Page 92 - Read Online

Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05 Page 101

3.3. Stability analysis

Lemma 2 [20] : If 0 < < 1, ( ) is a function in equation (23), then ( ) − ( ) − 1 + ( ) ≥ 0 holds
for any ∈ [0, 1].
1
( ) = 1 + 1− − 1− (23)
with 0 ≤ ≤ 1, 1 < ( ) < 2.

Proof: Let ( ) = ( ) − ( ) − 1 + ( ) , then to prove whether the minimum value of ( ) is greater

thanzero. Fromthepropertiesof ( ), weknowthat ( ) > 1, then (1) = −1+ ( ) > 0, and (0) = ( ).

Then, the extreme value of ( ) can be obtained from ( ) = 0. When ( ) = 0, = 1− ( ( )) can be
1
−1
¤
¤

1
1
1
obtained, and then the minimum value ( ) = ( ) − 1− − 1 + 1− = ( ) − 1 + ( − ) 1− > 0, then

the prove is completed.

Lemma 3 [20] : If 0 < < 1, ( ) is a function in equation (23), then ( ) + ( ) − 1 − ( ) ≥ 0 holds

for any ∈ [0, 1].
The proof of Lemma 3 is similar to the proof of Lemma 2.
Theorem 1: For system model (8) with uncertainties and external disturbances, the following sliding mode
motion properties can be guaranteed by using control law (21) :
1) The discrete-time sliding variable can be driven into the domain Ω within a finite number of steps , and
∗
Ω can be expressed as follows.
Ω = { ( )|| ( )| < ( ) } (24)

( )
1 1
1−
Φ ( )
= ,
( )Φ ( ) 1 −
(25)
( 1 )
1 1−

Φ ( )
= ,
( )
2 2
∗
+
∗ (0)−( ( ) ) c + 1, ∈ .

where, ( ) = ( )Φ ( ), = b
( Φ − ) 2
2) Once the sliding mode moves into the domain Ω, it will stay in the domain and will not escape, that is, when
| ( )| ≤ ( ) , then | ( + 1)| ≤ ( ) .
3) When the sliding mode variables move in the domain Ω, the errors will converge to the bounded region, as
follows:
( )
1 1
2 1−
| 1 ( )|≤ ( )· , (26)
2 (1 − 1 )
where is a bounded variable, which can be known from the following analysis.
Proof: Choose the discrete Lyapunov function as ( ) = ( ) ( ), then

4 ( ) = ( + 1) − ( )
2 (27)
Õ
= [ ( + 1) − ( )] [ ( + 1) + ( )] .
=1
To prove the reaching and existence of the control law, we discuss the following cases.