arXiv subjects

Pierre Rouchon

Publications and source records attributed to Pierre Rouchon.

At least 91 records · Page 5Linked to original sources

arXiv2012

Sensorless position estimation of Permanent-Magnet Synchronous Motors using a saturation model

Sensorless control of Permanent-Magnet Synchronous Motors (PMSM) at low velocity remains a challenging task. A now well-established method consists in injecting a high-frequency signal and use the rotor saliency, both geometric and magnetic-saturation induced. This paper proposes a clear and original analysis based on second-order averaging of how to recover the position information from signal injection; this analysis blends well with a general model of magnetic saturation. It also proposes a simple parametric model of the saturated PMSM, based on an energy function which simply encompasses saturation and cross-saturation effects. Experimental results on a surface-mounted PMSM and an interior magnet PMSM illustrate the relevance of the approach.

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arXiv2012

Signal injection and averaging for position estimation of Permanent-Magnet Synchronous Motors

Sensorless control of Permanent-Magnet Synchronous Motors at low velocity remains a challenging task. A now well-established method consists in injecting a high-frequency signal and use the rotor saliency, both geometric and magnetic-saturation induced. This paper proposes a clear and original analysis based on second-order averaging of how to recover the position information from signal injection; this analysis blends well with a general model of magnetic saturation. It also experimentally demonstrates the relevance for position estimation of a simple parametric saturation model recently introduced by the authors.

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arXiv2012

Stabilization of an arbitrary profile for an ensemble of half-spin systems

We consider the feedback stabilization of a variable profile for an ensemble of non interacting half spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H^1 neighborhood of this target profile. This convergence is shown for the weak H^1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the target profile.

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arXiv2011

Null controllability of the structurally damped wave equation with moving point control

We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.

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arXiv2011

SO(3)-invariant asymptotic observers for dense depth field estimation based on visual data and known camera motion

In this paper, we use known camera motion associated to a video sequence of a static scene in order to estimate and incrementally refine the surrounding depth field. We exploit the SO(3)-invariance of brightness and depth fields dynamics to customize standard image processing techniques. Inspired by the Horn-Schunck method, we propose a SO(3)-invariant cost to estimate the depth field. At each time step, this provides a diffusion equation on the unit Riemannian sphere that is numerically solved to obtain a real time depth field estimation of the entire field of view. Two asymptotic observers are derived from the governing equations of dynamics, respectively based on optical flow and depth estimations: implemented on noisy sequences of synthetic images as well as on real data, they perform a more robust and accurate depth estimation. This approach is complementary to most methods employing state observers for range estimation, which uniquely concern single or isolated feature points.

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arXiv2011

Design and Stability of Discrete-Time Quantum Filters with Measurement Imperfections

This work considers the theory underlying a discrete-time quantum filter recently used in a quantum feedback experiment. It proves that this filter taking into account decoherence and measurement errors is optimal and stable. We present the general framework underlying this filter and show that it corresponds to a recursive expression of the least-square optimal estimation of the density operator in the presence of measurement imperfections. By measurement imperfections, we mean in a very general sense unread measurement performed by the environment (decoherence) and active measurement performed by non-ideal detectors. However, we assume to know precisely all the Kraus operators and also the detection error rates. Such recursive expressions combine well known methods from quantum filtering theory and classical probability theory (Bayes' law). We then demonstrate that such a recursive filter is always stable with respect to its initial condition: the fidelity between the optimal filter state (when the initial filter state coincides with the real quantum state) and the filter state (when the initial filter state is arbitrary) is a sub-martingale.

math-ph↗

arXiv2011

Design of Strict Control-Lyapunov Functions for Quantum Systems with QND Measurements

We consider discrete-time quantum systems subject to Quantum Non-Demolition (QND) measurements and controlled by an adjustable unitary evolution between two successive QND measures. In open-loop, such QND measurements provide a non-deterministic preparation tool exploiting the back-action of the measurement on the quantum state. We propose here a systematic method based on elementary graph theory and inversion of Laplacian matrices to construct strict control-Lyapunov functions. This yields an appropriate feedback law that stabilizes globally the system towards a chosen target state among the open-loop stable ones, and that makes in closed-loop this preparation deterministic. We illustrate such feedback laws through simulations corresponding to an experimental setup with QND photon counting.

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arXiv2011

On stability of continuous-time quantum-filters

We prove that the fidelity between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters.

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arXiv2011

Approximate stabilization of an infinite dimensional quantum stochastic system

We propose a feedback scheme for preparation of photon number states in a microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field and a control signal consisting of a microwave pulse injected into the cavity are used to drive the system towards a desired target photon number state. Unlike previous work, we do not use the Galerkin approximation of truncating the infinite-dimensional system Hilbert space into a finite-dimensional subspace. We use an (unbounded) strict Lyapunov function and prove that a feedback scheme that minimizes the expectation value of the Lyapunov function at each time step stabilizes the system at the desired photon number state with (a pre-specified) arbitrarily high probability. Simulations of this scheme demonstrate that we improve the performance of the controller by reducing "leakage" to high photon numbers.

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arXiv2011

Real-time quantum feedback prepares and stabilizes photon number states

Feedback loops are at the heart of most classical control procedures. A controller compares the signal measured by a sensor with the target value. It adjusts then an actuator in order to stabilize the signal towards its target. Generalizing this scheme to stabilize a micro-system's quantum state relies on quantum feedback, which must overcome a fundamental difficulty: the measurements by the sensor have a random back-action on the system. An optimal compromise employs weak measurements providing partial information with minimal perturbation. The controller should include the effect of this perturbation in the computation of the actuator's unitary operation bringing the incrementally perturbed state closer to the target. While some aspects of this scenario have been experimentally demonstrated for the control of quantum or classical micro-system variables, continuous feedback loop operations permanently stabilizing quantum systems around a target state have not yet been realized. We have implemented such a real-time stabilizing quantum feedback scheme. It prepares on demand photon number states (Fock states) of a microwave field in a superconducting cavity and subsequently reverses the effects of decoherence-induced field quantum jumps. The sensor is a beam of atoms crossing the cavity which repeatedly performs weak quantum non-demolition measurements of the photon number. The controller is implemented in a real-time computer commanding the injection, between measurements, of adjusted small classical fields in the cavity. The microwave field is a quantum oscillator usable as a quantum memory or as a quantum bus swapping information between atoms. By demonstrating that active control can generate non-classical states of this oscillator and combat their decoherence, this experiment is a significant step towards the implementation of complex quantum information operations.

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arXiv2011

Semi-Global Approximate stabilization of an infinite dimensional quantum stochastic system

In this paper we study the semi-global (approximate) state feedback stabilization of an infinite dimensional quantum stochastic system towards a target state. A discrete-time Markov chain on an infinite-dimensional Hilbert space is used to model the dynamics of a quantum optical cavity. We can choose an (unbounded) strict Lyapunov function that is minimized at each time-step in order to prove (weak-$\ast$) convergence of probability measures to a final state that is concentrated on the target state with (a pre-specified) probability that may be made arbitrarily close to 1. The feedback parameters and the Lyapunov function are chosen so that the stochastic flow that describes the Markov process may be shown to be tight (concentrated on a compact set with probability arbitrarily close to 1). We then use Prohorov's theorem and properties of the Lyapunov function to prove the desired convergence result.

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arXiv2011

Fidelity is a sub-martingale for discrete-time quantum filters

Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for any discrete-time quantum filter: fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states.

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arXiv2011

Estimation of Saturation of Permanent-Magnet Synchronous Motors Through an Energy-Based Model

We propose a parametric model of the saturated Permanent-Magnet Synchronous Motor (PMSM) together with an estimation method of the magnetic parameters. The model is based on an energy function which simply encompasses the saturation effects. Injection of fast-varying pulsating voltages and measurements of the resulting current ripples then permit to identify the magnetic parameters by linear least squares. Experimental results on a surface-mounted PMSM and an interoir magnet PMSM illustrate the relevance of the approach.

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arXiv2011

Hamiltonian identification through enhanced observability utilizing quantum control

This paper considers Hamiltonian identification for a controllable quantum system with non-degenerate transitions and a known initial state. We assume to have at our disposal a single scalar control input and the population measure of only one state at an (arbitrarily large) final time T. We prove that the quantum dipole moment matrix is locally observable in the following sense: for any two close but distinct dipole moment matrices, we construct discriminating controls giving two different measurements. Such discriminating controls are constructed to have three well defined temporal components, as inspired by Ramsey interferometry. This result suggests that what may appear at first to be very restrictive measurements are actually rich for identification, when combined with well designed discriminating controls, to uniquely identify the complete dipole moment of such systems. The assessment supports the employment of quantum control as a promising means to achieve high quality identification of a Hamiltonian.

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arXiv2011

Non-classical field state stabilization in a cavity by reservoir engineering

We propose an engineered reservoir inducing the relaxation of a cavity field towards non-classical states. It is made up of two-level atoms crossing the cavity one at a time. Each atom-cavity interaction is first dispersive, then resonant, then dispersive again. The reservoir pointer states are those produced by an effective Kerr Hamiltonian acting on a coherent field. We thereby stabilize squeezed states and quantum superpositions of multiple coherent components in a cavity having a finite damping time. This robust method could be implemented in state-of-the-art experiments and lead to interesting insights into mesoscopic quantum state superpositions and into their protection against decoherence.

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arXiv2011

Observer-based quantum state estimation by continuous weak measurement

We propose to apply the Back and Forth Nudging (BFN) method used for geophysical data assimilations to estimate the initial state of a quantum system. We consider a cloud of atoms interacting with a magnetic field while a single observable is being continuously measured over time using homodyne detection. The BFN method relies on designing an observer forward and backwards in time. The state of the BFN observer is continuously updated by the measured data and tends to converge to the systems state. The proposed estimator seems to be globally asymptotically convergent when the system is observable. A detailed convergence proof and simulations are given in the 2-level case. A discussion on the extension of the algorithm to the multilevel case is also presented.

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arXiv2010

Adiabatic passage and ensemble control of quantum systems

This paper considers population transfer between eigenstates of a finite quantum ladder controlled by a classical electric field. Using an appropriate change of variables, we show that this setting can be set in the framework of adiabatic passage, which is known to facilitate ensemble control of quantum systems. Building on this insight, we present a mathematical proof of robustness for a control protocol -- chirped pulse -- practiced by experimentalists to drive an ensemble of quantum systems from the ground state to the most excited state. We then propose new adiabatic control protocols using a single chirped and amplitude shaped pulse, to robustly perform any permutation of eigenstate populations, on an ensemble of systems with badly known coupling strengths. Such adiabatic control protocols are illustrated by simulations achieving all 24 permutations for a 4-level ladder.

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arXiv2010

Stabilization of a delayed quantum system: the photon box case-study

We study a feedback scheme to stabilize an arbitrary photon number state in a microwave cavity. The quantum non-demolition measurement of the cavity state allows a non-deterministic preparation of Fock states. Here, by the mean of a controlled field injection, we desire to make this preparation process deterministic. The system evolves through a discrete-time Markov process and we design the feedback law applying Lyapunov techniques. Also, in our feedback design we take into account an unavoidable pure delay and we compensate it by a stochastic version of a Smith predictor. After illustrating the efficiency of the proposed feedback law through simulations, we provide a rigorous proof of the global stability of the closed-loop system based on tools from stochastic stability analysis. A brief study of the Lyapunov exponents of the linearized system around the target state gives a strong indication of the robustness of the method.

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