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

Classic linear control theory is applied to a three-dimensional, highly nonlinear model for the passivation dynamics of a metal surface exposed to an aqueous medium. For the specific set of parameter values selected for the model, the free (uncontrolled) dynamics consist of aperiodic or chaotic oscillations of the three system variables. It is first demonstrated that the model, when linearized about an unstable fixed point that corresponds to partial passivation of the metal surface, is completely controllable, and thus the fixed point can be stabilized by applying an as-yet-unspecified (but assumed-measurable) control to system parameters that are physically controllable. By contrast, a fixed point corresponding to total passivation of the metal surface is found to be not controllable using this approach. It is found that a variety of possibilities exists for stabilizing the fixed point that corresponds to partial passivation. One that is considered in detail is feedback, to the anodic potential, of the change of solution concentration, relative to its value at the fixed point. It is found, for this and various other linear feedback strategies, that stabilization of the fixed point also results in control or suppression of the chaotic dynamics. Implications of these results for control of chaos in experimental systems are also discussed.