On-line robust nonlinear state estimators for nonlinear bioprocess systems A. Iratni a,⇑ , R. Katebi b , M. Mostefai c a Centre Universitaire de B.B. Arreridj, B.B. Arreridj 34265, Algeria b Industrial Control Centre, Dept. of Electronic and Electrical Engineering, University of Strathclyde, 50 George Street, Glasgow G1 1QE, UK c Laboratoire d’Automatique de Setif, Department of Electrical Engineering, University of Ferhat Abbas, 19000 Setif, Algeria article info Article history: Received 22 April 2011 Received in revised form 6 August 2011 Accepted 22 September 2011 Available online 1 October 2011 Keywords: Nonlinear estimator Simulation study Wastewater systems ASM1 H-infinity State-dependent Riccati equation abstract This paper presents the design of a new robust nonlinear estimator for estimation of states of nonlinear systems. Two approaches are considered based on the state-dependent Riccati equation formulation and the technique of H-infinity control design. The proposed method differs from other well-known state estimators, because not only nonlinear dynamics but also the robustness is taken into account. The proposed method is implemented and tested on a biological wastewater system. The simulation study compares the Extended Kalman Estimator (EKE), the State-Dependent Riccati Estimator (SDRE), and the Extended H-infinity Estimator (EHE) with a new proposed State Dependent H-infinity Estimator (SDHE). The results are compared for different weather conditions, i.e. dry, rain and storm, showing a superior performance of the proposed method. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction The dynamic behavior of a process can be fully described by the evolution of its state variables. For the control and super- vision of a dynamic system, the knowledge of these variables is crucial. Unfortunately, these variables are generally not accessible for measurements. This problem can be solved, under certain conditions, by introducing a state observer or a state estimator whose task will be to provide an estimate of the state vector of the system depending on information available on system inputs and outputs, and a dynamic model of the process. The first observer dedicated to estimating the state of linear systems was developed by Luenberger for deterministic systems [22] and Kalman proposed a solution in a stochastic frame- work [17,18]. The latter had a significant impact in practice and has been applied to many problems such as target tracking, the navigation control, and fault detection in different industries. Both Luenberger and Kalman observers are widely used today, but linear systems covering only a small ratio of industrial processes, solutions specifically non-linear have been quickly addressed. These solutions are mainly based on extending the linear Kalman estimator (EKE) based on the lineari- zation technique of estimation error at each time instant [15]. In many practical cases, this approach gives relatively satis- factory results but the EKE is known to show instability and divergence in some cases. Note that, despite some restrictive conditions of application, EKE approach is often locally optimal, being very sensitive to initial conditions and modeling errors. In this work, we focus on compliance issues for state estimation of non-linear systems represented by differential equa- tions. These models represent the physical/chemical phenomena of the process and they are usually developed using first principle physical relationships. State estimation of nonlinear systems is a problem that is not fully solved. Several ap- proaches including extended Kalman filter, adaptive filters and particle filters are reported in the literature. In the linear case, 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.09.032 ⇑ Corresponding author. E-mail address: iratni@gmail.com (A. Iratni). Commun Nonlinear Sci Numer Simulat 17 (2012) 1739–1752 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns