Page 117 |
Save page Remove page | Previous | 117 of 289 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
i) Uniform Ellipticity: There exist as positive constant * such that:
>n
i=1
>n
j=1
2i,j(x, t).i.j $ *||.||2 (3.67)
holds for every . ! "n×1 and (t, x) ! [0,-)×"n×1.
ii) Boundness: The functions f (x, t), q(x, t), 2(x, t) are bounded in [0, T]×"n×1.
iii) Holder Continuity: The functions f (x, t), q(x, t), 2(x, t) and F(x, t) are uniformly
Holder continuous in [0, T]×"n×1.
iv) Polynomial Growth: the functions '(x(tN)) = .(x(tN)) and F(x, t) satisfy the (i)
and (iii) in (3.51)
Conditions (i),(ii) and (iii) can be relaxed by assuming that they are locally true.
Essentially, the Feynman- Kac lemma provides solution of the PDE (3.53) in a prob-abilistic
manner, if that solution exists, and it also tells us that this solution is unique.
The conditions above are sufficient conditions for the existence of the solution of (3.53).
With the goal to apply the Feynman- Kac lemma to learning approximate solution for
optimal planning and control problems, it is important to understand how the conditions
of this lemma are related to properties and characteristics of the dynamical systems under
consideration.
• The condition of linear growth, for the case of control, is translated as the require-ment
to deal with dynamical systems in which their vector field as a function of
state is bounded either by a linear function or by number. Therefore the Feyn-man
Kac lemma can be applied either to linear dynamical systems of the form
103
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 117 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | i) Uniform Ellipticity: There exist as positive constant * such that: >n i=1 >n j=1 2i,j(x, t).i.j $ *||.||2 (3.67) holds for every . ! "n×1 and (t, x) ! [0,-)×"n×1. ii) Boundness: The functions f (x, t), q(x, t), 2(x, t) are bounded in [0, T]×"n×1. iii) Holder Continuity: The functions f (x, t), q(x, t), 2(x, t) and F(x, t) are uniformly Holder continuous in [0, T]×"n×1. iv) Polynomial Growth: the functions '(x(tN)) = .(x(tN)) and F(x, t) satisfy the (i) and (iii) in (3.51) Conditions (i),(ii) and (iii) can be relaxed by assuming that they are locally true. Essentially, the Feynman- Kac lemma provides solution of the PDE (3.53) in a prob-abilistic manner, if that solution exists, and it also tells us that this solution is unique. The conditions above are sufficient conditions for the existence of the solution of (3.53). With the goal to apply the Feynman- Kac lemma to learning approximate solution for optimal planning and control problems, it is important to understand how the conditions of this lemma are related to properties and characteristics of the dynamical systems under consideration. • The condition of linear growth, for the case of control, is translated as the require-ment to deal with dynamical systems in which their vector field as a function of state is bounded either by a linear function or by number. Therefore the Feyn-man Kac lemma can be applied either to linear dynamical systems of the form 103 |
