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Page 6 of 14 Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14

Case 2 In the converse case 2 ≤ 1, one has

¤ 1 (14)
2 ≤ −2 12 | 1 || 1 | erf(| 1 |) + 2 11 1
2 − 0
2 0 1 = 2 . Considering the
As 1 + 0 ≥ 1 and | 1 | ≤ 1, it can be obtained that min | 1 | 1 ≥ min | 1 |
1
Lemma 3, then (14) is converted into the following form
¤ 1
2 ≤ −2 12 | 1 || 1 | tanh(| 1 |) + 2 11 1
1 1
≤ 2 12 | 1 || 1 | + 2 12 | 1 | 2 + 2 11 1
1
≤ −2 12 | 1 || 1 | + 2 12 2 + 2 11 1
1 +1
≤ −2 12 | 1 | + 2 11 1 + 2 12 2
(15)
− 0
≤ −2 12 2 | 1 | + 2 11 1 + 12 2
1

2
≤ − 1 + ˜
2
1 1
2

2
≤ − 13 1 − (1 − 13 ) 1 + ˜
2 2
1
− 0
with 1 = 2 12 2 , and ˜ = 2 11 1 +2 12 2. When 0 < 13 < 1, and ˜−(1− 13 ) 1 ≥ 0, (15) can be simplified

2
2
1
¤
2
as 2 ≤ − 13 1 . Then, the solution of 2 will reach a small set Δ 1, which is defined as Δ 1 = 1 | 1 ( 1 ) ≤
2
˜ 2 2
( ) within a settling time 2 ≤ .
1 (1− 13 ) 1 13

In view of the above two cases, the auxiliary variable 1 will converge into a small set Δ 1 = 1 | 1 ( 1 ) ≤
˜ 2
( ) within settling time = 1 + 2.
1 (1− 13 )
Then, the disturbance observation error
˜ ˆ
1 = 1 − 1
(16)

1
= 1 − 11 erf( 1 ) + 12 | 1 | erf( 1 )
The disturbance 1 is bounded according to Assumption 1. Thus, the disturbance observer (11) can estimate
n
˜

1 accurately, and the observation error 1 can remain in a small set Δ 2 = 1 | 1 | ≤ 1 + 11 erf(Δ 1 ) +
o
0 Δ 2
12 |Δ 1 | erf(Δ 1 ) after a fixed time, where ¯ 1 = 1 .
¯ 1
1+ 0 Δ 2
1
3.1.2. Fixed-time sliding mode controller
For the subsystem (7), define = − . A fixed-time integral sliding mode surface is introduced as
follows [22]
¹

2
1 1 2 + d c ) d (17)
1 = + 11 (d c + d c ) + 12 (d c
0
with 0 < < 1, > 1, and ( = 1, 2). For any ∈ R, ∈ R , the notation is defined as d c = | | sign( ).

+
Based on the sliding mode surface as (17), the fixed-time controller is designed as follows:

ˆ
1 1 2 2 3 3 + 3 erf( 1 ) − ¤ − 1 (18)
1 = − 11 (d c + d c ) + 12 (d c + d c ) + 1 d 1 c + 2 d 1 c
1 . In addition, , , ( = 1, 2, 3) are all
where , 1 , ( = 1, 2) are positive constants, 3 satisfies 3 ≥
positive odd integers with 0 < < 1, > 1.
Theorem 2 For the second-order system (7), if the fixed-time controller is constructed in the form of (18), then
the real sliding mode variable will converge into a small set within a fixed time.