226 research outputs found
Non-linear Symmetry-preserving Observer on Lie Groups
In this paper we give a geometrical framework for the design of observers on
finite-dimensional Lie groups for systems which possess some specific
symmetries. The design and the error (between true and estimated state)
equation are explicit and intrinsic. We consider also a particular case:
left-invariant systems on Lie groups with right equivariant output. The theory
yields a class of observers such that error equation is autonomous. The
observers converge locally around any trajectory, and the global behavior is
independent from the trajectory, which reminds of the linear stationary case.Comment: 12 pages. Submitted. Preliminary version publicated in french in the
CIFA proceedings and IFAC0
Symmetry-preserving Observers
This paper presents three non-linear observers on three examples of
engineering interest: a chemical reactor, a non-holonomic car, and an inertial
navigation system. For each example, the design is based on physical
symmetries. This motivates the theoretical development of invariant observers,
i.e, symmetry-preserving observers. We consider an observer to consist in a
copy of the system equation and a correction term, and we give a constructive
method (based on the Cartan moving-frame method) to find all the
symmetry-preserving correction terms. They rely on an invariant frame (a
classical notion) and on an invariant output-error, a less standard notion
precisely defined here. For each example, the convergence analysis relies also
on symmetries consideration with a key use of invariant state-errors. For the
non-holonomic car and the inertial navigation system, the invariant
state-errors are shown to obey an autonomous differential equation independent
of the system trajectory. This allows us to prove convergence, with almost
global stability for the non-holonomic car and with semi-global stability for
the inertial navigation system. Simulations including noise and bias show the
practical interest of such invariant asymptotic observers for the inertial
navigation system.Comment: To be published in IEEE Automatic Contro
Flatness-based control of a single qubit gate
This work considers the open-loop control problem of steering a two level
quantum system from an initial to a final condition. The model of this system
evolves on the state space SU(2), having two inputs that correspond to the
complex amplitude of a resonant laser field. A symmetry preserving flat output
is constructed using a fully geometric construction and quaternion
computations. Simulation results of this flatness-based open-loop control are
provided.Comment: Submitted to IEEE AC. Simulation code available at
http://cas.ensmp.fr/~rouchon/publications/PR2007/CodeMatlabScilabQubit.zi
A Time-Periodic Lyapunov Approach for Motion Planning of Controllable Driftless Systems on SU(n)
For a right-invariant and controllable driftless system on SU(n), we consider
a time-periodic reference trajectory along which the linearized control system
generates su(n): such trajectories always exist and constitute the basic
ingredient of Coron's Return Method. The open-loop controls that we propose,
which rely on a left-invariant tracking error dynamics and on a fidelity-like
Lyapunov function, are determined from a finite number of left-translations of
the tracking error and they assure global asymptotic convergence towards the
periodic reference trajectory. The role of these translations is to avoid being
trapped in the critical region of this Lyapunov-like function. The convergence
proof relies on a periodic version of LaSalle's invariance principle and the
control values are determined by numerical integration of the dynamics of the
system. Simulations illustrate the obtained controls for and the
generation of the C--NOT quantum gate.Comment: Submitte
Observing quantum state diffusion by heterodyne detection of fluorescence
A qubit can relax by fluorescence, which prompts the release of a photon into
its electromagnetic environment. By counting the emitted photons, discrete
quantum jumps of the qubit state can be observed. The succession of states
occupied by the qubit in a single experiment, its quantum trajectory, depends
in fact on the kind of detector. How are the quantum trajectories modified if
one measures continuously the amplitude of the fluorescence field instead?
Using a superconducting parametric amplifier, we have performed heterodyne
detection of the fluorescence of a superconducting qubit. For each realization
of the measurement record, we can reconstruct a different quantum trajectory
for the qubit. The observed evolution obeys quantum state diffusion, which is
characteristic of quantum measurements subject to zero point fluctuations.
Independent projective measurements of the qubit at various times provide a
quantitative validation of the reconstructed trajectories. By exploring the
statistics of quantum trajectories, we demonstrate that the qubit states span a
deterministic surface in the Bloch sphere at each time in the evolution.
Additionally, we show that when monitoring fluorescence, coherent
superpositions are generated during the decay from excited to ground state.
Counterintuitively, measuring light emitted during relaxation can give rise to
trajectories with increased excitation probability.Comment: Supplementary material can be found in the ancillary sectio
Using Spontaneous Emission of a Qubit as a Resource for Feedback Control
Persistent control of a transmon qubit is performed by a feedback protocol
based on continuous heterodyne measurement of its fluorescence. By driving the
qubit and cavity with microwave signals whose amplitudes depend linearly on the
instantaneous values of the quadratures of the measured fluorescence field, we
show that it is possible to stabilize permanently the qubit in any targeted
state. Using a Josephson mixer as a phase-preserving amplifier, it was possible
to reach a total measurement efficiency =35%, leading to a maximum of 59%
of excitation and 44% of coherence for the stabilized states. The experiment
demonstrates multiple-input multiple-output analog Markovian feedback in the
quantum regime.Comment: Supplementary material can be found as an ancillary objec
Aquifers survey in the context of source rocks exploitation: from baseline acquisition to long term monitoring.
International audienc
Controllability of spin-boson systems
In this paper we study the so-called spin-boson system, namely {a two-level
system} in interaction with a distinguished mode of a quantized bosonic field.
We give a brief description of the controlled Rabi and Jaynes--Cummings models
and we discuss their appearance in the mathematics and physics literature. We
then study the controllability of the Rabi model when the control is an
external field acting on the bosonic part. Applying geometric control
techniques to the Galerkin approximation and using perturbation theory to
guarantee non-resonance of the spectrum of the drift operator, we prove
approximate controllability of the system, for almost every value of the
interaction parameter
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