Finite-Time Control Analysis of Nonlinear Fractional-Order Systems Subject to Disturbances
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Finite-Time Control Analysis of Nonlinear Fractional-Order Systems Subject to Disturbances Mai V. Thuan1 · Piyapong Niamsup2 · Vu N. Phat3 Received: 1 October 2019 / Revised: 24 June 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract This paper deals with finite-time control problem for nonlinear fractional-order systems with order 0 < α < 1. We first derive sufficient conditions for finite-time stabilization based on Caputo derivative calculus and Lyapunov-like function method. Then, by introducing a new type of the cost control function, we study guaranteed cost control problem for such systems. In terms of linear matrix inequalities, an explicit expression for state and output feedback controllers is given to make the closed-loop system finite-time stable and to guarantee an adequate cost level of the performance. The proposed method is applied to analyze the finite-time control problem for a class of linear uncertain FOSs. Finally, numerical examples are given to illustrate the validity and effectiveness of the proposed results. Keywords Caputo derivative · Fractional-order system · Finite-time stabilization · Guaranteed cost control · Linear matrix inequalities. Mathematics Subject Classification 34D10 · 93D15 · 49M7 · 34K20
Communicated by See Keong Lee.
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Vu N. Phat [email protected] Mai V. Thuan [email protected] Piyapong Niamsup [email protected]
1
Department of Mathematics and Informatics, Thai Nguyen University of Science, Thai Nguyen, Vietnam
2
RCMAM, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3
ICRTM, Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam
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M. V. Thuan et al.
1 Introduction Over the past decades, the analysis and synthesis of fractional-order systems (FOSs) have received considerable attention for their significant applications in control engineering and a great number of excellent results have been obtained (see [1–6] and the references therein). Fractional calculus involves the computation of a derivative or integral of any real order, rather than just an integer. Fractional-order systems have been used in modeling control systems, fluid flow, viscoelasticity, control theory of dynamical systems, diffusive transport akin to diffusion, electrical networks, electrochemistry of corrosion, optics and signal processing, etc. Moreover, FOSs have found wide application in the control of dynamical systems, when the controlled system or/and the controller is described by a set of fractional-order differential equations. When the stability of FOSs is compared with integer-order systems, there are some main difficulties: (i) It is not easy to satisfy the existence and uniqueness of solutions; (ii) it is difficult to calculate and estimate the fractional-order derivatives of Lyapunov functions; and (iii) for the nonlinear FOSs, the stability analysis is much more difficult than that of the linear FOSs, etc. By using the M
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