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Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14 Page 7 of 14

and refer to Lemma 2 to Lemma 4, the time
Proof of Theorem 2 Choose a Lyapunov function as 3 = 1 2
2 1
derivative of 3 is

¤
1
3 = 1 ¤ + 11 (d c 1 + d c ) + 12 (d c 2 + d c 2

1
1
2
= 1 ( 1 + 1 ) − ¤ + 11 (d c 1 + d c ) + 12 (d c 2 + d c )

1 (19)
3 3 ˜
= 1 − 1 d 1 c − 2 d 1 c − 3 erf( 1 ) + 1

1
3 +1 3 +1 ˜
≤ − 1 | 1 | − 2 | 1 | − 3 1 tanh( 1 ) + | 1 || 1 |

≤ −2 1 ¯ 3 − 2 2 ¯ 3 ¯
¯ 3
¯ 3
2 2 + 1
¯

where ¯ 3 = 3 +1 , ¯ 3 = 3 +1 , 1 = 3 1 with 1 being a positive constant. By using Lemma 4, the second-
2 2
order system (7) is fixed-time stable. The sliding mode surface 1 will converge into a small region Δ 3 =
n o
1 1
1 | ( 1 ) ≤ min 2 ¯ 3 , 2 ¯ 3 around the origin in a fixed time 1 , which is determined by 1 ≤
¯
¯
1 2 3 2 2 3
1 1 . Then, one can obtain that variables and converge to zero along the real
¯ + ¯

1 2 3 1 (1− ¯ 3 ) 2 2 3 1 ( ¯ 3 −1)
sliding mode in a fixed time [23] .
3.2. Tracking control laws design for the third-order subsystem
After the angular error converges to zero according to Theorem 2, one can obtain that sin equals zero,
and cos equals 1. The system (8) can be simplified as
 ¤ = − +



¤ = − (20)



¤ = 2 + 2

3.2.1. Fixed-time disturbance observer
Introduce the following auxiliary variable for the simplified third-order subsystem (20)
(21)
2 = − 2
where 2 satisfies
1
¤ 2 = 2 + 21 erf( 2 ) + 22 | 2 | erf( 2 ) (22)
2

3 2
where 2 = 1 , and 3 and 3 are integers satisfying the constraints: 0 < 3 < 1, 1 + 3 < 3. The
1+ 3 2 1
parameters 21 and 22 are positive constants with 21 > 2 and 22 > 0.
Theorem 3 For the simplified third-order subsystem (20), a fixed-time disturbance observer is developed in the
form of

ˆ 2 (23)
2 = 21 erf( 2 ) + 22 | 2 | erf( 2 )
ˆ
˜
then it can estimate 2 in a fixed time, and the observation error 2 = 2 − 2 can converge into a small region
.
around the origin within a fixed time 2
Proof of Theorem 3 Similar to the proof of Theorem 1.
3.2.2. Fixed-time sliding mode controller
For the third-order subsystem (20), introduce the following auxiliary variable:
¹

3
= + 1 erf( ) − 2 erf( ) + 3 erf( ) + 1 | | erf d (24)
0 3