Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis
Open accessAbstract
This article explores the tracking control challenge within nonstrict-feedback nonlinear systems featuring input delay and unknown backlash-like hysteresis. A fixed-time adaptive control scheme is developed to address this problem. Radial basis function neural networks play a key role in identifying the unknown nonlinear functions during the design process. Pade approximation and an intermediate variable are strategically employed to mitigate the impact of input delay, while the compensation for the effect of unknown hysteresis input is achieved through approximating another intermediate variable. By combining the adaptive backstepping control technique with fixed-time stability theory, the article presents an adaptive fixed-time control method. The control scheme proposed ensures the boundedness of all signals in the closed-loop system, as analyzed through the Lyapunov stability theory. This guarantees accurate tracking of desired signals within a fixed time. The efficacy of the proposed approach is additionally validated through simulation studies. The effectiveness of the proposed controller is validated with a real-world example involving an electromechanical system.
AI snapshot
Written from the paper’s full text.
TL;DR
The paper develops a fixed-time adaptive backstepping controller for nonstrict-feedback nonlinear systems with unknown backlash-like hysteresis and input delay, using RBFNNs for unknown-function approximation and Padé approximation plus intermediate variables to handle delay and hysteresis. Lyapunov analysis and an electromechanical simulation show bounded closed-loop signals and fixed-time tracking, with improved error metrics over a finite-time scheme.
Key findings
- The proposed scheme guarantees boundedness of all closed-loop signals and tracking within a fixed time, with settling time independent of initial conditions (Remark 3).
- In the electromechanical simulation, the proposed controller achieved MSE 0.0009, RMSE 0.0300, MAE 0.2100, NMSE 0.0090, SSE 1.3269, and BFR 99.95%, compared with MSE 0.0012, RMSE 0.0346, MAE 0.2200, NMSE 0.0118, SSE 1.3969, and BFR 99.91% for the finite-time control scheme in [39].
- The simulation used input delay ς = 0.06, reference signal y_r = sin(t), and backlash-like hysteresis parameters ω = 1, κ = 3.1635, and = 0.345.
- The paper states that the proposed fixed-time strategy exhibits superior tracking performance and faster convergence than the compared finite-time method.
Methods
- Design: adaptive fixed-time backstepping control for nonstrict-feedback nonlinear systems with input delay and unknown backlash-like hysteresis; RBFNNs approximate unknown nonlinear functions, Padé approximation and an intermediate variable mitigate input delay, and another intermediate variable compensates unknown hysteresis input.
- Analysis: Lyapunov stability theory and fixed-time stability lemmas establish boundedness and convergence to a compact set, with settling time bound T ≤ T_m = 1/(cψ(σ−1)) + 1/(kψ(1−β)).
- Simulation: an electromechanical system model from [48] is used with listed parameters, design parameters, initial conditions [ζ1(0), ζ2(0), ζ3(0)]^T = [0.5, 0.5, 0.5]^T and [η1(0), η2(0), η3(0)]^T = [0, 0, 0]^T; performance is compared with finite-time control [39] using MSE, RMSE, MAE, NMSE, SSE, and BFR.
Limitations
- Stated in Remark 8: design parameters are selected by trial-and-error, which can be time-consuming because no optimal parameter selection method is provided.
- Stated in Remark 8: Padé approximation is only effective for small delays; for larger delays the approximation error increases, limiting applicability to systems with long delays.
- Evident from the text: validation is limited to simulation of one electromechanical example, with comparison against one finite-time control scheme [39]; no physical experiment or broader case study is reported.
Model-written summary. Check it against the paper before relying on it.
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Cites
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Claims the paper makes
4 of 52 cited sources checked · 4 supported · 0 flagged
- Premisereasoning
Fixed-time control methods were introduced in response to the challenge that finite-time settling time depends on initial states.
“In response to this challenge, fixed-time control methods were introduced [40].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [40] Mei Y, Wang J, Park JH, Shi K, Shen H (2022) Adaptive fixed-time control for nonlinear systems against time-varying actuator faults. Nonlinear Dyn 107(4):3629–3640 source
- Premisecitation
Fixed-time control methods offer a viable approach to address the issue of swift convergence without relying on knowledge of the initial states.
“The advent of fixed-time control methods offers a viable approach to address the issue of swift convergence without relying on knowledge of the initial states [41][42][43].”
Quote in the paperSubstantial claim1 of 3 sources checked- cites [41] Chen M, Wang H, Liu X (2019) Adaptive fuzzy practical fixed-time tracking control of nonlinear systems. IEEE Trans Fuzzy Syst 29(3):664–673 source
- cites [42] Ren P, Wang F, Zhu R (2021) Adaptive fixed-time fuzzy control of uncertain nonlinear quantized systems. Int J Fuzzy Syst 23:794–803 source
- cites [43] Zhao Y, Yao J, Tian J, Yu J (2023) Adaptive fixed-time stabilization for a class of nonlinear uncertain systems. Math Biosci Eng 20(5):8241–8260 source
- Findingreasoning
The proposed control scheme ensures the boundedness of all signals in the closed-loop system.
“The control scheme proposed ensures the boundedness of all signals in the closed-loop system, as analyzed through the Lyapunov stability theory.”
Quote in the paperSubstantial claim - Findingreasoning
The proposed control scheme guarantees accurate tracking of desired signals within a fixed time.
“This guarantees accurate tracking of desired signals within a fixed time.”
Quote in the paperSubstantial claim - Findingreasoning
The proposed approach ensures a bound on the settling time without imposing any restrictions on the initial conditions of the system states.
“This approach ensures a bound on the settling time without imposing any restrictions on the initial conditions of the system states.”
Quote in the paperSubstantial claim
- Findingreasoning
The suggested control method enables the output to accurately track desired signals within a fixed time.
“The suggested control method ensures that all signals in the closed-loop system are bounded, enabling the output to accurately track desired signals within a fixed time, based on Lyapunov stability analysis.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The desired trajectory y_r is continuously differentiable up to the nth order, and every derivative is bounded.
- Findingreasoning
The suggested control method ensures that all signals in the closed-loop system are bounded.
“The suggested control method ensures that all signals in the closed-loop system are bounded, enabling the output to accurately track desired signals within a fixed time, based on Lyapunov stability analysis.”
Quote in the paperSubstantial claim - Findingown data
Simulation results validate the effectiveness of the proposed scheme.
“Simulation results further validate the effectiveness of the proposed scheme.”
Quote in the paperSubstantial claim - Findingreasoning
V1 ≤ -c1 e1^{2σ} - k1 e1^{2β} + e2^2/2 + b1^2/2 + ε1^2/2 + γ1 λ1 η1 η1 + κ1 λ1 η1 η1.
“V1 ≤ -c 1 e 2σ 1 -k 1 e 2β 1 + e 2 2 2 + b 2 1 2 + 2 1 2 + γ 1 λ 1 η1 η1 + κ 1 λ 1 η1 η1 . (3.12”
Quote in the paperSubstantial claim - Findingown data
Table 1 shows an enhancement in the effectiveness of the proposed fixed-time control scheme compared to the finite-time control strategy discussed in [39].
“The information displayed in Table 1 shows an enhancement in the effectiveness of the proposed fixed-time control scheme compared to the finite-time control strategy discussed in [39].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [39] Wang H, Liu PX, Zhao X, Liu X (2019) Adaptive fuzzy finite-time control of nonlinear systems with actuator faults. IEEE Trans Cybernet 50(5):1786–1797 source
- Findingreasoning
V_i ≤ -Σ_{j=1}^i c_j e_j^{2σ} - Σ_{j=1}^i k_j e_j^{2β} + Σ_{j=1}^i b_j^2/2 + Σ_{j=1}^i ε_j^2/2 + Σ_{j=1}^i γ_j λ_j η_j η_j + Σ_{j=1}^i κ_j λ_j η_j η_j + e_{i+1}^2/2.
“Vi ≤ - i j=1 c j e 2σ j - i j=1 k j e 2β j + i j=1 b 2 j 2 + i j=1 2 j 2 + i j=1 γ j λ j η j η j + i j=1 κ j λ j η j η j + e 2 i+1 2 (3.23)”
Quote in the paperSubstantial claim - Findingreasoning
e_n κv ≤ c_n e_n^{2σ} - k_n e_n^{2β} + 1/(2b_n^2) e_n^2 η_n P_n^T(Z_n)P_n(Z_n).
“e n κv ≤ c n e 2σ n -k n e 2β n + 1 2b 2 n e 2 n ηn P T n (Z n )P n (Z n ) (3.34)”
Quote in the paperSubstantial claim - Findingreasoning
V_n ≤ Σ_{j=1}^{n-1} c_j e_j^{2σ} + Σ_{j=1}^{n-1} k_j e_j^{2β} + Σ_{j=1}^{n-1} γ_j λ_j η_j θ_j + Σ_{j=1}^{n-1} κ_j λ_j η_j η_j + Σ_{j=1}^{n-1} b_j^2/2 + Σ_{j=1}^{n-1} ε_j^2/2 + e_n φ_n(X_n) + e_n κv + 1/2 c^2 - 1/λ_n η_n η_n.
“Vn ≤ n-1 j=1 c j e 2σ j + n-1 j=1 k j e 2β j + n-1 j=1 γ j λ j η j θ j + n-1 j=1 κ j λ j η j η j + n-1 j=1 b j 2 2 + 2 j 2 + e n φn (X n ) + e n κv + 1 2 c2 - 1 λ n ηn ηn (3.29)”
Quote in the paperSubstantial claim - Findingcitation
e_n c(v) ≤ e_n^2/2 + 1/2 c^2.
“e n c(v) ≤ e 2 n 2 + 1 2 c2 .(3.28)”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Lemma 2
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingown data
Figure 2 demonstrates effective tracking of the system output y to the desired signal y_d.
“Figure 2 displays the system's output y = ζ 1 alongside the reference signal y r , demonstrating effective tracking of y to y d .”
Quote in the paperSubstantial claim - Interpretationreasoning
Attempting to directly apply the backstepping control method from strict feedback systems to nonstrict feedback systems (2.1) results in creating a virtual controller and an adaptive law that encompasses all state variables ζ1, ζ2, ..., ζn, leading to issues with algebraic loops.
“This is because attempting to directly apply the backstepping control method from strict feedback systems to nonstrict feedback systems (2.1) results in creating a virtual controller and an adaptive law that encompasses all state variables ζ 1 , ζ 2 , ..., ζ n , leading to issues with algebraic loops.”
Quote in the paperSubstantial claim - Interpretationnone given
The effects of unknown hysteresis input are counteracted by approximating an additional intermediate variable.
“Furthermore, the difficulties of input delay are tackled through Pade approximation and an intermediate variable, and the counteraction of the effects of unknown hysteresis input is realized by approximating an additional intermediate variable.”
Quote in the paperSubstantial claim - Premisecitation
Under the stated inequality condition, the equilibrium point of the system is practically fixed-time stable with a settling-time bound.
“Under this condition, the equilibrium point of system (2.13), i.e., ζ = 0, is practically fixed-time stable.The settling time T is given by T ≤ T max := 1 ρψ(τ -1) + 1 σ ψ(1 -ω) .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [53] Hu X, Li YX, Hou Z (2020) Event-triggered fuzzy adaptive fixed-time tracking control for nonlinear systems. IEEE Trans Cybernet 52(7):7206–7217 source
- Premisecitation
Traditional methods based on Lyapunov-Krasovskii functionals are generally less effective in managing input delays compared to state delays.
“Traditional methods based on Lyapunov-Krasovskii functionals are generally less effective in managing input delays compared to state delays [25][26][27].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [25] Zhao X, Yang H, Karimi HR, Zhu Y (2015) Adaptive neural control of MIMO nonstrict-feedback nonlinear systems with time delay. IEEE Trans Cybernet 46(6):1337–1349 source
- cites [26] Wang H, Sun W, Liu PX (2016) Adaptive intelligent control of nonaffine nonlinear time-delay systems with dynamic uncertainties. IEEE Trans Syst Man Cybernet Syst 47(7):1474–1485 source
- cites [27] Li DP, Li DJ, Liu YJ, Tong S, Chen CP (2017) Approximation-based adaptive neural tracking control of nonlinear MIMO unknown time-varying delay systems with full state constraints. IEEE Trans Cybernet 47(10):3100–3109 source
- Premisereasoning
RBFNNs are employed to approximate completely unknown nonlinear functions.
“Since the nonlinear functions in (2.1) are completely unknown, RBFNNs are employed to approximate these functions [14].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Ma L, Huo X, Zhao X, Niu B, Zong G (2019) Adaptive neural control for switched nonlinear systems with unknown backlash-like hysteresis and output dead-zone. Neurocomputing 357:203–214 source
- Premisecitation
The solution of the backlash-like hysteresis dynamics is u(v) = κv(t) + c(v).
“u(v) = κv(t) + c(v)”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [16] Liu Y, Zhu Q, Zhao N, Wang L (2021) Adaptive fuzzy backstepping control for nonstrict feedback nonlinear systems with time-varying state constraints and backlash-like hysteresis. Inf Sci 574:606–624 source
- Premisecitation
Traditional adaptive finite-time control methods depend on initial conditions, which makes the settling time difficult to predict in advance.
“Additionally, a significant limitation of traditional adaptive finite-time control methods is their dependency on initial conditions, which makes it difficult to predict the settling time in advance.”
- Also stated in: § 1 Introduction · plain
“This work overcomes the prevailing limitations associated with convergence time dependence on initial states in [38,39] by devising a fixed-time controller.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [38] Wang H, Xu K, Zhang H (2022) Adaptive finite-time tracking control of nonlinear systems with dynamics uncertainties. IEEE Trans Automat Control 68(9):5737–5744 source
- cites [39] Wang H, Liu PX, Zhao X, Liu X (2019) Adaptive fuzzy finite-time control of nonlinear systems with actuator faults. IEEE Trans Cybernet 50(5):1786–1797 source
- Also stated in: § 1 Introduction · plain
- Premisecitation
The function c(v) is bounded by a positive constant c.
“The function c(t) is bounded, such that |c(v)| ≤ c with c being a positive constant.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [16] Liu Y, Zhu Q, Zhao N, Wang L (2021) Adaptive fuzzy backstepping control for nonstrict feedback nonlinear systems with time-varying state constraints and backlash-like hysteresis. Inf Sci 574:606–624 source
- Study design and conductown data
The efficacy of the proposed approach is validated through simulation studies.
“The efficacy of the proposed approach is additionally validated through simulation studies.”
Quote in the paperSubstantial claim - Interpretationreasoning
Designing the nonstrict feedback system cannot follow the same backstepping control approach as the strict feedback system.
“It is important to note that designing the nonstrict feedback system cannot follow the same backstepping control approach as the strict feedback system.”
Quote in the paperSubstantial claim - Study design and conductnone given
The effectiveness of the proposed controller is validated with a real-world example involving an electromechanical system.
“The effectiveness of the proposed controller is validated with a real-world example involving an electromechanical system.”
Quote in the paperNot a substantial claim - Premisecitation
For any positive constant ε_i, there exists a neural network W_i^{*T} P_i(X_i) such that φ_i(X_i) = W_i^{*T} P_i(X_i) + δ_i(X_i) with |δ_i(X_i)| ≤ ε_i.
“In accordance with (2.19), given any positive constant 1 , ∃ a neural network denoted as W * T 1 P 1 (X 1 ), satisfying this condition φ1 (X 1 ) = W * T 1 P 1 (X 1 ) + δ 1 (X 1 ), |δ 1 (X 1 )| ≤ 1 (3.8)”
- Also stated in: § Remark 2 · plain
“In accordance with (2.19), given any positive constant i > 0, ∃ a neural network denoted as W * T i P i (X i ), satisfying this condition φi (X i ) = W * T i P i (X i ) + δ i (X i ), |δ i (X i )| ≤ i (3.19)”
- Also stated in: § Substituting (3.20)-(3.22) into (3.18), one has · plain
“In accordance with (2.19), given any positive constant n > 0, ∃ a neural network denoted as W * T n P n (X n ), satisfying this condition φn (X n ) = W * T n P n (X n ) + δ n (X n ), |δ n (X n )| ≤ n (3.30”
Quote in the paperSubstantial claim0 of 1 sources checked- cites (2.19)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Also stated in: § Remark 2 · plain
- Premisecitation
Radial basis function neural networks with Gaussian function as an activation functions satisfying 0 < P_i^T(•)P_i(•) ≤ N have been implemented to manage such complexities.
“To address this challenge, radial basis function neural networks with Gaussian function as an activation functions satisfying 0 < P T i (•)P i (•) ≤ N have been implemented to manage such complexities [28,47,49].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [28] Wang H, Liu S, Yang X (2020) Adaptive neural control for non-strict-feedback nonlinear systems with input delay. Inf Sci 514:605–616 source
- cites [47] Ni J, Wu Z, Liu L, Liu C (2020) Fixed-time adaptive neural network control for nonstrict-feedback nonlinear systems with deadzone and output constraint. ISA Trans 97:458–473 source
- cites [49] Ma L, Liu L (2019) Adaptive neural network control design for uncertain nonstrict feedback nonlinear system with state constraints. IEEE Trans Syst Man Cybernet Syst 51(6):3678–3686 source
- Rests on an assumption: Algebraic loop problems are unacceptable during backstepping control design.
- Interpretationreasoning
The proposed fixed-time controller overcomes the prevailing limitations associated with convergence time dependence on initial states in prior work.
“This work overcomes the prevailing limitations associated with convergence time dependence on initial states in [38,39] by devising a fixed-time controller.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [38] Wang H, Xu K, Zhang H (2022) Adaptive finite-time tracking control of nonlinear systems with dynamics uncertainties. IEEE Trans Automat Control 68(9):5737–5744 source
- cites [39] Wang H, Liu PX, Zhao X, Liu X (2019) Adaptive fuzzy finite-time control of nonlinear systems with actuator faults. IEEE Trans Cybernet 50(5):1786–1797 source
- Rests on an assumption: Fixed-time control overcomes the dependency of convergence or settling time on initial conditions.
- Interpretationnone given
The difficulties of input delay are tackled through Pade approximation and an intermediate variable.
“Furthermore, the difficulties of input delay are tackled through Pade approximation and an intermediate variable, and the counteraction of the effects of unknown hysteresis input is realized by approximating an additional intermediate variable.”
Quote in the paperSubstantial claim - Premisecitation
The design of nonlinear systems in nonstrict-feedback form incorporating input delay and backlash-like hysteresis remains an unresolved difficulty in prior work.
“The observation that the design of nonlinear systems in nonstrict-feedback form that incorporates input delay and hysteresis similar to backlash is still an unresolved difficulty in the previously discussed outcomes is what inspired this research.”
- Also stated in: § 1 Introduction · plain
“In contrast to established findings [13,18,19,28], this paper explores an adaptive fixedtime tracking control issue within nonstrict-feedback nonlinear systems, contending with both input delay and unknown backlash-like hysteresis.”
Quote in the paperSubstantial claim0 of 4 sources checked- cites [13] Ibrir S, Su CY (2016) Adaptive stabilization of a class of feedforward nonlinear systems subject to unknown backlash-hysteresis inputs. IEEE Trans Control Syst Technol 25(4):1180–1192 source
- cites [18] Zhu J, Li S (2024) Adaptive output dynamic feedback control for nonaffine pure-feedback time delay system with unknown backlash-like hysteresis. J Franklin Inst 361(4):106633 source
- cites [19] Wang H, Wei J, Wang Z (2024) Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis. Proc Inst Mech Eng Part I J Syst Control Eng 238(3):410–424 source
- cites [28] Wang H, Liu S, Yang X (2020) Adaptive neural control for non-strict-feedback nonlinear systems with input delay. Inf Sci 514:605–616 source
- Also stated in: § 1 Introduction · plain
- Interpretationreasoning
The tracking error converges to a narrow range around the origin within the time period T_m.
“The tracking error converges to a narrow range around the origin within the time period T m .”
Quote in the paperSubstantial claim - Interpretationreasoning
For the nonlinear system (2.1) under Assumption 1, global fixed-time stability is attained when the adaptive laws are formulated as (3.11), (3.22), (3.32), the virtual control laws are chosen as (3.10), (3.21), and the real control law is determined by (3.32).
“Theorem 1 For the nonlinear system (2.1) under the assumption 1, the global fixed-time stability of the system is attained when the adaptive laws are formulated as (3.11), (3.22), (3.32), the virtual control laws are chosen as (3.10), (3.21), and the real control law is determined by (3.32).”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
- Premisecitation
Traditional adaptive finite-time control methods like those discussed in [38][39][40] have a notable limitation: their reliance on initial conditions makes it challenging to predict the settling time in advance.
“Furthermore, traditional adaptive finitetime control methods like those discussed in [38][39][40] have a notable limitation: their reliance on initial conditions makes it challenging to predict the settling time in advance.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [38][39][40]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The proposed adaptive fixed-time technique guarantees tracking performance without imposing constraints on initial states.
“Consequently, the proposed adaptive fixed-time technique guarantees tracking performance without imposing constraints on initial states.”
Quote in the paperSubstantial claim - Interpretationreasoning
The convergence time T remains unaffected by initial conditions.
“Equation (3.44) demonstrates that the convergence time T remains unaffected by initial conditions.”
Quote in the paperSubstantial claim - Interpretationreasoning
The tracking error effectively converges to a compact neighborhood around the origin.
“This ensures that the tracking error effectively converges to a compact neighborhood around the origin.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
- Interpretationreasoning
Different settling time limits can be pre-configured based on particular performance requirements.
“Different settling time limits can be pre-configured based on particular performance requirements.”
Quote in the paperSubstantial claim - Interpretationcitation
In comparison to [14], this paper focuses on fixed-time stability for non-switched nonlinear systems.
“In comparison to [14], which presents an adaptive control scheme for switched nonlinear systems affected by backlash-like hysteresis and output dead-zone, this paper focuses on fixed-time stability for non-switched nonlinear systems.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [14] Ma L, Huo X, Zhao X, Niu B, Zong G (2019) Adaptive neural control for switched nonlinear systems with unknown backlash-like hysteresis and output dead-zone. Neurocomputing 357:203–214 source
- Interpretationreasoning
By thoughtfully selecting the right design parameters, the tracking error can be minimized to an optimal extent within a fixed time.
“By thoughtfully selecting the right design parameters, we can ensure that the tracking error is minimized to an optimal extent within a fixed time.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
- Interpretationreasoning
Fixed-time control, as explored in this paper, offers a promising alternative by eliminating the dependency on initial conditions for predicting settling time.
“In contrast, fixed-time control, as explored in this paper, offers a promising alternative by eliminating this dependency, thereby attracting considerable attention in recent research.”
Quote in the paperSubstantial claim- Rests on an assumption: The coexistence of backlash-like hysteresis and input delay in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.
- Interpretationreasoning
The tracking error |y - y_d| is bounded by 2 b0 / ((1-ψ)c)^{1/(2σ)}.
“|y -y d | ≤ 2 b 0 (1 -ψ)c 1/2σ .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
- Interpretationreasoning
The settling time T is bounded by T_m = 1/(cψ(σ-1)) + 1/(kψ(1-β)).
“Then, the settling time is T ≤ T m = 1 cψ(σ -1) + 1 kψ(1 -β) . (3.44)”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
- Interpretationcitation
This paper simultaneously considers the influence of both backlash-like hysteresis and input delay in nonstrict-feedback nonlinear systems.
“Unlike studies such as [16,17], which focus solely on backlash hysteresis, or [31,32], which primarily address input delay, this paper simultaneously considers the influence of both backlash-like hysteresis and input delay in nonstrict-feedback nonlinear systems.”
Quote in the paperSubstantial claim0 of 4 sources checked- cites [16] Liu Y, Zhu Q, Zhao N, Wang L (2021) Adaptive fuzzy backstepping control for nonstrict feedback nonlinear systems with time-varying state constraints and backlash-like hysteresis. Inf Sci 574:606–624 source
- cites [17] Sui S, Tong S, Chen CP, Sun K (2019) Fuzzy adaptive optimal control for nonlinear switched systems with actuator hysteresis. Int J Adapt Control Signal Process 33(4):609–625 source
- cites [31] Li DP, Liu YJ, Tong S, Chen CP, Li DJ (2018) Neural networks-based adaptive control for nonlinear state constrained systems with input delay. IEEE Trans Cybernet 49(4):1249–1258 source
- cites [32] Na J, Ren X, Shang C, Guo Y (2012) Adaptive neural network predictive control for nonlinear pure feedback systems with input delay. J Process Control 22(1):194–206 source
- Rests on an assumption: The coexistence of backlash-like hysteresis and input delay in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.
- Interpretationcitation
When compared to [46], this paper extends the approach by addressing input delay and backlash-like hysteresis in nonstrict-feedback systems.
“When compared to [46], which proposes an adaptive fuzzy control scheme for nonstrict-feedback nonlinear systems with input saturation and time-varying constraints, this paper extends the approach by addressing input delay and backlash-like hysteresis in nonstrict-feedback systems.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [46] Sun J, Yi J, Pu Z (2021) Fixed-time adaptive fuzzy control for uncertain nonstrict-feedback systems with time-varying constraints and input saturations. IEEE Trans Fuzzy Syst 30(4):1114–1128 source
- Interpretationcitation
This work simultaneously considers input delay and backlash-like hysteresis, which were not addressed together in [14].
“Additionally, this work simultaneously considers input delay and backlash-like hysteresis, which were not addressed together in [14].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Ma L, Huo X, Zhao X, Niu B, Zong G (2019) Adaptive neural control for switched nonlinear systems with unknown backlash-like hysteresis and output dead-zone. Neurocomputing 357:203–214 source
- May combine two claims. 12
- Classified as probably not a substantial claim. 23
May combine two claims.
Interpretationnone givenThis paper incorporates unknown backlash-like hysteresis alongside input delay in nonlinear systems, emphasizing fixed-time stability.
“In contrast, this paper incorporates unknown backlash-like hysteresis alongside input delay in nonlinear systems, emphasizing fixed-time stability.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [50] Kharrat M, Alhazmi H (2024) Neural networks-based adaptive fault-tolerant control for a class of nonstrict-feedback nonlinear systems with actuator faults and input delay. AIMS Math 9(6):13689–13711 source
- Rests on an assumption: The coexistence of backlash-like hysteresis and input delay in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.
Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.
PremisecitationCompared to other architectures like deep neural networks [54], RBFNNs offer simplicity, faster learning, and computational efficiency, making them ideal for real-time control applications in nonlinear systems.
“Compared to other architectures like deep neural networks [54], RBFNNs offer simplicity, faster learning, and computational efficiency, making them ideal for real-time control applications in nonlinear systems.”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [54] Kang B, Lee Y, Nguyen TQ (2018) Depth-adaptive deep neural network for semantic segmentation. IEEE Trans Multimedia 20(9):2478–2490 source
Classified as probably not a substantial claim.
FindingcitationFor any i, e_i e_{i+1} ≤ e_i^2/2 + e_{i+1}^2/2.
“e 1 e 2 ≤ e 2 1 2 + e 2 2 2 .(3.5)”
- Also stated in: § Remark 2 · plain
“e i e i+1 ≤ e 2 i 2 + e 2 i+1 2 .(3.17)”
Quote in the paperSubstantial claim0 of 2 sources checked- cites Lemma 4
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites Lemma 2
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Also stated in: § Remark 2 · plain
Probably not a substantial claim: may not pass S3 (load-bearing).
FindingreasoningThe design parameters used in the simulations are selected through a trial-and-error approach, which can be time-consuming due to the absence of an optimal parameter selection method.
“Firstly, the design parameters used in the simulations are selected through a trialand-error approach, which can be time-consuming due to the absence of an optimal parameter selection method.”
Quote in the paperNot a substantial claim- Rests on an assumption: There is no standardized method for selecting design parameters, leading to a reliance on the trial-and-error approach and system performance evaluation.
May combine two claims.
Findingown dataFigures 2-6 collectively indicate that the system's state variables ζ1, ζ2, and ζ3 remain bounded, the controller v is bounded, and the tracking error e1 is bounded.
“Figures 23456collectively indicate that the system's state variables ζ 1 , ζ 2 , and ζ 3 remain bounded, the controller v is bounded, and the tracking error e 1 is also bounded.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataThe proposed fixed-time control strategy exhibits superior tracking performance and faster convergence compared to the existing method.
“The results presented in Fig. 7 illustrate that the proposed fixed-time control strategy exhibits superior tracking performance and faster Fig. 7 Tracking error e 1 under two methods convergence compared to the existing method.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataThe adaptive parameters θ1, θ2, and θ3 are bounded, verifying the effective boundedness of all signals within the closed-loop system (4.1).
“Moreover, the adaptive parameters θ1 , θ2 , and θ3 are bounded, verifying the effective boundedness of all signals within the closed-loop system (4.1).”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationnone givenThe delay ς results in uncertainty in the parameter λ, which in turn results in uncertainty regarding the intermediate variable ζ_{n+1}.
“The delay ς results in uncertainty in the parameter λ, which in turn results in uncertainty regarding the intermediate variable ζ n+1 .”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationFor γ_i in R and 0 < p ≤ 1, the stated power-sum inequality holds.
“Lemma 2 [42] For γ i ∈ R, i = 1, . . ., m, where 0 < p ≤ 1, one has m i=1 |γ i | p ≤ m i=1 |γ i | p ≤ m 1-p m i=1 |γ i | p (2.16)”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [42] Ren P, Wang F, Zhu R (2021) Adaptive fixed-time fuzzy control of uncertain nonlinear quantized systems. Int J Fuzzy Syst 23:794–803 source
Classified as probably not a substantial claim.
PremisecitationAlthough many approaches have been developed for systems with state delays, addressing input delays introduces additional complexity.
“Although many approaches have been developed for systems with state delays, addressing input delays introduces additional complexity [24].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [24] Wang T, Qiu J, Gao H (2016) Adaptive neural control of stochastic nonlinear time-delay systems with multiple constraints. IEEE Trasn Syst Man Cybernet Syst 47(8):1875–1883 source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationSeveral adaptive controllers have been introduced to specifically tackle the complexities associated with unknown backlash-like hysteresis.
“In response to this, several adaptive controllers have been introduced to specifically tackle the complexities associated with unknown backlash-like hysteresis [15][16][17].”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [15] Bai L, Zhou Q, Wang L, Yu Z, Li H (2017) Observer-based adaptive control for stochastic nonstrict-feedback systems with unknown backlash-like hysteresis. Int J Adapt Control Signal Process 31(10):1481–1490 source
- cites [16] Liu Y, Zhu Q, Zhao N, Wang L (2021) Adaptive fuzzy backstepping control for nonstrict feedback nonlinear systems with time-varying state constraints and backlash-like hysteresis. Inf Sci 574:606–624 source
- cites [17] Sui S, Tong S, Chen CP, Sun K (2019) Fuzzy adaptive optimal control for nonlinear switched systems with actuator hysteresis. Int J Adapt Control Signal Process 33(4):609–625 source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationBacklash-like hysteresis is a highly intricate nonlinear phenomenon commonly encountered in industrial control processes.
“Among various input nonlinearities, backlash-like hysteresis stands out as a highly intricate nonlinear phenomenon commonly encountered in industrial control processes [13].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [13] Ibrir S, Su CY (2016) Adaptive stabilization of a class of feedforward nonlinear systems subject to unknown backlash-hysteresis inputs. IEEE Trans Control Syst Technol 25(4):1180–1192 source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationRecent research has introduced several adaptive control approaches tailored for nonlinear systems with input delays.
“Recent research has introduced several adaptive control approaches tailored for nonlinear systems with input delays [29,30].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [29] Zhang M, Jing X, Wang G (2020) Bioinspired nonlinear dynamics-based adaptive neural network control for vehicle suspension systems with uncertain/unknown dynamics and input delay. IEEE Trans Ind Electronics 68(12):12646–12656 source
- cites [30] Niu B, Li L (2017) Adaptive backstepping-based neural tracking control for MIMO nonlinear switched systems subject to input delays. IEEE Trans Neural Netw Learn Syst 29(6):2638–2644 source
Classified as probably not a substantial claim. May combine two claims.
InterpretationreasoningThe RBFNN strategy demonstrates potential in enhancing system stability and overall performance, offering a practical solution to the algebraic loop problem frequently encountered in nonstrict-feedback nonlinear systems.
“This strategy demonstrates potential in enhancing system stability and overall performance, offering a practical solution to the algebraic loop problem frequently encountered in nonstrict-feedback nonlinear systems.”
Quote in the paperSubstantial claim- Rests on an assumption: Algebraic loop problems are unacceptable during backstepping control design.
Classified as probably not a substantial claim.
PremisecitationThe presence of input delays can potentially degrade system performance or lead to instability.
“The presence of input delays can potentially degrade system performance or lead to instability [28].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [28] Wang H, Liu S, Yang X (2020) Adaptive neural control for non-strict-feedback nonlinear systems with input delay. Inf Sci 514:605–616 source
Classified as probably not a substantial claim.
Interpretationnone givenThe paper presents an adaptive fixed-time control approach that combines fixed-time stability theory with backstepping control technology.
“The paper presents an adaptive fixed-time control approach that combines fixed-time stability theory with backstepping control technology.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
PremisecitationFuzzy logic systems or neural networks provide estimates for unknown terms or parameters, enhancing the adaptability of the control approach due to their approximation capabilities.
“These methods play a crucial role in providing estimates for the unknown terms or parameters, enhancing the adaptability of the control approach due to their approximation capabilities [5][6][7].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [5] Li F, Wang F, Fan L, Zhou C, Zong Q (2023) Adaptive neural network fault-tolerant consensus control for multi-agent systems with time-delay and asymmetric error constraint. Neural Process Lett 55(8):10371–10392 source
- cites [6] Shan ZD, He WJ, Han YQ, Zhu SL (2023) Adaptive decentralized tracking control for a class of large-scale nonlinear systems with dynamic uncertainties using multi-dimensional taylor network approach. Neural Process Lett 55(3):3509–3531 source
- cites [7] Kharrat M (2024) Adaptive fault-tolerant control for a class of nonstrict-feedback nonlinear systems with unmodeled dynamics and dead-zone output using multi-dimensional Taylor networks. Nonlinear Dyn 112:13289–13306 source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationFor constants k > 0, b > 0, c > 0, and variables, the stated inequality holds.
“Lemma 3 [39] For constants k > 0, b > 0, c > 0, and variables and ω, one has | | k |ω| b ≤ k k + b c| | k+b + b k + b c -k b |ω| k+b (2.17)”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [39] Wang H, Liu PX, Zhao X, Liu X (2019) Adaptive fuzzy finite-time control of nonlinear systems with actuator faults. IEEE Trans Cybernet 50(5):1786–1797 source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationFor any given (ν_1, ν_2) in R^2 with α > 1, β > 1, and (α-1)(β-1) = 1, the stated Young-type inequality holds.
“Lemma 4 [42] For any given (ν 1 , ν 2 ) ∈ R 2 , where α > 1, β > 1, > 0, and (α-1)(β-1) = 1, we have ν 1 ν 2 ≤ α α |ν 1 | α + 1 β β |ν 2 | β . (2.18)”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [42] Ren P, Wang F, Zhu R (2021) Adaptive fixed-time fuzzy control of uncertain nonlinear quantized systems. Int J Fuzzy Syst 23:794–803 source
Classified as probably not a substantial claim.
PremisecitationWhen a system is influenced by hysteresis nonlinearity, it may exhibit undesirable inaccuracies, oscillations, and even instability, leading to severe consequences for the overall system performance.
“It is widely recognized that when a system is influenced by hysteresis nonlinearity, it may exhibit undesirable inaccuracies, oscillations, and even instability, leading to severe consequences for the overall system performance [14].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Ma L, Huo X, Zhao X, Niu B, Zong G (2019) Adaptive neural control for switched nonlinear systems with unknown backlash-like hysteresis and output dead-zone. Neurocomputing 357:203–214 source
Probably not a substantial claim: may not pass S3 (load-bearing).
Interpretationnone givenImproving the performance of the system as described by (3.42) can be achieved by theoretically decreasing b_i and increasing c_i, k_i, κ_i, and γ_i.
“In practice, improving the performance of the system as described by (3.42) can be achieved by theoretically decreasing b i and increasing c i , k i , κ i , and γ i .”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim. May combine two claims.
Premisenone givenThe weights in RBFNNs are trained in real-time using adaptive control laws, ensuring that the network can accurately approximate unknown functions based on the difference between desired and actual system outputs.
“The weights in RBFNNs are trained in real-time using adaptive control laws, ensuring that the network can accurately approximate unknown functions based on the difference between desired and actual system outputs.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThe efficiency of the controller is heavily impacted by the choice of suitable design parameters.
“The efficiency of the controller is heavily impacted by the choice of suitable design parameters.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningThe closed-loop system is guaranteed to be stable within a fixed time, and the convergence time is bounded by a specific upper bound.
“The closed-loop system is guaranteed to be stable within a fixed time, and the convergence time is bounded by a specific upper bound.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe centers and widths of the radial basis functions are often predefined or initialized based on the problem's characteristics to reduce computational complexity.
“The centers and widths of the radial basis functions are often predefined or initialized based on the problem's characteristics to reduce computational complexity.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningA trade-off exists because higher values of c_i and k_i may lead to stronger control signals.
“However, as emphasized in (3.32), a trade-off exists because higher values of c i and k i may lead to stronger control signals.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenFixed-time control offers the advantages of finite-time control while ensuring an upper bound on convergence time that depends on design parameters rather than initial conditions.
“Fixed-time control offers the advantages of finite-time control while ensuring an upper bound on convergence time that depends on design parameters rather than initial conditions.”
Quote in the paperNot a substantial claim- Rests on an assumption: The coexistence of backlash-like hysteresis and input delay in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.
Classified as probably not a substantial claim.
Premisenone givenRBFNNs are chosen for their ability to provide localized function approximation, meaning they respond more strongly to inputs in specific regions, which is essential for capturing complex nonlinear behaviors.
“Radial basis function neural networks (RBFNNs) are chosen for their ability to provide localized function approximation, meaning they respond more strongly to inputs in specific regions, which is essential for capturing complex nonlinear behaviors.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe Pade approximation is applied to handle input delay, simplifying the derivation process and reducing computational complexity, but it is only effective for small delays.
“Secondly, while the Pade approximation is applied to handle input delay, simplifying the derivation process and reducing computational complexity, it is only effective for small delays.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenEven when system states are unobservable, adaptive laws update the weights using available measurements or estimates.
“Even when system states are unobservable, adaptive laws update the weights using available measurements or estimates.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningFor larger delays, the approximation error increases, limiting the scheme's applicability to systems with long delays.
“For larger delays, the approximation error increases, limiting the scheme's applicability to systems with long delays.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningThe proposed control scheme ensures the boundedness of all signals in the closed-loop system, facilitating accurate tracking of desired signals within a fixed time.
“Utilizing Lyapunov stability analysis, the proposed control scheme ensures the boundedness of all signals in the closed-loop system, facilitating accurate tracking of desired signals within a fixed time.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The nonlinear system (2.1) satisfies Assumption 1.
The paper relies on this without support.
The simultaneous presence of backlash-like hysteresis and input delay in physical systems severely impairs tracking performance, degrades system efficiency, and potentially leads to instability.
“The simultaneous presence of backlash-like hysteresis and input delay in physical systems presents further complications, severely impairing tracking performance, degrading system efficiency, and potentially leading to instability.”
The authors state this as an assumption.
The desired trajectory y_r is continuously differentiable up to the nth order, and every derivative is bounded.
“Assumption 1 [39] The desired trajectory y r is a continuously differentiable signal up to the nth order, and every derivative is bounded.”
The paper relies on this without support.
Fixed-time control overcomes the dependency of convergence or settling time on initial conditions.
“To address this limitation, fixed-time control has emerged as a promising alternative, gaining considerable attention due to its ability to overcome this dependency.”
The paper relies on this without support.
Algebraic loop problems are unacceptable during backstepping control design.
“However, during the backstepping control design process, algebraic loop problems are unacceptable.”
The authors state this as an assumption.
The nonlinear system (2.1) satisfies Assumption 1.
“under the assumption 1”
The paper relies on this without support.
The coexistence of backlash-like hysteresis and input delay in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.
“This comprehensive approach is crucial because the coexistence of these factors in physical systems can significantly complicate matters, leading to reduced tracking performance, lower system efficiency, and potential instability.”
The paper relies on this without support.
There is no standardized method for selecting design parameters, leading to a reliance on the trial-and-error approach and system performance evaluation.
“However, there isn't a standardized method for selecting these parameters, leading to a reliance on the trial-and-error approach and system performance evaluation.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The transformed plant loses and then regains an intermediate-variable term
MinorEquation (2.8) gives the nth state equation without ζn+1. Equation (2.12), presented as a rewriting of (2.8) after substituting u(v)=κv+c(v), includes ζn+1 in that equation. Substitution alone cannot introduce the missing term, so the plant used in the controller derivation differs from (2.8).
“ζn = -u(v) + φ n (ζ ) ζn+1 = -λζ n+1 + 2λu(v)”
“ζn = ζ n+1 -κv -c(v) + φ n (ζ )”
Reported best-fit rates conflict with the defined NMSE relationship
MinorEquations (4.8) and (4.10) use the same error ratio, so BFR should be (1−NMSE)×100%. The table's NMSE values of 0.0090 and 0.0118 therefore imply 99.10% and 98.82%, respectively, not the reported 99.95% and 99.91%.
“| Proposed control scheme | 0.0009 | 0.0300 | 0.2100 | 0.0090 | 1.3269 | 99.95% | | Control scheme in [39] | 0.0012 | 0.0346 | 0.2200 | 0.0118 | 1.3969 | 99.91% |”
“6. Best Fit Rate (BFR): BFR = 1 -P t=1 (y i (t)y id (t)) 2 P t=1 (y i (t) -ȳ id ) 2 × 100%.(4.10)”
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“In the past few decades, considerable attention has been directed towards nonlinear adaptive control methods based on the backstepping technique [3, 4].”
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“Consequently, an escalating number of finite-time backstepping control methods have been progressively introduced [35].”
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“Recent research has introduced several adaptive control approaches tailored for nonlinear systems with input delays [29, 30].”
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Self-citationFuzzy adaptive optimal control for nonlinear switched systems with actuator hysteresis
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“Inresponsetothis,severaladaptivecontrollershavebeenintroduced tospecificallytacklethecomplexitiesassociatedwithunknownbacklash-likehysteresis [15–17].”
Ibrir, S., & Su, C.-Y. (2017) · IEEE Transactions on Control Systems Technology · doi:10.1109/tcst.2016.2597745 · Cited by 19 · cited 2× · see citing works
“The main achievements of this article are summarized as follows: (i) In contrast to established findings [13, 18, 19, 28], this paper explores an adaptive fixed-time tracking control issue within nonstrict-feedback nonlinear systems, contending with both input delay and unknown backlash-like…”
Kharrat, M. (2024) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2024.108478 · Cited by 19 · see citing works
Self-citationAdaptive Fixed-Time Fuzzy Control of Uncertain Nonlinear Quantized Systems
Ren, P., Wang, F., & Zhu, R. (2021) · International Journal of Fuzzy Systems · doi:10.1007/s40815-020-01018-1 · Cited by 15 · cited 1× · see citing works
“The settling time T is given by Lemma 2 [42] For Lemma 3 [39] For constants k > 0 , b > 0 , c > 0 , and variables (cid:15) and ω , one has Lemma 4 [42]Foranygiven (2.18) Assumption 1 [39] The desired trajectory y r is a continuously differentiable signal up to the n th order, and every derivative…”
InfluentialObserver-Based Adaptive NN Tracking Control for Nonstrict-Feedback Systems with Input Saturation
Liu, X., Tong, D., Chen, Q., et al. (2021) · Neural Processing Letters · doi:10.1007/s11063-021-10575-x · Cited by 13 · cited 1× · see citing works
“In the past few decades, considerable attention has been directed towards nonlinear adaptive control methods based on the backstepping technique [3, 4].”
Zhu, X., Huang, J. H., Ding, W., & Zhang, T. (2023) · Asian Journal of Control · doi:10.1002/asjc.3014 · Cited by 11 · cited 1× · see citing works
“For instance, for nonstrict-feedback state-constrained nonlinear systems, a finite-time control method using a command filter has been published to address unmodeled dynamics [36].”
Zhu, J., & Li, S. (2024) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2024.01.034 · Cited by 11 · see citing works
Kharrat, M., & Alhazmi, H. (2024) · AIMS Mathematics · doi:10.3934/math.2024668 · Cited by 11 · see citing works
Self-citationKharrat, M., Krichen, M., Alkhalifa, L., & Gasmi, K. (2023) · IEEE Access · doi:10.1109/access.2023.3338376 · Cited by 11 · see citing works
Self-citationChen, P., Luan, X., & Liu, F. (2023) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3572 · Cited by 9 · cited 1× · see citing works
“Similarly, for state-constrained nonstrict-feedback nonlinear stochastic systems with unmodeled dynamics, a finite-time control method has been documented [37].”
Adaptive fixed-time stabilization for a class of nonlinear uncertain systems
Zhao, Y., Yao, J., Tian, J., & Yu, J. (2023) · Mathematical Biosciences and Engineering · doi:10.3934/mbe.2023359 · Cited by 9 · see citing works
Sharghi, A., Karimi, B., & Dehghan, S. M. M. (2023) · Information Sciences · doi:10.1016/j.ins.2023.119503 · Cited by 9 · cited 1× · see citing works
“Additionally,researchershavedevelopedan adaptivefixed-timecontrolstrategytailoredfornonlinearmulti-agentsystemswithnon-affinedynamics,leveragingneuralnetworks[45].”
Neural Network Based Adaptive Inverse Optimal Control for Non-Affine Nonlinear Systems
Sancı, M. E., & Günel, G. Ö. (2024) · Neural Processing Letters · doi:10.1007/s11063-024-11456-9 · Cited by 7 · see citing works
Shan, Z.-D., He, W.-J., Han, Y.-Q., & Zhu, S.-L. (2023) · Neural Processing Letters · doi:10.1007/s11063-022-11020-3 · Cited by 2 · see citing works
Li, F., Wang, F., Fan, L., Zhou, C., & Zong, Q. (2023) · Neural Processing Letters · doi:10.1007/s11063-023-11331-z · Cited by 2 · cited 1× · see citing works
“Thesemethodsplayacrucialroleinprovidingestimatesfortheunknowntermsorparameters, enhancingtheadaptabilityofthecontrolapproachduetotheirapproximationcapabilities [5–7].”
Li, Z., Wang, F., Ren, P., et al. (2023) · Asian Journal of Control · doi:10.1002/asjc.3171 · Cited by 2 · see citing works
Wang, H., Wei, J., & Wang, Z. (2024) · Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering · doi:10.1177/09596518231208221
Adaptive Fixed-time Control for Nonlinear Systems Against Time-varying Actuator Faults
Mei, Y., Wang, J., Park, J. H., et al. (2021) · doi:10.21203/rs.3.rs-1023045/v1
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Kharrat, M., & Mercorelli, P. (2025) · Mathematics · doi:10.3390/math14010014 · Cited by 3 · see citing works
Self-citationLong, H. D. (2025) · Advanced Engineering Research · doi:10.23947/2687-1653-2025-25-4-2209
Kharrat, M. (2026) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2026.103927
Self-citationKharrat, M. (2026) · International Journal of General Systems · doi:10.1080/03081079.2026.2687865
Self-citationAlhazmi, H., Kharrat, M., Al-Jaser, A., & Mercorelli, P. (2026) · Mathematics · doi:10.3390/math14173159
Self-citation
- Authors
- Kharrat, M., & Alhazmi, H.
- Affiliations
- University of Sfax (TN)
- Princess Nourah bint Abdulrahman University (SA)
- Corresponding author
- Kharrat, Mohamed
- Venue
- Neural Processing Letters
- Published
- 2025-05-26
- Volume
- 57
- Issue
- 3
- ISSN
- 1370-4621 (ISSN-L)
- 1573-773X
- OpenAlex
- W4410739315
- Cited by
- 5
- References
- 53
- Citations by year
- 2026: 3 · 2025: 2
- FWCI
- 1.96
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Neural Processing Letters (publishedVersion, cc-by-nc-nd)
- Funders
- Princess Nourah Bint Abdulrahman University (PNURSP2025R528)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Adaptive Control of Nonlinear Systems
- Physical Sciences › Engineering › Control and Systems Engineering › Stability and Control of Uncertain Systems
- Physical Sciences › Computer Science › Computer Networks and Communications › Neural Networks Stability and Synchronization
- Keywords
- fixed-time control, adaptive control, nonstrict-feedback nonlinear systems, input delay, backlash-like hysteresis, radial basis function neural network, Padé approximants, adaptive backstepping control, tracking control, Lyapunov stability, electromechanical systems
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