Extended Kalman Filter Design for Semilinear Distributed Parameter Systems with Application to Anaerobic Digestion
Keywords:State estimation, Ricatti equation, Process monitoring, Biochemical systems
This paper is concerned with the online state estimation of a class of one-dimensional semilinear partial differential equation (PDE) systems considering piecewise measurements over the spatial domain. In the context of infinite-dimensional linear systems, it is well-known that the Kalman filter minimizes the mean square estimation error. For semilinear infinite-dimensional systems, the extended Kalman filter (EKF) is a widely used extension relying on successive linearizations of the estimation error dynamics. In this paper, we propose a computationally tractable implementation of the EKF using a sample-and-hold approach for which the optimal output injection operator associated to the proposed estimator is computed at each sampling time via the approximate solution of the infinite-dimensional Riccati equation. The performance of the observer is exemplified through numerical experiments which demonstrate the efficiency of the proposed approach.
I. Yupanqui Tello, A. Vande Wouver, and Daniel Coutinho. Extension of kalman filtering to semilinear pde systems-application to pulp and paper. In 2020 24th International Conference on System Theory, Control and Computing (ICSTCC), pages 813–818. IEEE, 2020.
L. Theodore. Chemical Reactor Analysis and Applications for the Practicing Engineer. Wiley Online Library, 2012.
A. Vande Wouwer Modeling and Simulation of Distributed Parameter Systems Control Systems, Robotics and Automation: Modeling and System Identification-I, page 85, 2009.
A. Vande Wouwer and M. Zeitz. State estimation in distributed parameter systems. Control Systems, Robotics and Automation–Volume XIV: Nonlinear, Distributed, and Time Delay Systems-III, page 92, 2009.
A. Vande Wouwer, C.Renotte, I. Queinnec and Ph. Bogaerts. Transient analysis of a wastewater treatment biofilter–distributed parameter modelling and state estimation. Mathematical and computer modelling of dynamical systems, 12(5):423-440, 2006.
E. Aguilar-Garnica, J.P. Garc´ıa-Sandoval and C. Gonz´alez-Figueredo. A robust monitoring tool for distributed parameter plug flow reactors. Computers & chemical engineering, 35(3):510-518, 2011.
A. Schaum, J.A. Moreno, E. Fridman, and J. Alvarez. Matrix inequalitybased observer design for a class of distributed transport-reaction systems. International Journal of Robust and Nonlinear Control, 24(16):2213-2230,2014.
A. Schaum, T. Meurer, and J. A. Moreno. Dissipative observers for coupled diffusion-convection-reaction systems. Automatica, 94:307–314, 2018.
H. Dimassi, J. Winkin, and A. Vande Wouwer. A sliding mode observer for a linear reaction–convection–diffusion equation with disturbances. Systems & Control Letters, 124:40-48,2019.
R. F. Curtain and H. Zwart. An introduction to infinite-dimensional linear systems theory, Springer Science & Business Media, 2012.
C. Delattre, D. Dochain, and J. Winkin. Sturm-liouville systems are riesz-spectral systems. International Journal of Applied Mathematics and Computer Science, 13:481–484, 2003.
O. Schoefs, D. Dochain, H. Fibrianto, and J.-P. Steyer. Modelling and identification of a partial differential equation model for an anaerobic wastewater treatment process. In Proceedings of the 10th World Congress on Anaerobic Digestion, Montreal, Canada, 1:343-347, 2004.
R. F. Curtain and A. J. Pritchard. Infinite-dimensional linear systems theory, Springer Verlag, Berlin, 1978.
M. Moarref and L. Rodrigues. Observer design for linear multi-rate sampled-data systems. In 2014 American Control Conference, Portland, OR, USA, 2014, pp. 5319-5324.
J. Winkin, D. Dochain, and P. Ligarius. Dynamical analysis of distributed parameter tubular reactors. Automatica, 36(3):349–361, 2000.
C.-K. Chen and S.-H. Ho. Application of differential transformation to eigenvalue problems. Applied mathematics and computation, 79(2-3):173–188, 1996.
Antonis Papachristodoulou, James Anderson, Giorgio Valmorbida, Stephen Prajna, Peter Seiler, and Pablo A Parrilo. Sum of squares optimization toolbox for matlab user’s guide, 2013.