Analysis of a nonlinear high-gain observer in the presence of measurement noise

Motivated by the sensitivity of high-gain observers to measurement noise, this paper presents a high-gain observer with a nonlinear gain that takes the form of a piecewise linear function. The piecewise function is designed to have three linear regions, with the two outermost sectors chosen to elicit the desired behavior in the transient and steady-state responses, respectively. In order to overcome the trade-off between fast state reconstruction and measurement noise attenuation, a larger observer gain is generated during the transient response than in the steady-state response. Thus, by reducing the observer gain after achieving satisfactory state estimates, the effect of noise on the steady-state performance is reduced. Moreover, the observer presented in this paper is shown to surpass the system performance achieved when using comparable observers. The proof argues boundedness and ultimate boundedness of the closed-loop system under the proposed output feedback.