Abstract
This thesis aims to find algorithms for optimal control of hybrid systems and explore them in sufficient detail to be able to implement the ideas in computational tools. By hybrid systems is meant systems with interacting continuous and discrete dynamics. Code for computations has been developed in parallel to the theory.
The optimal control methods studied in this thesis are global, i.e. the entire state space is considered simultaneously rather than searching for locally optimal trajectories. The optimal value function that maps each state of the state space onto the minimal cost for trajectories starting in that state is central for global methods. It is often difficult to compute the value function of an optimal control problem, even for a purely continuous system. This thesis shows that a lower bound of the value function of a hybrid optimal control problem can be found via convex optimization in a linear program. Moreover, a dual of this optimization problem, parameterized in the control law, has been formulated via general ideas from duality in transportation problems. It is shown that the lower bound of the value function is tight for continuous systems and that there is no gap between the dual optimization problems.
Two computational tools are presented. One is built on theory for piecewise affine systems. Various analysis and synthesis problems for this kind of systems are via piecewise quadratic Lyapunovlike functions cast into linear matrix inequalities. The second tool can be used for value function computation, control law extraction, and simulation of hybrid systems. This tool parameterizes the value function in its values in a uniform grid of points in the state space, and the optimization problem is formulated as a linear program. The usage of this tool is illustrated in a case study.
The optimal control methods studied in this thesis are global, i.e. the entire state space is considered simultaneously rather than searching for locally optimal trajectories. The optimal value function that maps each state of the state space onto the minimal cost for trajectories starting in that state is central for global methods. It is often difficult to compute the value function of an optimal control problem, even for a purely continuous system. This thesis shows that a lower bound of the value function of a hybrid optimal control problem can be found via convex optimization in a linear program. Moreover, a dual of this optimization problem, parameterized in the control law, has been formulated via general ideas from duality in transportation problems. It is shown that the lower bound of the value function is tight for continuous systems and that there is no gap between the dual optimization problems.
Two computational tools are presented. One is built on theory for piecewise affine systems. Various analysis and synthesis problems for this kind of systems are via piecewise quadratic Lyapunovlike functions cast into linear matrix inequalities. The second tool can be used for value function computation, control law extraction, and simulation of hybrid systems. This tool parameterizes the value function in its values in a uniform grid of points in the state space, and the optimization problem is formulated as a linear program. The usage of this tool is illustrated in a case study.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2003 May 26 
Publisher  
Publication status  Published  2003 
Bibliographical note
Defence detailsDate: 20030526
Time: 10:15
Place: Room M:B, the Mbuilding, Lund Institute of Technology
External reviewer(s)
Name: Vinter, Richard
Title: Professor
Affiliation: Department of Electrical and Electronic Engineering, Imperial College of Science Technology and Medicine, United Kingdom

Article: Hedlund, S. and M. Johansson, "A Toolbox for Computational Analysis of Piecewise Linear Systems", Proceedings of European Control Conference, 1999
Article: Hedlund, S. and A. Rantzer, "Hybrid Control Laws from Convex Dynamic Programming", IEEE Conference on Decision and Control, 2000
Article: Hedlund, S. and A. Rantzer, "Convex Dynamic Programming for Hybrid Systems", IEEE Transactions on Automatic Control, 2002
Article: Rantzer, A. and S. Hedlund, "Duality Between Cost and Density in Optimal Control", IEEE Conference on Decision and Control, 2003
Subject classification (UKÄ)
 Control Engineering
Keywords
 reglerteknik
 Automation
 robotics
 control engineering
 robotteknik
 Automatiska system