An Extremum Seeking Strategy Based on Block-Oriented Models: Application to Biomass Productivity Maximization in Microalgae Cultures

Abstract

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.

Experimental Study of Substrate Limitation and Light Acclimation in Cultures of the Microalgae Scenedesmus obliquus—Parameter Identification and Model Predictive Control

“…A full model development description can be found in [25,26]. This model has been implemented in several occasions [20,21,24,27,[29][30][31][32], sometimes including simplifications. The mass balance model of a photobioreactor for the main species in liquid and gaseous phases (including, substrates, products, and biomass concentration) is given by the general model representation [33].…”

“…A version of a model-based predictive control relying on an artificial neural network model (ANN) was developed by [22]. In [32], an extremum seeking control was developed based on a block-oriented model for on-line optimization of microalgae productivity, while in [53], the power consumption of the light source was considered within a productivity objective function, and the bioreactor was optimized for manipulating the dilution rate by an extremum seeking control. In [8], optimal values for the dilution rate and the incident photon flux density were provided to maximize the steady-state microalgal surface productivity in a continuous culture.…”

Model-Free Extremum Seeking Control of Bioprocesses: A Review with a Worked Example

“…A simple pole-placement procedure can be used to define the ES closed-loop dynamics, i.e., to design the integral constant k I . As discussed in [83], a controllable state-space representation of the gradient evolution can be inferred as:…”

“…In the field of biosystems, Feudjio et al [83] proposed an advanced auxiliary model-recursive prediction error method (AM-RPEM)-based ES algorithm for systems represented by block-oriented models. In this algorithm's structure, represented in Figure 6, a slope generator is coupled to the controller to reach any steady-state belonging to the static map, making ES a particular case of the slope-seeking strategy.…”

An experimental application of extremum seeking control to cultures of the microalgae Scenedesmus obliquus in a continuous photobioreactor

“…The gain K is chosen so as to ensure unitary steady‐state gain, that is, G (1) = 1, which implies . As the analytical or symbolic computation of the model parametric sensitivities in Equation () can be tedious, an algorithm extension involving an auxiliary model is proposed in Reference 14 in order to simplify the evaluation of the sensitivities.…”