Invariant Modes Research Articles

Partial inventory control of the CSTR via reaction-dependent generalized inventories

Motivated by considering structurally dynamical properties of reaction systems, we develop an approach to deal with the nonlinear control of an unstable CSTR. More precisely, this paper shows that the CSTR dynamics can be split into reaction variant and reaction invariant dynamics modes through the use of linear combinations of system variables (called inventories). Since the reaction invariant dynamics mode is exponentially stable, the control design for an exponential stabilization of the reactor is simply solved together with (partial) inventory control on the basis of controlling the reaction variant dynamics mode. An extension of the approach to multiple chemical reaction systems is briefly outlined. Simulation results for a first order chemical reaction with multiplicity behavior are given to illustrate the theoretical development.

Stabilization of a given set of equilibrium states of nonlinear systems

The problem of multiprogram stabilization of the given set of equilibrium states in the class of nonlinear time invariant systems is studied. A method for designing multiprogram control based on the idea of interpolation is proposed. Hermite's interpolation polynomial makes it possible to design a controller that provides the prescribed time invariant modes of operation for the closed-loop system and ensures their asymptotic stability. A theorem on sufficient conditions for the existence of a multiprogram control is proved. The results are illustrated by a practical example.

Majorana Modes in Time-Reversal Invariants-Wave Topological Superconductors

We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes. The multiband character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.

Strong coupling problem with time-varying sound speed

For a single scalar field with unit sound speed minimally coupled to Einstein gravity, there are exactly three distinct cosmological solutions which produce a scale invariant spectrum of curvature perturbations in a dynamical attractor background, assuming vacuum initial conditions: slow-roll inflation; a slowly contracting adiabatic ekpyrotic phase, described by a rapidly-varying equation of state; and an adiabatic ekpyrotic phase on a slowly expanding background. Of these three, only inflation remains weakly coupled over a wide range of modes, while the other scenarios can produce at most 12 e-folds of scale invariant and Gaussian modes. In this paper, we investigate how allowing the speed of sound of fluctuations to evolve in time affects this classification. While in the presence of a variable sound speed there are many more scenarios which are scale invariant at the level of the two-point function, they generically suffer from strong coupling problems similar to those in the canonical case. There is, however, an exceptional case with superluminal sound speed, which suppresses non-Gaussianities and somewhat alleviates strong coupling issues. We focus on a particular realization of this limit and show these scenarios are constrained and only able to produce at most 28 e-folds of scale invariant and Gaussian perturbations. A similar bound should hold more generally---the condition results from the combined requirements of matching the observed amplitude of curvature perturbations, demanding that the Hubble parameter remain sub-Planckian and keeping non-Gaussianities under control. We therefore conclude that inflation remains the unique cosmological scenario, assuming a single degree of freedom on an attractor background, capable of producing arbitrarily many scale invariant modes while remaining weakly coupled. Alternative mechanisms must inevitably be unstable or rely on multiple degrees of freedom.

Static length scales of [formula omitted] Chern–Simons plasma

Using gravity description, we compute various static length scales of N=6 Chern–Simons plasma in a strongly coupled regime. For this, we consider the CP3 compactification of the type IIA supergravity down to four dimensions, and identify all the low-lying bosonic modes up to masses corresponding to the operator dimension 3 together with all the remaining CP3 invariant modes. We find the true mass gap, the Debye screening mass and the corresponding dual operators to be probed in the field theory side.

A new non‐linear sliding‐mode torque and flux control method for an induction machine incorporating a sliding‐mode flux observer

AbstractIn this paper a novel sliding‐mode control algorithm, based on the differential geometry state‐co‐ordinates transformation method, is proposed to control motor torque directly. Non‐linear feedback linearization theory is employed to decouple the control of rotor flux magnitude and motor torque. The advantages of this method are: (1) The rotor flux and the generated torque can be accurately controlled. (2) Robustness with respect to matched and mismatched uncertainties is obtained. Additionally, a varying continuous control term is proposed. As a result, chattering is eliminated without sacrificing robustness and precision. The control strategy is based on all motor states being available. In practice the rotor fluxes are not usually measurable, and a sliding‐mode observer is derived to estimate the rotor flux. The observer is designed to possess invariant dynamic modes which can be assigned independently to achieve the desired performance. Furthermore, it can be shown that the observer is robust against model uncertainties and measurement noise. Simulation and practical results are presented to confirm the characteristics of the proposed control law and rotor flux observer. Copyright © 2004 John Wiley & Sons, Ltd.

Process State Estimation Using Neural Network-Rate Function Model with Application to Batch Reactors

In this work, both the mechanistic and semi-mechanistic models are used as a basis for an on-line state estimation technique. The known parts of the semi-mechanistic model are based on first principles, and black-box models consisting of neural networks model the remaining unknown parts. A nonlinear reduced state-observer with variable gain is developed to accommodate both mechanistic and semi-mechanistic models. The observer is designed to possess invariant dynamic modes that can be assigned independently to achieve the desired performance. Convergence of the estimating algorithm can be formulated using Lyapunov stability theorems. Application of the designed observer is demonstrated in a simulation study on a free-radical polymerization reactor system.

Drag reduction by polymers in turbulent channel flows: Energy redistribution between invariant empirical modes.

We address the phenomenon of drag reduction by a dilute polymeric additive to turbulent flows, using direct numerical simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows, respectively. The modes are obtained empirically using the Karhunen-Loéve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular, there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes, as proposed in some previous theories.

Effects of extra Wilson fermions with large negative mass

We study effects of extra Wilson fermion with large negative mass coupled to domain-wall fermions (DWF) for improving the chiral properties. The relation between unwanted violation of the chiral symmetry and the translational invariant modes in the 5th direction is discussed in detail.

Chiral properties of domain wall fermions with improved gauge actions

We study the chiral properties of quenched domain wall fermions with several gauge actions. We demonstrate that the nearly translationally invariant modes in the fifth dimension that dominate the residual mass for Wilson gauge action can be substantially suppressed using improved gauge actions. In particular, we study the Symanzik action, the Iwasaki action, the DBW2 action and compare them to the Wilson action.

Comparative study of a sliding-mode observer and Kalman filters for full state estimation in an induction machine

The paper compares a sliding-mode observer with the standard and extended versions of the Kalman filter for full state estimation in an induction machine. The design method for the sliding-mode observer is presented and shown to possess invariant dynamic modes, which can be assigned independently to achieve the desired performance. The chattering concerned with the sliding-mode method can be eliminated by modifying the observer gain, without sacrificing robustness and precision. The indices of comparison are dynamic performance, robustness against parameter uncertainties, noise sensitivity, stability and computational complexity. Simulation and experimental results are presented in this paper. Important conclusions, together with recommendations for observer selection, regarding the effectiveness of the various schemes are given.

Homogeneous modes of cosmological instantons

We discuss the $O(4)$ invariant perturbation modes of cosmological instantons. These modes are spatially homogeneous in Lorentzian spacetime and thus not relevant to density perturbations. But their properties are important in establishing the meaning of the Euclidean path integral. If negative modes are present, the Euclidean path integral is not well defined, but may nevertheless be useful in an approximate description of the decay of an unstable state. When gravitational dynamics is included, counting negative modes requires a careful treatment of the conformal factor problem. We demonstrate that for an appropriate choice of coordinate on phase space, the second order Euclidean action is bounded below for normalized perturbations and has a finite number of negative modes. We prove that there is a negative mode for many gravitational instantons of the Hawking-Moss or Coleman--De Luccia type, and discuss the associated spectral flow. We also investigate Hawking-Turok constrained instantons, which occur in a generic inflationary model. Implementing the regularization and constraint proposed by Kirklin, Turok and Wiseman, we find that those instantons leading to substantial inflation do not possess negative modes. Using an alternate regularization and constraint motivated by reduction from five dimensions, we find a negative mode is present. These investigations shed new light on the suitability of Euclidean quantum gravity as a potential description of our universe.

Invariant sub-manifolds and modes of nonlinear autonomous systems

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

A sliding observer for nonlinear process control

Sliding observers are considered as nonlinear state estimators with good robustness to bounded modeling errors. In this paper we have developed sliding observers for process control. The observer is hence designed so as to possess invariant dynamic modes which can be assigned independently to achieve the desired performance. Convergence of the estimating algorithm is formulated by using Lyapunov stability theorems. Conditions for robustness to modeling errors are derived by analyzing the norms of estimation errors. For process control, servo-tracking and disturbance rejection for chemical reactors have been discussed by making use of this sliding observer. Simulation examples to demonstrate the construction and performance of this proposed sliding observer for chemical process control are also presented.

Resonant frequencies and field distributions for the shielded uniaxially anisotropic dielectric resonator by the FD-SIC method

New formulations for resonant modes of a shielded uniaxially anisotropic dielectric resonator (DR), such as sapphire, are proposed. They are solved by the finite-difference and simultaneous iteration with the Chebyshev (FD-SIC) acceleration method. Like an isotropic DR cavity, one azimuthal field is used for azimuthally invariant TM or TE modes and two TM fields are used for azimuthally variant hybrid modes. It is shown that the governing equation for TE modes is the same as that for the isotropic DR case. For TM and hybrid modes, more general /spl psi/(=rH/sub /spl phi//) and H/sub r/-H/sub z/ formulations than those for the isotropic DR are derived, respectively. Cylindrical cavities loaded with a rod or ring DR can be easily modeled and analyzed by the present method. Resonant frequencies and field distributions can be accurately and efficiently obtained. Numerical results of resonant frequencies of rod sapphire DR cavities are compared to those by the mode-matching method in the literature to verify the present approach. The electric- and magnetic-field distributions are also presented for hybrid modes of the uniaxially anisotropic DR cavity.

Model of Nonlocal Effects in Tokamaks

Recent experiments have demonstrated strong nonlocal effects in tokamaks, that apparently cannot be explained by conventional transport theories. In this letter the observed rapid changes in temperature are interpreted on dynamical grounds, by analyzing the adiabatic response, under an appropriate global constraint, of a certain family of axially invariant modes to an externally produced change of the current density at the edge.

On the Theory of Isotropic Turbulence

The theory of isotropic turbulence is investigated. In particular it is established that, for self-similar turbulence, energy transfer occurs in two distinct invariant modes. Implications of this result are discussed. The related decay of turbulence intensity is determined.

Isotropic analysis of grid-turbulence

The principles of self-similar isotropic turbulence are applied to the analysis of grid-decay. In particular it is theoretically established that energy-transfer occurs in only two distinct invariant modes. The decay constant of third-order correlations is posed in terms of the “amounts” of each mode present in the decay. This constant is determined empirically, and is found to be relatively insensitive to changes in Re. The related decay of turbulence intensity is computed.

Iterative technique for the synthesis of distortion-invariant optical correlation filters

A procedure is presented for designing distortion-invariant correlation filters. Optical correlation filters designed using this technique retain full position invariance. The filter design begins by finding the distortion-invariant modes (eigenfunctions) for a particular image. The input image, filter, and correlation response are all spectrally expanded in terms of these orthogonal eigenfunctions. An iterative technique between the spatial domain and the spectral domain is used to rephase the invariant modes so that a filter composed of a linear combination of modes has the proper overall invariance. The iterative technique also controls the information content of the filter by maintaining specified amplitudes for each invariant mode in the filter. Targets are detected by spanning the filter to determine points of constant amplitude.

Gross electrocortical activity as a linear wave phenomenon with variable temporal damping, regulated by ascending catecholamine neurones.

We report critical tests for a theory of electrocortical wave processes, in which telencephalic dendritic potentials reflect the mass action of coupled oscillatory circuits exhibiting complicated and unspecified non-linearities, the whole system being driven by active cell firing. Specific assumptions were: stochastic independence for instantaneous coupling parameters in the system, an individual central tendency to the cycle time for each circuit, and the maintenance of steady state conditions. Application of the central limit theorem to the state transition matrix shows that the gross electrocortical waves should be linear waves, exhibiting a multitude of invariant resonant modes, with the natural frequencies of all the modes being clustered about a smaller number of center values. Ascending brain-stem neurones of at least the dopaminergic and noradrenergic classes should regulate both the power of noise-like signals driving the telencephalic resonant patterns, and the temporal damping of each resonance. We devised tests which involved between hemisphere comparisons of electrocortical spectra, before and after unilateral lesion of transhypothalamic ascending fibres, thus obtaining ratio power changes attributable to post lesion asymmetry of damping and driving, in modes of equivalent left-right center-frequencies. These ratio spectra were curve-fitted to an approximate theoretical expression, and the parameters obtained enabled tests of several specific predictions. Estimates of the center values for resonant mode frequencies, comparison of the relative changes in left/right phase with that expected from the ratio changes in power, and estimates of the surface-to-signal transformation of left and right signals made by a back-calculation, all conform to expectation from the theory, and are consistent across lesion of different types of ascending neurone.