In control theory and engineering, it is often desirable to obtain full state information for control and diagnostic purposes. In many practical applications, however, some of states are not directly measurable. Instead, these states must be estimated using available system output measurements, which are related to the states by some dynamic model. For example, in an anaerobic digester in biological waste water treatment plants [1], the concentrations of biomass are difficult, if not impossible, to identify in real time. Therefore, estimation of the actual states based on output measurements has to be done instead.

For doing this, there are many available tools for linear models, as described, for example, by Chernousko [2]. However, in the nonlinear, continuous-time context, the methodology is far less developed [3,4]. Uncertainties in the initial conditions, model parameters, and inputs, as well as measurement noise, pose considerable challenges to the development and application of nonlinear state estimators.

In this study, we present a new guaranteed nonlinear state estimator based on a type of predictor-corrector approach. The prediction step aims at computing a set of values for the state vector that are attainable, based on the uncertainties in the system, while the correction step retains only the parts of the set which are compatible with measurements. The result is a set guaranteed to enclose all values of the state vector that are consistent with the output measurements and uncertainties. A key feature of the approach is that a new validating solver for parametric ODEs [5], VSPODE, will be used in the prediction step. The result of this step is a representation of the state vector by a Taylor model [6] in terms of the uncertain quantities (initial conditions, parameters, inputs, etc.). In the correction step, a constraint satisfaction procedure will then be employed to delete the parts of the domains of the uncertain variables that are incompatible with the measurements. The approach will be demonstrated using data from an anaerobic digestion process in waste water treatment plants [1].

References

[1] Bernard, O.; Hadi-Sadok, Z.; Dochain, D. Genoveso, A.; Steyer, J. P. Biotechnol. Bioeng. 2001, 75, 424-438.

[2] Chernousko, F. L. State Estimation for Dynamic System; CRC Press, Boca Raton, 1994.

[3] Jaulin, L. Automatica 2002, 38 1079-1082.

[4] Raissi, T.; Ramdani, N.; Candau, Y. Automatica 2004, 40, 1771-1777.

[5] Lin, Y; Stadtherr, M. A. 2005, submitted.

[6] Makino, K.; Berz, M. Remainder differential algebras and their applications. In Computational Differentiation: Techniques, Applications, and Tools; M. Berz, C. Bishof, G. Corliss, A. Griewank (Eds); SIAM, Philadelphia, 1996.