41,338 research outputs found
The averaged control system of fast oscillating control systems
For control systems that either have a fast explicit periodic dependence on
time and bounded controls or have periodic solutions and small controls, we
define an average control system that takes into account all possible
variations of the control, and prove that its solutions approximate all
solutions of the oscillating system as oscillations go faster. The dimension of
its velocity set is characterized geometrically. When it is maximum the average
system defines a Finsler metric, not twice differentiable in general. For
minimum time control, this average system allows one to give a rigorous proof
that averaging the Hamiltonian given by the maximum principle is a valid
approximation.Comment: (2012
Multi-phase averaging of time-optimal low-thrust transfers
International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve
Multi-phase averaging of time-optimal low-thrust transfers
International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve
Robust and fast sliding-mode control for a DC-DC current-source parallel-resonant converter
Modern DC-DC resonant converters are normally built around a voltage-source series-resonant converter. This study aims to facilitate the practical use of current-source parallel-resonant converters due to their outstanding properties. To this end, this study presents a sliding-mode control scheme, which provides the following features to the closed-loop system: (i) high robustness to external disturbances and parameter variations and (ii) fast transient response during large and abrupt load changes. In addition, a design procedure for determining the values of the control parameters is presented. The theoretical contributions of this study are experimentally validated by selected tests on a laboratory prototype.Peer ReviewedPreprin
The economic basis of periodic enzyme dynamics
Periodic enzyme activities can improve the metabolic performance of cells. As
an adaptation to periodic environments or by driving metabolic cycles that can
shift fluxes and rearrange metabolic processes in time to increase their
efficiency. To study what benefits can ensue from rhythmic gene expression or
posttranslational modification of enzymes, I propose a theory of optimal enzyme
rhythms in periodic or static environments. The theory is based on kinetic
metabolic models with predefined metabolic objectives, scores the effects of
harmonic enzyme oscillations, and determines amplitudes and phase shifts that
maximise cell fitness. In an expansion around optimal steady states, the
optimal enzyme profiles can be computed by solving a quadratic optimality
problem. The formulae show how enzymes can increase their efficiency by
oscillating in phase with their substrates and how cells can benefit from
adapting to external rhythms and from spontaneous, intrinsic enzyme rhythms.
Both types of behaviour may occur different parameter regions of the same
model. Optimal enzyme profiles are not passively adapted to existing substrate
rhythms, but shape them actively to create opportunities for further fitness
advantage: in doing so, they reflect the dynamic effects that enzymes can exert
in the network. The proposed theory combines the dynamics and economics of
metabolic systems and shows how optimal enzyme profiles are shaped by network
structure, dynamics, external rhythms, and metabolic objectives. It covers
static enzyme adaptation as a special case, reveals the conditions for
beneficial metabolic cycles, and predicts optimally combinations of gene
expression and posttranslational modification for creating enzyme rhythms
Optimal Subharmonic Entrainment
For many natural and engineered systems, a central function or design goal is
the synchronization of one or more rhythmic or oscillating processes to an
external forcing signal, which may be periodic on a different time-scale from
the actuated process. Such subharmonic synchrony, which is dynamically
established when N control cycles occur for every M cycles of a forced
oscillator, is referred to as N:M entrainment. In many applications,
entrainment must be established in an optimal manner, for example by minimizing
control energy or the transient time to phase locking. We present a theory for
deriving inputs that establish subharmonic N:M entrainment of general nonlinear
oscillators, or of collections of rhythmic dynamical units, while optimizing
such objectives. Ordinary differential equation models of oscillating systems
are reduced to phase variable representations, each of which consists of a
natural frequency and phase response curve. Formal averaging and the calculus
of variations are then applied to such reduced models in order to derive
optimal subharmonic entrainment waveforms. The optimal entrainment of a
canonical model for a spiking neuron is used to illustrate this approach, which
is readily extended to arbitrary oscillating systems
Rapidly-oscillating scatteringless non-Hermitian potentials and the absence of Kapitza stabilization
In the framework of the ordinary non-relativistic quantum mechanics, it is
known that a quantum particle in a rapidly-oscillating bound potential with
vanishing time average can be scattered off or even trapped owing to the
phenomenon of dynamical (Kapitza) stabilization. A similar phenomenon occurs
for scattering and trapping of optical waves. Such a remarkable result stems
from the fact that, even though the particle is not able to follow the rapid
external oscillations of the potential, these are still able to affect the
average dynamics by means of an effective -albeit small- nonvanishing potential
contribution. Here we consider the scattering and dynamical stabilization
problem for matter or classical waves by a bound potential with oscillating ac
amplitude in the framework of a non-Hermitian extension of the
Schr\"odinger equation, and predict that for a wide class of imaginary
amplitude modulations possessing a one-sided Fourier spectrum the
oscillating potential is effectively canceled, i.e. it does not have any effect
to the particle dynamics, contrary to what happens in the Hermitian caseComment: 7 pages, 3 figure
Onset, timing, and exposure therapy of stress disorders: mechanistic insight from a mathematical model of oscillating neuroendocrine dynamics
The hypothalamic-pituitary-adrenal (HPA) axis is a neuroendocrine system that
regulates numerous physiological processes. Disruptions in the activity of the
HPA axis are correlated with many stress-related diseases such as
post-traumatic stress disorder (PTSD) and major depressive disorder. In this
paper, we characterize "normal" and "diseased" states of the HPA axis as basins
of attraction of a dynamical system describing the inhibition of peptide
hormones such as corticotropin-releasing hormone (CRH) and adrenocorticotropic
hormone (ACTH) by circulating glucocorticoids such as cortisol (CORT). In
addition to including key physiological features such as ultradian oscillations
in cortisol levels and self-upregulation of CRH neuron activity, our model
distinguishes the relatively slow process of cortisol-mediated CRH biosynthesis
from rapid trans-synaptic effects that regulate the CRH secretion process.
Crucially, we find that the slow regulation mechanism mediates external
stress-driven transitions between the stable states in novel, intensity,
duration, and timing-dependent ways. These results indicate that the timing of
traumatic events may be an important factor in determining if and how patients
will exhibit hallmarks of stress disorders. Our model also suggests a mechanism
whereby exposure therapy of stress disorders such as PTSD may act to normalize
downstream dysregulation of the HPA axis.Comment: 30 pages, 16 figures, submitted to BMC Biology Direc
- …