Page 35 |
Save page Remove page | Previous | 35 of 289 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
if H is stationary and convex the minimum principle is satisfied. If it is the one but
no the other then the minimum principle is not satisfied . A stronger condition for the
minimum principle is formulated as follows:
J(u" + *u) − J(u") = (2.27)
=
" tN
t0
.
H"
$
x"(t), u"(t) + *u,#"(t), t
%
−H"
$
x"(t), u"(t),#"(t), t
%/
dt $ 0 (2.28)
2.3 Iterative optimal control algorithms
There is a variety of optimal control algorithms depending on 1) the order of the expansion
of the dynamics, 2) the order of the expansion of the cost function and 3) the existence
of noise.
More precisely, if the dynamics under consideration are linear in the state and the
controls, deterministic, and the cost function is quadratic with respect to states and
controls, we can use one of the most established tools in control theory: the Linear
Quadratic Regulator (Stengel 1994). For such type of optimal control problems the dy-namics
are formulated as f(x, u) = Ax + Bu, F(x, u) = 0 and the immediate cost
l (', x(' ), u(', x(' ))) = xTQx + uTRu. Under the presence of stochastic dynamics
F(x, u) )= 0, the resulting algorithm is called the Linear Gaussian Quadratic Regula-tor
(LQG).
For nonlinear deterministic dynamical systems, expansion of the dynamics is per-formed
and the optimal control algorithm is solved in iterative fashion. Under a first
21
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 35 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | if H is stationary and convex the minimum principle is satisfied. If it is the one but no the other then the minimum principle is not satisfied . A stronger condition for the minimum principle is formulated as follows: J(u" + *u) − J(u") = (2.27) = " tN t0 . H" $ x"(t), u"(t) + *u,#"(t), t % −H" $ x"(t), u"(t),#"(t), t %/ dt $ 0 (2.28) 2.3 Iterative optimal control algorithms There is a variety of optimal control algorithms depending on 1) the order of the expansion of the dynamics, 2) the order of the expansion of the cost function and 3) the existence of noise. More precisely, if the dynamics under consideration are linear in the state and the controls, deterministic, and the cost function is quadratic with respect to states and controls, we can use one of the most established tools in control theory: the Linear Quadratic Regulator (Stengel 1994). For such type of optimal control problems the dy-namics are formulated as f(x, u) = Ax + Bu, F(x, u) = 0 and the immediate cost l (', x(' ), u(', x(' ))) = xTQx + uTRu. Under the presence of stochastic dynamics F(x, u) )= 0, the resulting algorithm is called the Linear Gaussian Quadratic Regula-tor (LQG). For nonlinear deterministic dynamical systems, expansion of the dynamics is per-formed and the optimal control algorithm is solved in iterative fashion. Under a first 21 |
