A Technical Result for the Study of High-gain Observers with Sign-indefinite Gain Adaptation

We address the problem of state observation for a system whose dynamics may involve poorly known, perhaps even nonlocally Lipschitz functions and whose output measurement may be corrupted by noise. It is known that one way to cope with all these uncertainties and noise is to use a high-gain observer with a gain adapted on-line. As a difference from most previous results, we study such a solution with an adaptation law allowing both increase and decrease of the gain. The proposed method, while presented for a particular case, relies on a “generic” analysis tool based on the study of differential inequalities involving quadratic functions of the error system in two coordinate frames plus the gain adaptation law. We establish that, for bounded system solutions, the estimated state and the gain are bounded. Moreover, we provide an upper bound for the mean value of the error signals as a function of the observer parameters.