###### Abstract  
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. Various approaches have been proposed to improve the convergence rate. In [7], a proportional integral scheme is introduced to improve the transient behavior while [8] uses a recursive least square (RLS) algorithm to estimate online the parameters of a linear model approximating the process input/output static map. However, in [9], the authors point out that the latter approach generally leads to a biased estimation of the optimal operating point when dealing with dynamic systems. It is therefore necessary to account for those dynamics, and in order to preserve simplicity and avoid the need for extensive prior knowledge about the process, a black-box approach is preferable, for instance using Wiener-Hammerstein models [10; 11]. Extension of ESC to multiple-inputs-single-output (MISO) process optimization has also provided interesting results both in simulation and experiments. In [12], the classical scheme is extended to achieve maximum power tracking of PV panels with an experimental validation. In [13], an extended Kalman filter is used to estimate the gradient and Hessian of a static quadratic dual-input-single-output (DISO) map to remove thermo-acoustic instabilities in an atmospheric combustor test rig. In applications such as PV panels or wind turbine power generation, it is sometimes required to operate at a given sub-optimal operating point so as to adjust the production to the actual load/demand. In this regard, slope seeking control (SSC) [14] provides the possibility of optimal and suboptimal setpoint tracking. In [15; 16], the authors propose an adaptive SSC for a class of SISO dynamic processes, and a numerical validation is provided with biomass productivity optimization in microalgae cultures. In this work, an extension of adaptive slope seeking a DISO dynamic processes is proposed. To this end, the process under consideration is assumed to be approachable at least locally with a quadratic block-oriented model in the form of a Hammerstein model. Adaptive slope seeking allows driving the process either at the extremum of the measurable performance index or an intermediate operating point specified via the slope of the performance index. The degrees of freedom offered by the additional input can be used in various ways, for instance, to minimize the input energy. A recursive extended least squares (RELS) algorithm is used to estimate online the model parameters. To alleviate the tedious and generally heuristic integrator gain tuning procedure of the classical filter-based ES/SOP control, a pole placement controller (PPC) is used, which was introduced by the authors in [15; 16].