This paper focuses on observer design for a dynamic system with non-monotonic nonlinear measurement equations. A Lyapunov-analysis-based observer design method for exponentially stable state estimation and a corresponding LMI for computing the observer design gains are developed. This observer design method is applied to linear position estimation using magnetic fields for smart industrial actuators. The output equations for magnetic field as a function of position are nonlinear and non-monotonic. A single constant gain that can satisfy the observer stability condition over the entire range of operating conditions does not exist, but a constant stabilizing gain does exist in each piecewise monotonic region. A methodology for discretizing the operating range into piecewise regions and using a finite state machine for switching between piecewise regions with stabilizing observer gains is developed. Experimental results are presented on the performance of the observer in accurately estimating linear position.