Abstract
Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method.
Highlights
Fractional calculus almost appeared at the same time as classic calculus, but it has not been paid more attention to due to its lack of application background and difficult calculation
Our main contributions include: (1) The sufficient conditions that ensure the state constraint set and/or the control constraint set are positive invariant sets (PIS) are established by using the invariant set theory; (2) A new Lyapunov function is constructed on the basis of the constraints, and the asymptotic stability conditions for delayed linear fractional-order systems (DLFS) are obtained; and (3) A feedback controller and its solution algorithm are proposed to make DLFS under the state and control input constraints asymptotically stable
The controller u(t) = Kx (t), K ∈ Rm×n is the solution of the asymptotic stabilization problem of system (1) if (i)
Summary
Fractional calculus almost appeared at the same time as classic calculus, but it has not been paid more attention to due to its lack of application background and difficult calculation. Fractional calculus has experienced rapid development during the last few decades both in mathematics and applied sciences. It has been recognized as an excellent tool to describe modern complex dynamics [1,2]. From this perspective, some models governing physical phenomena have been reformulated in light of fractional calculus to better reflect their non-local, frequency- and history-dependent properties. Time delay often occurs in different practical systems, and the delayed fractional-order system can better describe these phenomena [6,7,8]
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