Page 18 - Read Online

Zhang et al. Intell. Robot. 2025, 5(2), 333-54 I http://dx.doi.org/10.20517/ir.2025.17 Page 343

and nonlinear components. The linear term 1 provides stable feedback control, which helps maintain system
stability, while the nonlinear term 2 offers additional control authority when the error approaches zero,
2
ensuring that the system can accurately reach the desired value. The design of the nonlinear term Ψ ( ) takes
into account the system’s rapid convergence, stability, and robustness. Using the same nonlinear control form

( ) in both equilibrium and large error phases simplifies the control strategy design while ensuring that
the system can provide effective control signals in different stages. This design effectively reduces the system’s
convergence time and prevents oscillations or instability near the equilibrium point.

Differentiating Equation (20), and considering Equation (6) yield:

¤
¤ = ¥ + 1 Ψ ( ) + 2 ¤

¤

= ¥ − ¥ + 1 Ψ ( ) + 2 ¤ (21)
−1
−1

¤
= ¥ + ¤ + − − + 1 Ψ ( ) + 2 ¤
where
−1
∗
∗
| | ¤ , = 0/ ≠ 0, | | ≥
¤
Ψ ( ) =

1 ¤ + 2 2 | | ¤, ≠ 0, | | <
∗
The NDO-based NFSMC controller is then given by:

ˆ

= ¥ + ¤ + − + 3 ( ) + 4 − 5 sgn ( ) + 1 Ψ ( ) + 2 ¤ (22)
¤
where 3 = ( 31 , . . . , 3 ), 4 = ( 41 , . . . , 4 ), 5 = ( 51 , . . . , 5 ), ∈ (0, 1).
Considering the uncertain nonlinear system represented by Equation (5), which is subject to an external dis-
turbance, a terminal sliding mode disturbance observer is designed in Equation (14). Under the NFTSMC is
constructed as Equation (22), all signals of the closed-loop system are shown to converge in a finite time.
Choose the Lyapunov function V as:
1
2 = (23)
2
Substituting Equation (22) into the derivative of the Lyapunov function, we obtain:

¤
2 = ¤

−1

= ¥ + −1 ¤ + − − + 1 Ψ ( ) + 2 ¤
¤

ˆ
= − 3 ( ) − 4 − 5 sgn ( ) + −

= − 3 ( ) − 4 − 5 sgn ( ) + ˜ (24)

≤ (− 3 ( ) − 4 ) − 5 k k + k k

˜

≤ − 3 ( ) − 4
+1 2
≤ − 3 k k 2 − 4 k k

where 3 = min ( 3 ) and 4 = min ( 4 ), 5 = min ( 5 ) and 5 ≥ .

˜
Invoking Equations (23) and (24) are written as:
+1 +1
¤ (25)
2 ≤ −2 2 3 2 2 − 2 4 2
Based on Equation (25), we can conclude that the system is stable and the state of the system will converge to
the stable point as the 2 decreases. The value of 3 and 4 has a significant effect on the rate of convergence,