Dual-Input Slope Seeking Control of Continuous Micro-Algae Cultures with Experimental Validation

Featured Application The production of algal biomass or of a product associated with biomass growth can be optimized by manipulating the dilution rate and the incident light intensity in indoor photo-bioreactors. However, model-based optimization is delicate in view of the time and experimental efforts required to develop sufficiently accurate dynamic models. Extremum seeking provides an alternative real-time optimization approach, where prior model knowledge is not required, and only a measurable (or estimable) performance index is needed online. This paper investigates the application of adaptive slope-seeking strategies to dual-input single output dynamic processes. While the classical objective of extremum seeking control is to drive a process performance index to its optimum, this paper also considers slope seeking, which allows driving the performance index to a desired level (which is thus sub-optimal). Moreover, the consideration of more than one input signal allows minimizing the input energy thanks to the degrees of freedom offered by the additional inputs. The actual process is assumed to be locally approachable by a Hammerstein model, combining a nonlinear static map with a linear dynamic model. The proposed strategy is based on the interplay of three components: (i) a recursive estimation algorithm providing the model parameters and the performance index gradient, (ii) a slope generator using the static map parameter estimates to convert the performance index setpoint into slope setpoints, and (iii) an adaptive controller driving the process to the desired setpoint. The performance of the slope strategy is assessed in simulation in an application example related to lipid productivity optimization in continuous cultures of micro-algae by acting on both the incident light intensity and the dilution rate. It is also validated in experimental studies where biomass production in a continuous photo-bioreactor is targeted.