Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field

In this paper we consider the minimum time population transfer problem for the $z$-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|\Omega(t)|\leq M$ the bound on the field amplitude. For each couple of values $E$ and $M$, we determine the time optimal synthesis starting from the level $-E$ and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For $M/E<<1$, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency $\omega_R=2E$. On the other side, for $M/E>1$, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. In the case $M/E>1$ there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the $x$ and $y$ directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte

Unknown input observer for Takagi-Sugeno implicit models with unmeasurable premise variables

Recent years have seen a great deal of interest in implicit nonlinear systems, which are used in many different engineering applications. This study is dedicated to presenting a new method of fuzzy unknown inputs observer design to estimate simultaneously both non-measurable states and unknown inputs of continuous-time nonlinear implicit systems defined by Takagi-Sugeno (T-S) models with unmeasurable premise variables. The suggested observer is based on the singular value decomposition approach and rewritten the continuous-time T-S implicit models into an augmented fuzzy system, which gathers the unknown inputs and the state vector. The exponential convergence condition of the observer is established by using the Lyapunov theory and linear matrix inequalities are solved to determine the gains of the observer. Finally, the effectiveness of the suggested method is then assessed using a numerical application. It demonstrates that the estimated variables and the unknown input converge to the real variables accurately and quickly (less than 0.5 s)

A fuzzy observer synthesis to state and fault estimation for Takagi-Sugeno implicit systems

The present paper inquire a fuzzy observer design issue for continuous time Takagi-Sugeno aiming at estimating both of fault and state in case of unmeasurable premise variables. Thanks to singular value decomposition (SVD) approach, the fuzzy observer is synthesized in explicit form. The developed method is based on augmented structure that assemble state and actuator fault, sensor fault as well as their sequential derivatives is enjoined to institute the destined observer. The Lyapunov function is analyzed to assume the exponential stability of the studied observer, furthermore convergence conditions are expressed as linear matrix inequalities (LMIs) form. The applicability of the nominated theoritical result is elucidated and confiremed through numerical simulation using an example of an implicit fuzzy model

Discrete-time Takagi-Sugeno singular systems with unmeasurable decision variables: state and fault fuzzy observer design

The studied problem in this paper, treat the issue of state and fault estimation using a fuzzy observer in the case of unmeasurable decision variable for Discrete-Time Takagi-Sugeno Singular Sytems (DTSSS). First, an augmented system is introduced to gather state and fault into a single vector, then on the basis of Singular Value Decomposition (SVD) approach, this observer is designed in explicit form to estimate both of state and fault of a nonlinear singular system. The exponential stability of this observer is studied using Lyapunov theory and the convergence conditions are solved with Linear Matrix Inequalities (LMIs). Finally a numerical example is simulated, and results are given to validate the offered approach

Observer Design for a Class of Nonlinear Descriptor Systems: A Takagi-Sugeno Approach with Unmeasurable Premise Variables

The Takagi-Sugeno (T-S) fuzzy observer for dynamical systems described by ordinary differential equations is widely discussed in the literature. The aim of this paper is to extend this observer design to a class of T-S descriptor systems with unmeasurable premise variables. In practice, the computation of solutions of differential-algebraic equations requires the combination of an ordinary differential equations (ODE) routine together with an optimization algorithm. Therefore, a natural way permitting to estimate the state of such a system is to design a procedure based on a similar numerical algorithm. Beside some numerical difficulties, the drawback of such a method lies in the fact that it is not easy to establish a rigorous proof of the convergence of the observer. The main result of this paper consists in showing that the state estimation problem for a class of T-S descriptor systems can be achieved by using a fuzzy observer having only an ODE structure. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of linear matrix inequalities (LMIs). Finally, an application to a model of a heat exchanger pilot process is given to illustrate the performance of the proposed observer

Investigation of an IC Engine Intake Flow Based on Highly Resolved LES and PIV

To reduce emissions and fuel consumption, the current generation of gasoline engines uses technologies such as direct injection, downsizing and supercharging. All of them require a strong vortical in-cylinder charge motion, usually described as “tumble”, to improve fuel-air mixing and enhance flame propagation. The tumble development strongly depends on the flow field during the intake stroke. This flow field is dominated by the intake jet, which has to be captured well in the simulation. This work investigates the intake jet on a steady-state flow bench, especially in the vicinity of the intake valve. At first, the general flow dynamics of the intake jet for three different valve lifts and three different mass flows were investigated experimentally. For the smallest valve lift (3 mm), flow-field measurements using Particle Image Velocimetry (PIV) show that the orientation of the intake jet significantly depends on the air flow rate, attaching to the pent roof for low flow rates. This phenomenon is less pronounced for higher valve lifts. An intermediate valve lift and flow rate were chosen for further investigations by scale-resolving simulations. Three different meshes (coarse, medium and fine) and two turbulence models (Sigma and Detached Eddy Simulation-Shear Stress Transport (DES-SST)) are applied to consider their effect on the numerical results. An ad-hoc post-processing methodology based on the ensemble-averaged velocity field is presented capturing the jet centerline’s mean velocity and velocity fluctuations as well as its orientation, curvature and penetration depth. The simulation results are compared to each other as well as to measurements by PIV

Cartan decompositions and semigroups of simple Lie groups

http://www.heldermann.de/JLT/JLT28/JLT281/jlt28010.htmInternational audienceLet G be a split real connected simple Lie group and S a semigroup of G that contains a subgroup G(α) for an arbitrary root α, isomorphic to SL(2,R). We present a Cartan decomposition of the Lie algebra of G, related to α, invariant by the adjoint action of the Lie algebra sl(2,R) that allows to characterize some properties of the Lie saturate of the semigroup S. We give necessary and sufficient conditions for S to be equal to the whole group G

Adjoint Orbits of sl(2, ℝ) on Real Simple Lie Algebras and Controllability

International audienceThis paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root ź, there exists a Cartan decomposition of L, related to ź, which characterizes some controllability properties by using the adjoint orbits of sl(2, ź). For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient

Accessibilite par des champs de vecteurs invariants sur un groupe de Lie et ses espaces homogenes

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