Global Optimization using Interval Analysis: Interval Optimization for Aerospace Applications

Optimization is an important element in aerospace related research. It is encountered for example in trajectory optimization problems, such as: satellite formation flying, spacecraft re-entry optimization and airport approach and departure optimization; in control optimization, for example in adaptive control algorithms; and in system identification problems, such as online aircraft model identification or human perception modeling. The main goal of this thesis is to investigate how Interval Analysis (IA) can be used as a tool for aerospace related optimization problems; to examine its theoretical and practical limitations, and to explore the ways in which optimization algorithms can benefit from interval analysis. A subset of goals is to improve the solutions for a number of aerospace related optimization problems. The scientific contribution of this thesis consists of the design and implementation of interval optimization algorithms for four important aerospace problems. The first contribution concerns finding the trim points for a nonlinear aircraft model. Trim points, defined as the combination of control settings for which all linear and rotational accelerations on the aircraft are zero, are important for flight control system design, since they provide information about the flight envelope and stability properties of the aircraft. Unlike other trim algorithms, the interval based method can guarantee that all trim points are found. In the second application, an interval optimization algorithm is developed for fitting pilot input/output data from an experiment in the SIMONA Research Simulator to a multi-modal human perception model. Perception models improve the understanding of how humans perceive motion and are an essential tool in the design of flight simulators. Results show that the minimum of the cost function found by the interval method is lower than the one previously found, resulting in an improved human perception model. This second application particularly demonstrates the capabilities of IA optimization as a parameter identification tool. The third contribution is an interval based algorithm for solving the integer ambiguity problem related to Global Navigation Satellite Systems (GNSS). Phase measurements of the carrier wave of a GNSS signal are used to estimate the length and orientation of baselines between two or more antennas. This estimation procedure contains an optimization problem in which the integer number of carrier wavelengths between antennas has to be determined. The new interval method provides guarantees that correct solutions are found when the measurement noise is encapsulated by an interval number. The final contribution is an interval optimization algorithm that minimizes fuel consumption during rendezvous and docking procedures of satellites in circular orbits. To avoid integration of interval functions, an analytical solution to the system of differential equations that describes the relative motion of the satellites is used to generate trajectories resulting from a set of thruster pulses of varying amplitudes. Introduction of obstacles, in the form of forbidden areas in the path between the two satellites, makes the problem nonlinear, such that gradient-based optimization algorithms can fail to obtain the globally optimal solution. The interval algorithm always converges to the trajectory that avoids all obstacles and results in minimum fuel consumption. It can be concluded that IA is an excellent tool for solving nonlinear optimization algorithms, providing guarantees on obtaining the global minimum of the cost function.