This paper treats the problem of estimating simultaneously the state and the unknown inputs of a class of nonlinear discrete-time systems. An observer design method for nonlinear Lipschitz discrete-time systems is proposed. By assuming that the linear part of this class of systems is time-varying, the state estimation problem of nonlinear system is transformed into a state estimation problem for LPV system. The stability analysis is performed using a Lyapunov function that leads to the solvability of linear matrix inequalities (LMIs). Performances of the proposed observer are shown through the application to an activated sludge process model.

No abstract

An estimation of the state and the unknown inputs of the reduced nonlinear model of an activated sludge process using the Extended Kalman Filter (EKF) is proposed. First, we present the reduced nonlinear model. This model contained five state variables and four unknown inputs. For satisfying the rank condition for the construction of an EKF, one unknown input has been approximated and the daily mean value of another unknown input has been used. Then, to estimate conjointly the state and the unknown inputs, the reduced nonlinear system is transformed to a nonlinear singular system. High performances of the proposed observer will be shown through the simulation results.

A model for the simulation of a moving bed bioreactor (MBBR) used for the treatment of municipal wastewater is proposed. The model includes attachment of particulates to the biofilm and detachment of biofilm into the bulk liquid. The growth kinetics are modelled with the activated sludge model no. 1 (ASM1). Respirometry was used for the estimation of kinetic parameters. The resulting respirograms featured the typical endogenous and exogenous respiration phases and the respirogram shapes were as expected from analogous respirometry with activated sludge. The estimated parameter set was used for modelling and simulation of the pilot-scale MBBR. The main proportion of biomass in the MBBR was found to be attached as biofilm on the carrier elements (4.1 -4.6 g dm-3) and only a small amount was suspended in the bulk liquid (0.15gdm(-3)). Attachment and detachment rates were estimated to be 4.8-7.5g m(-2) d(-1) 1for attachment and 6.5-7.5g m(-2) d(-1) for detachment. The biofilm age was estimated to be 1.8-2.7d. The model was used to predict effluent quality parameters and a good fit of the simulated data to the measured data originating from a four-days-long measurement campaign was obtained.

This paper treats the problem of estimating simultaneously the state and the unknown inputs of a class of nonlinear discrete-time systems. An observer design method for nonlinear Lipschitz discrete-time systems is proposed. By assuming that the linear part of this class of systems is timevarying, the state estimation problem of nonlinear system is transformed into a state estimation problem for LPV system. The stability analysis is performed using a Lyapunov function that leads to the solvability of linear matrix inequalities (LMIs). Performances of the proposed observer are shown through the application to an activated sludge process model.