A discrete interval observer for linear system: Application on a real process

Indirect Measurement of Variables in a Heterogeneous Reaction for Biodiesel Production

“…In addition, no undesirable oscillations are observed with this observer, as found in another report [19]. In addition, the performance index is comparable with that obtained for multiobservers [9,22,23]. It is important to remember that heterogeneous transesterification is considered in this work instead of the homogeneous one considered in the other reported works.…”

“…State observers have already been used as a strategy to estimate variables [5,9,[19][20][21][22], parameters [21], and even the reaction heat [10] in the homogeneous transesterification of oils for biodiesel production. The estimated variables are used as soft sensors in supervision systems or as elements in control strategies [21]; the final objective is to enhance the performance of reactors for biodiesel production.…”

“…In comparison with other methods such as the EKF [5,10,38], fuzzy methods [19,20], and multimodel methods [22,23], the computational load of Luenberger observers is lower since no recurrent calculations are required. The reported observers consider six [5,10,20] or four [19,21] differential equations; multiobservers consider two equations or set of models with two equations [9,22], but these are reduced models, and only two variables are estimated. The Luenberger observer presented in this paper considers only two differential equations instead of six or four as usual; the estimation of the concentrations of products is performed by algebraic equations.…”

“…They also allow for formalizing and incorporating the The goal of the observer is to obtain an accelerated process dynamic, allowing us to predict (estimate) the behavior of variables; then, a mathematical model is required. Usually, the models considered are based on the general equation of the transesterification reaction [7] since it describes the different steps to transform triglycerides in biodiesel and crude glycerol, regardless of the reaction phase (homogeneous or heterogeneous transesterification); kinetics, conversion rate, reaction mechanism, and mass and energy balance are usually included in the models for variables' estimation [5,[8][9][10]. Once the models are formulated, it is necessary to adapt their structure according to the state space theory, since this is the basis for observer development.…”

“…An interval observer was implemented for a lab-scale reactor, achieving an estimation error of around 5%; this kind of observer provides variable estimations within an interval of values, assuming that initial conditions and uncertainties are unknown but bounded. The main issue is that the interval estimation error dynamic is required to be positive, which is not always accurate [9]. An evolutionary algorithm was used to minimize the estimation error in an algorithm devoted to estimating uncertain parameters; this algorithm was integrated into an automatic control strategy [21].…”