Page 130 - IJOCTA-15-4

Anjum et al. / IJOCTA, Vol.15, No.4, pp.670-685 (2025)
(i) Utilizing the error model, an FVECDO corresponding error dynamics for the system can
is designed to estimate lumped distur- be expressed as in Equation (2).
bances. Unlike finite-time Dos, this ob-
server guarantees lumped disturbance es- ˙
Ξ 1 = Ξ 2
timation within a fixed timeframe, re- (2)
˙
gardless of initial conditions. Addition- Ξ 2 = G (x) + h (x) u + D a
ally, compared to fixed-time DOs 38,39 that
where G (x) = g (x) − ¨y des , u = τ(t) and D a =
rely on known bounds of lumped distur-
∆g (x) + d a (t) represent the lumped disturbances
bances and their derivatives, the proposed
in the second-order system.
FVECDO imposes fewer constraints on
Assumption 1: The function h (x) in Equa-
these disturbances, making it more effec-
tion (1) is assumed to be greater than zero. That
tive for practical scenarios.
is, there exists a real positive value b such that
(ii) To achieve stable, rapid, and precise trajec- 2
h (x) > b, ∀x ∈ ℜ .
tory tracking in nonlinear dynamical sys-
tems, a novel FTTSMC utilizing a reaching Assumption 2: The lumped disturbances in
strategy with a state-dependent exponent the second-order system are typically unknown
coefficient is proposed. This controller, but constrained by a positive constant η, as in
leveraging disturbance estimation infor- Equation (3):
mation, ensures rapid convergence both
near and far from the system’s equilibrium |D a | = |∆g (x) + d a (t)| ≤ η (3)
point, while avoiding singularity issues.
(iii) Furthermore, fixed-time stability of the Assumption 3: The desired trajectory y des is
closed-loop system is analyzed using the twice continuously differentiable with respect to
Lyapunov method. time.
The structure of this paper is as follows: Sec-
tion 2 introduces the foundational concepts and
outlines the problem. Section 3 presents the 2.2. Mathematical foundations
observer-based fixed-time control method and de-
Consider a general nonlinear system as given in
tails the fixed-time stability of the closed-loop sys-
Equation (4):
tem. In Section 4, the proposed control strategy
is validated through simulations. Finally, Section
5 concludes the study. ˙ y = f (y (t)) , y (0) = y 0 (4)
n
n
2. Problem formulation and where y ∈ ℜ and f (y (t)) : D → ℜ are nonlin-
mathematical foundations ear functions.
Definition 1. 40 The equilibrium point y = 0
2.1. Problem formulation
of System IV is said to be finite-time stable if it
Consider the following second-order nonlinear dy- satisfies Lyapunov stability, and there exists an
namical system with model uncertainties and ex- open region D ⊆ D 0 containing the origin and a
ternal disturbances Equation (1): function ˆy : D\ {0} → (0, ∞) such that, for all
y 0 ∈ D, the outcome y (t, y 0 ) for t ∈ (0, ˆy (y 0 ))
˙ x 1a = x 2a satisfies lim y (t, y 0 ) = 0 and y (t, y 0 ) = 0 for
t→ˆy(y 0 )
˙ x 2a = g (x) + ∆g (x) + h (x) τ(t) + d a (t) (1) n
any t > ˆy (y 0 ). Specifically, when D ∈ ℜ , the
y a = x 1a system is globally finite-time stable.
where y a denotes the output signal, x 1a and x 2a Definition 2. 41 The equilibrium point y = 0 of
signify the system’s state variables. The func- System IV is said to be fixed-time stable if it is
tions g (x) and h (x) are assumed to be smooth finite-time stable and its settling time T is con-
T
and nonlinear, depending on x = [x 1a , x 2a ] . The strained by a value T max > 0, ensuring T ≤ T max .
term ∆g (x) represents model uncertainties, d a (t) Definition 3. 42 The equilibrium point y = 0 of
denotes external disturbances, and τ(t) is the con- System IV is said to be practically fixed-time sta-
trol input. ble if it satisfies the fixed-time stability property,
n
Let Ξ 1 = y a − y des be the difference between and there exists a specified region Φ ∈ ℜ and a
the actual output and the desired output, and let constant T > 0 such that, for all initial conditions
n
y 0 ∈ ℜ , the state y(t) enters the region Φ within
Ξ 2 = ˙y a − ˙y des be its time derivative. Here, y des
and ˙y des refer to the desired trajectory and its de- time T and remains in Φ for all t > T, where T
rivative with respect to time, respectively. The does not depend on the initial conditions.
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