Variable structure and singular perturbation control of elastic dynamical systems

This thesis treats the question of control of flexible dynamical systems for space applications. Two elastic dynamical systems are considered; The multibody system developed in Phillips Laboratory, Edwards Air Force Base, CA consists of two elastic links actuated by electric motors at the joints and rotate on a smooth horizontal granite table. An air bearing is used, which allows the whole system to float on the air so that the frictional forces do not exist between the support plate and the granite table. The controlled output is judiciously chosen such that the zero dynamics are stable or almost stable. For the control of the end point, two kinds of parameterizations of end effector position are considered. A variable structure control (VSC) law is derived for the end point trajectory control of each chosen output. Stability of zero dynamics associated with end point control is examined. Although, the VSC law accomplishes precise end point tracking, elastic modes are excited during the maneuver of the arm. A linear stabilizer is designed for the final capture of the terminal state; The second flexible system considered in this thesis is the elastic space vehicle. For the attitude control and vibration stabilization of the elastic space craft (spacecraft-beam-tip body configuration), singular perturbation technique is used. Based on nonlinear inversion, a control law is derived to decouple the attitude angle and the dominant flexible modes from the remaining elastic modes. The inverse control law decomposes the spacecraft dynamics into a slow and a fast subsystem. Based on singular perturbation theory, controllers are designed for each lower-order subsystem. Then a composite state feedback control is obtained by combining the slow and the fast control laws. (Abstract shortened by UMI.)

Hidden dynamics in models of discontinuity and switching

AbstractSharp switches in behaviour, like impacts, stick–slip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewise-smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity

Condensation and equilibration in an urn model

After reviewing the general scaling properties of aging systems, we present a numerical study of the slow evolution induced in the zeta urn model by a quench from a high temperature to a lower one where a condensed equilibrium phase exists. By considering both one-time and two-time quantities we show that the features of the model fit into the general framework of aging systems. In particular, its behavior can be interpreted in terms of the simultaneous existence of equilibrated and aging degrees with different scaling properties.Comment: 13 pages, 4 figures, Proceedings of the International Conference on Statistical Physics SigmaPhi, Rhodes 2014. v2: a footnote and one reference added, few typos correcte

Control of hierarchical polymer mechanics with bioinspired metal-coordination dynamics.

In conventional polymer materials, mechanical performance is traditionally engineered via material structure, using motifs such as polymer molecular weight, polymer branching, or block copolymer design. Here, by means of a model system of 4-arm poly(ethylene glycol) hydrogels crosslinked with multiple, kinetically distinct dynamic metal-ligand coordinate complexes, we show that polymer materials with decoupled spatial structure and mechanical performance can be designed. By tuning the relative concentration of two types of metal-ligand crosslinks, we demonstrate control over the material's mechanical hierarchy of energy-dissipating modes under dynamic mechanical loading, and therefore the ability to engineer a priori the viscoelastic properties of these materials by controlling the types of crosslinks rather than by modifying the polymer itself. This strategy to decouple material mechanics from structure is general and may inform the design of soft materials for use in complex mechanical environments. Three examples that demonstrate this are provided

Particle filtering in high-dimensional chaotic systems

We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. The purpose of the present paper is to show that the homogenization method developed in Imkeller et al. (2011), which is applicable to high dimensional multi-scale filtering problems, along with important sampling and control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.Comment: 28 pages, 12 figure

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method

On the evolution of mean motion resonances through stochastic forcing: Fast and slow libration modes and the origin of HD128311

Aims. We clarify the response of extrasolar planetary systems in a 2:1 mean motion commensurability with masses ranging from the super Jovian range to the terrestrial range to stochastic forcing that could result from protoplanetary disk turbulence. The behaviour of the different libration modes for a wide range of system parameters and stochastic forcing magnitudes is investigated. The growth of libration amplitudes is parameterized as a function of the relevant physical parameters. The results are applied to provide an explanation of the configuration of the HD128311 system. Methods. We first develop an analytic model from first principles without making the assumption that both eccentricities are small. We also perform numerical N-body simulations with additional stochastic forcing terms to represent the effects of putative disk turbulence. Results. Systems are quickly destabilized by large magnitudes of stochastic forcing but some stability is imparted should systems undergo a net orbital migration. The slow mode, which mostly corresponds to motion of the angle between the apsidal lines of the two planets, is converted to circulation more readily than the fast mode which is associated with oscillations of the semi-major axes. This mode is also vulnerable to the attainment of small eccentricities which causes oscillations between periods of libration and circulation. Conclusions. Stochastic forcing due to disk turbulence may have played a role in shaping the configurations of observed systems in mean motion resonance. It naturally provides a mechanism for accounting for the HD128311 system.Comment: 15 pages, 8 figures, added discussion in h and k coordinates, recommended for publicatio

Fault-Tolerant Dissipative Preparation of Atomic Quantum Registers with Fermions

We propose a fault tolerant loading scheme to produce an array of fermions in an optical lattice of the high fidelity required for applications in quantum information processing and the modelling of strongly correlated systems. A cold reservoir of Fermions plays a dual role as a source of atoms to be loaded into the lattice via a Raman process and as a heat bath for sympathetic cooling of lattice atoms. Atoms are initially transferred into an excited motional state in each lattice site, and then decay to the motional ground state, creating particle-hole pairs in the reservoir. Atoms transferred into the ground motional level are no longer coupled back to the reservoir, and doubly occupied sites in the motional ground state are prevented by Pauli blocking. This scheme has strong conceptual connections with optical pumping, and can be extended to load high-fidelity patterns of atoms.Comment: 12 pages, 7 figures, RevTex