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

Abstract

The problem of designing a continuous control to guarantee finite-time tracking based on output feedback for a system subject to a Hölder disturbance has remained elusive. The main difficulty stems from the fact that such disturbance stands for a function that is continuous but not necessarily differentiable in any integer-order sense, yet it is fractional-order differentiable. This problem imposes a formidable challenge of practical interest in engineering because (i) it is common that only partial access to the state is available and, then, output feedback is needed; (ii) such disturbances are present in more realistic applications, suggesting a fractional-order controller; and (iii) continuous robust control is a must in several control applications. Consequently, these stringent requirements demand a sound mathematical framework for designing a solution to this control problem. To estimate the full state in finite-time, a high-order sliding mode-based differentiator is considered. Then, a continuous fractional differintegral sliding mode is proposed to reject Hölder disturbances, as well as for uncertainties and unmodeled dynamics. Finally, a homogeneous closed-loop system is enforced by means of a continuous nominal control, assuring finite-time convergence. Numerical simulations are presented to show the reliability of the proposed method.

General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms

“…In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system (see [15]). Then, various techniques are used such as LaSalle's invariance principle and multiplier method mixed with frequency domain, (see [2], [3], [6], [7], [8], [16], [19]).…”