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While state estimation techniques are routinely applied to systems represented by ordinary differential equation (ODE) models, it remains a challenging task to design an observer for a distributed parameter system described by partial differential equations (PDEs). Indeed, PDE systems present a number of unique challenges related to the space-time dependence of the states, and well-established methods for ODE systems do not translate directly. However, the steady progresses in computational power allows executing increasingly sophisticated algorithms, and the field of state estimation for PDE systems has received revived interest in the last decades, also from a theoretical point of view. This paper provides a concise overview of some of the available methods for the design of state observers, or software sensors, for linear and semilinear PDE systems based on both early and late lumping approaches.

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.

The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments.

Climate change and the efficient use of freshwater for irrigation pose a challenge for sustainable agriculture. Traditionally, the prediction of agricultural production is carried out through crop-growth models and historical records of the climatic variables. However, one of the main flaws of these models is that they do not consider the variability of the soil throughout the cultivation area. In addition, with the availability of new information sources (i.e., aerial or satellite images) and low-cost meteorological stations, it is convenient that the models incorporate prediction capabilities to enhance the representation of production scenarios. In this work, an agent-based model (ABM) that considers the soil heterogeneity and water exchanges is proposed. Soil heterogeneity is associated to the combination of individual behaviours of uniform portions of land (agents), while water fluxes are related to the topography. Each agent is characterized by an individual dynamic model, which describes the local crop growth. Moreover, this model considers positive and negative effects of water level, i.e., drought and waterlogging, on the biomass production. The development of the global ABM is oriented to the future use of control strategies and optimal irrigation policies. The model is built bottom-up starting with the definition of agents, and the Python environment Mesa is chosen for the implementation. The validation is carried out using three topographic scenarios in Colombia. Results of potential production cases are discussed, and some practical recommendations on the implementation are presented.