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cases one could use PDEs, on the one hand, to predict the outcome of the force field, on
the other hand, to find the control policy which when applied meets the desired behavior.
So, both communities are using PDEs but for different purposes. This fact results also
in different terminology. What is called a ”force field”, for physics, it can be renamed as
”control policy” in control theory.
The observations above are not necessarily objective, but they are very much related
to our experiences as we were trying to understand and bring together concepts from
physics and control theory. With this background in our mind, in this chapter our goal
is to bring together concepts from physics and control theory with emphasis on the
connection between PDEs, SDEs and Path Integrals. More precisely, section 3.1 is a
short journey in the world of quantum mechanics and the work on Path Integrals by one
of the most brilliant intellectuals in the history of sciences, Dr. Richard Feynman. By no
means, this section is not a review his work. This section just aims to show that the core
concepts of this thesis, which is the Path Integral, has its historical origins in the work
by Richard Feynman.
In sections 3.2 and 3.4 we highlight the convection between the forward Fokker Planck
PDEs and the underlying SDE for both the Itˆo and the Stratonovich calculus. With
the goal to establish the connection between the path integral formalism and SDEs, in
section 3.3, we derive the path integral for the general stochastic integration scheme
for 1-dimensional SDE and then we specialize for the cases of Itˆo and the Stratonovich
calculus. In section 3.4 the derivation of the Itˆo path integral for multi-dimensional SDEs
is presented. The last two sections are aiming to show how forwards PDEs are connected
to Path Integrals and SDEs.
61
Object Description
| Title | Iterative path integral stochastic optimal control: theory and applications to motor control |
| Author | Theodorou, Evangelos A. |
| Author email | etheodor@usc.edu; theo0027@umn.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2011-01-11 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-29 |
| Advisor (committee chair) | Schaal, Stefan |
| Advisor (committee member) |
Valero-Cuevas, Francisco Sukhatme, Gaurav S. Todorov, Emo Schweighofer, Nicolas |
| Abstract | Motivated by the limitations of current optimal control and reinforcement learning methods in terms of their efficiency and scalability, this thesis proposes an iterative stochastic optimal control approach based on the generalized path integral formalism. More precisely, we suggest the use of the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton Jacobi Bellman (HJB) equation, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The new algorithm, Policy Improvement with Path Integrals (PI2), demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Applications to high dimensional robotic systems are presented for a variety of tasks that require optimal planning and gain scheduling.; In addition to the work on generalized path integral stochastic optimal control, in this thesis we extend model based iterative optimal control algorithms to the stochastic setting. More precisely we derive the Differential Dynamic Programming algorithm for stochastic systems with state and control multiplicative noise. Finally, in the last part of this thesis, model based iterative optimal control methods are applied to bio-mechanical models of the index finger with the goal to find the underlying tendon forces applied for the movements of, tapping and flexing. |
| Keyword | stochastic optimal control; reinforcement learning,; robotics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3804 |
| Contributing entity | University of Southern California |
| Rights | Theodorou, Evangelos A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Theodorou-4581 |
| Archival file | uscthesesreloadpub_Volume14/etd-Theodorou-4581.pdf |
Description
| Title | Page 75 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | cases one could use PDEs, on the one hand, to predict the outcome of the force field, on the other hand, to find the control policy which when applied meets the desired behavior. So, both communities are using PDEs but for different purposes. This fact results also in different terminology. What is called a ”force field”, for physics, it can be renamed as ”control policy” in control theory. The observations above are not necessarily objective, but they are very much related to our experiences as we were trying to understand and bring together concepts from physics and control theory. With this background in our mind, in this chapter our goal is to bring together concepts from physics and control theory with emphasis on the connection between PDEs, SDEs and Path Integrals. More precisely, section 3.1 is a short journey in the world of quantum mechanics and the work on Path Integrals by one of the most brilliant intellectuals in the history of sciences, Dr. Richard Feynman. By no means, this section is not a review his work. This section just aims to show that the core concepts of this thesis, which is the Path Integral, has its historical origins in the work by Richard Feynman. In sections 3.2 and 3.4 we highlight the convection between the forward Fokker Planck PDEs and the underlying SDE for both the Itˆo and the Stratonovich calculus. With the goal to establish the connection between the path integral formalism and SDEs, in section 3.3, we derive the path integral for the general stochastic integration scheme for 1-dimensional SDE and then we specialize for the cases of Itˆo and the Stratonovich calculus. In section 3.4 the derivation of the Itˆo path integral for multi-dimensional SDEs is presented. The last two sections are aiming to show how forwards PDEs are connected to Path Integrals and SDEs. 61 |
