Use of linear feedback to control relaxation oscillations in a metal-passivation model

Abstract

A method of linear feedback is shown to be applicable to the control or suppression of spontaneously occurring relaxation oscillations in a two-dimensional model for the passivation of a metal surface exposed to an aqueous medium. For the operating conditions selected, these oscillations are characterized by a large limit cycle within the state space of the electrochemical system. A physically plausible control is used, involving feedback of a linear combination of two measurable quantities to a controllable parameter. The control acts to stabilize an otherwise unstable fixed point without changing its state space coordinates. Addition of the control to the system's dynamics also results in the appearance of two new fixed points, one of which is stable, the other unstable. The control is found to be extremely robust; when applied, even when the system is far from a fixed point, stabilization onto one or the other of the two stable fixed points occurs quickly. Which of the two it approaches depends upon the state space coordinates of the system on the limit cycle at the time the control is initiated. Stabilization onto the fixed point of the free system, from any point on the limit cycle, can be achieved using a combination of free and controlled dynamics. Release of the control causes the system to return to its oscillatory dynamics. Other linear-feedback-based approaches to control also are briefly considered, such as feeding back the time derivative of a measurable quantity to a controllable parameter.