  The development of disturbance estimators using extended state observers
(ESOs) typically assumes that the system is observable. This paper introduces
an improved method for systems that are initially unobservable, leveraging
Taylor expansion to approximate the integral of disturbance dynamics. A new
extended system is formulated based on this approximation, enabling the design
of an observer that achieves exponential stability of the error dynamics. The
proposed method's efficacy is demonstrated through a practical example,
highlighting its potential for robust disturbance estimation in dynamic
systems.
