Page 133 - IJOCTA-15-4

Fixed-time sliding mode control with disturbance observer and variable exponent coefficient for nonlinear
systems
to lumped disturbances. This controller utilizes a Theorem 2. Applying the FTTSMC law in
fixed-time sliding variable and a fixed-time reach- Equation (20) to the nonlinear system in Equa-
ing method with a state-dependent exponent coef- tion (2) facilitates the convergence of trajectory
ficient. Furthermore, it is demonstrated through tracking errors to a minimal residual set, with set-
Lyapunov-based analysis that the closed-loop sys- tling time satisfying Equations (23) and (24):
tem’s tracking errors, irrespective of the initial
conditions, settle in a fixed time to an area close ′ 2 β 5 2
T r ≤ ln 1 + + (23)
to the origin. Consequently, a fixed-time sliding
β 5 β 6 β 4 (p 2 − 1)
variable for the system in Equation (2) is em-
2 2
′
′
ployed as follows: T ≤ T r + + (24)
χ 1 (Ω + 1) χ 2
Proof. The controller constructed using slid-
Ω
s = Ξ 2 + ϑ 1 |Ξ 1 | sign (Ξ 1 ) + ϑ 2 tanh (Ξ 1 /υ 1 ) ing mode control theory has the characteristic
(16) that the control process can be divided into two
where tanh (·) represents the hyperbolic tangent stages. Accordingly, the proof will be carried out
function, and υ 1 ∈ (0, 1), ϑ 1 , ϑ 2 > 0, and Ω > 1 in two stages.
denote the control parameters. Differentiating Stage 1. By selecting a Lyapunov candidate
2
the sliding variable in Equation (16) and employ- V 2 = 0.5s and using Equation (17), the time de-
ing Equation (2), we arrive at Equation (17): rivative of this Lyapunov candidate yields the ex-
pression in Equation (25):
˙ s = G (x) + h (x) u + D a + ϑ 1 Ω|Ξ 1 | Ω−1 Ξ 2
(17) ˙ ˆ Ω−1
2 V 2 = s G (x) + h (x) u + D a + ϑ 1 Ω|Ξ 1 | Ξ 2
+ (ϑ 2 /υ 1 ) 1 − tanh (Ξ 1 /υ 1 ) Ξ 2
2
A state-dependent exponent coefficient reach- + (ϑ 2 /υ 1 ) 1 − tanh (Ξ 1 /υ 1 ) Ξ 2
ing law is designed as Equation (18): (25)
If Assumption 2 holds, Equation (26) can be
obtained
˙ s = −α 4 χ(s) ϕ(s) sign (s) − α 5 s − α 6 sign (s) (18)
ϕ(s)
˙
˜
+
where α 4 ∈ ℜ , α 5 ∈ ℜ + are controller gains, V 2 = s −α 4 χ(s) sign (s) − α 5 s − α 6 sign (s) + D
χ (s) is employed to represent the function ϕ(s) 2
˜
≤ −α 4 |s| (|s| + |sign (s)|) − α 5 |s 1 | − α 6 − D |s|

χ (s) = |s + sign (s)| , and χ (s) = 0 if and

˜
2
only if s = 0, otherwise χ (s) > 1. More- ≤ −α 4 |s| (|s| + |sign (s)|) p 2 − α 5 |s| − α 6 − D |s|

over, ϕ (s) the time varying exponent is ϕ (s) = ≤ −α 4 |s| p 2 +1 − α 5 |s| − α 6 − D |s|

2
˜

p 2 + λ 2 s 2 / 1 + µ 2 s 2 , with p 2 > 1, λ 2 , µ 2 > 0
(26)
and λ 2 > p 2 µ 2 . Furthermore, α 6 is selected as
by using Equation (19). For 0 ≤ t ≤ T Obs , the
shown in Equation (19):
estimation error of the lumped disturbances sat-

∗
ˆ
˜
˜
isfies D a+η ≥ D, which implies α 6 −D ≥ α .


6
 ∗ ˆ
α + D a + η, 0 ≤ t ≤ T Obv Furthermore, Theorem 1 states that the estima-
6
α 6 = (19)
 ∗ tion error of the lumped disturbances is zero for
α , t > T Obv
6
∗
˜
t ≥ T Obs , implying that α 6 −D = α . According

6
+
∗
where α ∈ ℜ . By integrating the observer es-
6
timation, the fixed-time sliding variable in Equa- to this approach, Equation (26) can be recast as
follows:
tion (19), and the reaching law in Equation (18),
the FTTSMC law is formulated as given in Equa-
p 2 +1 p 2 +1 √ 1
˙
tion (20): V 2 ≤ −α 4 2 2 V 2 − 2α 5 V 2 − α ∗ 2V 2
2 6 2
(27)
p 2 +1 1
u = −h(x) −1 [u a + u b ] (20) = −β 4 V 2 2 − β 5 V 2 − β 6 V 2 2
p 2 +1 √
where where β 4 = α 4 2 2 , β 2 = 2α 5 , and β 6 = α ∗ 2.
6
Based on earlier analysis, Equation (27) replicates
ˆ
u a = G (x) + D a + ϑ 1 Ω|Ξ 1 | Ω−1 Ξ 2 the structure of Equation (11). As a result, will
(21) be attained, regardless of its initial value, within a
2
+ (ϑ 2 /υ 1 ) 1 − tanh (Ξ 1 /υ 1 ) Ξ 2 ′ 2 β 5 2
specific timeframe T r ≤ ln 1 + + .
β 5 β 6 β 4 (p 2 −1)
Stage 2. The convergence of the tracking er-
u b = α 4 χ(s) ϕ(s) sign (s) + α 5 s + α 6 sign (s) (22) ror is demonstrated in this stage. As previously
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