Bypass transition to sustained thermoacoustic oscillations in a linearly stable rijke tube

In this paper we examine triggering in a simple linearly-stable thermoacoustic system using techniques from flow instability and optimal control. Firstly, for a noiseless system, we find the initial states that have highest energy growth over given times and from given energies. Secondly, by varying the initial energy, we find the lowest energy that just triggers to a stable periodic solution. We show that the corresponding initial state grows first towards an unstable periodic solution and, from there, to the stable periodic solution. This exploits linear transient growth, which arises due to nonnormality in the governing equations and is directly analogous to bypass transition to turbulence. Thirdly, we introduce noise that has similar spectral characteristics to this initial state. We show that, when triggering from low noise levels, the system grows to high amplitude self-sustained oscillations by first growing towards the unstable periodic solution of the noiseless system. This helps to explain the experimental observation that linearly-stable systems can trigger to self-sustained oscillations even with low background noise. © 2010 by University of Cambridge. Published by the American Institute of Aeronautics and Astronautics, Inc.

Enhanced poisson sum representation for alpha-stable processes

In this paper we present Poisson sum series representations for α-stable (αS) random variables and a-stable processes, in particular concentrating on continuous-time autoregressive (CAR) models driven by α-stable Lévy processes. Our representations aim to provide a conditionally Gaussian framework, which will allow parameter estimation using Rao-Blackwellised versions of state of the art Bayesian computational methods such as particle filters and Markov chain Monte Carlo (MCMC). To overcome the issues due to truncation of the series, novel residual approximations are developed. Simulations demonstrate the potential of these Poisson sum representations for inference in otherwise intractable α-stable models. © 2011 IEEE.

Functional significance of stiffness in adaptation of multijoint arm movements to stable and unstable dynamics.

This study compared the mechanisms of adaptation to stable and unstable dynamics from the perspective of changes in joint mechanics. Subjects were instructed to make point to point movements in force fields generated by a robotic manipulandum which interacted with the arm in either a stable or an unstable manner. After subjects adjusted to the initial disturbing effects of the force fields they were able to produce normal straight movements to the target. In the case of the stable interaction, subjects modified the joint torques in order to appropriately compensate for the force field. No change in joint torque or endpoint force was required or observed in the case of the unstable interaction. After adaptation, the endpoint stiffness of the arm was measured by applying displacements to the hand in eight different directions midway through the movements. This was compared to the stiffness measured similarly during movements in a null force field. After adaptation, the endpoint stiffness under both the stable and unstable dynamics was modified relative to the null field. Adaptation to unstable dynamics was achieved by selective modification of endpoint stiffness in the direction of the instability. To investigate whether the change in endpoint stiffness could be accounted for by change in joint torque or endpoint force, we estimated the change in stiffness on each trial based on the change in joint torque relative to the null field. For stable dynamics the change in endpoint stiffness was accurately predicted. However, for unstable dynamics the change in endpoint stiffness could not be reproduced. In fact, the predicted endpoint stiffness was similar to that in the null force field. Thus, the change in endpoint stiffness seen after adaptation to stable dynamics was directly related to changes in net joint torque necessary to compensate for the dynamics in contrast to adaptation to unstable dynamics, where a selective change in endpoint stiffness occurred without any modification of net joint torque.

Different mechanisms involved in adaptation to stable and unstable dynamics.

Recently, we demonstrated that humans can learn to make accurate movements in an unstable environment by controlling magnitude, shape, and orientation of the endpoint impedance. Although previous studies of human motor learning suggest that the brain acquires an inverse dynamics model of the novel environment, it is not known whether this control mechanism is operative in unstable environments. We compared learning of multijoint arm movements in a "velocity-dependent force field" (VF), which interacted with the arm in a stable manner, and learning in a "divergent force field" (DF), where the interaction was unstable. The characteristics of error evolution were markedly different in the 2 fields. The direction of trajectory error in the DF alternated to the left and right during the early stage of learning; that is, signed error was inconsistent from movement to movement and could not have guided learning of an inverse dynamics model. This contrasted sharply with trajectory error in the VF, which was initially biased and decayed in a manner that was consistent with rapid feedback error learning. EMG recorded before and after learning in the DF and VF are also consistent with different learning and control mechanisms for adapting to stable and unstable dynamics, that is, inverse dynamics model formation and impedance control. We also investigated adaptation to a rotated DF to examine the interplay between inverse dynamics model formation and impedance control. Our results suggest that an inverse dynamics model can function in parallel with an impedance controller to compensate for consistent perturbing force in unstable environments.

Adaptation to stable and unstable dynamics achieved by combined impedance control and inverse dynamics model.

This study compared adaptation in novel force fields where trajectories were initially either stable or unstable to elucidate the processes of learning novel skills and adapting to new environments. Subjects learned to move in a null force field (NF), which was unexpectedly changed either to a velocity-dependent force field (VF), which resulted in perturbed but stable hand trajectories, or a position-dependent divergent force field (DF), which resulted in unstable trajectories. With practice, subjects learned to compensate for the perturbations produced by both force fields. Adaptation was characterized by an initial increase in the activation of all muscles followed by a gradual reduction. The time course of the increase in activation was correlated with a reduction in hand-path error for the DF but not for the VF. Adaptation to the VF could have been achieved solely by formation of an inverse dynamics model and adaptation to the DF solely by impedance control. However, indices of learning, such as hand-path error, joint torque, and electromyographic activation and deactivation suggest that the CNS combined these processes during adaptation to both force fields. Our results suggest that during the early phase of learning there is an increase in endpoint stiffness that serves to reduce hand-path error and provides additional stability, regardless of whether the dynamics are stable or unstable. We suggest that the motor control system utilizes an inverse dynamics model to learn the mean dynamics and an impedance controller to assist in the formation of the inverse dynamics model and to generate needed stability.