36,629 research outputs found
Final-State Constrained Optimal Control via a Projection Operator Approach
In this paper we develop a numerical method to solve nonlinear optimal
control problems with final-state constraints. Specifically, we extend the
PRojection Operator based Netwon's method for Trajectory Optimization (PRONTO),
which was proposed by Hauser for unconstrained optimal control problems. While
in the standard method final-state constraints can be only approximately
handled by means of a terminal penalty, in this work we propose a methodology
to meet the constraints exactly. Moreover, our method guarantees recursive
feasibility of the final-state constraint. This is an appealing property
especially in realtime applications in which one would like to be able to stop
the computation even if the desired tolerance has not been reached, but still
satisfy the constraints. Following the same conceptual idea of PRONTO, the
proposed strategy is based on two main steps which (differently from the
standard scheme) preserve the feasibility of the final-state constraints: (i)
solve a quadratic approximation of the nonlinear problem to find a descent
direction, and (ii) get a (feasible) trajectory by means of a feedback law
(which turns out to be a nonlinear projection operator). To find the (feasible)
descent direction we take advantage of final-state constrained Linear Quadratic
optimal control methods, while the second step is performed by suitably
designing a constrained version of the trajectory tracking projection operator.
The effectiveness of the proposed strategy is tested on the optimal state
transfer of an inverted pendulum
Minimum-time trajectory generation for quadrotors in constrained environments
In this paper, we present a novel strategy to compute minimum-time
trajectories for quadrotors in constrained environments. In particular, we
consider the motion in a given flying region with obstacles and take into
account the physical limitations of the vehicle. Instead of approaching the
optimization problem in its standard time-parameterized formulation, the
proposed strategy is based on an appealing re-formulation. Transverse
coordinates, expressing the distance from a frame path, are used to
parameterise the vehicle position and a spatial parameter is used as
independent variable. This re-formulation allows us to (i) obtain a fixed
horizon problem and (ii) easily formulate (fairly complex) position
constraints. The effectiveness of the proposed strategy is proven by numerical
computations on two different illustrative scenarios. Moreover, the optimal
trajectory generated in the second scenario is experimentally executed with a
real nano-quadrotor in order to show its feasibility.Comment: arXiv admin note: text overlap with arXiv:1702.0427
The Significance of the -Numerical Range and the Local -Numerical Range in Quantum Control and Quantum Information
This paper shows how C-numerical-range related new strucures may arise from
practical problems in quantum control--and vice versa, how an understanding of
these structures helps to tackle hot topics in quantum information.
We start out with an overview on the role of C-numerical ranges in current
research problems in quantum theory: the quantum mechanical task of maximising
the projection of a point on the unitary orbit of an initial state onto a
target state C relates to the C-numerical radius of A via maximising the trace
function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one
may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii)
in restricting the dynamics to {\em local} operations on each qubit, i.e. to
the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2).
Interestingly, the latter then leads to a novel entity, the {\em local}
C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither
star-shaped nor simply connected in contrast to the conventional C-numerical
range. This is shown in the accompanying paper (math-ph/0702005).
We present novel applications of the C-numerical range in quantum control
assisted by gradient flows on the local unitary group: (1) they serve as
powerful tools for deciding whether a quantum interaction can be inverted in
time (in a sense generalising Hahn's famous spin echo); (2) they allow for
optimising witnesses of quantum entanglement. We conclude by relating the
relative C-numerical range to problems of constrained quantum optimisation, for
which we also give Lagrange-type gradient flow algorithms.Comment: update relating to math-ph/070200
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