No-motion State Research Articles

Feedback control of Marangoni convection in a thin film heated from below

We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

An Analysis of Motion Transition in Subtle Errors using Inductive Logic Programming

This article seeks to utilize the data collected from virtual reality (VR)-based software and a leap-motion device used for learning of subtle errors in mild cognitive impairment (MCI) cases to enable early detection of MCI by analyzing the classification rules for errors (action slips) based on finger-action transitions when performing instrumental activities of daily living (IADL). Finger motion was recorded as a time-series database. An induction technique known as Inductive-Logic Programming (ILP), which uses logical and clausal language to represent the training data, was then used to discover a concise classification rule using logical programming. The content within this article was able to generate rules on how action transitions of the finger in the experiments were related to the pattern of micro-errors that indicate the difference of error regarding the length of the no-motion state of the finger.

Active Control of Thermal Convection in a Rectangular Loop by Changing its Spatial Orientation

The problem of the automatic control of the fluid flow in a rectangular convective loop heated from below is studied theoretically and experimentally. The control is performed by using a feedback subsystem which changes the convection regimes by introducing small discrete changes in the spatial orientation of the loop with respect to gravity. We focus on effects that arise when the feedback controller operates with an unavoidable time delay, which is cause by the thermal inertia of the medium. The mathematical model of the phenomenon is developed. The dynamic regimes of the convection in the thermosyphon loop under control are studied. It is shown that the proposed control method can successfully stabilize not only a no-motion state of the fluid, but also time-dependent modes of convection including the irregular fluid flow at high values of the Rayleigh number. It is shown that the excessive gain of the proportional feedback can result in oscillations in the loop orientation exciting the unsteady convection modes. The comparison of the experimental data obtained for dielectric oil and dodecane with theory is given, and their good agreement is demonstrated.

Suppression of Rayleigh-Bénard convection with proportional-derivative controller

We study theoretically (linear stability) and experimentally the use of proportional and derivative controllers to postpone the transition from the no-motion state to the convective state in a circular cylinder heated from below and cooled from above. The heating is provided with an array of individually controlled actuators whose power is adjusted in proportion to temperatures measured in the cylinder’s interior. As the proportional controller’s gain increases, so does the critical Rayleigh number for the onset of convection. Relatively large proportional controller gains lead to oscillatory convection. The oscillatory convection can be suppressed with the application of a derivative controller, allowing further increases in the critical Rayleigh number. The experimental observations are compared with theoretical predictions.

Limitations of linear control of thermal convection in a porous medium

The ability of linear controllers to stabilize the conduction (no-motion) state of a saturated porous layer heated from below and cooled from above is studied theoretically. Proportional, suboptimal robust (H∞) and linear quadratic Gaussian (H2) controllers are considered. The proportional controller increases the critical Rayleigh number for the onset of convection by as much as a factor of 2. Both the H2 and H∞ controllers stabilize the linearized system at all Rayleigh numbers. Although all these controllers successfully render negative the real part of the linearized system’s eigenvalues, the linear operator of the controlled system is non-normal and disturbances undergo substantial growth prior to their eventual, asymptotic decay. The dynamics of the nonlinear system are examined as a function of the disturbance’s amplitude when the system is subjected to the “most dangerous disturbances.” These computations provide the critical amplitude of the initial conditions above which the system can no longer be stabilized. This critical amplitude decreases as the Rayleigh number increases. To facilitate extensive computations, we examine two-dimensional convection in a box containing a saturated porous medium, heated from below and cooled from above, as a model system. The heating is provided by a large number of individually controlled heaters. The system’s state is estimated by measuring the temperature distribution at the box’s midheight. All the controllers considered here render the linearized, controlled system’s operator non-normal. The transient amplification of disturbances limits the “basin of attraction” of the nonlinear system’s controlled state. By appropriate selection of a controller, one can minimize, but not eliminate, the controlled, linear system’s non-normality.

Gain-scheduled controller for the suppression of convection at high Rayleigh number

Recent studies in the feedback control of Rayleigh-Bénard convection indicate that one can sustain the no-motion state at a moderate supercritical Rayleigh number (Ra) using only proportional compensation. However, stabilization occurs at a much higher Rayleigh number using linear-quadratic-Gaussian (LQG) control synthesis. The restriction is that the convection model is linear. In this paper, we show that a comparable degree of stabilization is achievable for a fully nonlinear convection state. The process is demonstrated in two stages using a fully nonlinear, 3D Boussinseq model, compensated by a reduced-order LOQ controller and a gain-schedule table. In the first stage a fully-developed convective state is suppressed through the control action at a moderate supercritical Ra. After the residual convection decays to a sufficiently small amplitude, in the second stage, we increase the Ra by a large step and switch the compensator gains using the gain-schedule table. During this change the control action is in place. Our nonlinear simulation results suggest that the nonlinear system can be stabilized to the limit predicted by the linear analysis. The simulation shows that the large Ra jump induces a large transient temperature in the conductive component, which appears to have very small impact on the stabilization.

Nonlinear stabilization of a thermal convection loop by state feedback

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop (widely known for its “Lorenz system” approximation) is presented. The loop consists of viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled, which makes the no-motion steady state for the uncontrolled case unstable for values of the non-dimensional Rayleigh number R a >1. The objective is to stabilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in space using finite difference method and get a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we transform the original coupled system into two uncoupled systems that are asymptotically stable in l 2-norm with homogeneous Dirichlet boundary conditions. The resulting boundary controls actuate velocity and temperature in the original coordinates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (using just a few measurements of the flow and temperature fields) can successfully stabilize the actual system for a very wide range of the Rayleigh number.

Robust feedback control of Rayleigh–Bénard convection

We investigate the application of linear-quadratic-Gaussian (LQG) feedback control, or, in modern terms, [Hscr ]2 control, to the stabilization of the no-motion state against the onset of Rayleigh–Bénard convection in an infinite layer of Boussinesq fluid. We use two sensing and actuating methods: the planar sensor model (Tang & Bau 1993, 1994), and the shadowgraph model (Howle 1997a). By extending the planar sensor model to the multi-sensor case, it is shown that a LQG controller is capable of stabilizing the no-motion state up to 14.5 times the critical Rayleigh number. We characterize the robustness of the controller with respect to parameter uncertainties, unmodelled dynamics. Results indicate that the LQG controller provides robust performances even at high Rayleigh numbers.

Numerical investigation of the stabilization of the no-motion state of a fluid layer heated from below and cooled from above

The feasibility of controlling flow patterns of Rayleigh–Bénard convection in a fluid layer confined in a circular cylinder heated from below and cooled from above (the Rayleigh–Bénard problem) is investigated numerically. It is demonstrated that, through the use of feedback control, it is possible to stabilize the no-motion (conductive) state, thereby postponing the transition from a no-motion state to cellular convection. The control system utilizes multiple sensors and actuators. The actuators consist of individually controlled heaters positioned on the bottom surface of the cylinder. The sensors are installed at the fluid’s midheight. The sensors monitor the deviation of the fluid’s temperatures from preset desired values and direct the actuators to act in such a way so as to eliminate these deviations. The numerical predictions are critically compared with experimental observations.

Experiments on the stabilization of the no-motion state of a fluid layer heated from below and cooled from above

It is demonstrated experimentally that through the use of feedback control, it is possible to stabilize the no-motion (conductive) state of a fluid layer confined in a circular cylinder heated from below and cooled from above (the Rayleigh–Bénard problem), thereby postponing the transition from a no-motion state to cellular convection. The control system utilizes multiple sensors and actuators. The actuators consist of individually controlled heaters microfabricated on a silicon wafer which forms the bottom of the test cell. The sensors are diodes installed at the fluid's midheight. The sensors monitor the deviation of the fluid temperatures from preset, desired values and direct the actuators to act in such a way as to eliminate these deviations.

Active control of Rayleigh–Bénard convection

We report on stabilizing the unstable no-motion state in a moderate aspect ratio one-dimensional Rayleigh–Bénard convection experiment. A linear proportional control algorithm uses shadowgraphic convection images to determine heat flux perturbations which are applied to the lower boundary by a network of local heaters. We show that simple linear control stabilizes the otherwise unstable no-motion (conduction) state over a substantial range of supercritical Rayleigh numbers.

Control of Rayleigh-Bénard convection in a small aspect ratio container

Active control of Rayleigh-Benard convection in small aspect ratio horizontal layer, by perturbation of the lower thermal boundary condition, is shown. These experiments use a novel shadowgraphic system to measure the departure of a high Prandtl number fluid from the no-motion state. A proportional control scheme uses the shadowgraphic information as feedback to determine the lower boundary heat flux distribution. An array of individually controlled heaters imposes the heating distribution. Significant suppression of the convection amplitude is possible to at least 10 times the critical Rayleigh number. This convection control method provides an economical alternative to magnetic convection suppression.

Stabilization of the No-Motion State of a Horizontal Fluid Layer Heated From Below With Joule Heating

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Be´nard problem of an infinite fluid layer heated from below with Joule heating and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data. The bottom of the layer is heated by a network of heaters whose power supply is modulated in proportion to the deviations of the temperatures at various locations in the fluid from the conductive, no-motion temperatures. Similar control strategies can also be used to induce complicated, time-dependent flows at relatively low Rayleigh numbers.

Stabilization of the no-motion state in the Rayleigh–Bénard problem

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Bénard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of a feedback controller effectuating small perturbations in the boundary data. The controller consists of sensors which detect deviations in the fluid’s temperature from the motionless, conductive values and then direct actuators to respond to these deviations in such a way as to suppress the naturally occurring flow instabilities. Actuators which modify the boundary’s temperature or velocity are considered. The feedback controller can also be used to control flow patterns and generate complex dynamic behaviour at relatively low Rayleigh numbers.

Feedback control stabilization of the no-motion state of a fluid confined in a horizontal porous layer heated from below

We consider a horizontal three-dimensional saturated porous layer, confined in an upright cubic box, heated from below and cooled from above. In the absence of a controller, the fluid maintains a no-motion state for subcritical Rayleigh numbers R<R c , where R c depends on the box's aspect ratio. Once this critical number is exceeded, fluid motion ensues. We demonstrate that, with the use of feedback control strategies which suppress flow instabilities, one can maintain a stable no-motion state for Rayleigh numbers far exceeding the classical critical one for the onset of convection. To preserve the equilibrium no-motion state of the classical problem, the controller alters the system's dynamics so as to stabilize an otherwise non-stable state

Stabilization of the no-motion state in Rayleigh-Bénard convection through the use of feedback control.

It is demonstrated theoretically that the critical Rayleigh number for transition from the no-motion (conduction) to the motion state in the Rayleigh-B\'enard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data.

Active control of convection

It is demonstrated theoretically that active (feedback) control can be used to alter the characteristics of thermal convection in a toroidal, vertical loop heated from below and cooled from above. As the temperature difference between the heated and cooled sections of the loop increases, the flow in the uncontrolled loop changes from no motion to steady, time-independent motion to temporally oscillatory, chaotic motion. With the use of a feedback controller effecting small perturbations in the boundary conditions, one can maintain the no-motion state at significantly higher temperature differences than the critical one corresponding to the onset of convection in the uncontrolled system. Alternatively, one can maintain steady, time-independent flow under conditions in which the flow would otherwise be chaotic. That is, the controller can be used to suppress chaos. Likewise, it is possible to stabilize periodic nonstable orbits that exist in the chaotic regime of the uncontrolled system. Finally, the controller also can be used to induce chaos in otherwise laminar (fully predictable), nonchaotic flow.