Page 134 - IJOCTA-15-4

Anjum et al. / IJOCTA, Vol.15, No.4, pp.670-685 (2025)
stated, the state-dependent exponent coefficient rate. Conversely, as the sliding mode variable ap-
reaching law in Equation (18) ensures that s = 0 proaches zero, ψ (s) approaches ι = c 1 c 4 < 1, re-
is attained within a specific timeframe. Therefore, sulting in reduced chattering. However, this may
from Equation (16) it follows that: slow the convergence rate. This issue is addressed
by inserting the linear term α 5 s. Therefore, the
proposed scheme ensures rapid convergence dur-

Ω
Ξ 2 = − ϑ 1 |Ξ 1 | sign (Ξ 1 ) + ϑ 2 tanh (Ξ 1 /υ 1 ) ing the transient state and reduces chattering in
(28) the steady state.
Consider the Lyapunov function defined as By substituting Equation (22) with Equation
2
V 3 = 0.5Ξ . By differentiating it with respect (30), the inequality in Equation (26) from the
1
to time, we get: proof of Theorem 2 can be rewritten as shown
in Equation (31):

Ω
V 3 = Ξ 1 −ϑ 1 |Ξ 1 | sign (Ξ 1 ) − ϑ 2 tanh (Ξ 1 /υ 1 ) ϕ(s)
˙
˜
V 2 = s −ψ (s) χ(s) sign (s) − α 5 s − α 6 tanh (s/υ 1 ) + D
= −ϑ 1 |Ξ 1 | Ω+1 − ϑ 2 Ξ 1 tanh (Ξ 1 /υ 1 ) ≤ −ιχ(s) ϕ(s) |s| − α 5 |s 1 | − α 6 − D |s| + α 6 ϖυ 1

˜
2

≤ −ϑ 1 |Ξ 1 | Ω+1 − ϑ 2 |Ξ 1 | + ϑ 2 ϖυ 1 (31)
Given that ϕ (s) = p 2 + λ 2 s 2 / 1 + µ 2 s 2 ≥
Ω+1 1
≤ −χ 1 V 2 − χ 2 V 2 + ϑ 2 ϖυ 1
3 3 p 2 is true when p 2 > 1, λ 2 , µ 2 > 0 and λ 2 > p 2 µ 2 ,
(29) it follows that:
√
Ω+1
where χ 1 = ϑ 1 2 2 and χ 2 = 2ϑ 2 . According
2
2
to Lemma 1, the trajectory tracking error Ξ 1 ap- − ιχ(s) ϕ(s) |s| = −ι|s + sign (s)| (p 2 +λ 2 s )/(1+µ 2 s ) |s|
proaches an area near zero within a fixed-time ≤ −ι(|s| + |sign (s)|) |s|
p 2
′
duration T s ≤ 2 + 2 . Moreover, the closed-
χ 1 (Ω+1) χ 2 p 2 +1
loop system will reach a vicinity of zero within a ≤ −ι|s|
′
′
′
fixed time T ≤ T r + T s . (32)
Remark 2: The fixed-time sliding mode vari- By substituting Equation (32) into Equation
able in Equation (16) employs a hyperbolic tan- (31) and based on the previous analysis, we ob-
tain:
gent function rather than a signum function to
mitigate chattering. A smaller increases the ap-

˜
˙
2
proximation accuracy to the sign function. How- V 2 ≤ −ι|s| p 2 +1 − α 5 |s 1 | − α 6 − D |s| + α 6 ϖυ 1

ever, setting this parameter too low can cause
p 2 +1 1
spikes in the control output, adversely affecting ≤ −χ 3 V 2 2 − χ 4 V 2 − χ 5 V 2 2 + α 6 ϖυ 1
system performance. Additionally, as shown in (33)
Equation (29), the parameter ϑ 2 ϖυ 1 negatively p 2 +1 ∗ √
impacts error convergence performance. There- where χ 3 = ι2 2 , χ 4 = 2α 5 and χ 5 = α 6 2.
fore, the gain ϑ 2 should be chosen as a small con- Equation (33) can take three forms:
stant to optimize the trade-off between reducing
p 2 +1 p 2 +1
˙
chattering and ensuring timely convergence. V 2 ≤ − (1 − δ 0 ) χ 3 V 2 2 − δ 0 χ 3 V 2 2
To reduce chattering in the control generated 1 (34)
by α 6 sign (s), the sign function can be substi- − χ 4 V 2 − χ 5 V 2 2 + α 6 ϖυ 1
tuted with a hyperbolic tangent function. Addi-
p 2 +1 1
tionally, to mitigate the chattering resulting from V 2 ≤ −χ 3 V 2 − χ 4 V 2 − (1 − δ 0 ) χ 5 V 2
˙
α 4 χ(s) ϕ(s) , a variable gain represented as ψ (s) is 2 2 (35)
1
designed. Consequently, the control law in Equa- − δ 0 χ 5 V 2 + α 6 ϖυ 1
2
tion (22) is modified as follows:
p 2 +1
˙
V 2 ≤ −χ 3 V 2 2 − (1 − δ 0 ) χ 4 V 2 − δ 0 χ 4 V 2
u b = ψ (s) χ(s) ϕ(s) sign (s) + α 5 s + α 6 tanh (s/υ 1 ) 1 (36)
− χ 5 V 2 2 + α 6 ϖυ 1
(30)
c 3
where ψ (s) = [1 + c 1 − 1/ (1 + c 2 |s| ) ] c 4 . c 2 , c 3 > where δ 0 ∈ (0, 1). In the case of inequality in
p 2 +1
0, c 4 > 1, and c 1 < 1/c 4 represent real values. 2
Equation (34), if − (1 − δ 0 ) χ 3 V 2 +α 6 ϖυ 1 ≤ 0
The stability analysis is unaffected by ψ (s), as
is true, it can be rewritten as:
it is positive. When the sliding mode variable in
Equation (16) is far from zero, ψ (s)approaches p 2 +1 1
˙
(1 + c 1 ) c 4 > 1, which improves the convergence V 2 ≤ −δ 0 χ 3 V 2 2 − χ 4 V 2 − χ 5 V 2 2 (37)
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