A sufficient condition for the stability of the Third-order Extended State Observer

In this paper, the stability of the Third-order Extended State Observer (ESO) as an important component of the Active Disturbance Rejection Controller (ADRC) is investigated. First, the Lyapunov function with particular form is constructed, and the sufficient condition for the asymptotic stability of the Third-order ESO is derived, based on the symmetry and the positive definiteness of Lyapunov function matrix, the nature of the nonlinear function and the related theory. This condition only includes the parameters of the observer, so it can be easily used as both stability criterion and design of ESO. Compared with the existing results, this stability condition proposed in this paper is more general since it can be considered as an extension of the existing second-order ESO stability results. At last, the effectiveness of the condition is verified by a simulation example.