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Computational Methods for Optimal Control of Hybrid Systems

Hedlund, Sven LU (2003) In PhD Theses TFRT-1068.
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... (More)
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 Lyapunov-like 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. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Professor Vinter, Richard, Department of Electrical and Electronic Engineering, Imperial College of Science Technology and Medicine, United Kingdom
organization
publishing date
type
Thesis
publication status
published
subject
keywords
reglerteknik, Automation, robotics, control engineering, robotteknik, Automatiska system
in
PhD Theses
volume
TFRT-1068
pages
133 pages
publisher
Department of Automatic Control, Lund Institute of Technology (LTH)
defense location
Room M:B, the M-building, Lund Institute of Technology
defense date
2003-05-26 10:15
ISSN
0280-5316
language
English
LU publication?
yes
id
33d4396f-391f-4222-ae5d-ae6fdde97da4 (old id 465830)
date added to LUP
2007-09-07 13:50:36
date last changed
2016-09-19 08:44:52
@phdthesis{33d4396f-391f-4222-ae5d-ae6fdde97da4,
  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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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 Lyapunov-like 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.},
  author       = {Hedlund, Sven},
  issn         = {0280-5316},
  keyword      = {reglerteknik,Automation,robotics,control engineering,robotteknik,Automatiska system},
  language     = {eng},
  pages        = {133},
  publisher    = {Department of Automatic Control, Lund Institute of Technology (LTH)},
  school       = {Lund University},
  series       = {PhD Theses},
  title        = {Computational Methods for Optimal Control of Hybrid Systems},
  volume       = {TFRT-1068},
  year         = {2003},
}