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'(x, t) has been approximated, the optimal control are found according to the equation
that follows:
u(x, t) = %R−1G(x)T 'x'(x, t)
'(x, t)
(4.17)
Clearly optimal controls in the equation above act such that the stochastic dynamical
system visits regions of the state space with high exponentiated values function '(x, t)
while in the optimal control formulation (4.5) controls will move the system towards part
of the state space with minimum cost-to-go V (x, t). This observation is in complete agree-ment
with the exponentiation of value function '(x, t) = exp
A
−1+
V (x, t)
B
. Essentially,
the resulting value function '(x, t) can be thought as a probability of the state and thus
states with high cost to go V (x, t) will be less probable(= small '(x, t)) while state with
small cost to go will be most probable. In that sense the stochastic optimal control has
been transformed from a minimization to maximization optimization problem. Finally
the intuition behind the condition %G(x, t)R−1G(x, t)T = B(x, t)!wB(x, t)T is that,
since the weight control matrix R is inverse proportional to the variance of the noise, a
high variance control input implies cheap control cost, while small variance control inputs
have high control cost. From a control theoretic stand point such a relationship makes
sense due to the fact that under a large disturbance (= high variance) significant control
authority is required to bring the system back to a desirable state. This control authority
can be achieved with corresponding low control cost in R.
With the goal to find the '(x, t) in equation (4.16), in the next section we derive the
distribution p (" i|xi) based on the passive dynamics. This is a generalization of results
in (Kappen 2007, Broek et al. 2008).
125
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 139 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | '(x, t) has been approximated, the optimal control are found according to the equation that follows: u(x, t) = %R−1G(x)T 'x'(x, t) '(x, t) (4.17) Clearly optimal controls in the equation above act such that the stochastic dynamical system visits regions of the state space with high exponentiated values function '(x, t) while in the optimal control formulation (4.5) controls will move the system towards part of the state space with minimum cost-to-go V (x, t). This observation is in complete agree-ment with the exponentiation of value function '(x, t) = exp A −1+ V (x, t) B . Essentially, the resulting value function '(x, t) can be thought as a probability of the state and thus states with high cost to go V (x, t) will be less probable(= small '(x, t)) while state with small cost to go will be most probable. In that sense the stochastic optimal control has been transformed from a minimization to maximization optimization problem. Finally the intuition behind the condition %G(x, t)R−1G(x, t)T = B(x, t)!wB(x, t)T is that, since the weight control matrix R is inverse proportional to the variance of the noise, a high variance control input implies cheap control cost, while small variance control inputs have high control cost. From a control theoretic stand point such a relationship makes sense due to the fact that under a large disturbance (= high variance) significant control authority is required to bring the system back to a desirable state. This control authority can be achieved with corresponding low control cost in R. With the goal to find the '(x, t) in equation (4.16), in the next section we derive the distribution p (" i|xi) based on the passive dynamics. This is a generalization of results in (Kappen 2007, Broek et al. 2008). 125 |
