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Page 100 Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05

3.2. Controller design
To synthesize the advantages of the fractional order sliding mode surface and the fast terminal sliding mode
controllawtoconstructthecontroller, theappropriatefractionalorderslidingmodesurfaceshouldbeselected.
Several fractional order sliding mode surfaces have been described in the literature [8,12–14,22] . Inspired by the
above strategies, the discrete fractional order sliding mode surface selected are as follows:
(15)
( ) = 1 1 ( ) + 2 ( ) + 2 [| 1 ( )| ( 1 ( ))]
where 1 ( ) = [ 1 ( ) − 1 ( ), 2 ( ) − 2 ( )] is the tracking error between the desired position and real

position, 2 ( ) = [ 3 ( ) − 3 ( ), 4 ( ) − 4 ( )] is the tracking error between desired velocity and real

velocity, ( ) ∈ 4×1 is the reference signal vector, 1 = ( 1 ) ( = 1, 2), 2 = ( 2 ) ( = 1, 2) are

selected constant matrices, 0 < = < 1 with , being both odd positive integers.

Remark 1: For a nonlinear system, when the system state is far from the equilibrium point, the fractional
order terminal sliding mode surface proposed by Sun et al. [22] can ensure that the system converges in a finite
time. However, considering that the system state is close to the equilibrium point, the terminal attractor can-
not guarantee the fast convergence of the system. In this paper, a linear term 1 1 ( ) is introduced into the
sliding mode surface, when the system state is close to the equilibrium point, the convergence time is mainly
determined by the linear term 1 1 ( ), which can accelerate the convergence of the system. Therefore, the
sliding mode surface designed in this paper not only makes the system state converge in a finite time but also
preserves the rapidity of the linear sliding mode when it is close to the equilibrium point.

To make the system stable, for the system model (8), the ideal quasi-sliding mode band should meet the fol-
lowing requirements: ( + 1) = 0, then the controller can be obtained as follows:

11 0 1 0
1 = (16)
0 12 0 1

1 [ ( + 1) − ( ) − ( ) ( ) − ( )] + 2 [| 1 ( )| ( 1 ( ))] = 0. (17)
The equivalent control law is:

−1
( ) = [ 1 ( )] [ 1 ( ( + 1) − ( )) + 2 [| 1 ( )| ( 1 ( ))]]. (18)
Toeliminatetheinfluencebroughtbysystemparameteruncertaintyandexternaldisturbance,thenewadaptive
terminal sliding mode reaching law used for system model (8) is:

( + 1) = ( ) − Φ| ( )| [ ( )] (19)

( ) 2
where, = ( ) with = 1 − −[ ] , = ( ) with = 1 − , Φ = (Φ ) with
Φ = | ( )|, > 0, 0 < 1 − < 1, 0 < < 1, and are positive real numbers, = 1, 2.

Then the switching control law is as follows:

−1
( ) = −[ 1 ( )] [ ( ) − Φ| ( )| [ ( )]]. (20)
Combining (8), (15) and the reaching law (19), an AFOFTSMC law is obtained, and the corresponding control
input can be expressed as
( ) = ( ) + ( ). (21)
Substituting equation (21) into the sliding mode function ( + 1), which can be written as:

( + 1) = ( ) − Φ| ( )| [ ( )] − ( ) (22)
where ( ) = 1 ( ), | ( )| ≤ , | ( )| ≤ .