High-gain observers for nonlinear systems with trajectories close to unobservability

High-gain observers can be easily designed for nonlinear systems and they present very good convergence properties, when the observability mapping of the system is injective in the whole. If the observability mapping is injective in the whole, the system can be transformed into a normal form, from which the estimation of the successive output derivatives can be easily carried out. If the observability mapping is not injective in the whole and the state trajectory of the system passes close to some set where the property of observability is lost (about which the observability mapping is not injective), the resulting high-gain observers may be subject to significant numerical errors. The objective of the paper is to propose an improvement of the classical procedure for the high-gain observer design, so that the nonlinear system is transformed into a modified normal form, through a transformation that is obtained from the observability mapping, by removing singularities belonging to a certain class. Hence, a new high-gain structure, containing two sets of small parameters, is proposed for designing a high-gain observer for this modified normal form. The new observer is applied to a Bullard dynamo, to show that the new observer obtains good state estimates even for trajectories very close to unobservability, for which the classical high-gain observer is affected by significant numerical errors.