1,559 research outputs found
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
The property that every control system should posses is stability, which
translates into safety in real-life applications. A central tool in systems
theory for synthesizing control laws that achieve stability are control
Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting
closed-loop state trajectory is contained within a cone with a fixed,
predefined shape, and which is centered at and converges to a desired
converging point. However, such a requirement often proves to be
overconservative, which is why most of the real-time controllers do not have a
stability guarantee. Recently, a novel idea that improves the design of CLFs in
terms of flexibility was proposed. The focus of this new approach is on the
design of optimization problems that allow certain parameters that define a
cone associated with a standard CLF to be decision variables. In this way
non-monotonicity of the CLF is explicitly linked with a decision variable that
can be optimized on-line. Conservativeness is significantly reduced compared to
classical CLFs, which makes \emph{flexible CLFs} more suitable for
stabilization of constrained discrete-time nonlinear systems and real-time
control. The purpose of this overview is to highlight the potential of flexible
CLFs for real-time control of fast mechatronic systems, with sampling periods
below one millisecond, which are widely employed in aerospace and automotive
applications.Comment: 2 figure
Explicit approximate controllability of the Schr\"odinger equation with a polarizability term
We consider a controlled Schr\"odinger equation with a dipolar and a
polarizability term, used when the dipolar approximation is not valid. The
control is the amplitude of the external electric field, it acts non linearly
on the state. We extend in this infinite dimensional framework previous
techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in
finite dimension. We consider a highly oscillating control and prove the
semi-global weak stabilization of the averaged system using a Lyapunov
function introduced by Nersesyan. Then it is proved that the solutions of the
Schr\"odinger equation and of the averaged equation stay close on every finite
time horizon provided that the control is oscillating enough. Combining these
two results, we get approximate controllability to the ground state for the
polarizability system
Asymptotic ensemble stabilizability of the Bloch equation
In this paper we are concerned with the stabilizability to an equilibrium
point of an ensemble of non interacting half-spins. We assume that the spins
are immersed in a static magnetic field, with dispersion in the Larmor
frequency, and are controlled by a time varying transverse field. Our goal is
to steer the whole ensemble to the uniform "down" position. Two cases are
addressed: for a finite ensemble of spins, we provide a control function (in
feedback form) that asymptotically stabilizes the ensemble in the "down"
position, generically with respect to the initial condition. For an ensemble
containing a countable number of spins, we construct a sequence of control
functions such that the sequence of the corresponding solutions pointwise
converges, asymptotically in time, to the target state, generically with
respect to the initial conditions. The control functions proposed are uniformly
bounded and continuous
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