Boundary disturbance rejection for fractional distributed parameter systems via the sliding mode and Riesz basis approach

• Published in: Nonlinear dynamics Vol. 111; no. 2; pp. 1355 - 1367
• Main Authors: Cai, Rui-Yang, Zhou, Hua-Cheng, Kou, Chun-Hai
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.01.2023; Springer Nature B.V
• Subjects: Automotive Engineering; Boundary conditions; Classical Mechanics; Closed loops; Control; Controllers; Design; Distributed parameter systems; Dynamical Systems; Engineering; Hypotheses; Mathematical analysis; Mechanical Engineering; Original Paper; Partial differential equations; Random walk theory; Sliding mode control; Vibration; Wave equations; Riesz basis approach; Mittag-Leffler stability; Fractional distributed parameter systems; Sliding mode control; Boundary disturbance rejection
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-022-07897-3

We study the sliding mode control (SMC) design for unstable fractional heat and wave equations involving unknown external disturbances, respectively. In the case that the disturbance vanishes, a backstepping transform is constructed at first to stabilize the considered system in Mittag-Leffler (M-L) sense. When the disturbance flows into the boundary, sliding mode controllers are provided and the reaching conditions are also verified for fractional heat and wave systems, respectively. In the light of the Riesz basis approach, the well-posedness and closed-loop algebraic stability conclusions are established for fractional partial differential inclusion systems with discontinuous boundary conditions. As a by-product, a longtime unsolved problems raised in [ Nonlinear Dynam , 38 (2004), 339-354], which is the first contribution about the boundary feedback stabilization control for fractional partial differential equations (PDEs), are completely solved with rigorous proof.