Observer design via interconnections of second‐order mixed sliding‐mode/linear differentiators

Abstract

High-gain observers and sliding mode observers are two of the most common techniques to design observers (or differentiators) for lower triangular nonlinear dynamics. While sliding mode observers can handle globally bounded nonlinearities, high-gain linear techniques can deal with globally Lipschitz nonlinearities. To gain in generality and avoid the usual assumption that the plant's solutions are bounded with known bound, we propose here to mix both designs in the more general case where the nonlinearities satisfy a global incremental affine bound. We inspire from the recently-developed low-power high-gain observer technique, which relies on the interconnection of several second order high-gain observers. Adding slidingmode correction terms into this low-power structure enables to guarantee global convergence of the estimation error in finite-time with gains depending only on the parameters of the incremental affine bound of the nonlinearities. The estimation error is also proved to be uniformly stable along solutions starting from any compact sets of initial conditions.

Observer design for continuous-time dynamical systems

“…Finally, it is interesting to know that D. Luenberger initially obtained in [157] the observer (14) for observable autonomous linear systems by linearly transforming (11) into an Hurwitz form with output injection and showing the invertibility of this transformation thanks to observability. Unlike the direct design of (14), this method can actually be extended to nonlinear systems leading to the so-called nonlinear Luenberger or KKL observers presented in Section 7.…”

“…recovering the gains of the well-known two-order sliding-mode observer, see, e.g., [147,14] and references therein. For the case r = −1, the observer dynamics with the correction term (76) must be understood as a differential inclusion.…”

“…Nonlinear gains: As proposed in [14] the gains k i, j in (90) can be designed following a mixed linear/homogeneous approach as k i,1 (s) = q(κs), k i,2 (s) = sign(s) + q(κs), q(s) = sign(κs)|s| 1 2 + s.…”

On nonlinear robust state estimation for generalized Persidskii systems

“…Example 1 Consider a perturbed two-mass spring damper system on a horizontal plane in the form of (15) from [1]:…”

Active disturbance rejection control for the stabilization of a linear hyperbolic system

“…The contribution of this paper is to apply the ADRC strategy to design a feedback-law which allows to reject the disturbance asymptotically and to ensure that the resulting closed-loop system is globally asymptotically stable. The proposed ADRC is based on the observer proposed in 25 which allows to estimate the disturbance in finite time. It should be noted that this observer combines high-gain observers and sliding mode observers.…”