Nonlinear observer design for the slow states of a singularly perturbed system

The present work proposes a new approach to the nonlinear observer design problem in the presence of two-time-scale multiplicity. In particular, nonlinear processes are considered that exhibit fast and unmeasurable slow dynamic modes, and the latter need to be accurately reconstructed through the use of a state observer. The proposed observer is designed on the basis of the reduced-order process dynamics that evolve on the system's slow manifold, and the dynamic behavior of the estimation error is analyzed and mathematically characterized in the presence of the unmodeled fast process dynamics. It is shown, that within the proposed nonlinear observer design framework, the observation error generated by neglecting the fast process dynamics is of order O(ɛ), where ɛ is the perturbation parameter and a measure of the relative speed/time-constant of the fast and the slow component of the process dynamics. Furthermore, the analysis conducted establishes robustness of the proposed observer design method with respect to fast unmodeled process dynamics. Finally, the performance of the proposed method and the convergence properties of the reduced-order nonlinear observer designed are evaluated in an illustrative biological reactor example.