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Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14 Page 9 of 14

Choose a Lyapunov function as 5 = , the time derivative of 5 is
2

¤
5 = 2 ¤

2 3
= −2 1 erf( ) − 2 erf( ) + 3 erf( ) + 1 | | erf( )

3
(30)

3
≤ −2 1 | | − 2 | | − 1 4 + 1 | | erf( )
3

3
≤ −2 1 3 | || | erf(| |) − 1 4
3
where 4 isapositiveconstant. TherestoftheproofissimilartotheproofofTheorem2. Thereexistsaconstant
0 < 3 < 1 such that the variable will reach and keep in a small region Δ 5 around the origin within a fixed
time 2:
1 1
2 ≤ + (31)
4 4
1 3 − 2 2 1 3 ( 1+ 4 − 1)
Then, it can be obtained that the ¤ is a uniformly continuous form (29). Employ Barbalat Lemma [25] to prove

¤ → 0 as → ∞, then ¤ is bounded after the variable converges. Hence, there exists a small region Δ 5
around the origin that can converge into Δ 5.
Step 3 Before the angular error converges to zero, ≠ 0, such that subsystem (13) cannot be simplified
as (19). It should be proved that system state variables , , and are bounded before the angular error
converges to zero.
Consider the following bounded function:

1 2 1 2
6 = + + | | (32)

2 2
The time derivative of 6 is

¤
6 ≤ | || ¤ | + | || ¤ | + |¤ |

¤
¤
¤
≤ | || ¤ | + | || ¤ | + ¤ + |ℎ( , )| + 21 | | 4 + | | 4 + 22 (| | 5 + | | 5
˜
| 2 |
6 6
+ 1 | | + 2 | | +
2 2

¤
¤
≤ | || ¤ | + | || ¤ | + | ¤ | + 21 | | 4 + | | 4 + 22 | | 5 + | | 5 + 1 | | 6 + 2 | | 6
2 2

1 4 | || ¤ | 2| |
+ 3 | ¤ || | + | || ¤ | + | || || ¤ | + | | 3 1 +
1 + 0 2 1 + 4 2 (33)
˜
2 1 | 2 |
3
+ √ | | | ¤ | +
2

¤
¤
≤ | || ¤ | + | || ¤ | + | ¤ | + 21 | | 4 + | | 4 + 22 | | 5 + | | 5 + 1 | | 6 + 2 | | 6
2 2
1 4 4 −1

+ 3 | ¤ || | + | || ¤ | + | || || ¤ | + (1 + 2| |)| | 4 | |
4
˜
4 | 2 |
2 1

+ √ | | 4 | | +
2
p
2
2
Let 1 = + + | | ≥ > 1 , then one has the following inequalities: | | ≤ 1 , | | ≤ 1 , | | ≤ 1.
4
Furthermore, there exist positive constants ( = 3, 4), , , ( = 4, 5, ..., 9), which satisfy | | 4 ≤ 3 1,

4 −1

¤ 5
| | 4 ≤ 4 1, | 2 | 6 < 4 + 4 1, | 2 | 6 < 5 + 5 1, | | 4 < 6 + 6 1, | | 4 < 7 + 7 1, | | < 8 + 8 1,