Process identification using second order Volterra models for nonlinear model predictive control design of flotation circuits

Abstract

The control of flotation circuits is a complicated problem, since flotation circuits are nonlinear multivariable processes with a significant degree of interaction between the variables. Isolated PID controllers usually do not perform adequately. The application of a nonlinear model predictive algorithm based on second order Volterra models was investigated. Volterra series models are a higher order extension of linear impulse response models. The nonlinear model predictive control algorithm can also be seen as a linear model predictive controller with higher order correction terms. A dynamic model of a flotation circuit based on the governing continuity equations was developed. The responses obtained represented the qualitative relationships between the model inputs and the controlled variables. This model exhibited strong nonlinearities, including asymmetrical responses to symmetrical inputs and gain sign changes. This dynamic model was treated as the plant to be identified and from which second order Volterra models were obtained. Full Volterra models required excessively large data sets, but significant reductions in the size of the required data set could be achieved if some of the second order coefficients were constrained to zero. These "pruned" Volterra models represented the plant dynamics significantly better than linear models. In particular, these second order Volterra models were able to model asymmetrical responses including gain sign changes. A special case of "pruned" second order Volterra models are diagonal second order models, where all the off-diagonal coefficients (hij where i ≠ j) are constrained to zero. These models required less data than pruned Volterra models containing off-diagonal coefficients, but were less accurate. The performance of nonlinear model predictive controllers based on a pruned second order and diagonal second order Volterra models was evaluated. The performance of these controllers was also compared to the performance obtained with a first order (linear) Volterra model. All three controllers gave equivalent results for large manipulated variable weights. However, when the controllers were tuned more aggressively, results obtained from the three controllers differed considerably. The pruned nonlinear controller performed well even when tuned aggressively while the performance of the linear controller deteriorated. For the case of disturbance rejection, the linear controller performed slightly better than the nonlinear controllers.