PERGAMON Control Engineering Practice 9 (2001) 267-281 CONTROL ENGINEERING PRACTICE www.elsevier.com/locate/conengprac Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations V.M. Becerra":", P.D. Roberts", G.W. Griffiths" "Department of Cybernetics, University of Reading, P.O. Box 225, Whiteknights, Reading, Berkshire RG6 2AY, UK "City University, London, Control Engineering Research Centre, Northampton Square, London ECl V OHB, UK "Aspen'Iech Ltd., Castle Park, Cambridge CB3 OAX UK Received 25 November 1999; accepted 14 August 2000 Abstract This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: State estimation; Generalised state space; Large-scale systems; Extended Kalman filters; Process models; Nonlinear systems; Batch reactors 1. Introduction Systems of coupled differential and algebraic equations (DAEs) often occur as differential equations that are subject to constraints. The constraints may be linear or nonlinear. DAE systems arise frequently as initial value problems in the computer-aided design and modelling of mechanical systems subject to constraints (multi-body systems), circuit simulation, chemical process modelling, and in many other applications. The numerical treatment of DAEs may be more complicated than the numerical integration of ordinary differential equations (ODEs) (Petzold, 1982). Although sometimes it is possible to eliminate the algebraic equations in a DAE to transform it into an ODE, it is not always convenient to do so. The Kalman filter is a stochastic filter that allows the estimation of the states of a system based on a linear state space model. The extended Kalman filter (EKF) uses local linearisation to extend the scope of the Kalman * Corresponding author. + 44(0)118-9318220. E-mail addresses:v.m.becerra@reading.ac.uk (V.M. Becerra), p.d.roberts@city.ac.uk (P.D. Roberts), graham.griffiths@aspentech. com (G.W. Griffiths). filter to systems described by nonlinear ordinary differen- tial equations (Maybeck, 1982). The state estimation problem for linear DAEs using the Kalman filter has been studied by several researchers (Nikoukhah, Wi1lsky, & Levy, 1992; Chisci & Zappa, 1992). State estimation for nonlinear DAEs has been studied, for instance, by Albuquerque and Biegler (1997), who used a moving-horizon approach together with model decomposition and nonlinear programming tech- niques, and by Cheng, Mongkhonsi, and Kershenbaum (1997), who used a variational approach. Moving-horizon estimation techniques use the mea- sured trajectories over a given period of time and adjust the states of the model along the estimation horizon so that the model trajectories are optimally close in a given sense to the measured trajectories. Parameter estimation for nonlinear DAEs has been studied, for instance, by Tjoa and Biegler (1991), who used simultaneous solution and optimisation via collocation methods and nonlinear programming. The main advantages of the moving-horizon approach to state estimation over the extended Kalman filter are its capability of handling constraints on the variables and its flexibility with respect to the choice of the estimation criterion. On the other hand, for on-line implementation, 0967-0661/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0967-0661(00)00110-6