Continuous fractional sliding mode-like control for exact rejection of non-differentiable Hölder disturbances

Abstract

Exploiting algebraic and topological properties of differintegral operators as well as a proposed principle of dynamic memory resetting, a uniform continuous sliding mode controller for a general class of integer order affine non-linear systems is proposed. The controller rejects a wide class of disturbances, enforcing in finite-time a sliding regime without chattering. Such disturbance is of Hölder type that is not necessarily differentiable in the usual (integer order) sense. The control signal is uniformly continuous in contrast to the classical (integer order) discontinuous scheme that has been proposed for both fractional and integer order systems. The proposed principle of dynamic memory resetting allows demonstrating robustness as well as: (i) finite-time convergence of the sliding manifold, (ii) asymptotic convergence of tracking errors, and (iii) exact disturbance observation. The validity of the proposed scheme is discussed in a representative numerical study.

Output Feedback Finite-Time Stabilization of Systems Subject to Hölder Disturbances via Continuous Fractional Sliding Modes

“…Secondly, recent results in [26][27][28] are adopted to design a fractional-order sliding mode control that rejects exactly (not just equivalently) the matched Hölder disturbances in finite-time, by invoking the topological properties of differintegral operators [17]. Finally, a nominal controller is designed to enforce finite-time convergence of the whole (pseudo)state [10].…”

“…The interested reader is referred to [14,17,18,[26][27][28], and for completeness, in this section, the differintegral operators and some properties of Hölder functions are introduced. Consider ∈ (0, 1] and the following differintegral operators:…”

“…, }, with Ω the timeinterval at which the system is studied, output = 1 stands for the measured variable, is the control input, and : Ω → R represents matched disturbances and uncertainties. Examples of Hölder effects in physical systems modeled throughout , are given in [27][28][29][30][31].…”

“…where the continuous nominal control 0 stabilizes the ideal system without disturbance [10], and the differintegral sliding mode control compensates the matched Hölder disturbance [26][27][28]. Let be a penalization variable that measures the deviation from the ideal system (without disturbances) to the real system defined by (7).…”

“…The proof is built upon [34] and our previous proposals [28][29][30] in conjunction with the finite-time state estimator. Consider the fractional-order differintegral dynamics resulting from applying control (18) into (17):…”