18,745 research outputs found
Optimal Navigation Functions for Nonlinear Stochastic Systems
This paper presents a new methodology to craft navigation functions for
nonlinear systems with stochastic uncertainty. The method relies on the
transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear
partial differential equation. This approach allows for optimality criteria to
be incorporated into the navigation function, and generalizes several existing
results in navigation functions. It is shown that the HJB and that existing
navigation functions in the literature sit on ends of a spectrum of
optimization problems, upon which tradeoffs may be made in problem complexity.
In particular, it is shown that under certain criteria the optimal navigation
function is related to Laplace's equation, previously used in the literature,
through an exponential transform. Further, analytical solutions to the HJB are
available in simplified domains, yielding guidance towards optimality for
approximation schemes. Examples are used to illustrate the role that noise, and
optimality can potentially play in navigation system design.Comment: Accepted to IROS 2014. 8 Page
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
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