A CONTINUOUS FIRST-ORDER SLIDING MODE CONTROL LAW Keyvan Mohammadi School of Aerospace and Mechanical Engineering The University of Oklahoma Norman, OK 73019 Email: keyvan.mohammadi@ou.edu Andrea L’Afflitto * School of Aerospace and Mechanical Engineering The University of Oklahoma Norman, OK 73019 Email: a.lafflitto@ou.edu ABSTRACT Sliding mode control is a technique to design robust feed- back control laws. In its classical formulation, this approach in- volves discontinuous controls that arise several theoretical and practical challenges, such as the existence of non-unique solu- tions of nonlinear differential equations and chattering. Numero- us variations of the sliding mode control architecture, such as the higher-order sliding mode method, have been presented to over- come these issues. In this paper, we present an alternative slid- ing mode control architecture that involves H¨ older continuous feedback control laws, is simpler to implement than other non- classical nonlinear robust control techniques, guarantees robust- ness and uniform asymptotic stability of the closed-loop system, and ensures both existence and uniqueness of the closed-loop system’s trajectory. Our results are applied to design a robust nonlinear observer in the same form as the Walcott and ˙ Zak ob- server. Moreover, a numerical example illustrates our theoretical results and compares the proposed control law to the classical sliding mode control, the second order sliding mode control, and the super-twisting control. 1 INTRODUCTION In most cases of practical interest, it is difficult to accurately model dynamical systems and estimate the parameters charac- terizing such models. Consequently, designing feedback con- trol laws that guarantee closed-loop asymptotic stability, satis- factory command following, and robustness to unmodeled dy- namics may result in a daunting task. The robust control problem * Address all correspondence to this author. for linear dynamical systems has been extensively studied in the 1970s and 1980s within the framework of H ∞ and H 2 /H ∞ con- trol theory [1–4], whereas notorious robust control techniques for nonlinear dynamical systems are adaptive control [5, 6] and sliding mode control [7, 8], [9, Ch. 7], [10, Ch. 14], [11]. The sliding mode control architecture, which was first de- vised in the 1960s by Emel’yanov and Barbashin [8], consists in steering in finite-time the system’s trajectory to a subspace of the state space, known as sliding manifold, which has been de- signed so that if the system’s trajectory reaches this manifold, then the system’s state asymptotically converges to zero. The control law that drives the system’s trajectory to the sliding man- ifold involves the signum function, and hence is discontinuous. It is well known that solutions of ordinary differential equations with discontinuous right-hand sides may not exist or may not be unique [12, Ch. 2], [13]. Furthermore, in most cases of practical interest discontinuous control inputs induce an undesired effect known as chattering, which consists in high-frequency oscilla- tions of the system’s state about the sliding manifold [10, Ch. 14]; a detailed characterization of chattering is provided in [14]. In spite of the theoretical and practical challenges concern- ing sliding mode control, this technique has drawn considerable interest in aerospace [15], chemical [16], electrical [17], marine [18], and mechanical engineering [19] for its ease of implemen- tation [20] and ability to compensate for disturbances and un- certainties [21]; for further details, see [22, 23] and the numerous references therein. Several approaches have been proposed to de- sign sliding mode control laws that are not affected by chattering. The most popular of these chattering-free techniques consists in modifying the classical sliding mode control in an arbitrarily small neighborhood of the sliding manifold, known as boundary layer, where the signum function is approximated by the satura- Proceedings of the ASME 2017 Dynamic Systems and Control Conference DSCC2017 October 11-13, 2017, Tysons, Virginia, USA DSCC2017-5082 1 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/19/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use