Probabilistic observers for a class of uncertain biological processes

Abstract

SUMMARYIn this paper, probabilistic observers are considered for a class of continuous biological processes described by mass-balance-based models. It is assumed that the probability density functions (PDFs) of the uncertain parameters and inputs of the model, as well as the PDFs of the missing initial conditions are known. Then, the PDFs of the unmeasured state variables are obtained, at any time, by considering the image of these initial PDFs by the flow of the dynamic model (differential system). In comparison to classical open-loop asymptotic and interval observers, the method provides information on the confidence level of the estimates rather than simple upper and lower bounds. Moreover, unlike Kalman filters, probabilistic observers are not restricted to Gaussian distributions for the uncertain parameters. The design and application of a probabilistic observer to an industrial wastewater treatment plant is presented. Finally, a number of practical considerations is discussed in connection to both implementation and utilization issues.

State Estimation for Wastewater Treatment Processes

“…Note that these ideas find their origin in the theory of positive systems [21]. More recently, probabilistic observers have been formulated for a class of uncertain biological processes [5]; these observers take advantage of the knowledge of probability density functions (PDFs) for the uncertain parameters to calculate the PDFs of the unmeasured sate variables.…”

Fuzzy observers for anaerobic WWTP: Development and implementation

“…This observer gives good estimations but the interval selection and estimation convergence need to be improved. Other approaches have been tested as shown in Deza, Bossanne, Busvelle, Gauthier, and Rakotopara (1993), Alcaraz-Gonzalez et al (1999), Lemesle and Gouze (2005) and Chachuat and Bernard (2005). All these observers propose particular solutions; however, some disadvantages are noted, such as difficulties to design, tune and implementation, and numerical instability due to ill conditioning of the process dynamics, estimation errors due to model uncertainties, etc.…”

Monitoring of fed-batch E. coli fermentations with software sensors

“…Several works have been published concerning the application of such observers, mainly the Extended Kalman Observer, to biological processes [12][13][14][15][16][17]. Nevertheless, in spite of the satisfactory results reported, an uncertainty in the model parameters can generate a large bias in the estimation of unmeasured state(s) with these methodologies [2,10]. The second class of observers, the Asymptotic Observers [1], do not require the knowledge of the process kinetics showing, however, a potential problem concerning the dependence of the estimation convergence rate on the operating conditions [1,13,18].…”

“…In fact, in many practical applications, only some of the state variables involved are available for on-line measurement. Therefore, the development of methodologies, namely software sensors [1,2], which can provide accurate estimation of process variables that are not measurable in real time, based on on-line available data while overcoming the significant model uncertainty and the non-linear and time-varying nature of the system, is of great interest [3][4][5][6][7][8][9][10][11].…”