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Page 106 Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05

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where |Υ ( )| ≤ , Lemma 1 tells us, = ( ) + , ≥ | 1 ( − )| , = (−1) =
=1
− 1, based on Lemma 4,
Γ( +1− )
Γ(1− )Γ( +1)
( )
1 1
2 1−
| 1 ( )|≤ ( )· , . (48)
2 (1 − 1 )

According to the above analysis, the system errors will also converge within the bounded region when the
sliding variables enter the domain.

3.4. Selection of control parameters
Through detailed control input exhibition and stability proof accomplished so far, choosing befitting control
parameters concerning the factors including control input smoothness and measuring noises is also important
in the acquisition of outstanding performance. Hence, a parameter selection guideline is provided here.

Selection of 1 : When the sliding variables enter the equilibrium states, the parameter 1 can make the sliding
variables decay exponentially rapidly to ensure that the system states converge in a finite number of steps and
realizetheslidingvariablesconvergequicklyandpreciselytotheequilibriumstates. Increasing 1 canimprove
the rapidness of the system convergence, but too large a value will lead to serious system chattering. Given this
trade-off, we set 11 = 15 and 12 = 10.

Selection of 2 : In equation (47), if 2 is too large or too small, the convergence limit of errors will be affected
and chattering will occur in the system. To achieve a balance, we set 21 = 100, 22 = 100.

Selection of : The smaller the parameter is, the higher the tracking accuracy will be, but too small will
seriously make the system chattering problem. For better performance, we set = −1.7.

Selection of : = , the selection of and must satisfy the odd number of > > 0, making ( )

have no switching item, which can effectively eliminate chattering. we set = 5 and = 3.

Selection of , , : In equation (19), the parameter , , are the system overcomes the main parameters per-
turbation and external disturbance, but the parameter selection inappropriate tends to cause system chattering.
In order to get better performance, we set = 500, = 2, = 160.

Selection of : The larger the parameter is, the faster the system reaches the sliding mode surface, but it
also causes chattering. Taking this tradeoff into consideration, we set 1 = 0.6 and 2 = 0.6.

Selection of : The smaller the parameter is, the smaller the sliding mode bandwidth will be obtained. But
it is better to set = 0.5 so that the boundary layer of the sliding variable is ( ).
2

Remark 2: For the method proposed in this paper, the stability of convergence in finite steps has been proved
in the above parts. This method combines the global memory characteristics of fractional order operators,
accelerates the convergence rate of the system, and makes the system errors approach zero quickly. Moreover,
adaptive law is added to adjust the width of the quasi-sliding mode band in real time to reduce the chattering
of the system. Compared with CSMC algorithm [26] and FTSMC algorithm [27] , it can improve the dynamic
response ability of the system.