(675e) Modular Design of Nonlinear Observers for State and Disturbance Estimation

Technical limitations and/or high cost of sensors result in the non-availability of all state variables for direct on-line measurement, and this creates the need for on-line state estimation. Furthermore, the operation of a process or plant is subject to time-varying disturbances, associated with changes in key process parameters or improper operation of sensing instruments. For this reason, in addition to monitoring the state variables, there is a definite practical need for detection and estimation of disturbances.

The problem of combined state and disturbance estimation can be conceptually formulated as a state estimation problem for an extended system. In the case of linear systems, the well-known Luenberger observer offers a comprehensive solution. More specifically, in industrial applications of combined state and disturbance estimation, the Luenberger observer is designed and implemented in a modular configuration, consisting of an observer for the disturbance-free part of the system and, on top of it, a disturbance estimator and a state-estimate corrector ([1]).

The purpose of the present work is to develop a systematic nonlinear observer design method for state and disturbance estimation, so that the resulting observer possesses the modular configuration that is sought for in practice. The nonlinear observer design problem will be formulated within the general framework of exact observer linearization and in particular, following the invariant-manifold formulation, originally proposed in [2] and further developed in [3], [4].

After a review of recent results on state and disturbance observers in non-modular form ([5]), the modular observer will be defined and characterized in terms of appropriate invariance conditions. Necessary and sufficient conditions for exact linearization with eigenvalue assignment will be derived, leading to a step-by-step design procedure for the modular observer.

The theoretical developments will be applied to a bioreactor case study, where biomass is continuously measured, and the observer must estimate the substrate concentration as well as the external disturbance affecting the measuring device.

References:

[1] B. Friedland, Control System Design. An Introduction to State-Space Methods. New York: McGraw-Hill, 1986. [2] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov's auxiliary theorem,” Systems & Control Letters, Vol. 34, pp. 241-247, 1998. [3] A. J. Krener and M. Xiao, “Nonlinear observer design in the Siegel domain,” SIAM J. Control and Optimization, Vol. 41, No. 3, pp. 932-953, 2002. [4] V. Andrieu and L.Praly, “On the existence of a Kazantzis-Kravaris/Luenberger observer,” SIAM J. Control and Optimization, Vol. 45, No.2, pp. 432-456, 2006. [5] C. Kravaris, V. Sotiropoulos, C. Georgiou, N. Kazantzis, M. Xiao and A. J. Krener, “Nonlinear observer design for state and disturbance estimation,” in Proc. 2004 ACC, Boston, Massachusetts, pp. 2931-2936, 2004.