Aceleración de algoritmos intervalares de ajuste automático del lazo en QFT

Abstract

This paper deals with the acceleration of interval analysis global search algorithms whose goal is to automatically solve the loop shaping problem in QFT. This design technique for robust controllers in the frequency domain is proposed as a succession of stages. Each step generates information for the following stage. The last stage, the most important one, is the loop shaping, where the controller is generated. Having algorithms which can automatically (without user intervention) solve the QFT loop shaping problem isn't an easy task. Given that it's a non-linear and non-convex optimization problem, the algorithmic solutions that can be applied pose several difficulties. Throughout time, different proposals have been made to solve the problem but, until now, no satisfactory solutions has been achieved. Given the complexity of the process, the problem has been faced from various perspectives. An interesting approach is interval global search algorithms. This type of proposals try to ensure an optimum solution using interval arithmetic, but they have the problem that, being a global search, they have a very high computational cost, and because of that, an exponential curve of the execution time grows exponentially with the size of the problem. This work has focused on carefully analyzing this kind of algorithms and the effort has been centered on understanding where more possibilities for improvement were. Once this task has been completed, the work has been focused on the proposal of alternatives that improve the existing algorithms and it culminates in a series of novel strategies that aim to reduce the computational cost and the execution time. A first group of strategies has the purpose of delimiting the algorithm's search space and its objective is to detect both the feasible and unfeasible subrange of each controller's variable, using the phase and magnitude in the Nichols plane. In this way, on the one hand, it's possible to eliminate the subranges identified as unfeasible, and on the other hand, to find solutions in the viable subrange. Within this same group, it's proposed a strategy which seeks to quickly find solutions to the problem and whose performance is similar to the bounding described although it has the ability to find better solutions, which can be used to prune unpromising nodes. A second group has a more transversal disposition. The first one contains several options to perform the bisection of the node. As a main option, the proposal is an advanced bisection which uses the information generated from the current node to make the best decision. Three more basic options are proposed to complement the previous one, to carry out the bisection by the most influential variable on the area, by the most influential one on the magnitude or by the most influential one on the phase, all of them in the Nichols plane, as appropiate depending on each moment’s conditions. Following, it's proposed a second strategy that aims to adapt the algorithm behaviour to every situation along the execution, in other words, it modifies the behavior of the rest of strategies and adapts them to each specific circumstance of the algorithm at any time. Thereby it improves the performance of every developed strategies. Finally, this thesis proposes a new interval global search QFT Automatic Loop Shaping algorithm which involves all these strategies. This algorithm achieves a significant reduction of the computational cost with respect to previous solutions and, in this way, it achieves a two orders of magnitude improvement in execution time for typical controllers.