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Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14 Page 5 of 14

and a new fixed-time sliding mode surface are proposed for the second-order subsystem (7). On this basis, a
fixed-time controller is constructed to make the error variables, and , converge into a small region around
¤
the origin. Then, a fixed-time controller is developed for the third-order subsystem (8), which guarantees that
the system state variables, , , and , are all uniformly ultimately bounded, and the tracking errors, and
, can converge into a small region around the origin in a fixed time.

3.1. Tracking control laws design for the second-order subsystem
3.1.1. Fixed-time disturbance observer
Firstly, for the attitude error subsystem (7), define an auxiliary variable as

(9)
1 = − 1
where 1 satisfies
1
1
¤ 1 = 1 + 11 erf( 1 ) + 12 | 1 | erf( 1 ) (10)

The parameters 11 and 12 are positive constants with 11 > 1 / . Let variable exponential coefficient 1 =
0 2
1 with 0 and 0 satisfying 0 < 0 < 1 and 1 + 0 < 0.
1+ 0 2
1
Theorem 1 For the second-order subsystem (7), if the disturbance observer is constructed as

ˆ 1 (11)
1 = 11 erf( 1 ) + 12 | 1 | erf( 1 )
then it can estimate 1 accurately in a fixed time. That is to say, the observation error 1 = 1 − 1 can converge
˜
ˆ
into a small region within a fixed time.
Proof of Theorem 1 Select a Lyapunov function as 2 = , differentiating it, one has
2
1
¤
2 = 2 1 ( ¤ − ¤ 1 )

1 1
1
= 2 1 ( 1 + 1 ) − 1 + 11 erf( 1 ) + 12 | 1 | erf( 1 )

1
1
= 2 1 − 11 erf( 1 ) − 12 | 1 | erf( 1 ) + 1

1
1
= −2 11 1 erf( 1 ) − 1 1 + 12 1 | 1 | erf( 1 )
(12)

1
1
≤ −2 11 1 tanh( 1 ) − 1 1 + 12 1 | 1 | erf( 1 )

1
1
≤ −2 11 | 1 | − 11 1 − | 1 | + 12 1 | 1 | erf( 1 )

1
≤ −2 12 | 1 || 1 | erf(| 1 |) + 2 11 1
1 +1
= −2 12 | 1 | erf(| 1 |) + 2 11 1
where 1 is a positive constant.
Case 1 When 2 > 1 and | 1 | > 1, one has erf(| 1 |) > erf(1) and 0 2 1 ≥ 0 > 1. Then (12) can be
1+ 0 1 2 1+ 0
rewritten as
0 +1
¤
2 ≤ −2( 12 erf(1) − 11 1 )| 1 | 1+ 0
(13)
0 + 0 +1
≤ −2( 12 erf(1) − 11 1 ) 2(1+ 0 )
2
0 + 0 +1
As 12 erf(1) − 11 1 > 0 and ¯ 1 = > 1, then all the solutions of { 2 > 1} will reach the set { 2 ≤ 1}
2(1+ 0 )
1
within a fixed time 1 ≤ .
2( 12 erf(1)− 11 1 )( ¯ 1 −1)