Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance

Abstract

The pseudo-state stabilization problem of commensurate fractional-order nonlinear systems is investigated. The concerned fractional-order nonlinear system is of parametric strict-feedback form with both unknown parameters and the additive disturbance. To solve this problem, a new nonlinear adaptive control law is constructed via fractional-order backstepping scheme. The developed fractional-order controller does not require the knowledge about both the interval of uncertain parameters and the upper bound of the additive disturbance. The asymptotic pseudo-state stability of the closed-loop system is proved in terms of fractional Lyapunov stability. Several examples are performed finally, and the efficiency is verified.

Robust mixed order backstepping control of non‐linear systems

“…Since e is uniformly continuous, for any ϵ > 0, intervals of unbounded increasing length will exist where e 2 (t) > ϵ and therefore, I α [λe 2 ](t) diverges as t α but the integral I α − β [ϕ T ϕ(0)](t) diverges as t α − β contradicting inequality (31). Thus, e is bounded.…”

“…In [29], non-mixed systems with order in (0, 1) are controlled with backstepping method but the convergence is proved for the transformed variables rather than for the system's original ones. In [29][30][31], the adaptive case for non-mixed systems with the orders in (0, 1) is studied with a fractional Lyapunov direct method which is unsuited, since the Lyapunov function depends on the parametric error, which does not appear in the bound of its derivative.…”

Adaptive Stabilization of a Fractional-Order System with Unknown Disturbance and Nonlinear Input via a Backstepping Control Technique

“…Next, the main control strategy will be introduced in detail, the frequency distributed model is used so that the indirect Lyapunov theory can be applied to demonstrate the feasibility of the proposed control scheme [23][24][25][26][27][28]. In simulation, the numerical method [29] is used to solve the fractional order equations, two examples about fractional-order Chua system and fractional-order Rossler's [30] system are given to verify the correctness of the presented control method.…”

Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients

“…Especially, stabilization problem of fractional-order systems is a very interesting and important research topic. In recent years, more and more researchers and scientists have begun to address this problem [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. With the help of the Lyapunov direct method, Mittag-Leffler stability and generalized Mittag-Leffler stability was studied [10,20].…”