Page 132 - IJOCTA-15-4

Anjum et al. / IJOCTA, Vol.15, No.4, pp.670-685 (2025)
+ + ∈ ℜ +
where ℓ 3 ∈ ℜ , α 1 ∈ ℜ , α 2 and
α 3 ∈ ℜ + are observer gains. 0 < Υ < p 1 +1 2 Υ+1
˜ ˜ ˜

˜ ≤ −α 1Θ 1 − α 2Θ 1 − α 3Θ 1
1, χ Θ 1 is employed to represent the function
p 1 +1 p 1 +1 Υ+1
˜ ˜ ˜ ˜ Υ+1
χ Θ 1 = Θ 1 + sign Θ 1 , where χ Θ 1 = 0 ≤ −α 1 2 2 V 2 − 2α 2 V 1 − α 3 2 2 V 2
1 1

˜ ˜ p 1 +1 Υ+1
if and only if Θ 1 = 0, otherwise χ Θ 1 >
= −β 1 V 2 − β 2 V 1 − β 3 V 2
1 1
˜ ˜
1. Moreover, ϕ Θ 1 is expressed as ϕ Θ 1 = (12)
p 1 +1
Υ+1
˜ 2
˜ 2
p 1 + λ 1 Θ 1 / 1 + µ 1 Θ 1 , with p 1 > 1, λ 1 where β 1 = α 1 2 2 , β 1 = 2α 2 , and β 3 = α 3 2 2 .
ˆ
ˆ ∗ ˆ
, µ 1 > 0, and λ 1 > p 1 µ 1 . Θ 1 , d , D a represents the Next, we solve Equation (12) and demonstrate

˜
∗
estimations of Θ 1 , d , D a respective disturbances, that V 1 Θ 1 = 0 can be achieved in fixed time.
ˆ
˜
and Θ 1 = Θ 1 − Θ 1 denotes the estimation error. The following expression, denoted as Equation
ˆ
˜
The observer error is defined as D = D a − D a , (13), can be obtained:
which leads to the establishment of the following
˜
theorem. Z V 1(Θ 1 0 ) dV 1
lim
˜
Theorem 1. The estimation error D of the T Obs ≤ V 1(Θ 1 0 )→0 0 β 1 V p 1 +1 + β 2 V 1 + β 3 V Υ+1
˜
2
2
observer will reach zero within a predetermined 1 1
Z 1
time. In other words, there exists a positive con- ≤ dV 1 Υ+1
˜
stant T Obs ∈ ℜ such that D = 0 for t > T Obs . 0 β 2 V 1 + β 3 V 1 2
˜
Proof. The proof consists of two steps. Step Z V 1(Θ 1 0 ) dV 1
˜
1 provides the stability analysis of Θ 1 using Lya- + lim p 1 +1
˜
V 1(Θ 1 0 )→0 0 β 1 V 2
punov theory, while Step 2 demonstrates the 1 + β 2 V 1 ˜
˜
fixed-time convergence of the observer error D. ≤ Z 1 dV 1 + lim Z V 1(Θ 1 0 ) dV 1
˜
˜
Step 1. The Lyapunov function for Θ 1 is se- 0 β 2 V 1 + β 3 V 1 Υ+1 V 1(Θ 1 0 )→0 0 β 1 V 1 p 1 +1
2
2
lected as 2 β 2 2
= ln 1 + +
β 2 (1 − Υ) β 3 β 1 (p 1 − 1)
˜ 2
V 1 = 0.5Θ 1 (11) (13)
Step 2. Referring to Equations (7) and (10),
Taking the time derivative of Equation (11) the observer error D takes the form of
˜
and applying Equations (9) and (10) yields:
ˆ ∗
∗
˜
D = d − G (x) − ℓ 1 Ξ 2 − d + ℓ 1 Ξ 2 + G (x)
−1 ˙
ˆ
˙
˜
˙
˜
V 1 = Θ 1 Θ 1 + ℓ 2 ℓ 3 Θ 1 − ℓ 2 Θ 2 − ℓ 3 Θ 2 − α 2 Θ 1 = d − d ˆ ∗
∗
˜
ϕ(Θ 1) Υ (14)
˜ ˜ ˜
−α 1 χ Θ 1 sign Θ 1 − α 3 sig Θ 1
Subsequently, Equation (14) can be further
−1 reduced to Equation (15) by applying the estima-
˜
˙
ˆ
˜
˙
= Θ 1 Θ 1 + ℓ 2 ℓ 3 Θ 1 − ℓ 2 ℓ 2 Θ 1 − ℓ 2 ℓ 3 Θ 1 − α 2 Θ 1 ∗
tion of d as specified in Equation (10):
˜
ϕ(Θ 1) Υ
˜ ˜ ˜
−α 1 χ Θ 1 sign Θ 1 − α 3 sig Θ 1

ˆ
˜
∗
˙
D = d − ℓ −1 ℓ 1 ℓ 2 Θ 1 + Θ 2
˜ 2
ϕ(Θ 1)
˜ ˜ ˜ ˜ ˜
= Θ 1 −ℓ 2 ℓ 3 Θ 1 − α 1 χ Θ 1 sign Θ 1 − α 2 Θ 1 ∗ −1 ∗
ˆ
= d − ℓ ℓ 1 ℓ 2 Θ 1 − ℓ 1 ℓ 2 Θ 1 + ℓ 2 d (15)
2

Υ
˜
−α 3 sig Θ 1 ˜
= ℓ 1 Θ 1
˜
ϕ(Θ 1) ˜
˜ ˜ ˜ ˜ Given that Θ 1 (t) = 0 holds for t > T Obs as es-
≤ −Θ 1 α 1 χ Θ 1 sign Θ 1 + α 2 Θ 1
tablished in Step 1, it follows from Equation (15)
˜

Υ that D (t) = 0 is also true fort > T Obs . As a re-
˜
+α 3 sig Θ 1 ˆ
sult, D a can be reliably approximated by D a in a
˜
ϕ(Θ 1) 2 fixed time, finalizing the proof.
˜ ˜ ˜ ˜
≤ −α 1 Θ 1 Θ 1 + sign Θ 1 − α 2Θ 1
Υ+1 3.2. Fixed-time trajectory tracking sliding
˜
mode control law design
− α 3Θ 1
p 1 2
˜ ˜ ˜ ˜
≤ −α 1 Θ 1 Θ 1 + sign Θ 1 − α 2Θ 1 In this section, employing the obtained distur-
bance estimation, a novel robust FTTSMC is pro-
Υ+1
˜
posed for a nonlinear dynamical system subjected
− α 3Θ 1
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