https://orcid.org/0000-0002-2302-705X
Figures
Abstract
For nonlinear systems subjected to external disturbances, an adaptive terminal sliding mode control (TSM) approach with fixed-time convergence is presented in this paper. The introduction of the fixed-time TSM with the sliding surface and the new Lemma of fixed-time stability are the main topics of discussion. The suggested approach demonstrates quick convergence, smooth and non-singular control input, and stability within a fixed time. Existing fixed-time TSM schemes are often impacted by unknown dynamics such as uncertainty and disturbances. Therefore, the proposed strategy is developed by combining the fixed-time TSM with an adaptive scheme. This adaptive method deals with an uncertain dynamic system when there are external disturbances. The stability of a closed-loop structure in a fixed-time will be shown by the findings of the Lyapunov analysis. Finally, the outcomes of the simulations are shown to evaluate and demonstrate the efficacy of the suggested method. As a result, examples with different cases are provided for a better comparison of suggested and existing control strategies.
Citation: Ahmed S, Azar AT, Wang H (2024) Adaptive fixed-time TSM for uncertain nonlinear dynamical system under unknown disturbance. PLoS ONE 19(8): e0304448. https://doi.org/10.1371/journal.pone.0304448
Editor: Jun Ma, Lanzhou University of Technology, CHINA
Received: February 19, 2024; Accepted: May 13, 2024; Published: August 21, 2024
Copyright: © 2024 Ahmed et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Networked systems, robotic arm systems, bio-mathematical systems, renewable energy systems, chaotic systems, and servo control systems are just a few examples of the numerous applications in applied mathematics that commonly deal with nonlinear systems influenced by outside perturbations [1–5]. Therefore, the performance and stability of these systems are commonly affected by uncertainties, nonlinearities, and disturbances [6–10]. To cope with these issues, researchers have developed various control strategies, including adaptive control and sliding mode control [11]. Thus, adaptive control and sliding mode control schemes improve transient performance and provide robustness for the entire dynamical system by combining the advantages of these control techniques [12]. As a result, a robust control approach for nonlinear perturbed dynamical system is obtained as well as adaptive scheme deals with unknown uncertain dynamics [13].
Due to its ability to effectively manage uncertainties and disturbances, the sliding mode scheme has seen extensive use in the development of nonlinear control systems, resulting in overall system robustness [14–16]. Nevertheless, this method has an issue with chattering occurring in the control input. Chattering can cause undesired noise to be produced inside the system, which could cause performance instability [17]. Therefore, employing an integral sliding mode, incorporating a boundary layer, or applying a saturation function are some of the approaches that have been proposed to reduce chattering [13, 18, 19]. This scheme has also been applied to various applications such as robotic manipulator [20], unmanned aerial vehicle (UAV) [21], suspension systems [22, 23], biomathematical models [24], etc. Furthermore, finite-time terminal nonsingular (FTNSMC) has been developed in recent years to achieve superior tracking, nonsingularity, and rapid state convergence [25, 26]. Nonetheless, the convergence performance of finite-time approach is considerably affected by the beginning values of the states [27]. Therefore, a substitute that may be utilized to calculate the convergence time that is not dependent on the initial values is a fixed-time control technique [28].
By adjusting the controller parameters according to the present scenario of the system, the method is known as adaptive control [29]. It is a popular control strategy that is commonly used to deal with unknown parameters in nonlinear systems. Therefore, by adjusting the controller’s parameters in real time to compensate for the impacts of nonlinearity, uncertainty, as well as disturbances, this method can enhance a system’s performance and stability [30]. Furthermore, this control strategy is widely used to regulate the dynamics of linear and nonlinear systems [29]. Several classical and advanced control methods have been found to combine well with adaptive and TSM controllers. The new hybrid algorithm with PID-TSM has been proposed to significantly reduce vibration in active suspension systems [22]. An adaptive differential evolution tuned hybrid fuzzy PD-PI controller for power system automatic generation control has been proposed. Evaluations via simulations, stability analysis, and hardware-in-the-Loop real-time validation confirm its effectiveness [31, 32]. An adaptive fixed-time TSM has been developed for the application of Euler-Lagrange systems under unknown dynamics [13]. A TSM based time delay estimation scheme has been designed for the unknown dynamics of robotic manipulator [20]. Moreover, model reference adaptive H∞ control scheme has been proposed to control dynamics of the robotic exoskeleton [33]. An innovative adaptive MPPT approach is proposed for photovoltaic system, combining the incremental conductance method with a fuzzy self-tuning PID controller. Evaluations demonstrate superior efficiency (up to 99.80%) and precise control compared to conventional MPPT techniques under diverse climate scenarios [34]. Hence, the combined application of adaptive and sliding mode control strategies has emerged as a powerful approach for nonlinear system control. This synergistic integration offers several advantages, including enhanced robustness to system uncertainties, improved tracking performance, expedited convergence to desired states, the elimination of non-smooth control inputs, and the capability to handle systems with unknown parameters [35].
Motivated by the schemes mentioned in literature, such as adaptive control and fixed-time TSM, this work is to design the new adaptive fixed sliding mode control scheme. Fixed-time control, which is not affected by the initial conditions of the states, has been employed in this work as the convergence speed of a finite-time scheme varies as the initial value changes. Moreover, an adaptive scheme is used to control the unknown dynamics in the system. Therefore, a new adaptive sliding mode control is proposed that offers to handle the control of a class of nonlinear systems affected by external disturbance with fixed-time convergence, better tracking, and transient responses, as well as obtaining a robust performance closed-loop system. The key points are given as
- The new fixed time sliding mode scheme is designed to achieve fast convergence within a fixed time.
- The adaptive law is combined with fixed-time TSM and is used to deal with the unknown bounded dynamics.
- The investigation of overall system stability with fixed-time convergence is carried out by the Lyapunov synthesis.
- The comparative simulation analyses are provided to validate the efficacy of the proposed scheme.
The outline of this paper is arranged as follows: The proposed fixed-time TSM scheme with closed-loop stability analyses is given in Section 2. The key findings, together with the recommended adaptive sliding mode control strategy and overall stability analysis, are provided in Section 3. Numerical results of the computational simulation are presented in Section 4. The evaluation and discussion are given in Section 5. In the end, conclusion is given in Section 6.
2. Proposed fixed-time sliding control method
In this section, the designed time-varying fixed-time sliding surface is given. Then, the proposed control fixed-time sliding mode scheme is developed, and a closed-loop system is provided using a nonlinear dynamical system with external disturbances given as
(1)
where
,
and
are nonlinear functions,
is uncertainty,
is the unknown disturbance, and
is the control input.
The tracking error will be utilized in the suggested control method and stability analysis; therefore, it is given in (1) as follows:
(2)
where
and
is the reference input, and
with the bounded condition
, κ > 0 is unknown constant.
Lemma 1 [36]: The following nonlinear dynamics are considered:
(3)
where
denotes the continuous function. Thus, the Lyapunov function
that holds:
with
,
. The settling time is computed as
(4)
2.1 Design of sliding surface
The design of a fixed-time sliding surface is given in this subsection. It offers robust performance and precise trajectory tracking for nonlinear systems in fixed-time. The proposed sliding surface is provided as
(5)
where γ1, γ2, γ3 are positive constants, α1 > 1 and α2 ∈ (0, 1). For brevity, (5) can be rewritten as
(6)
with
.
From (6), can be computed as
(7)
Then (7) can be expressed as
(8)
Tracking error (2) substituted into (8), we can have
(9)
Now that the sliding manifold design has been finalized, the next phase will be designing the proposed scheme using fixed-time terminal sliding mode control for the nonlinear system to achieve robust tracking and fast convergence despite dealing with external disturbances.
2.2 Design fixed-time control scheme
The following formulation (u(t) = u1(t) + u2(t)) of the suggested scheme using fixed-time TSM can be used to effectively control the nonlinear system when bounded disturbances are present
(10)
and
(11)
where ξ1, ξ2 and ξ3 are positive constants, β1 > 1 and β2 ∈ (0, 1). From (10) and (11), u(t) substituted into (9), we can obtain
(12)
The design of control method is completed, now the stability analysis of the proposed scheme will be given in next subsection.
2.3 Stability analysis
Fixed-time stability investigations of the suggested control scheme have been carried out in this subsection. Moreover, the settling time is computed using the corresponding Lemma 1. The following Lyapunov function equation is selected for the stability analysis of tracking error
(13)
The expression from (5) is achieved as when σ = 0. So, the following equation is obtained by putting
into (14) as
(15)
(16)
The above expression can be written as
(17)
This expression represents an equation that converges in a fixed amount of time, hence Lemma 1 can be used to obtain the convergence time as
(18)
The fixed-time stability of a closed-loop nonlinear system will now be investigated in Theorem 1 by applying the Lyapunov theorem.
Theorem 1: By considering the nonlinear system as presented in (1), the sliding surface discussed in (5), and the designed control scheme provided in (10)-(11), then it becomes attainable for the nonlinear dynamics to converge states within a fixed time, assuming condition of bounded disturbance is known.
Proof: The candidate for the Lyapunov function is selected as:
(19)
From (12), is putting into (20), we can obtain
(21)
By using bounded condition of disturbance to simplify the preceding expression, one can derive
(22)
The following are two potential scenarios: i) , and ii)
. The Lyapunov function exhibits fixed-time stability when the following condition is met:
for
. If
, substitution of u(t) from (10)-(11) into (2) yields
It is obvious that for σ > 0 then , and for σ < 0 then
. This indicates that
is unable to be an attractor. Consequently, no trajectory can stay on
; all trajectories will cross
and arrive at the sliding surface within a fixed amount of time.
(23)
According to Lemma 1, the fixed settling time is formulated as
(24)
Hence, within a specified duration, the system’s states attain the value of σ. According to Lemma 1, the fixed settling time can easily be calculated, and then total time can be obtained as
.
3. Proposed adaptive fixed-time TSM scheme
The following scheme provides a thorough explanation of the proposed control input (u(t) = u2(t) + u3(t)) and provides an adaptive law with TSM to handle the uncertain dynamics caused by an unknown disturbance.
(25)
whereas
is the estimation of κ, u2(t) is similar to (11). The following adaptive law is given to compensate the unknown dynamics:
(26)
where υ is known + ve constant.
Remark 1. Eq (1), when applied, provides a successful solution to the compensation of uncertain dynamics. As a result, the proposed method is used to attain tracking performance in nonlinear systems while disturbances are present.
By putting u2(t) and u3(t) into (9), we can get
(27)
Theorem 2: This discussion pertains to the nonlinear system (1) and its susceptibility to various challenges, such as external disturbance. The sliding surface (5), the proposed control approach (11) and (25), and the adaptive control law (26), will cause the required state convergence within a fixed time.
Proof: The appropriate Lyapunov function is given as
(28)
where
is the estimation error.
By substitution of given in Eq (27) in (29), Eq (30) can be constructed as follows
(30)
Considering the bounded disturbance condition and adaptive rule Eq (26), it is feasible to describe Eq (30) as
(31)
To calculate the settling time, first the above equation is written as
(32)
Then Eq (33) is obtained as
(33)
where
.
Thus, the settling time is computed by Lemma 1 as
(34)
Using , one may find the equation for the total settling time. As a result, the dynamical system is used to precisely control the system’s states and maintain overall stability over a fixed period of time.
Remark 2. It is stated that state convergence within a fixed-time interval is attained by applying the suggested approach, which integrates the sliding surface, adaptive fixed-time TSM scheme, and adaptive rule, to the nonlinear dynamical system. In Fig 1, the complete developed model is depicted. Furthermore, Lemma 1 suggests that the fixed-time, denoted by , can be strongly impacted by the selection of parameters, such as γi and ξi. The speed of convergence will be affected when these parameters are tuned. The computer simulation findings are going to be presented in the following section.
4. Numerical results
In this section, four cases are given to control the nonlinear uncertain systems with disturbance to analyze the effectiveness of the simulation and validate the suggested scheme. The numerical simulations are intended to demonstrate the performance in the absence and presence of uncertainty, parameter variation, and unknown disturbance, and to be compared with the existing control schemes. The given examples are provided and simulated using MATLAB/Simulink to verify the findings of this study.
Case 1
In this case, the comparative simulations are shown in the absence of the disturbance. For this analysis, intended trajectory, dynamical system model, and parameters of the nonlinear model are given. Now, we may express the following second-order nonlinear system as [37]:
(35)
with the parameters ϕ = 2. The parameters of the proposed scheme are given as follows: γ1 = 100, γ2 = 30, γ3 = 30, ξ1 = 300, ξ2 = 100, ξ3 = 30, α1 = 1.95, α2 = 0.9, β1 = 1.5 and β2 = 0.7. In addition, the initial parameters of
and
are provided as
and
. Making sure the system’s trajectories meet with the chosen desired input
is one of the main objectives of the proposed scheme. And constant with adaptive rule is υ = 1 and the initial parameter is given as κ(0) = 2.
The parameters are suitably tuned, and the tracking error performance of the proposed fixed-time scheme at different initial conditions is given in Fig 2. Then, a detailed comparison between the suggested adaptive fixed-time TSM and adaptive finite-time TSM (AFTSMC) [35] is now being performed. It is clear from examining Figs 3–5 that the suggested strategy performs better in terms of tracking, minimum error, and reaches convergence faster. Moreover, Fig 6 depicts the control input and illustrates that the suggested solution results in a lessened chattering problem.
Case 2
A comparison with the results of the fast nonsingular sliding mode control scheme [35] is done to highlight the advantages of the suggested adaptive fixed-time TSM method in the face of the unknown external disturbance. The details about the uncertain dynamics and external disturbance are given by and d(t) = 30sin(t) + 7.5sin(10t), respectively. Since the goal of this work is to establish an efficient control scheme for second-order nonlinear dynamics, a study is conducted to demonstrate the efficacy of the proposed control approach in comparison with the results provided by [35]. The states tracking, tracking error, and control performance are depicted in Figs 7 to 10.
Figs 7–9 illustrate graphs that show the states ,
, and tracking error ε. The satisfactory performance of states tracking has been successfully achieved. We may observe that the proposed and compared schemes rapidly reduce the tracking error to zero. Nonetheless, the control method defined in [35] struggles to accomplish precise tracking, and one can see that the proposed scheme yields swift convergence. Fig 10 shows the control input. The recommended control input u is smooth and performs well enough in tracking to effectively reduce external disturbance. Moreover, compared to the control law provided in [35], it is observed that the proposed scheme takes less effort.
Case 3
This case presents the simulations of the robotic system; therefore, the dynamics of a single-link robotic manipulator are taken into account while uncertainties and outside disturbance are present. The comparative simulations between the proposed scheme and fixed-time TSM (FxTSM) [28] are illustrated. The single-link manipulator dynamics its parameters, desired trajectory, applied uncertain dynamics, and disturbances are given as [38]:
The parameters of the suggested method are selected as: γ1 = 100, γ2 = 40, γ3 = 30, ξ1 = 300, ξ2 = 100, ξ3 = 30, α1 = 1.95, α2 = 0.9, β1 = 1.5 and β2 = 0.7. ,
, d(t) = 2sin(t) + 0.5cos(10t). The initial value is set as
. Moreover, adaptive constant is υ = 1 and the initial parameter is given as κ(0) = 0.3. The performance of the proposed control strategy is comprehensively evaluated through simulations. Figs 11–14 each depict a specific aspect of the system’s response. Fig 11 illustrates the achieved position tracking performance, allowing for graphical assessment of how closely the system’s output follows the desired trajectory. Fig 12 focuses on the tracking error, providing quantitative information about the difference between the reference and actual positions. Fig 13 presents the control torque exerted by the system to achieve the desired motion. Finally, Fig 14 explores the behavior of the adaptive parameters within the controller, offering insights into how they adjust to optimize system performance.
Simulation outcomes from Figs 11–14, provide compelling evidence that the proposed control scheme outperforms the FxTSM approach in terms of tracking accuracy and convergence speed. This reflects the system’s ability to more precisely follow the desired trajectory. Additionally, the proposed scheme achieves this desired state in a short time compared to FxTSM, indicating its efficiency. These combined improvements in tracking performance and convergence characteristics establish the proposed scheme as a more effective solution for the investigated control problem.
Case 4
This case presents the simulations of the dynamics of a single-link robotic manipulator under the uncertainties, external disturbance and parameter variation. The robustness and adaptiveness of the proposed control approach was rigorously assessed through simulations. Thus, a key aspect of this evaluation involved varying the mass of the robotic manipulator during the simulations. This variation in mass represents a common source of uncertainty in real-world manipulator applications, such as manipulator arms with payloads that can change throughout simulation. The comparative performances between the proposed method and FxTSM under parameter variation (Fig 15), such as position tracking, error, and control torques, are demonstrated in Figs 16–18, respectively.
The simulations show that the controller can continue to work well even when there are unforeseen changes in the dynamics of the system, as seen by its capacity to sustain performance in spite of these mass variations.
5. Discussions and analysis
The simulated results of the proposed fixed time TSM method have been presented. This section now offers an in-depth investigation into the limitations given by the recommended controller. This paper conducts a thorough analysis of the limitations related to the recommended controller gain values and stability demonstrations. Furthermore, this theoretical discourse will go further into the possible uses of the suggested methodology in a nonlinear dynamical system. As a result, a nonlinear system has been used to evaluate and validate the suggested scheme’s effectiveness, and the analysis also shows that it performs better than the comparison scheme.
The parameters that have been chosen in compliance with the provided range as the appropriate ones for the suggested method are γ1 > 0, γ2 > 0, γ3 > 0, ξ1 > 0, ξ2 > 0, ξ3 > 0, α1 > 1, β1 > 1, and α2, β2 ∈ (0, 1). Although the recommended scheme in this instance remains unchanged, ignoring these requirements may cause the system’s closed-loop stability to become unstable. According to the settling time equation, there is an inversely proportional relationship between γi and , as well as between ξi and
, except for the directly proportional relationship between γi, ζi and u(t). The values of γi and ξi must be adequately adjusted to achieve both closed-loop stability and error convergence within a fixed time. As a result, these values will be a significant element in determining the system’s stability. To some extent, it is possible to choose a reasonable value when details about the precise ranges that each parameter falls within are known. This simplifies the process of determining a suitable value of the proposed control scheme.
6. Conclusion
An adaptive fixed-time TSM scheme has been developed to achieve the high states’ trajectory tracking of an uncertain nonlinear dynamical system under external disturbance. The ability to deal with the unknown bounds of system dynamics requires the application of the suggested adaptive scheme. Using this method, the fixed-time TSM scheme demonstrates convergence in a fixed amount of time and provides sufficient trajectory tracking performance. Utilizing the proposed adaptive fixed-time TSM for a nonlinear system with unknown disturbance, we demonstrate the effectiveness of the developed technique by utilizing one of its applications on a second-order system. The results indicate that, in comparison to the AFTSMC and FxTSM approaches, the adaptive fixed-time TSM method performs better in terms of quicker response times, lower tracking errors, and improved control over unknown dynamics. In addition, the tracking performances are given of the proposed εrms = 0.0043 in comparison with εrms = 0.0101 of AFTSMC in case 2, and the proposed εrms = 0.0054 in comparison with εrms = 0.0066 of FxTSM in case 3. For future work, this scheme can be modified with the fractional-order scheme and applied to the higher-order system.
Acknowledgments
The authors would like to thank Prince Sultan University, Riyadh, Saudi Arabia for their support. Special acknowledgment to Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh, Saudi Arabia. In addition, the authors wish to acknowledge the editor and anonymous reviewers for their insightful comments, which have improved the quality of this publication.
References
- 1. Liu J, Wu ZG, Yue D, Park JH. Stabilization of networked control systems with hybrid-driven mechanism and probabilistic cyber attacks. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 2019; 51(2):943–953.
- 2. Khan AQ, Alsulami IM, Hamdani SKA. Controlling the chaos and bifurcations of a discrete prey-predator model. AIMS Mathematics. 2024; 9(1):1783–1818.
- 3. Awad M, Said A, Saad MH, Farouk A, Mahmoud MM, Alshammari MS, et al. A review of water electrolysis for green hydrogen generation considering PV/wind/hybrid/hydropower/geothermal/tidal and wave/biogas energy systems, economic analysis, and its application. Alexandria Engineering Journal. 2024; 87:213–239.
- 4. Ibrahim NF, Mahmoud MM, Al Thaiban AMH, Barnawi AB, Elbarbary ZMS, Omar AI, et al. Operation of grid-connected PV system with ANN-based MPPT and an optimized LCL filter using GRG algorithm for enhanced power quality. IEEE Access. 2023.
- 5. Kamel OM, Diab AAZ, Mahmoud MM, Al-Sumaiti AS, Sultan HM. Performance enhancement of an islanded microgrid with the support of electrical vehicle and STATCOM systems. Energies. 2023; 16(4):1577.
- 6. Chen WH. Disturbance observer based control for nonlinear systems. IEEE/ASME transactions on mechatronics. 2004; 9(4): 706–710.
- 7. Ali A, Shah K, Abdeljawad T, Khan H, Khan A. Study of fractional order pantograph type impulsive antiperiodic boundary value problem. Advances in Difference Equations. 2020; 2020:1–32.
- 8. Ahmad S, Ullah A, Al-Mdallal QM Khan H, Shah K, Khan A. Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons & Fractals. 2020; 139:110256. pmid:32905156
- 9. Shah K, Khan ZA, Ali A, Amin R, Khan H, Khan A. Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative. Alexandria Engineering Journal. 2020; 59(5):3221–3231.
- 10. Mahmoud MM, Atia BS, Esmail YM, Bajaj M, Mbadjoun Wapet DE, Ratib MK, et al. Evaluation and comparison of different methods for improving fault ride-through capability in grid-tied permanent magnet synchronous wind generators. International Transactions on Electrical Energy Systems. 2023; 2023:1–22.
- 11. Mobayen S, Alattas KA, Fekih A, El-Sousy FFM, Bakouri M. Barrier function-based adaptive nonsingular sliding mode control of disturbed nonlinear systems: A linear matrix inequality approach. Chaos, Solitons & Fractals. 2022; 157:111918.
- 12. Ahmed S, Azar AT, Ibraheem IK. Nonlinear system controlled using novel adaptive fixed-time SMC. AIMS Mathematics. 2024; 9(4):7895–7916.
- 13. Ahmed S, Azar AT. Adaptive fractional tracking control of robotic manipulator using fixed-time method. Complex & Intelligent Systems. 2023:1–14.
- 14. Feng Y, Yu X, Man Z. Non-singular terminal sliding mode control of rigid manipulators. Automatica. 2002; 38(12): 2159–2167.
- 15. Yang L, Yang J. Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems. International Journal of Robust and Nonlinear Control. 2011; 21(16): 1865–1879.
- 16. Ton C, Petersen C. Continuous fixed-time sliding mode control for spacecraft with flexible appendages. IFAC-PapersOnLine. 2018; 51(12): 1–5.
- 17. Musmade BB, Patre BM. Robust sliding mode control of uncertain nonlinear systems with chattering alleviating scheme. International Journal of Modern Physics B. 2021; 35(14-16):2140042.
- 18. Cheng NB, Guan LW, Wang LP, Han J. Chattering reduction of sliding mode control by adopting nonlinear saturation function, Advanced Materials Research. 2011; 143: 53–61.
- 19. Suryawanshi PV, Shendge PD, Phadke SB. A boundary layer sliding mode control design for chatter reduction using uncertainty and disturbance estimator. International Journal of Dynamics and Control. 2016; 4:456–465.
- 20. Ahmed S, Azar AT, Ibraheem IK. Model-free scheme using time delay estimation with fixed-time FSMC for the nonlinear robot dynamics. AIMS Mathematics. 2024; 9(4):9979–10009.
- 21. Labbadi M, Iqbal J, Djemai M, Boukal Y, Bouteraa Y. Robust tracking control for a quadrotor subjected to disturbances using new hyperplane-based fast Terminal Sliding Mode. Plos one. 2023; 18(4):e0283195. pmid:37093830
- 22. Nguyen TA. Applying a PID-SMC synthetic control algorithm to the active suspension system to ensure road holding and ride comfort. Plos one. 2023; 18(10): e0283905. pmid:37856506
- 23. Nguyen DN, Nguyen TA. Evaluate the stability of the vehicle when using the active suspension system with a hydraulic actuator controlled by the OSMC algorithm. Scientific reports. 2022; 12(1):19364. pmid:36371531
- 24. Ahmed S, Azar AT, Abdel-Aty M, Khan H, Alzabut J. A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy. Ain Shams Engineering Journal. 2024:102566.
- 25. Zhao XM, Gong YW, Jin HY, Xu C. Adaptive super-twisting-based nonsingular fast terminal sliding mode control of permanent magnet linear synchronous motor. Transactions of the Institute of Measurement and Control. 2023; 01423312231162782.
- 26. Nasim U, Bhatti AR, Farhan M, Rasool A, Butt AD. Finite-time robust speed control of synchronous reluctance motor using disturbance rejection sliding mode control with advanced reaching law. Plos one. 2023; 18(9):e0291042. pmid:37695775
- 27. Hou Z, Lu P, Tu Z. Nonsingular terminal sliding mode control for a quadrotor UAV with a total rotor failure. Aerospace Science and Technology. 2020; 98:105716.
- 28.
Mishra J. Finite-time sliding mode control strategies and their applications. 2019; RMIT University.
- 29. Tao G. Multivariable adaptive control: A survey. Automatica. 2014; 50(11):2737–2764.
- 30.
Lavretsky E, Wise KA. Robust adaptive control. Robust and adaptive control: With aerospace applications. Springer, 2012: 317–353.
- 31. Chintu J, Mohana R, Sahu RK Panda S. Adaptive differential evolution tuned hybrid fuzzy PD-PI controller for automatic generation control of power systems. International Journal of Ambient Energy. 2022; 43(1): 515–530.
- 32. Mahmoud MM, Atia BS,Esmail YM, Ardjoun SEEM, Anwer N, Omar AI, et al. Application of whale optimization algorithm based FOPI controllers for STATCOM and UPQC to mitigate harmonics and voltage instability in modern distribution power grids. Axioms. 2023 12(5); 420.
- 33. Ahmed S. Robust model reference adaptive control for five-link robotic exoskeleton. International Journal of Modelling, Identification and Control. 2021; 39(4):324–331.
- 34. Ibrahim NF Mahmoud MM, Alnami H, Mbadjoun Wapet DE, Ardjoun SAEM, Mosaad MI, Hassan AM, et al. A new adaptive MPPT technique using an improved INC algorithm supported by fuzzy self-tuning controller for a grid-linked photovoltaic system. Plos one. 2023; 18(11): e0293613. pmid:37922271
- 35. Ahmed S, Wang HP, Tian Y. Fault tolerant control using fractional-order terminal sliding mode control for robotic manipulators. Studies in Informatics and Control. 2018; 27(1):55–950.
- 36. Chen C, Li L, Peng H, Yang Y, Mi L,Zhao H. A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks. Neural networks. 2020; 123: 412–419. pmid:31945620
- 37. Golestani M, Mobayen S, Richter H. Fast robust adaptive tracker for uncertain nonlinear second-order systems with time-varying uncertainties and unknown parameters. International Journal of Adaptive Control and Signal Processing. 2018; 32(12): 1764–1781.
- 38. Bingi K, Rajanarayan PB, Pal SA. A review on fractional-order modelling and control of robotic manipulators. Fractal and Fractional. 2023; 7(1):77.