TY - THES

T1 - Investigating the possibility to solve the Hamilton-Jacobi-Bellman Equation by the Finite Element Method to enable feedback control of nonlinear dynamical systems

AU - Flachberger, Wolfgang

N1 - no embargo

PY - 2021

Y1 - 2021

N2 - Even though Optimal Control Theory was developed in the 1950s its usage in feedback control systems is restricted to linear dynamical systems. In General PID and LQR controllers are still the state of the art which is due to their simple implementation and reliability. There is, however, a wide range of problems in robotics and mechatronics that exceed the ability of these practices due to nonlinear dynamical behaviour. To make Optimal Control Theory feasible in these fields, its application has to be unified and simplified within a software that can reliably solve these problems. It is possible to formulate the Optimal Control Problem in such a way that a solution to the resulting boundary value problem does not only reveal one optimal trajectory but a feedback control law or strategy, as it is called in Game Theory. The PDE which could provide the strategy is called Hamilton-Jacobi-Bellman Equation and it was the aim of this thesis to find out whether or not it is possible to solve it via the Finite Element Method. The first chapters were used to outline the applications of Optimal Control Theory and to derive the needed mathematical concepts. Next, various ways to prepare a problem formulation for the Finite Element Analysis were presented. One of the major challenges of this thesis was to develop a Finite Element software that can solve PDEs in a space with arbitrary dimension. Finally, a problem formulation was developed that features a convex variational form with a unique solution which promises reliable solvability by numerical methods. The results showed that it is fact possible to generate feedback control laws for nonlinear dynamical systems by the Finite Element Method. The author is convinced that the developed methods will shape the future of fields such as control engineering, robotics, automation and AI.

AB - Even though Optimal Control Theory was developed in the 1950s its usage in feedback control systems is restricted to linear dynamical systems. In General PID and LQR controllers are still the state of the art which is due to their simple implementation and reliability. There is, however, a wide range of problems in robotics and mechatronics that exceed the ability of these practices due to nonlinear dynamical behaviour. To make Optimal Control Theory feasible in these fields, its application has to be unified and simplified within a software that can reliably solve these problems. It is possible to formulate the Optimal Control Problem in such a way that a solution to the resulting boundary value problem does not only reveal one optimal trajectory but a feedback control law or strategy, as it is called in Game Theory. The PDE which could provide the strategy is called Hamilton-Jacobi-Bellman Equation and it was the aim of this thesis to find out whether or not it is possible to solve it via the Finite Element Method. The first chapters were used to outline the applications of Optimal Control Theory and to derive the needed mathematical concepts. Next, various ways to prepare a problem formulation for the Finite Element Analysis were presented. One of the major challenges of this thesis was to develop a Finite Element software that can solve PDEs in a space with arbitrary dimension. Finally, a problem formulation was developed that features a convex variational form with a unique solution which promises reliable solvability by numerical methods. The results showed that it is fact possible to generate feedback control laws for nonlinear dynamical systems by the Finite Element Method. The author is convinced that the developed methods will shape the future of fields such as control engineering, robotics, automation and AI.

KW - Optimale Steuerung

KW - Regelungstechnik

KW - nichtlineare dynamische systeme

KW - Spieltheorie

KW - Optimal Control

KW - Game Theory

KW - Differential Games

KW - Feedback Control

KW - Control Theory

KW - nonlinear dynamical systems

M3 - Master's Thesis

ER -