Backstepping control for fractional discrete-time systems

Highlights：

• Compared with the direct derivation of time in an integer-order time system, this paper utilizes the inequality (10) (see page 3) to perform fractional difference processing on the V(t) function, and iteratively designs the V(t) function step by step, and finally achieves the asymptotic stability of the system. Thus, the back-step recursive approach and fractional discrete-time systems are effectively combined well.

• For discrete-time systems, conventional nonlinear continuous-time methods cannot be applied. In this paper, using inequality (10) (see page 3) the backstepping approach can also be applied to fractional discrete-time systems. Finally, through the backstepping method, ∇bυVn(t)≤−∑j=1nkjzj2(t)≤0 is gained, and Theorem 2.7 is satisfied, so that the system succeeds asymptotic stability.

摘要

•Because there is (t−μ(r))−υ−1¯ in inequality (7) (see page 3), it is inconvenient to apply numerical formula instantly. In this paper, the following equivalent formula needs to be used to obtain numerical solutions to verify the effectiveness and feasibility of the presented control design approach. For υ∈(0,1) and t=b+n,n≥1, inequality (7) (see page 3) can also be defined in the following form∇bυω(t)=1Γ(−υ)∑j=1nΓ(n−j−υ)Γ(n−j+1)ω(b+j).•Compared with the direct derivation of time in an integer-order time system, this paper utilizes the inequality (10) (see page 3) to perform fractional difference processing on the V(t) function, and iteratively designs the V(t) function step by step, and finally achieves the asymptotic stability of the system. Thus, the back-step recursive approach and fractional discrete-time systems are effectively combined well.•For discrete-time systems, conventional nonlinear continuous-time methods cannot be applied. In this paper, using inequality (10) (see page 3) the backstepping approach can also be applied to fractional discrete-time systems. Finally, through the backstepping method, ∇bυVn(t)≤−∑j=1nkjzj2(t)≤0 is gained, and Theorem 2.7 is satisfied, so that the system succeeds asymptotic stability.