It is well known that the super-twisting algorithm is robust to matched perturbation but is sensitive to measurement noise. Contrarily to this, the classical linear algorithm is less sensitive to measurement noise but less robust to perturbation. To combine both the good accuracy of super-twisting algorithm with respect to perturbation and the good performance of linear algorithm with respect to measurement noise, this paper proposes a new differentiator/observer with a varying exponent gain ${\alpha}$ whose variation depends on the magnitude of measurement noise (high frequency signal). When the magnitude of measurement noise increases (respectively decreases) ${\alpha}$ tends to 1 (respectively tends to 0.5) and the proposed differentiator/observer behaves as a linear algorithm (respectively as a super-twisting algorithm). Thus, by one parameter ${\alpha}$ the differentiator/observer can take care of high frequency noise and matched perturbations. A complete stability analysis of the proposed differentiator/observer is provided. To highlight the applicability of the proposed methodology, the dedicated differentiator/observer is respectively validated on electropneumatic actuator and electric machine testbenches. These experimental results are compared to those of linear and super-twisting algorithms.

Controlling chaotic systems is very often investigated by using empirical laws, without taking advantage of the structure of the governing equations. There are two concepts, observability and controllability, which are inherited from control theory, for selecting the best placement of sensors and actuators. These two concepts can be combined (extended) into flatness, which provides the conditions to fulfill for designing a feedback linearization or another classical control law for which the system is always fully observable and fully controllable. We here design feedback linearization control laws using flatness for the three popular chaotic systems, namely, the Rössler, the driven van der Pol, and the Hénon–Heiles systems. As developed during the last two decades for observability, symbolic controllability coefficients and symbolic flatness coefficients are introduced here and their meanings are tested with numerical simulations. We show that the control law works for every initial condition when the symbolic flatness coefficient is equal to 1.