Model-independent nonlinear control algorithm with application to a liquid bridge experiment

Abstract

We present a control method for high-dimensional nonlinear dynamical systems that can target remote unstable states without a priori knowledge of the underlying dynamical equations. The algorithm constructs a high-dimensional look-up table based on the system's responses to a sequence of random perturbations. The method is demonstrated by stabilizing unstable flow of a liquid bridge surface-tension-driven convection experiment that models the float zone refining process. Control of the dynamics is achieved by heating or cooling two thermoelectric Peltier devices placed in the vicinity of the liquid bridge surface. The algorithm routines along with several example programs written in the MATLAB language can be found at

Route to aperiodicity followed by high Prandtl-number liquid bridge. 1-g case

“…2b). It is known [3,5] that for the medium and high Prandtl number liquid bridges with =1 under absence of gravity the critical wave number is m = 2. The appearance of the second wave could be caused by the fact that for the Reynolds number values beyond 1957 the gravitational convection does not dominate the thermo-capillary one any more, and the two different wave number solutions start to "compete" with one another.…”

“…The latter is achieved by organizing the process itself in such a manner that there is no contact of the liquid with the vessel walls that minimizes the introducing impurity striations in grown crystals. That is why study of dynamical systems, which model the real technological process, through experiments [1][2][3] or analysis of partial differential equations [4][5][6][7] is currently an active field of physical and mathematical research.…”

Control of oscillatory thermocapillary convection with local heating

“…Since applying only one controller could result in a standing wave with nodes at the sensor/heater positions [15], two controllers were used. The third sensor was installed to capture the oscillation structure during the control.…”

“…For a half-zone model, a nonlinear control was performed by Petrov et al [14,15] to stabilize the oscillation by using local temperature measurements close to the free surface and modifying the temperature on the free surface at different azimuthal locations. They construct a look-up table based on the system's response to a sequence of random perturbations.…”

Interfacial Convection in Multilayer Systems

“…The nonlinear control algorithm has been developed in order to bring the system into the target state characterized by a state vector independent of time (i.e., a periodic, one-frequency, flow). An extended version of the approach described above has been used by Petrov et al (1998) for the complete suppression of the time-dependent behavior, i.e., the stabilization of an unstable steady state.…”