A novel feedback control system to study the stability in stationary states

The unstable branch of stationary states in an illuminated thermochemical bistable system is stabilized with an external feedback loop which does not otherwise alter the autonomous system. The technique allows the measurement of the entire set of stationary states available to the system. Trajectories of transition between branches of stationary states (relaxation) are investigated theoretically and experimentally. Critical slowing down in the speed of the trajectories is predicted and measured as the points of marginal stability are approached. Trajectories are observed to spread as they depart from the unstable stationary state and then to coalesce as they approach a stable stationary state. These results are obtained due to experimental spread in initial conditions and not from noise along the relaxation trajectory.

This paper reports the detailed parameter space study of the nonlinear dynamical behaviors and their control in a time-delay digital tanlock loop (TDTL). At first, we explore the nonlinear dynamics of the TDTL in parameter space and show that beyond a certain value of loop gain parameter the system manifests bifurcation and chaos. Next, we consider two variants of the delayed feedback control (DFC) technique, namely, the time-delayed feedback control (TDFC) technique, and its modified version, the extended time-delayed feedback control (ETDFC) technique. Stability analyses are carried out to find out the stable phase-locked zone of the system for both the controlled cases. We employ two-parameter bifurcation diagrams and the Lyapunov exponent spectrum to explore the dynamics of the system in the global parameter space. We establish that the control techniques can extend the stable phase-locked region of operation by controlling the occurrence of bifurcation and chaos. We also derive an estimate of the optimum parameter values for which the controlled system has the fastest convergence time even for a larger acquisition range. The present study provides a necessary detailed parameter space study that will enable one to design an improved TDTL system.

The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.

The global environmental imperative demands urgent actions on ecological stabilization, yet the global scale of such actions is persistently insufficient. This calls for investigating why the world economy appears to be so fearful of any potential environmental expenditure. Using the formalism of Lyapunov potential function it is shown that the stability principles for biomass in the ecosystem and for employment in economics are mathematically similar. The ecosystem has a stable and unstable stationary state with high (forest) and low (grasslands) biomass, respectively. In economics, there is a stable stationary state with high employment in mass production of conventional goods sold at low cost price, and an unstable stationary state with lower employment in production of novel products of technological progress sold at higher prices. An additional stable state is described for economics with very low employment in production of life essentials, such as energy and raw materials that are sold at greatly inflated prices. In this state the civilization pays 10% of global GDP for energy produced by a negligible minority of the working population (currently ∼0.2%) and sold at prices exceeding the cost price by 40 times, a state when any extra expenditures of whatever nature appear intolerable. The reason lies in the fundamental shortcoming of economic theory, which allows for economic ownership over energy sources. This is shown to be equivalent to equating measurable variables of different dimensions (stores and fluxes), which leads to effective violation of the laws of energy and matter conservation in modern economics.

Results concerning the middle-latitude atmospheric flow in barotropic and quasi-geostrophic approximations are presented. With no dissipation, two semi-infinite regions of stability bound a finite region of instability, ensuring that the solutions are limited. In the presence of dissipation, above a critical value of forcing below which there is one stationary stable state, there are three stationary states, one unstable and two stable. Numerically we find that the two basins of attraction in phase space are strongly asymmetric, the one, corresponding to the solution resonant with the mountain, being much larger than the other. The model used for this study is the Charney-DeVore one.

On stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls

Nonlinear chains inside walls

In the present paper, we examine the Model Feedback Control System. Since the Model Feedback Control System is simple, the ModeI Feedback ControI System has been applied to many applications such as the trajectory control of robot manipulator, level control of serially connected tanks and so on. Since the control structure of the ModeI Feedback Control System has constrained, it is likely that the Model Feedback Control System can not represent all of stabilizing controllers for the plant. Nothing has been reported whether or not, the ModeI Feedback Control System can represent all of stabilizing controllers. The purpose of the present paper is to give a solution to the question whether or not, all of stabilizing controllers for the plant are written as the Model Feedback Control structure. Since Model Feedback Control System is applied to many minimum phase systems, in the present paper, we especially clarify the relationship between the ModeI Feedback Control System and the parametrization of all stabilizing controllers for the minimum phase system. First, the relationship between the Model Feedback Control system and the parametrization by Glaria and Goodwin is clarified. Next, for the strictly proper system, the relationship between the Model Feedback Control System and the parametrization by Yamada and Moki is presented. Simple design method of the Model Feedback ControI System to specify control characteristics is shown.

Advantages and limitations of washout filters in feedback control of both continuous-time and discrete-time systems are discussed, and generalizations that alleviate the limitations are presented. The generalized washout filters presented involve feedback through auxiliary state variables that are intercoupled. This is in contrast to the more traditional washout filter-aided control, in which these variables couple to the system state but not to each other. The generalized washout filter results obtained in the paper include a systematic feedback control design procedure, which is not available for traditional washout filter-aided control. Some previously unpublished results in the Ph.D. dissertation of one of the authors (Lee, 1991) are presented in the context of their relation to the generalized results and to recent publications on delayed feedback control. We observe that delayed feedback control for discrete-time systems, used in a number of controls of chaos studies, is a special case of washout filter-aided feedback. Moreover, the limitations of delayed feedback control can be overcome by the use of washout filter-aided feedback, which gives rise to the possibility of stabilizing a much larger class of systems.

The first-passage failure of Duffing oscillator with the delayed feedback control under the combined harmonic and white-noise excitations is investigated. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Ito stochastic differential equations for the system are derived by using the stochastic averaging method. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of the first-passage time are established. Finally, the conditional reliability function, the conditional probability density and moments of the first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. The effects of time delay in feedback control force on the conditional reliability function, conditional probability density and moments of the first-passage time are analyzed. The validity of the proposed method is confirmed by digital simulation.

In various kinds of feedback control, delayed control is an important topic for chaos control, which deserves more thorough researches. However, only a few researchers take in to account that whether the delayed feedback control (DFC) can be employed to control chaotic systems with time-delay. To investigate the control strategy, a stabilization problem of unstable fixed points in the discrete time-delay system is taken into considerations in this paper. Based on our conclusion, it is obvious that the odd number limitation property existing in the system without delay also exists in the time-delay one while the DFC is employed to stabilize the unstable fixed points. Second, based on the property of the root-locus diagram, a developed DFC strategy is proposed to release the limitation. The numerical simulation results validate the effectiveness of our design and are in agreement of our analysis.

Universite´Libre de Bruxelles, Service de Chimie-Physique, Case Postale 231, Boulevard du Triomphe, 1050 Bruxelles, Belgium~Received 28 May 1996!The numerical resolution of the Langevin equations, with specific internal noises deduced from masterequations, exhibits two qualitatively different behaviors for the reaction-diffusion wave fronts associated witheither a cubic or a quadratic chemical rate. In the case of a wave front between two stable stationary states,illustrated by the Schlo¨gl model, the effect of fluctuations in the vicinity of a bifurcation induces perturbativedeviations from the deterministic predictions on observable properties, like the propagation velocity, the profilewidth, and the value of the highest plateau. These deviations obey power laws that are determined. For wavefronts propagating into an unstable stationary state, such as in the Fisher model, a nonperturbative fluctuationeffect on velocity and profile width is observed, in relation to the selection, in the presence of noise, of aparticular solution in the continuum of linearly stable deterministic solutions. @S1063-651X~96!03010-3#PACS number~s!: 47.70.Fw, 82.20.2w, 05.40.1jI. INTRODUCTION

The subject of the present chapter is the front propagation in reaction-diffusion systems. We start with the consideration of the basic model, Fisher-Kolmogorov equation, and discuss the linear and nonlinear criteria of the velocity selection for fronts between a stable and an unstable stationary states. Then we consider the dynamics of plane and curved fronts between stable stationary states. The last section of this chapter contains the description of combustion fronts and their stability.

This work is concerned with the existence of asymptotically stable 2-D (2-dimensional) systems by means of a feedback control model represented by the system of partial difference equations and their Lagrange solutions. Thus, the goal is to establish a controller that provides a feedback control system with state variables depending solely on its Lagrange solution in the sense that the solution to the variable state is not a linear combination of other Lagrange solutions. Roughly speaking, the results showed that, to achieve such a control system, the controller has to diagonalize the block matrices of the matrices composing the system description model. Finally, a numerical example is presented to show how the controller is designed in order to generate a stable feedback control with given Lagrange solutions.

PurposeTo assess whether the use of a patented, novel feedback device intended to accurately control phacoemulsification tip elongation is effective under varying machine settings and material resistance.MethodsSculpt mode phaco (550-mm Hg Venturi pump; elongations, 35 and 70 µm) and quadrant settings (550-mm Hg Venturi pump; elongations, 15, 30, and 60 µm) were used in agar gel of incremental density (1%, 2%, 3%, and 6% in demineralized water). Dispersed lens fragments were also simulated with 6% agar gel spherules (2–5 mm in diameter; 550-mm Hg vacuum, and 60-µm elongation). Actual phaco tip elongation was measured on voltage readings from the piezoelectric crystals and compared to nominal elongation with feedback control off and on.ResultsMismatch between nominal and actual elongation when feedback control was off in sculpt mode varied between –13.51 µm and –23.07 µm of nominal elongation; in quadrant mode, mismatch varied between –2.79 µm and –20.41 µm. When the feedback control system was switched on, mismatch varied between –0.02 µm and +0.43 µm (P < 0.001 for all matchings). When the feedback system was off, the elongation mismatch among the 1%, 3%, and 6% agar was also statistically significant (P < 0.001). Elongation was 44.72 ± 4.16 µm with feedback control off and 60.02 ± 1.63 µm with it on (nominal elongation 60 µm; P < 0.001) when emulsifying agar 6% gel fragments. Dispersion of elongation data was also significantly wider when feedback control was turned off.ConclusionsA novel feedback control system effectively controls elongation accuracy regardless of the resistance offered by incremental agar gel concentrations.Translational RelevanceImplementing feedback control in phaco handpieces dramatically improves surgical accuracy. The translational value of this research relies on its immediate applicability to routine cataract surgery, resulting in a more appropriate use of ultrasound energy.