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Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances
Open accessAbstract
This paper deals with the predefined time trajectory tracking control of quadrotor unmanned aerial vehicle (UAV) system in the presence of input delay and external disturbances. By combining the backstepping with the predefined time nonlinear disturbance observer, the uncertainty caused by the external disturbance is eliminated, enabling the UAV to reach the control target within a predefined time. Moreover, in order to compensate for the impact of the control input delay, an appropriate auxiliary system is introduced. Simulation and hardware‐in‐the‐loop (HIL) experimental are carried out to show the effectiveness of the proposed control scheme.
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TL;DR
This paper proposes a predefined-time trajectory tracking control scheme for a quadrotor UAV subject to input delay and external disturbances, combining backstepping, predefined-time nonlinear disturbance observers, and auxiliary systems for delay compensation. Simulations and Pixhawk hardware-in-the-loop experiments indicate tracking and disturbance estimation within the predefined times.
Key findings
- Predefined-time nonlinear disturbance observers estimate external disturbances in each loop; simulation results show observer estimation errors converge around 0.6 s.
- In simulation, the UAV tracks the desired flight trajectory and attitude in about 0.7√2 s, with position and attitude tracking errors converging within 0.7√2 s.
- The auxiliary system compensates input delay; as input delay increases, tracking speed slows and tracking error increases.
- Compared with ordinary NDO and ESO, the proposed predefined NDO lets the z channel track z_d = 3.5 m more quickly within the predefined time with smaller tracking-error fluctuation. In HIL tests, the UAV flies smoothly and maintains target pitch, roll, and yaw angles of 0.26°, 0.35°, and 176.66°, respectively.
Methods
- Design: backstepping position and attitude controllers with predefined-time nonlinear disturbance observers and auxiliary dynamical systems for input delay; Lyapunov-based stability proofs are given in Theorems 1–4.
- Data/sample: numerical simulation of an F450 quadrotor with m = 2.5 kg, I_x = I_y = 0.082 kg·m², I_z = 0.149 kg·m², l = 0.325 m, τ = 0.01, initial position (0.3, 0.3, 0.3) m, initial Euler angles (0, 0, 0.3) rad, desired position (2, 4, 3.5) m, desired yaw 0 rad, and disturbances such as d₁ = 5 sin 2t.
- Analysis: Lyapunov stability arguments for predefined-time convergence, plus numerical simulations and hardware-in-the-loop experiments using the RflySim platform and Pixhawk flight control.
Limitations
- Stated: future work will focus on predefined-time flight tracking control to overcome state constraints and actuator failures, implying these are not addressed in the present work.
- Stated assumptions: the geometric center coincides with the center of gravity; the UAV is a rigid body with uniform symmetry; mass and moment of inertia are constant; disturbances are differentiable with bounded derivatives; and the input delay is known and non-negative.
- Evident from the text: validation is limited to numerical simulation and hardware-in-the-loop experiments, not real outdoor flight, and robustness to unknown delay or actuator faults is not demonstrated.
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In the paper
“(3) The proposed predefined time tracking controller has strong anti-disturbance ability, which can guarantee that the quadrotor UAV tracks the required signal within the predefined time, and the stability time is not affected by the initial conditions and other design parameters.”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 1 INTRODUCTION
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“(3) The proposed predefined time tracking controller has strong anti-disturbance ability, which can guarantee that the quadrotor UAV tracks the required signal within the predefined time, and the stability time is not affected by the initial conditions and other design parameters.”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 1 INTRODUCTION
boostedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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In the paper
“Moreover, in order to compensate for the impact of the control input delay, an appropriate auxiliary system is introduced.”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > Abstract
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“(3) The proposed predefined time tracking controller has strong anti-disturbance ability, which can guarantee that the quadrotor UAV tracks the required signal within the predefined time, and the stability time is not affected by the initial conditions and other design parameters.”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 1 INTRODUCTION
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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In the paper
“(2) Predefined time nonlinear disturbance observers are designed for each loop to efficiently estimate the external disturbances of the system, and the upper bound of the convergence time of the disturbance observer error depends on one design parameter.”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 1 INTRODUCTION
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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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 3.1 Controller design of quadrotor position subsystem
“With the help of Lemma 1 and ( 9), one obtains ẽz = 0 for t ≥ T z .”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 3.1 Controller design of quadrotor position subsystem
“where 0 < 𝜂 zd < 1, T zd > 0 is the predefined time.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 𝜓.
“d𝜃 = 𝛽 𝜃 ê𝜃 + ̇e𝑦”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 𝜓.
“e 𝜙 = x 𝜙2q 𝜙 , e 𝜃 = x 𝜃2q 𝜃 , e 𝜓 = x 𝜓2q 𝜓”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 𝜓.
“U 3 = I 𝑦 l ( -p 𝜃2 𝜆 z2 + . 𝛼 𝜃 -𝜉 𝜃1”
§ Predefined time trajectory tracking control and hardware in loop verification for quadrotor UAV under input delay and disturbances > 𝜓.
“𝜆 𝜃2 = l I 𝑦 U 3 (t -𝜏) - l I 𝑦 U 3 (t) -p 𝜃2 𝜆 𝜃2”
Citations
Cites
Introduction to Multicopter Design and Control
Quan, Q. (2017) · Springer eBooks · doi:10.1007/978-981-10-3382-7 · Cited by 470
Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems
Krstić, M. (2009) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2009.2034923 · Cited by 399
Neural Networks-Based Adaptive Control for Nonlinear State Constrained Systems With Input Delay
Li, D.-P., Liu, Y.-J., Tong, S., Chen, C. L. P., & Li, D.-J. (2019) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2018.2799683 · Cited by 382 · cited 1×
“Whereafter, Pade approximation approach was developed in the literature [32–34] to remove the effect of input delay.”
Predefined-time stability of dynamical systems with sliding modes
Sánchez‐Torres, J. D., Sanchez, E. N., & Loukianov, A. G. (2015) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2015.7172255 · Cited by 346
Predefined-Time Predefined-Bounded Attitude Tracking Control for Rigid Spacecraft
Ye, D., Zou, A., & Sun, Z. (2021) · IEEE Transactions on Aerospace and Electronic Systems · doi:10.1109/taes.2021.3103258 · Cited by 221
Xie, S., Chen, Q., & Yang, Q. (2022) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2022.3223253 · Cited by 166
Active disturbance rejection control for uncertain time-delay nonlinear systems
Ran, M., Wang, Q., Dong, C., & Xie, L. (2020) · Automatica · doi:10.1016/j.automatica.2019.108692 · Cited by 136
Wang, X., Sun, S., Kampen, E.-J. V., & Chu, Q. P. (2019) · Aerospace Science and Technology · doi:10.1016/j.ast.2019.03.001 · Cited by 136
Adaptive Fuzzy Fixed-Time Control for High-Order Nonlinear Systems With Sensor and Actuator Faults
Wang, H., Ma, J., Zhao, X., Niu, B., Chen, M., & Wang, W. (2023) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2023.3235395 · Cited by 127
Planning and Tracking in Image Space for Image-Based Visual Servoing of a Quadrotor
Zheng, D., Wang, H., Chen, W., & Wang, Y. (2017) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2017.2752124 · Cited by 89
Research works printed but not linked: 3
- Liu H. T., 2023, Event‐triggered predefined‐time H∞$$ {H}_{\infty } $$ formation control for multiple underactuated surface vessels with error constraints and input quantization, Ocean Eng., 114294, 277
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- Zhang Z. S., 2021, Fuzzy sampled‐data H∞$$ {H}_{\infty } $$ sliding‐mode control for active hysteretic suspension of commercial vehicles with unknown actuator‐disturbance, Ocean Eng., 117, 104940
Cited by
Attitude Controller for UAV First-Order Acceleration Coefficients Tracking
Xu, D., & Pei, H. (2026) · Sensors · doi:10.3390/s26092727
Self-citationHigh-precision trajectory tracking control for quadrotor UAV with time -varying delay
Dou, J., Zhang, T., Wu, Y., & Xie, D. (2026) · Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering · doi:10.1177/09544100261451364
Predefined-Time Anti-Saturation Formation Control for Unmanned Aerial Vehicles
Han, Y., Ma, Z., & Ma, A. (2026) · Chinese Control and Decision Conference (CCDC) · doi:10.1109/ccdc69976.2026.11560682
Zhang, T. A., Chen, J., Peng, J., & Kou, W. (2026) · International Conference on Communication, Remote Sensing and Information Technology (CRSIT) · doi:10.1109/crsit69981.2026.11605387
Metadata
- Authors
- Li, S., Duan, N., & Pei, H.
- Affiliations
- Key Laboratory of Autonomous Systems and Network Control, Ministry of Education (CN)
- Jiangsu Normal University (CN)
- Corresponding author
- Pei, Hailong
- Venue
- Asian Journal of Control
- Publisher
- Wiley
- Published
- 2025-01-14
- Volume
- 27
- Issue
- 4
- Pages
- 1950-1963
- ISSN
- 1561-8625 (ISSN-L)
- 1934-6093
- OpenAlex
- W4406376734
- Semantic Scholar
- 9ecb1d32b9374370bf6788305270928396f93e20
- Cited by
- 4
- References
- 39
- Citations by year
- 2026: 4
- FWCI
- 1.07
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Asian Journal of Control (publishedVersion)
- Funders
- Aeronautical Science Foundation of China (20220056060001)
- National Key Research and Development Program of China (2023YFB4704900)
- Fundamental Research Funds for the Central Universities
- 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 › Engineering › Aerospace Engineering › Aerospace Engineering and Control Systems
- Keywords
- quadrotor, predefined-time control, trajectory tracking, input delay, external disturbances, backstepping control, nonlinear disturbance observer, auxiliary systems, hardware-in-the-loop testing
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