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Page 10 of 16 Wang et al. Intell Robot 2023;3(3):479-94 I http://dx.doi.org/10.20517/ir.2023.26

ˆ
ˆ
( ) + ( ) + sgn( ( )) − + = 0
¤
ˆ ˆ
¤ ( ) = −ℑ sgn( ( )) − (− + )
The Lyapunov stability theory is used to analyze the stability of the system, and the candidate Lyapunov func-
tion is selected as
˜ 1 1 ˜ ˜ 1 1 1 1 ˜ 2
e
( , , ˜ , ˜ , ˜ , ℑ) = + tr + ˜ ˜ + ˜ ˜ + ˜ ˜ +
2 2 2 2 2 2 ℑ

˜
e ¤
˜

ℑ
¤
¤
¤
¤
e
( , , ˜ , ˜ , ˜ , ℑ) = ¤ + 1 tr + 1 ˜ ˜ + 1 ˜ ˜ + 1 ˜ ˜ + 1 ˜ ˜ ¤

ˆ ˆ

¤ = (−ℑ sgn( ) − (− + ))

e ˆ
ˆ
ˆ
ˆ
ˆ

= − sgn( ) − Φ + Φ e + Φ e + Φ e+ Θ

˜ ˆ
ˆ
ˆ
ˆ
ˆ

= − k k − Θ − Φ − Φ ˜ − Φ ˜ − Φ ˜
Í Í
˜
ˆ
˜ ˆ
e ¤

¤
Considering the structural characteristics of RNNs, tr = e e , and Φ = e Φ, we
=1 =1
can obtain that
¤ e e ˆ ˆ ˆ ˆ
e
( , , e,e , ˜ , =) = − k k − Θ − Φ − Φ e − Φ e − Φ e
=

e ˜
1 1 1 1 1 ˜ ¤
˜
˜ ¤
¤
¤
¤
+ tr + ˜ ˜ + ˜ ˜ + ˜ ˜ + ==
ℑ
ˆ Í ˆ ˆ ˆ ˆ
= −=k k − Θ − e Φ − Φ e − Φ e − Φ e
=1
1 Í 1 1 ¤ 1 1 ˜ ¤ (26)

¤
¤
¤
+ e ˜ + e ˜ + e e + ˜ ˜ + e˜
=1

Í ¤
ˆ
= − Θ
= − e Φ − 1 ¤ = 1 ˆ e
e − k k +
=1
ℑ
ˆ
e− Φ
e − Φ + e
1 ¤ ˆ 1 ¤ ˜ − Φ + ˜ 1 ¤ ˆ
+e

¤
˜
¤
ˆ
ˆ ¤

¤ ¤
According to the basic principle of SMC, there is ( ) → 0 when → ∞, thus, e = − , ˜ ˜ = − ˆ , ˜ = −ˆ , ˜ =
¤ ¤

¤
− ˆ , = = − .
¤
ˆ
¤ e
Let the upper bound error be defined as ℑ = ℑ − ℑ; on the other hand, by substituting the adaptive regulation
b
e
law (23) into (26), we can obtain that

¤
e
e
( , , e,e , e, =)

Í 1 ˆ 1 e
ˆ
ˆ
e
= − e Φ − − =k k + = k k − Θ
=1 =
(27)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ

+e 1 Φ − Φ + e 1 Φ − Φ + e 1 Φ − Φ

ˆ
= (= − =) | k − Θ ≤ 0
e
Hence, from (41), it can be seen that the Lyapunov function is not increasing but bounded in its domain of defi-
nition;thatis, (0) = ( (0), (0), e(0),e (0), e(0), =(0)),and ( ) = ( ( ), ( ), e( ),e ( ), e( ), =( )).

e
e
e
e
Let ( ) = (kΘk − ) ( ). Considering ( ) is bounded, there exists

( ) ≤ kΘΘk − =)k( )k ≤ − ( , , e,e , ˜ , =).
¤
e
e
¯ ¯ (28)

¤
e
e
( ) ≤ − ( , , e,e , e, ∃) = (0) − ( ).
0 0