Perturbation Framework Research Articles

Natural Observer Design for Singularly Perturbed Vector Second-Order Systems

Our aim in the present research study is to develop a systematic natural observer design framework for vector second-order systems in the presence of time-scale multiplicity. Specifically, vector second-order mechanical systems are considered along with fast sensor dynamics, and the primary objective is to obtain accurate estimates of the unmeasurable slow system state variables that are generated by an appropriately designed model-based observer. Within a singular perturbation framework, the proposed observer is designed on the basis of the system dynamics that evolves on the slow manifold, and the dynamic behavior of the estimation error that induces is analyzed and mathematically characterized in the presence of the unmodeled fast sensor dynamics. It is shown, that the observation error generated by neglecting the (unmodeled) fast sensor dynamics is of order O(ε), where ε is the singular perturbation parameter and a measure of the relative speed/time constant of the fast (sensor) and the slow component (vector second-order system) of the overall instrumented system dynamics. Finally, the performance of the proposed method and the convergence properties of the natural observer designed are evaluated in an illustrative example of a two-degree of freedom mechanical system.

Lagrangian perturbation theory of non-relativistic rotating superfluid stars

We develop a Lagrangian perturbation framework for rotating non-relativistic superfluid neutron stars. This leads to the first generalization of classic work on the stability properties of rotating stars to models that account for the presence of potentially weakly coupled superfluid components. Our analysis is based on the standard two-fluid model expected to be relevant for the conditions that prevail in the outer core of mature neutron stars. We discuss the implications of our results for dynamic and secular instabilities of a simple neutron star model in which the two fluids are allowed to assume different (uniform) rotation rates.

Approach to the extremal limit of the Schwarzschild–de Sitter black hole

The quasinormal-mode spectrum of the Schwarzschild-de Sitter black hole is studied in the limit of near-equal black-hole and cosmological radii. It is found that the mode_frequencies_ agree with the P"oschl-Teller approximation to one more order than previously realized, even though the effective_potential_ does not. Whether the spectrum approaches the limiting one uniformly in the mode index is seen to depend on the chosen units (to the order investigated). A perturbation framework is set up, in which these issues can be studied to higher order in future.

Stability of Higher-Dimensional Schwarzschild Black Holes

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. We show that, for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

We consider the Dirac operator with a long-range potential V(x). Scalar, pseudo-scalar and vector components of V(x) may have arbitrary power-like decay at infinity. We introduce wave operators with time-independent modifiers. These modifiers are pseudo-differential operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper. This allows us to fit the long-range scattering theory for the Dirac operator into the framework of smooth perturbations. Finally, we find the asymptotics for large times t of solutions u(x, t) of the time-dependent Dirac equation.

Low shear asymptotics for elastic seabeds

This paper deals with the low shear asymptotics of the sound field propagating in a fluid layer with a homogeneous elastic seabed. Using the displacement formulation and applying WKB, we show that although the problem is formally a singular perturbation problem, it can be reduced to a regular one, and we construct a regular expansion for the elastodynamic field in the bottom. The same results come out using alternative formulations, based either on Lamè potentials or on the Love function. The later formulation appears to have independent interest, as it explains why the problem under investigation does not fall in the existing framework of coercive singular perturbations which appear in one-dimensional elasticity.

Dynamics of a string coupled to gravitational waves: Gravitational wave scattering by a Nambu-Goto straight string

We study the perturbative dynamics of an infinite gravitating Nambu-Goto string within the general-relativistic perturbation framework. We develop the gauge invariant metric perturbation on a spacetime containing a self-gravitating straight string with a finite thickness and solve the linearized Einstein equation. In the thin string case, we show that the string does not emit gravitational waves by its free oscillation in the first order with respect to its oscillation amplitude; nevertheless the string actually bends when the incidental gravitational waves go through it.

Molecular wall effects: are conditions at a boundary "boundary conditions"?

This paper addresses and answers "no" to the question of whether the literal molecular-dynamically-derived species velocities prevailing at a solid surface bounding a two-component fluid continuum undergoing molecular diffusion constitute the appropriate species-velocity boundary conditions to be imposed upon the fluid continuum. In a broader context, generic boundary condition issues arising from the presence of different length scales in continuum-mechanical descriptions of physical phenomena are clarified. This is achieved by analyzing a model problem involving the steady-state diffusion of a dilute system of Brownian spheres (the latter envisioned as tractable models of solute "molecules") through a quiescent viscous solvent continuum bounded laterally by solid plane walls. Both physicochemical (potential energy) and hydrodynamic (steric) wall interaction effects experienced by the Brownian spheres are explicitly accounted for in our refined, microscale continuum model of the diffusion process. Inclusion of these "solid-wall-fluid" (s-f ) boundary-generated forces [above and beyond the usual "fluid-fluid" (f-f ) intermolecular forces implicit in the conventional Fick's law macroscale continuum description] serves to simulate the comparable s-f molecular boundary forces modeled in molecular dynamics simulations of the diffusional process. A singular perturbation framework is used to clarify the physical interpretation to be ascribed to "continuum-mechanical boundary conditions." In this same spirit we also clearly identify the origin of the physical concept of a "surface field" as well as of the concomitant surface transport conservation equation for strongly adsorbed species at solid walls. Our analysis of such surface phenomena serves to emphasize the fact that these are asymptotic, surface-excess, macroscale concepts assigned to a surface, rather than representing literal molecular material entities physically confined to the surface. Overall, this paper serves to illustrate the manner in which molecular simulations need to account for these different length scales and corresponding scale-dependent concepts if such analyses are to avoid drawing incorrect inferences regarding the molecular origins of continuum-mechanical boundary conditions.

Robust control of particulate processes using uncertain population balances

AbstractA general method is proposed for the synthesis of robust, nonlinear controllers for spatially homogeneous particulate processes described by population balances including time‐varying uncertain variables. The controllers are synthesized via Lyapunov's direct method on the basis of finite‐dimensional approximations of the population balances which are obtained by using the method of weighted residuals. The controllers enforce stability in the closed‐loop system and attenuation of the effect of uncertain variables on the output, and achieve particle‐size distributions with desired characteristics. The robustness of the controllers with respect to unmodeled dynamics is also addressed within the singular perturbation framework. The controllers enforce the desired stability and performance specifications in the closed‐loop system, provided that the unmodeled dynamics are stable and sufficiently fast. The proposed control method is applied to a continuous crystallizer with fines trap in which the nucleation rate and the crystal density change arbitrarily with time, and the actuator and sensor dynamics are explicitly considered in the process model, but not included in the model used for controller synthesis. Simulation runs of the closed‐loop system clearly demonstrate that the controller attenuates uncertainty, achieves a crystal‐size distribution with desired characteristics, and is superior to nonlinear controllers that do not account for the presence of uncertainty.

Output Feedback Control of Nonlinear Two-Time-Scale Processes

This paper focuses on dynamic output feedback control of a broad class of nonlinear two-time-scale processes modeled within the mathematical framework of singular perturbations. A sequential procedure is employed to synthesize a nonlinear well-conditioned two-time-scale output feedback controller, which guarantees stability and enforces output tracking in the closed-loop system, provided that the separation of the slow and fast dynamics of the two-time-scale process is sufficiently large. The proposed controller is successfully applied to two-time-scale chemical processes, a series of two chemical reactors and a fluidized catalytic cracker, and is shown to outperform output feedback controller design methods that do not account for the presence of time-scale multiplicity.

Robust control of hyperbolic PDE systems

This work concerns systems of quasi-linear first-order hyperbolic partial differential equations (PDEs) with uncertain variables and unmodeled dynamics. For systems with uncertain tain variables, the problem of complete elimination of the effect of uncertainty on the output via distributed feedback (uncertainty decoupling) is initially considered; a necessary and sufficient condition for its solvability as well as explicit controller synthesis formulas are derived. Then, the problem of synthesizing a distributed robust controller that achieves asymptotic output tracking with arbitrary degree of attenuation of the effect of uncertain variables on the output of the closed-loop system is addressed and solved. Robustness with respect to unmodeled dynamics is studied within a singular perturbation framework. It is established that controllers which are synthesized on the basis of a reduced-order slow model, and achieve uncertainty decoupling or uncertainty attenuation, continue to enforce these objectives in the presence of unmodeled dynamics, provided that they are stable and sufficiently fast. The developed controller synthesis results are successfully implemented through simulations on a fixed-bed reactor, modeled by two quasi-linear first-order hyperbolic PDEs, where the reactant wave propagates through the bed with significantly larger speed than the heat wave, and the heat of reaction is unknown and time varying.

Robust control of multivariable two-time-scale nonlinear systems

Abstract This paper focuses on the synthesis of well-conditioned multivariable robust controllers for a broad class of multi-input multi-output two-time-scale nonlinear systems with time-varying uncertain variables, modeled within the framework of singular perturbations. The proposed controller stabilizes the fast dynamics, guarantees boundedness of trajectories, and ensures that the ultimate discrepancy between the outputs and the external reference inputs in the closed-loop system can be made arbitrarily small by an appropriate choice of controller parameters, provided that the singular perturbation parameter is sufficiently small. The developed control method is tested, through simulations, on a fluidized catalytic cracking reactor.

Q Methodology, Textuality, and Tectonics

Along with the paper from Rex Stainton Rogers, and that from Simon Watts and Paul Stenner, this paper seeks to explain and illustrate to and why the Q work we are doing here (mainly in the UK, but also with some colleagues elsewhere) differs from the work with a much longer history which has been undertaken by Q scholars in the USA and their students and proteges (again, this includes work outside the primary geographical locus). While our work has much in common with the latter, there are significant differences both in our approach to carrying out Q studies and in our objectives. Our interest in Q arose from a theoretical location within a climate of perturbation framework, drawing from French Theory. Q met our need for an idiographic methodology, and offered us a highly effective means to conduct discourse analytic work. We use Q to address issues of power and knowledge, and their interplay, using analytics of textuality and tectonics. These terms are defined and an illustration is provided to show how Q can be used in this way.

Q Methodology and "Going Critical": Some Reflections on the British Dialect

In seeking to explain what Q methodology has made possible for those of us working within a critical climate of perturbation framework, we have employed the device of inventing a British dialect of Q. While I would still hold to the argument that there is something distinct about our approach, that distinctiveness really requires a more encompassing tag than might be taken from the anodyne term dialect. My ambition in this paper is to spell out our position (and mine within that) and to leave readers to make up their own minds as to whether we are; still operating in the Q-methodology community; schismatics; or, even, heretics.

Compensation of measurable disturbances for two-time-scale nonlinear systems

This paper deals with a class of two-time-scale nonlinear systems with time-varying disturbances, modeled within the framework of singular perturbations. Systems in both standard and nonstandard form are considered. For these systems, we synthesize well-conditioned control laws that utilize feedback of the full state vector and feedforward compensation of disturbances to exponentially stabilize the fast dynamics and enforce a pre-specified input/output behavior independently of the disturbances in the closed-loop slow subsystem. Singular perturbation methods are employed to establish that the discrepancy between the output of the closed-loop full-order system and the output of the closed-loop slow subsystem is proportional to the value of the singular perturbation parameter. The stability of the closed-loop system is analyzed using Lyapunov's direct method and precise conditions that guarantee boundedness of the trajectories, for sufficiently small values of the singular perturbation parameter, are derived. The application of the developed methodology is illustrated through a catalytic continuous stirred tank reactor example, modeled as a singularly perturbed system in nonstandard form.

Stability analysis of self-gravitating skyrmions

>We investigate the stability of our recently constructed self-gravitating skyrmions in the framework of small time-dependent perturbations. It is shown that the frequency spectrum for radial perturbations of the coupled Einstein-Skyrme equations can be reduced to the energy spectrum of a p-wave Schrödinger equation with a bounded effective potential, which is determined by the equilibrium solution. Bound states of this Schrödinger equation correspond to exponentially growing modes. It turns out that there are no such modes, as long as the relevant dimensionless coupling constant (κ) is less than a critical value, κc. For larger values of κ there is exactly one unstable mode. Therefore, the Skyrme “stars” become unstable (in the sense of Liapunov) for κ>κc, in spite of the fact that their topological winding number is conserved. The linear stability for κ<κc (with respect to radial modes) is, of course, only a necessary condition for (non-linear) stability. Similar results are expected to hold for our new black hole solutions of the Einstein-Skyrme system.

Von Weizsäcker coefficient for high-temperature electron plasmas.

The coefficient of the von Weizs\acker term in the Thomas\char21{}Fermi\char21{}von Weizs\acker energy functional is determined within a perturbation framework for high-temperature plasmas. A notable deviation in comparison to the corresponding zero-temperature result is found.

Instability of a colored black hole solution

We analyze the stability of a black hole solution of the Einstein-Yang-Mills equations that was recently found numerically by Bizon, and which provides a counter example for the “no hair” conjecture for non-abelian black holes. In the framework of small time-dependent perturbations it is shown that there is at least one exponentially growing radial mode with the correct boundary conditions at the horizon and at infinity.

Instability of the Bartnik-McKinnon solution of the Einstein-Yang-Mills equations

We analyse the stability of the (ground state) Bartnik-McKinnon solution of the Einstein-Yang-Mills equations in the frame-work of small time-dependent perturbations. It is shown that the frequency spectrum of a class of radial perturbations is determined by the spectrum of a radial p-wave Schrödinger equation with a bounded effective potential. Bound states of this Schrödinger equation correspond to exponentially growing radial modes and it turns out that there is exactly one such bound state.

A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group

For pt.I see ibid., vol.1, p.621-32, 1984. Unitary representations of the canonical group C for quantum gravity, discussed in an earlier paper, are studied producing concrete operator realisations of the basic canonical fields eAB, pi CD of the theory together with its spatial diffeomorphism group covariance. The appropriate state functionals are concentrated on distributional triads and not smooth ones. In addition, for a class of these representations the distributional triads are also degenerate, a feature does not appear for the conventional canonical commutation relations and the weak-field perturbation framework.