Control (MPC) trajectory tracking problem of an unmanned quadrotor with input and output constraints is addressed. In this article, the dynamic models of the quadrotor are obtained purely from operational data in the form of probabilistic Gaussian Process (GP) models. This is different from conventional models obtained through Newtonian analysis. A hierarchical control scheme is used to handle the trajectory tracking problem with the translational subsystem in the outer loop and the rotational subsystem in the inner loop. Constrained GP based MPC are formulated separately for both subsystems. The resulting MPC problems are typically nonlinear and non-convex. We derived 15 a GP based local dynamical model that allows these optimization problems to be relaxed to convex ones which can be efficiently solved with a simple active-set algorithm. The performance of the proposed approach is compared with an existing unconstrained Nonlinear Model Predictive Control (NMPC). Simulation results show that the two approaches exhibit similar trajectory tracking performance. However, our approach has the advantage of incorporating constraints on the control inputs. In addition, our approach only requires 20% of the computational time for NMPC.Comment: arXiv admin note: text overlap with arXiv:1612.0121