This paper proposes an adaptive slope seeking strategy targeting any reachable operating pointincluding extremumon the input/output map of a general dynamic single input single output (SISO) system approximated either by a quadratic Hammerstein, Wiener, or Wiener−Hammerstein model. The proposed control strategy is based on a recursive estimation algorithm which is used to estimate the model parameters, a slope reference generator, and a controller. A new algorithm called auxiliary model-recursive prediction error method (AM-RPEM) is used for the first task, and it is shown that the three model structures are equivalent from an identification point of view. Using the estimated parameters, a slope reference generator is proposed together with a self-governed pole placement controller with integral action, which advantageously replaces the heuristic integrator gain tuning in classical extremum-seeking schemes. Finally, the proposed control strategy is tested in simulation, first with a numerical example and then, using a dynamic model of Isochrysis galbana cultures so as to achieve concurrently extremum-seeking and suboptimal control. Simulation results using a recursive least squares algorithm and the proposed AM-RPEM are discussed.

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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.