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Keywords = fixed-time sliding mode control

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17 pages, 13297 KiB  
Article
Speed Control of Permanent Magnet Synchronous Motor Based on Variable Fractional-Order Fuzzy Sliding Mode Controller
by Liping Chen, Haoyu Liu, Ze Cao, António M. Lopes, Lisheng Yin, Guoquan Liu and Yangquan Chen
Actuators 2025, 14(1), 38; https://doi.org/10.3390/act14010038 - 18 Jan 2025
Viewed by 383
Abstract
A variable fractional-order (VFO) fuzzy sliding mode controller is designed to control the speed of a permanent magnet synchronous motor (PMSM). First, a VFO sliding mode surface is established. Then, a VFO fuzzy sliding mode controller is designed, capable of suppressing the effects [...] Read more.
A variable fractional-order (VFO) fuzzy sliding mode controller is designed to control the speed of a permanent magnet synchronous motor (PMSM). First, a VFO sliding mode surface is established. Then, a VFO fuzzy sliding mode controller is designed, capable of suppressing the effects of parameter uncertainties and disturbances to achieve precise PMSM speed control. The global stability and finite time convergence of the controlled system state are demonstrated using Lyapunov stability theory. The numerical and experimental results validate the effectiveness of the controller, showing better immunity to disturbances and a smaller overshoot compared to PID and fixed-order fuzzy sliding mode controllers. Full article
(This article belongs to the Section Control Systems)
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<p>The membership functions of input/output.</p>
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<p>The system block diagram.</p>
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<p>Speed response of the PMSM under CFOFSMC for orders 0.8–1.2.</p>
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<p>Speed response of the PMSM with the three control methods.</p>
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<p>Torque under the PID method.</p>
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<p>Torque under the COFSMC method.</p>
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<p>Torque under the VFOFSMC method.</p>
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<p>Current under the PID method.</p>
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<p>Current under the COFSMC method.</p>
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<p>Current under the VFOFSMC method.</p>
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<p>Experimental platform.</p>
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<p>Phase plane using controller without fuzziness.</p>
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<p>Phase plane using controller with fuzziness.</p>
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<p>The experimental speed response of the PID.</p>
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<p>The experimental speed response under the COFSMC.</p>
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<p>The experimental speed response under the VFOFSMC.</p>
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<p>The experimental torque response under the PID.</p>
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<p>The experimental torque response under the COFSMC.</p>
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<p>The experimental torque response under the VFOFSMC.</p>
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25 pages, 3925 KiB  
Article
Finite-Time Path-Following Control of Underactuated AUVs with Actuator Limits Using Disturbance Observer-Based Backstepping Control
by MohammadReza Ebrahimpour and Mihai Lungu
Drones 2025, 9(1), 70; https://doi.org/10.3390/drones9010070 - 18 Jan 2025
Viewed by 326
Abstract
This paper presents a three-dimensional (3D) robust adaptive finite-time path-following controller for underactuated Autonomous Underwater Vehicles (AUVs), addressing model uncertainties, external disturbances, and actuator magnitude and rate saturations. A path-following error system is built in a path frame using the virtual guidance method. [...] Read more.
This paper presents a three-dimensional (3D) robust adaptive finite-time path-following controller for underactuated Autonomous Underwater Vehicles (AUVs), addressing model uncertainties, external disturbances, and actuator magnitude and rate saturations. A path-following error system is built in a path frame using the virtual guidance method. The proposed cascaded closed-loop control scheme can be described in two separate steps: (1) A kinematic law based on a finite-time backstepping control (FTBSC) is introduced to transform the 3D path-following position errors into the command velocities; (2) The actual control inputs are designed in the dynamic controller using an adaptive fixed-time disturbance observer (AFTDO)-based FTBSC to stabilize the velocity tracking errors. Moreover, the adverse effects of magnitude and rate saturations are reduced by an auxiliary compensation system. A Lyapunov-based stability analysis proves that the path-following errors converge to an arbitrarily small region around zero within a finite time. Comparative simulations illustrate the effectiveness and robustness of the proposed controller. Full article
(This article belongs to the Special Issue Advances in Autonomy of Underwater Vehicles (AUVs))
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<p>Diagram of the 3D path-following.</p>
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<p>The block diagram of the cascaded control architecture.</p>
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<p>3D straight path-following in Scenario 1.</p>
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<p>Position, attitude, and velocity tracking errors in Scenario 1.</p>
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<p>Control force and moments in Scenario 1.</p>
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<p>Disturbance estimation errors and adaptive parameter estimation for the proposed control in Scenario 1.</p>
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<p>Helix path-following in Scenario 2.</p>
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<p>Position, attitude, and velocity tracking errors in Scenario 2.</p>
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<p>Control force and moments in Scenario 2.</p>
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<p>Disturbance estimation errors in Scenario 2.</p>
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<p>Control inputs for tighter constraints on input rates in Scenario 3.</p>
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26 pages, 6106 KiB  
Article
Design of an Adaptive Fixed-Time Fast Terminal Sliding Mode Controller for Multi-Link Robots Actuated by Pneumatic Artificial Muscles
by Hesam Khajehsaeid, Ali Soltani and Vahid Azimirad
Biomimetics 2025, 10(1), 37; https://doi.org/10.3390/biomimetics10010037 - 8 Jan 2025
Viewed by 513
Abstract
Pneumatic artificial muscles (PAMs) are flexible actuators that can be contracted or expanded by applying air pressure. They are used in robotics, prosthetics, and other applications requiring flexible and compliant actuation. PAMs are basically designed to mimic the function of biological muscles, providing [...] Read more.
Pneumatic artificial muscles (PAMs) are flexible actuators that can be contracted or expanded by applying air pressure. They are used in robotics, prosthetics, and other applications requiring flexible and compliant actuation. PAMs are basically designed to mimic the function of biological muscles, providing a high force-to-weight ratio and smooth, lifelike movement. Inflation and deflation of these muscles can be controlled rapidly, allowing for fast actuation. In this work, a continuum mechanics-based model is developed to predict the output parameters of PAMs, like actuation force. Comparison of the model results with experimental data shows that the model efficiently predicts the mechanical behaviour of PAMs. Using the actuation force–air pressure–contraction relation provided by the proposed mechanical model, a dynamic model is derived for a multi-link PAM-actuated robot manipulator. An adaptive fixed-time fast terminal sliding mode control is proposed to track the desired joint position trajectories despite the model uncertainties and external disturbances with unknown magnitude bounds. Furthermore, the performance of the proposed controller is compared with an adaptive backstepping fast terminal sliding mode controller through numerical simulations. The simulations show faster convergence and more precise tracking for the proposed controller. Full article
(This article belongs to the Special Issue Bioinspired Structures for Soft Actuators: 2nd Edition)
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<p>(<b>Left</b>) The designed PAM and (<b>Right</b>) the rest and deformed states.</p>
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<p>Schematic representation of a PAM and a single thread.</p>
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<p>Free-body diagram of a pneumatic artificial muscle.</p>
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<p>Antagonistic joint actuation by a pair of PAMs.</p>
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<p>The Neo-Hookean model in comparison with the uniaxial tensile test data.</p>
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<p>Pressure–contraction curve, comparison of the test data with the model results.</p>
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<p>Actuation force–contraction curve at different air pressures.</p>
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<p>A two-link robot manipulator actuated by PAMs.</p>
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<p>Joint angles of the manipulator.</p>
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<p>Tracking errors of the joints.</p>
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<p>Time responses of the adaptive gains.</p>
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<p>Time responses of the adaptive gains.</p>
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<p>Time response of <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi>d</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Pressure variations in the muscles.</p>
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<p>Actuation forces of the muscles.</p>
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<p>Actuation forces of the muscles.</p>
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<p>Pressure variation in the muscles of the first joint with <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>200</mn> <mo> </mo> <mi>kPa</mi> </mrow> </semantics></math>.</p>
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<p>Actuation forces of the muscles with <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>400</mn> <mo> </mo> <mi>kPa</mi> </mrow> </semantics></math>.</p>
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23 pages, 7214 KiB  
Article
Sliding Mode Control for Variable-Speed Trajectory Tracking of Underactuated Vessels with TD3 Algorithm Optimization
by Shiya Zhu, Gang Zhang, Qin Wang and Zhengyu Li
J. Mar. Sci. Eng. 2025, 13(1), 99; https://doi.org/10.3390/jmse13010099 - 7 Jan 2025
Viewed by 481
Abstract
An adaptive sliding mode controller (SMC) design with a reinforcement-learning parameter optimization method is proposed for variable-speed trajectory tracking control of underactuated vessels under scenarios involving model uncertainties and external environmental disturbances. First, considering the flexible control requirements of the vessel’s propulsion system, [...] Read more.
An adaptive sliding mode controller (SMC) design with a reinforcement-learning parameter optimization method is proposed for variable-speed trajectory tracking control of underactuated vessels under scenarios involving model uncertainties and external environmental disturbances. First, considering the flexible control requirements of the vessel’s propulsion system, the desired navigation speed is designed to satisfy an S-curve acceleration and deceleration process. The rate of change of the trajectory parameters is derived. Second, to address the model uncertainties and external disturbances, an extended state observer (ESO) is designed to estimate the unknown bounded disturbances and to provide feedforward compensation. Moreover, an adaptive law is designed to estimate the upper bound of the unknown disturbances, ensuring system stability even in the presence of asymptotic observation errors. Finally, the Twin-Delayed Deep Deterministic Policy Gradient (TD3) algorithm is employed for real-time controller parameter tuning. Numerical simulation results demonstrate that the proposed method significantly improves the trajectory tracking accuracy and dynamic response speed of the underactuated vessel. Specifically, for a sinusoidal trajectory with an amplitude of 200 m and a frequency of 0.01, numerical results show that the proposed method achieves convergence of the longitudinal tracking error to zero, while the lateral tracking error remains stable within 1 m. For the circular trajectory with a radius of 300 m, the numerical results indicate that both the longitudinal and lateral tracking errors are stabilized within 1 m. Compared with the fixed-value sliding mode controller, the proposed method demonstrates superior trajectory tracking accuracy and smoother control performance. Full article
(This article belongs to the Section Ocean Engineering)
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<p>The inertial coordinate system and the ship body coordinate system.</p>
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<p>Illustration of the LADLOS guidance law.</p>
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<p>S-curve acceleration and deceleration.</p>
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<p>TD3 parameter optimization structure.</p>
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<p>Control system structure.</p>
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<p>Maneuverability validation. (<b>a</b>) Forward speed under different thrusts. (<b>b</b>) Trajectory under different torques.</p>
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<p>Effect of the drift angle on the reference angle.</p>
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<p>Control parameters and trajectory comparison of the sine trajectory. (<b>a</b>) Control parameters. (<b>b</b>) Trajectory comparison.</p>
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<p>Longitudinal tracking error and lateral tracking error of the sine trajectory.</p>
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<p>Speed tracking error and yaw tracking error of the sine trajectory.</p>
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<p>Variable speed tracking of the sine trajectory.</p>
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<p>Control force of the sine trajectory.</p>
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<p>Control parameters and trajectory comparison of the circular trajectory. (<b>a</b>) Control parameters. (<b>b</b>) Trajectory comparison.</p>
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<p>Longitudinal tracking error and lateral tracking error of the circular trajectory.</p>
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<p>Speed tracking error and yaw tracking error of the circular trajectory.</p>
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<p>Variable speed tracking of the circular trajectory.</p>
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<p>Control force of the circular trajectory.</p>
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25 pages, 7487 KiB  
Article
A Novel Time Delay Nonsingular Fast Terminal Sliding Mode Control for Robot Manipulators with Input Saturation
by Thanh Nguyen Truong, Anh Tuan Vo and Hee-Jun Kang
Mathematics 2025, 13(1), 119; https://doi.org/10.3390/math13010119 - 31 Dec 2024
Viewed by 608
Abstract
Manipulator systems are increasingly deployed across various industries to perform complex, repetitive, and hazardous tasks, necessitating high-precision control for optimal performance. However, the design of effective control algorithms is challenged by nonlinearities, uncertain dynamics, disturbances, and varying real-world conditions. To address these issues, [...] Read more.
Manipulator systems are increasingly deployed across various industries to perform complex, repetitive, and hazardous tasks, necessitating high-precision control for optimal performance. However, the design of effective control algorithms is challenged by nonlinearities, uncertain dynamics, disturbances, and varying real-world conditions. To address these issues, this paper proposes an advanced orbit-tracking control approach for manipulators, leveraging advancements in Time-Delay Estimation (TDE) and Fixed-Time Sliding Mode Control techniques. The TDE approximates the robot’s unknown dynamics and uncertainties, while a novel nonsingular fast terminal sliding mode (NFTSM) surface and novel fixed-time reaching control law (FTRCL) are introduced to ensure faster convergence within a fixed time and improved accuracy without a singularity issue. Additionally, an innovative auxiliary system is designed to address input saturation effects, ensuring that system states converge to zero within a fixed time even when saturation occurs. The Lyapunov-based theory is employed to prove the fixed-time convergence of the overall system. The effectiveness of the proposed controller is validated through simulations on a 3-DOF SAMSUNG FARA AT2 robot manipulator. Comparative analyses against NTSMC, NFTSMC, and GNTSMC methods demonstrate superior performance, characterized by faster convergence, reduced chattering, higher tracking accuracy, and a model-free design. These results underscore the potential of the proposed control strategy to significantly enhance the robustness, precision, and applicability of robotic systems in industrial environments. Full article
(This article belongs to the Special Issue Advancements in Nonlinear Control Strategies)
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<p>Comparison of convergence behavior across fixed-time control methods under different initial conditions.</p>
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<p>Proposed sliding surface.</p>
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<p>Structure of the proposed control system.</p>
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<p>Three-dimensional SOLIDWORKS model of SAMSUNG FARA AT2 manipulator.</p>
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<p>Time evolution of the auxiliary system variables.</p>
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<p>Trajectory tracking performance of the robot end-effector across four control methods.</p>
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<p>Joint-level trajectory tracking performance across four control methods.</p>
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<p>Tracking error comparison under four different control methods.</p>
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<p>RMSEs across four control methods.</p>
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<p>Control input comparison under four different control methods.</p>
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32 pages, 12104 KiB  
Article
RBFNN-Based Adaptive Fixed-Time Sliding Mode Tracking Control for Coaxial Hybrid Aerial–Underwater Vehicles Under Multivariant Ocean Disturbances
by Mingqing Lu, Wei Yang, Zhenyu Xiong, Fei Liao, Shichong Wu, Yumin Su and Wenhua Wu
Drones 2024, 8(12), 745; https://doi.org/10.3390/drones8120745 - 10 Dec 2024
Viewed by 613
Abstract
In this study, the design of an adaptive neural network-based fixed-time control system for a novel coaxial trans-domain hybrid aerial–underwater vehicle (HAUV) is investigated. A radial basis function neural network (RBFNN) approximation strategy-based adaptive fixed-time terminal sliding mode control (AFTSMC) scheme is proposed [...] Read more.
In this study, the design of an adaptive neural network-based fixed-time control system for a novel coaxial trans-domain hybrid aerial–underwater vehicle (HAUV) is investigated. A radial basis function neural network (RBFNN) approximation strategy-based adaptive fixed-time terminal sliding mode control (AFTSMC) scheme is proposed to solve the problems of the dynamic nonlinearity, model parameter perturbation, and multiple external disturbances of coaxial HAUV trans-media motion. A complete six-degrees-of-freedom model for a continuous water–air cross-domain model is first established based on the hyperbolic tangent transition function, and, subsequently, based on a basic framework of FTSMC, a fixed-time and fast-convergence controller is designed to track the target position and attitude signals. To reduce the dependence of the control scheme on precise model parameters, an RBFNN approximator is integrated into the sliding mode controller for the online model identification of the aggregate uncertainties of the coaxial HAUV, such as nonlinear unmodeled dynamics and external disturbances. At the same time, an adaptive technique is used to approximate the upper bound of the robust switching term gain in the controller, which further offsets the estimation error of the RBFNN and effectively attenuates the chattering effect. Based on Lyapunov stability theory, it is proven that the tracking error can converge in a fixed time. The effectiveness and superiority of the proposed control strategy are verified by several sets of simulation results obtained under typical working conditions. Full article
(This article belongs to the Special Issue Advances in Autonomous Underwater Drones)
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<p>Structure diagram of coaxial HAUV.</p>
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<p>Flowchart of deployment and operation of coaxial HAUV.</p>
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<p>Main outline drawing and coordinate system of coaxial HAUV.</p>
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<p>Three types of switching coefficients.</p>
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<p>RBFNN structure diagram.</p>
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<p>Sensor measurement noises of the coaxial HAUV.</p>
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<p>Tracking results of the coaxial HAUV in x, y, and z channels for Case 1.</p>
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<p>The 3D view position tracking results in Case 1.</p>
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<p>Attitude angle tracking results in Case 1.</p>
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<p>Attitude angle tracking error evolutionary results in Case 1.</p>
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<p>Attitude angle tracking error evolutionary results in Case 1.</p>
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<p>Attitude angle tracking results in Case 1.</p>
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<p>Identification of evolution of nonlinear functions of RBFNN in position and attitude loops in Case 1.</p>
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<p>Evolution of the designed controller robust switching terms in Case 1.</p>
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<p>RBFNN norm evolution results of weight coefficient estimates for each DOF.</p>
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<p>Variation in the control signal in Case 1.</p>
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<p>Variation in the control signal in Case 1.</p>
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<p>Tracking results of position channels with different controllers.</p>
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<p>Tracking results for the 2D plane with different controllers.</p>
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<p>The 3-D trajectory tracking results with different controllers.</p>
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<p>Attitude angle tracking results with different controllers.</p>
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<p>Evolution of attitude angle tracking error with different controllers.</p>
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<p>Evolution of position tracking error with different controllers.</p>
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<p>Adaptive parameter evolution results of NNASMC and traditional ASMC.</p>
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<p>Results of the evolution of the control inputs for the different controllers.</p>
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20 pages, 1947 KiB  
Article
Pressure Control of Multi-Mode Variable Structure Electro–Hydraulic Load Simulation System
by He Hao, Hao Yan, Qi Zhang and Haoyu Li
Sensors 2024, 24(22), 7400; https://doi.org/10.3390/s24227400 - 20 Nov 2024
Viewed by 764
Abstract
During the loading process, significant external position disturbances occur in the electro–hydraulic load simulation system. To address these position disturbances and effectively mitigate the impact of uncertainty on system performance, this paper first treats model parameter uncertainty and external disturbances as lumped disturbances. [...] Read more.
During the loading process, significant external position disturbances occur in the electro–hydraulic load simulation system. To address these position disturbances and effectively mitigate the impact of uncertainty on system performance, this paper first treats model parameter uncertainty and external disturbances as lumped disturbances. The various states of the servo valve and the pressures within the hydraulic cylinder chambers are then examined. Building on this foundation, the paper proposes a nonlinear multi-mode variable structure independent load port electro–hydraulic load simulation system that is tailored for specific loading conditions. Secondly, in light of the significant motion disturbances present, this paper proposes an integral sliding mode active disturbance rejection composite control strategy that is based on fixed-time convergence. Based on the structure of the active disturbance rejection control framework, the fixed-time integral sliding mode and active disturbance rejection control algorithms are integrated. An extended state observer is designed to accurately estimate the lumped disturbance, effectively compensating for it to achieve precise loading of the independent load port electro–hydraulic load simulation system. The stability of the designed controller is also demonstrated. The results of the simulation research indicate that when the command input is a step signal, the pressure control accuracy under the composite control strategy is 99.94%, 99.86%, and 99.76% for disturbance frequencies of 1 Hz, 3 Hz, and 5 Hz, respectively. Conversely, when the command input is a sinusoidal signal, the pressure control accuracy remains high, measuring 99.94%, 99.8%, and 99.6% under the same disturbance frequencies. Furthermore, the simulation demonstrates that the influence of sensor random noise on the system remains within acceptable limits, highlighting the effective filtering capabilities of the extended state observer. This research establishes a solid foundation for the collaborative control of load ports and the engineering application of the independent load port electro–hydraulic load simulation system. Full article
(This article belongs to the Section Electronic Sensors)
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<p>Schematic view of independent load port electro–hydraulic load simulation system model (1—tank, 2—hydraulic pump, 3—relief valve, 4—electro–hydraulic servo valve, 5—independent load port electro–hydraulic load simulation system controller, 6—pressure sensor, 7—hydraulic cylinder, 8—displacement sensor, 9—linear actuator, 10—linear actuator controller).</p>
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<p>Block diagram of fixed-time integral sliding mode active disturbance rejection composite control.</p>
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<p>Closed-loop pressure with a disturbance frequency of 1 Hz.</p>
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<p>Closed-loop pressure error with a disturbance frequency of 1 Hz.</p>
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<p>Controller output with a disturbance frequency of 1 Hz.</p>
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<p>Closed-loop pressure with a disturbance frequency of 3 Hz.</p>
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<p>Closed-loop pressure error with a disturbance frequency of 3 Hz.</p>
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<p>Controller output with a disturbance frequency of 3 Hz.</p>
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<p>Closed-loop pressure with a disturbance frequency of 5 Hz.</p>
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<p>Closed-loop pressure error with a disturbance frequency of 5 Hz.</p>
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<p>Controller output with a disturbance frequency of 5Hz.</p>
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<p>Closed-loop pressure with a disturbance frequency of 1 Hz.</p>
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<p>Closed-loop pressure error with a disturbance frequency of 1 Hz.</p>
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<p>Controller output with a disturbance frequency of 1 Hz.</p>
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<p>Closed-loop pressure with a disturbance frequency of 3 Hz.</p>
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<p>Closed-loop pressure error with disturbance frequency of 3 Hz.</p>
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<p>Controller output with a disturbance frequency of 3 Hz.</p>
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<p>Closed-loop pressure with a disturbance frequency of 5 Hz.</p>
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<p>Closed-loop pressure error with a disturbance frequency of 5 Hz.</p>
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<p>Controller output with a disturbance frequency of 5 Hz.</p>
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<p>Pressure curve under random noise of 50 mV.</p>
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<p>Pressure curve under random noise of 100 mV.</p>
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26 pages, 16654 KiB  
Article
Adaptive Fast Smooth Second-Order Sliding Mode Fault-Tolerant Control for Hypersonic Vehicles
by Lijia Cao, Lei Liu, Pengfei Ji and Chuandong Guo
Aerospace 2024, 11(11), 951; https://doi.org/10.3390/aerospace11110951 - 18 Nov 2024
Viewed by 521
Abstract
In response to control issues in hypersonic vehicles under external disturbances, model uncertainties, and actuator failures, this paper proposes an adaptive fast smooth second-order sliding mode fault-tolerant control scheme. First, a system separation approach is adopted, dividing the hypersonic vehicle model into fast [...] Read more.
In response to control issues in hypersonic vehicles under external disturbances, model uncertainties, and actuator failures, this paper proposes an adaptive fast smooth second-order sliding mode fault-tolerant control scheme. First, a system separation approach is adopted, dividing the hypersonic vehicle model into fast and slow loops for independent design. This ensures that the airflow angle tracking error and sliding mode variables converge to the vicinity of the origin within a finite time. A fixed-time disturbance observer is then designed to estimate and compensate for the effects of model uncertainties, external disturbances, and actuator failures. The controller parameters are dynamically adjusted through an adaptive term to enhance robustness. Furthermore, first-order differentiation is used to estimate differential terms, effectively avoiding the issue of complexity explosion. Finally, the convergence of the controller within a finite time is rigorously proven using the Lyapunov method, and the perturbation of aerodynamic parameters is tested using the Monte Carlo method. Simulation results under various scenarios show that compared with the terminal sliding mode method, the proposed method outperforms control accuracy and convergence speed. The root mean square errors for the angle of attack, sideslip angle, and roll angle are reduced by 65.11%, 86.71%, and 45.51%, respectively, while the standard deviation is reduced by 81.78%, 86.80%, and 45.51%, demonstrating that the proposed controller has faster convergence, higher control accuracy, and smoother output than the terminal sliding mode controller. Full article
(This article belongs to the Section Aeronautics)
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<p>Geometric parameters of the HSV model.</p>
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<p>The structure diagram of the control system.</p>
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<p>Angle of bank.</p>
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<p>Angle of attack.</p>
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<p>Sideslip angle and error.</p>
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<p>Error of bank angle.</p>
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<p>Error of attack.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>a</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>e</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Angle of bank.</p>
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<p>Angle of attack.</p>
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<p>Sideslip angle and error.</p>
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<p>Error of bank angle.</p>
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<p>Error of attack.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>a</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>e</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Aerodynamic uncertainty scatter plot.</p>
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<p>Bank angle of TSMFTC.</p>
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<p>Attack angle of TSMFTC.</p>
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<p>Sideslip angle of TSMFTC.</p>
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<p>Bank angle of AFSSOSMFTC.</p>
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<p>Attack angle of AFSSOSMFTC.</p>
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<p>Sideslip angle of AFSSOSMFTC.</p>
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<p>Angle of bank.</p>
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<p>Angle of attack.</p>
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<p>Sideslip angle and error.</p>
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<p>Error of bank angle.</p>
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<p>Error of attack.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>a</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>e</mi> </msub> </mrow> </semantics></math>.</p>
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<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> </mrow> </semantics></math>.</p>
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16 pages, 3155 KiB  
Article
Fixed-Time Consensus Multi-Agent-Systems-Based Speed Cooperative Control for Multiple Permanent Magnet Synchronous Motors with Complementary Sliding Mode Control
by Limin Hou and Xiaoru Lan
Electronics 2024, 13(22), 4407; https://doi.org/10.3390/electronics13224407 - 11 Nov 2024
Viewed by 796
Abstract
To improve the tracking performance and robustness of traditional multi-motor speed cooperative control, this paper proposes a speed cooperative control method for multiple permanent magnet synchronous motors (multi-PMSMs) based on the fixed-time consensus protocol for multi-agent systems (MASs) combined with CSMC. Firstly, the [...] Read more.
To improve the tracking performance and robustness of traditional multi-motor speed cooperative control, this paper proposes a speed cooperative control method for multiple permanent magnet synchronous motors (multi-PMSMs) based on the fixed-time consensus protocol for multi-agent systems (MASs) combined with CSMC. Firstly, the speed regulation system of multi-PMSMs is regarded as a MAS. By designing a distributed consensus protocol based on an undirected communication topology, the system achieves fixed-time consensus convergence. Then, a terminal integral sliding mode observer (TISMO) is designed to estimate disturbances, and feedforward compensation is introduced into the consensus protocol to obtain the desired q-axis current. Furthermore, within the framework of the vector control speed cooperative system of PMSMs, a CSMC is designed to track the q-axis reference current. Meanwhile, the stability of the above controllers and observers is theoretically proven using the Lyapunov functions. Finally, comparative experiments are conducted on a multi-PMSM speed regulation experimental platform to verify the proposed control method against the traditional deviation coupling control (DCC) method. The results indicate that under the new control method proposed in this paper, the chattering phenomenon is reduced by about 2 r/min compared to the traditional DCC method. During sudden load and sudden relief load conditions, the speed fluctuation is reduced by approximately 4%, demonstrating good tracking performance and strong robustness. Full article
(This article belongs to the Section Systems & Control Engineering)
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<p>Schematic diagram of the traditional DCC method for multi-PMSMs.</p>
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<p>Schematic diagram of multi-PMSM control method.</p>
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<p>CSMC schematic diagram.</p>
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<p>Multi-motor speed control and load integration experimental platform.</p>
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<p>Comparative experimental results of speed-up and speed-down as well as forward and reverse operation. (<b>a</b>) Comparative curve of speed response. (<b>b</b>) Comparative curve of speed tracking error. (<b>c</b>) Comparative curve of speed synchronization error.</p>
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<p>Comparative experimental results of load addition and reduction. (<b>a</b>) Comparative curve of speed response. (<b>b</b>) Comparative curve of speed tracking error. (<b>c</b>) Comparative curve of speed synchronization error.</p>
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<p>Comparative experimental results of low-speed operation. (<b>a</b>) Comparative curve of speed response. (<b>b</b>) Comparative curve of speed tracking error. (<b>c</b>) Comparative curve of speed synchronization error.</p>
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16 pages, 17925 KiB  
Article
Linear Disturbance Observer-Enhanced Continuous-Time Predictive Control for Straight-Line Path-Following Control of Small Unmanned Aerial Vehicles
by Weiwei Qi, Mingbo Tong, Xubo Li, Qi Wang and Wei Song
Aerospace 2024, 11(11), 902; https://doi.org/10.3390/aerospace11110902 - 2 Nov 2024
Viewed by 775
Abstract
This paper studies the straight-line path-following problem on the lateral plane for fixed-wing unmanned aerial vehicles (FWUAVs) which are susceptible to uncertainties. Firstly, based on the natural frame’s location on the prescribed reference paths, the command yaw angle (which is the basis for [...] Read more.
This paper studies the straight-line path-following problem on the lateral plane for fixed-wing unmanned aerial vehicles (FWUAVs) which are susceptible to uncertainties. Firstly, based on the natural frame’s location on the prescribed reference paths, the command yaw angle (which is the basis for yaw angle control system design) is solved analytically by combining it with the errors of path following, attack angle, sideslip angle, attitude angles, and geometric parameters of the prescribed reference paths. Secondly, by considering complicated dynamic characteristics, a linear extended state observer is designed to estimate uncertainties such as nonlinearities, couplings, and unmodeled dynamics whose estimated values are incorporated into the continuous-time predictive controllers for feedback compensation. Finally, numerical simulations are conducted to demonstrate the advantages of the proposed method, including reduced tracking errors and enhanced robustness in the closed-loop system, as compared to the conventional nonlinear dynamic inversion and sliding mode control approaches. Full article
(This article belongs to the Section Aeronautics)
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<p>Straight-line path following.</p>
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<p>Schematic of the control scheme.</p>
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<p>Path-following results.</p>
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<p>Path-following errors.</p>
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<p>Uncertainty estimation results.</p>
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<p>Yaw angles: proposed approach.</p>
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<p>Yaw angles: the NDI approach.</p>
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<p>Yaw angles: the SMC approach.</p>
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<p>Yaw rates.</p>
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<p>Control laws.</p>
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<p>Path-following results.</p>
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<p>Path-following errors.</p>
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<p>LESO estimation results: +30% perturbation.</p>
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<p>LESO estimation results: −30% perturbation.</p>
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<p>Yaw angles: +30%.</p>
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<p>Yaw angles: −30%.</p>
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<p>Yaw rates.</p>
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<p>Control laws.</p>
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20 pages, 3324 KiB  
Article
A Disturbance Observer-Based Fractional-Order Fixed-Time Sliding Mode Control Approach for Elevators
by Zhe Sun, Huaqing Liu, Ke Li, Wanbin Su, Yefeng Jiang and Bo Chen
Actuators 2024, 13(11), 438; https://doi.org/10.3390/act13110438 - 1 Nov 2024
Cited by 1 | Viewed by 743
Abstract
For elevators, appropriate speed control is pivotal for ensuring the safety and comfort of passengers and optimizing energy efficiency, system stability, and service life. Therefore, the design and implementation of effective speed control strategies are crucial for the operation and management of modern [...] Read more.
For elevators, appropriate speed control is pivotal for ensuring the safety and comfort of passengers and optimizing energy efficiency, system stability, and service life. Therefore, the design and implementation of effective speed control strategies are crucial for the operation and management of modern elevator systems. In response to this issue, this paper establishes a dynamic model of an elevator through mechanism analysis. Then, a novel fractional-order sliding mode control strategy with the assistance of a fixed-time adaptive sliding mode observer is proposed. The designed observer can effectively monitor and counteract external perturbations, thereby enhancing the stability and precision of the control system. The fractional-order sliding mode controller can realize a fixed-time convergence property, which is rigorously proven in the sense of Lyapunov. Finally, the effectiveness and superiority of the control scheme are validated by simulations compared with benchmark controllers. Full article
(This article belongs to the Section Control Systems)
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<p>Schematic of elevator structure.</p>
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<p>Velocity and acceleration references of elevator system. (<b>a</b>) Velocity profiles; (<b>b</b>) acceleration profiles.</p>
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<p>Simulation results in Case 1. (<b>a</b>) Tracking profiles; (<b>b</b>) tracking errors; (<b>c</b>) control inputs.</p>
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<p>Comparison of MAE and RMSE in Case 1. (<b>a</b>) MAE; (<b>b</b>) RMSE.</p>
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<p>Simulation results in Case 2. (<b>a</b>) Tracking profiles; (<b>b</b>) tracking errors; (<b>c</b>) control inputs.</p>
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<p>Comparison of MAE and RMSE in Case 2. (<b>a</b>) MAE; (<b>b</b>) RMSE.</p>
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<p>Simulation results in Case 3. (<b>a</b>) Tracking profiles; (<b>b</b>) tracking errors; (<b>c</b>) control inputs.</p>
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<p>Comparison of MAE and RMSE in Case 3. (<b>a</b>) MAE; (<b>b</b>) RMSE.</p>
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<p>Simulation results in Case 4. (<b>a</b>) Tracking profiles; (<b>b</b>) tracking errors; (<b>c</b>) control inputs.</p>
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<p>Comparison of MAE and RMSE in Case 4. (<b>a</b>) MAE; (<b>b</b>) RMSE.</p>
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20 pages, 9244 KiB  
Article
Fixed-Time Backstepping Sliding-Mode Control for Interleaved Boost Converter in DC Microgrids
by Hang Wang, Yanfei Dong, Guofeng He and Wenbin Song
Energies 2024, 17(21), 5377; https://doi.org/10.3390/en17215377 - 29 Oct 2024
Viewed by 767
Abstract
Interleaved boost converters (IBCs) are commonly used as interface converters for DC microgrids (MGs) due to their high efficiency and low output ripple. However, the MGs system can easily become unstable due to the negative impedance characteristics of constant power load (CPL) and [...] Read more.
Interleaved boost converters (IBCs) are commonly used as interface converters for DC microgrids (MGs) due to their high efficiency and low output ripple. However, the MGs system can easily become unstable due to the negative impedance characteristics of constant power load (CPL) and rapid power fluctuations. This paper proposes a fixed-time backstepping sliding-mode controller (FTBSMC) aimed at stabilizing the MGs system and achieving fixed-time tracking of the DC bus voltage. Firstly, the fixed-time disturbance observer (FxTDO) estimates the load disturbance at a fixed time, which improves the fast disturbance resistance of the system. Then, based on the dis-turbance estimation, the FTBSMC is designed, which combines the fast dynamic response of the sliding-mode control with the global stability of the backstepping control, avoiding the singularity problem of the conventional sliding-mode control. In addition, a first-order nonlinear filter is employed to avoid the direct differentiation of conventional backstepping control and at the same time to ensure global fixed-time stability. The fixed-time convergence of the proposed FTBSMC is rigorously demonstrated by using Lyapunov stability analysis. Finally, the FTBSMC proposed is verified by simulation and experiment in terms of faster dynamic response and stronger robustness. Full article
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<p>DC microgrid system using IBC.</p>
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<p>Simplified three-phase IBC-interfaced MGs.</p>
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<p>Current-sharing compensator.</p>
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<p>The general control block diagram of FTBSMC.</p>
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<p>Dynamic response of the FxTDO under different parameter variations. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> variation; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> variation; (<b>c</b>) <math display="inline"><semantics> <mi>m</mi> </semantics></math> and <math display="inline"><semantics> <mi>n</mi> </semantics></math> variation.</p>
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<p>Dynamic response of the DC bus voltage under different parameters variation. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math> variation; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math> variation.</p>
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<p>Parameter selection flowchart.</p>
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<p>FxTDO dynamic response when CPL changes.</p>
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<p>The DC bus voltage response under FTBSMC control with FxTDO and without FxTDO.</p>
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<p>Output current response: (<b>a</b>) current-sharing compensator being disconnected at 0.03 s; (<b>b</b>) current-sharing compensator working normally.</p>
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<p>The waveform of the actual control law and the compensation duty cycle. (<b>a</b>) CSC working normally and CPL power takes a 5 kW step change at 0.04 s. (<b>b</b>) Local enlargement of (<b>a</b>).</p>
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<p>Dynamic response of IBC system under different disturbances. (<b>a</b>) CPL power variation disturbance. (<b>b</b>) DC bus voltage reference value change disturbance. (<b>c</b>) Input voltage variation disturbance.</p>
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<p>DC bus voltage of IBC under CPL variation.</p>
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<p>DC bus voltage of IBC under DC bus reference voltage variation.</p>
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<p>DC bus voltage of IBC under input voltage variation.</p>
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<p>Experimental platform.</p>
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<p>Experimental waveforms when CPL changes: (<b>a</b>) CPL power change from 20 W to 80 W; (<b>b</b>) CPL power change from 80 W to 20 W.</p>
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<p>Experimental waveforms when the input voltage changes: (<b>a</b>) input voltage change from 15 V to 20 V; (<b>b</b>) input voltage change from 20 V to 15 V.</p>
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<p>Experimental waveforms when the output voltage reference is changed.</p>
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18 pages, 3597 KiB  
Article
Safety-Critical Fixed-Time Formation Control of Quadrotor UAVs with Disturbance Based on Robust Control Barrier Functions
by Zilong Song and Haocai Huang
Drones 2024, 8(11), 618; https://doi.org/10.3390/drones8110618 - 28 Oct 2024
Viewed by 1026
Abstract
This paper focuses on the safety-critical fixed-time formation control of quadrotor UAVs with disturbance and obstacle collision risk. The control scheme is organized in a distributed manner, with the leader’s position and velocity being estimated simultaneously by a fixed-time distributed observer. Meanwhile, a [...] Read more.
This paper focuses on the safety-critical fixed-time formation control of quadrotor UAVs with disturbance and obstacle collision risk. The control scheme is organized in a distributed manner, with the leader’s position and velocity being estimated simultaneously by a fixed-time distributed observer. Meanwhile, a disturbance observer that combines fixed-time control theory and sliding mode control is designed to estimate the external disturbance. Based on these techniques, we design a nominal control law to drive UAVs to track the desired formation in a fixed time. Regarding obstacle avoidance, we first construct safety constraints using control barrier functions (CBFs). Then, obstacle avoidance can be achieved by solving an optimization problem with these safety constraints, thus minimally affecting tracking performance. The main contributions of this process are twofold. First, an exponential CBF is provided to deal with the UAV model with a high relative degree. Moreover, a robust exponential CBF is designed for UAVs with disturbance, which provides robust safety constraints to ensure obstacle avoidance despite disturbance. Finally, simulation results show the validity of the proposed method. Full article
(This article belongs to the Special Issue Flight Control and Collision Avoidance of UAVs)
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<p>Block diagram of the control law.</p>
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<p>The moving trajectories of UAVs.</p>
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<p>Snapshots during obstacle avoidance. (<b>a</b>) Obstacles 1 and 2; (<b>b</b>) obstacles 3 and 4; (<b>c</b>) obstacle 5; (<b>d</b>) obstacle 6.</p>
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<p>The tracking error.</p>
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<p>The value of <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="bold-italic">x</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="bold-italic">x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow> <mi>o</mi> <mi>b</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </msup> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow> <mi>o</mi> <mi>b</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>d</mi> <mrow> <mi>o</mi> <mi>b</mi> <mo>,</mo> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>, where subscript <span class="html-italic">i</span> denotes the <span class="html-italic">i</span>-th UAV and <span class="html-italic">j</span> denotes the <span class="html-italic">j</span>-th obstacle.</p>
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<p>The control input.</p>
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26 pages, 7622 KiB  
Article
Design and Implementation of Small Modular Amphibious Robot System
by Fushen Ren and Zhongyang Wang
Processes 2024, 12(11), 2355; https://doi.org/10.3390/pr12112355 - 27 Oct 2024
Viewed by 925
Abstract
Various marine engineering facilities have been eroded by marine organisms and wind waves for a long time, resulting in different types of damage to the surface of marine engineering facilities, such as the pile legs of offshore platforms. Therefore, in order to carry [...] Read more.
Various marine engineering facilities have been eroded by marine organisms and wind waves for a long time, resulting in different types of damage to the surface of marine engineering facilities, such as the pile legs of offshore platforms. Therefore, in order to carry out safety inspections and other work on marine engineering facilities, a small amphibious robot structure system and a set of control systems adapted to it are independently developed. Various problems such as the modular design of the structure, composite motion mode, adsorption stability, wall adaptability of the crawling mode, and flaw localization have been solved by means of three-dimensional modeling, mechanical analysis, simulation, and electronic design. At the same time, a set of control systems including hardware and software is developed for the amphibious robot. In order to improve the stability and efficiency of the amphibious robot working underwater, a sliding mode control algorithm based on the exponential reaching law and saturation function is designed. For the fixed depth and fixed heading control functions, the sliding mode control algorithm and the PID control algorithm are simulated and compared. Finally, several types of experiments are carried out for the amphibious robot. The simulation and experimental results show that all the functions of the amphibious robot meet work requirements, such as the motion performance of the composite motion mode. Compared with the PID control algorithm, the sliding mode control algorithm has a faster response speed and better stability, which is conducive to the efficient and stable work of the amphibious robot underwater. Full article
(This article belongs to the Section Manufacturing Processes and Systems)
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<p>Ocean platform.</p>
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<p>The amphibious robot system.</p>
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<p>Three-dimensional model of the floating robot system. 1—the bottom half of the buoyancy module; 2—the top half of the buoyancy module (transparent state); 3—horizontal thruster; 4—vertical thruster; 5—control cabin; 6—underwater floodlight; 7—screw rod; 8—underwater camera.</p>
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<p>Three-dimensional model of the control cabin. 1—transparent protective window; 2—pan–tilt camera; 3—connecting circular plate; 4—cylindrical cabin; 5—connecting flange; 6—sealing hollow bolt.</p>
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<p>Simulation result diagram of the control cabin. (<b>a</b>) Stress result diagram; (<b>b</b>) deformation result diagram.</p>
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<p>Three-dimensional model of the crawler chassis. 1—horizontal connection plate; 2—protective plate; 3—magnetic adsorption module; 4—spring shock absorber; 5—vertical connection plate; 6—tightening mechanism; 7—synchronous belt; 8—waterproof motor; 9—synchronous wheel.</p>
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<p>Control system block diagram.</p>
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<p>Control main board.</p>
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<p>FreeRTOS flow chart.</p>
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<p>Remote upgrade program software.</p>
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<p>Force analysis diagram.</p>
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<p>The relation curve between F<sub>m</sub> and θ.</p>
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<p>Simulation of and experiment on adsorption force.</p>
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<p>Diagram of coordinate system.</p>
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<p>Graphs of functions. (<b>a</b>) Sign function; (<b>b</b>) saturation function.</p>
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<p>Simulation model block diagram of sliding mode control system.</p>
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<p>Simulation model block diagram of PID control system.</p>
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<p>The simulation result of the fixed depth controller.</p>
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<p>The simulation result of the fixed heading controller.</p>
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<p>The amphibious robot system and experimental environment.</p>
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<p>Crawling experiment. (<b>a</b>) Forward state; (<b>b</b>) stationary state; (<b>c</b>) backward state; (<b>d</b>) turning state.</p>
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<p>The test experiment of buoyancy performance. (<b>a</b>) The roll angle in the static state; (<b>b</b>) the pitch angle in the static state; (<b>c</b>) the state of zero or minimal buoyancy.</p>
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<p>Floating motion experiment. (<b>a</b>) Backward state; (<b>b</b>) diving state; (<b>c</b>) forward state; (<b>d</b>) turning state.</p>
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<p>Comparison of simulation data and experimental data. (<b>a</b>) Depth change curve; (<b>b</b>) heading angle change curve.</p>
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<p>Application scene simulation experiment. (<b>a</b>) swimming process; (<b>b</b>) steel pipe defect location; (<b>c</b>) climbing process; (<b>d</b>) steel plate defect location.</p>
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23 pages, 3336 KiB  
Article
Insensitive Mechanism-Based Nonlinear Model Predictive Guidance for UAVs Intercepting Maneuvering Targets with Input Constraints
by Danpeng Huang, Mingjie Zhang, Taideng Zhan and Jianjun Ma
Drones 2024, 8(11), 608; https://doi.org/10.3390/drones8110608 - 24 Oct 2024
Viewed by 1081
Abstract
This paper proposed an innovative guidance strategy, denoted as NMPC-IM, which integrates the Insensitive Mechanism (IM) with Nonlinear Model Predictive Control (NMPC) for Unmanned Aerial Vehicle (UAV) pursuit-evasion scenarios, with the aim of effectively intercepting maneuvering targets with consideration of input constraints while [...] Read more.
This paper proposed an innovative guidance strategy, denoted as NMPC-IM, which integrates the Insensitive Mechanism (IM) with Nonlinear Model Predictive Control (NMPC) for Unmanned Aerial Vehicle (UAV) pursuit-evasion scenarios, with the aim of effectively intercepting maneuvering targets with consideration of input constraints while minimizing average energy expenditure. Firstly, the basic principle of IM is proposed, and it is transformed into an additional cost function in NMPC. Secondly, in order to estimate the states of maneuvering target, a fixed-time sliding mode disturbance observer is developed. Thirdly, the UAV’s interception task is formulated into a comprehensive Quadratic Programming (QP) problem, and the NMPC-IM guidance strategy is presented, which is then improved by the adjustment of parameters and determination of maximum input. Finally, numerical simulations are carried out to validate the effectiveness of the proposed method, and the simulation results show that the NMPC-IM guidance strategy can decrease average energy expenditure by mitigating the impact of the target’s maneuverability, optimizing the UAV’s trajectory during the interception process. Full article
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<p>The UAV pursuit-evasion scenario. The blue and orange conical areas are the variable range of orientation angles of the UAV and the target, respectively, and represent their maneuvering capabilities, and also the constraints of control input. Larger top angles of the cones mean more maneuverability.</p>
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<p>Kinematics between UAV (<span class="html-italic">M</span>) and target (<span class="html-italic">T</span>).</p>
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<p>The structure of NMPC-IM in guidance.</p>
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<p>The basic idea of IM.</p>
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<p>Step-by-step schematic of Insensitive Mechanism.</p>
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<p>Different <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> in NMPC-IM: (<b>a</b>) Trajectories. (<b>b</b>) Control input. Green represents <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mi>d</mi> </msub> <mo>=</mo> <msqrt> <mi>r</mi> </msqrt> </mrow> </semantics></math> and blue represents <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mi>d</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mi>r</mi> </msqrt> </mrow> </semantics></math>.</p>
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<p>Comparisons of PNG and pure NMPC-IM (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>20</mn> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> g): (<b>a</b>) Trajectories. (<b>b</b>) Control input. Green represents PNG and purple represents pure NMPC-IM.</p>
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<p>Comparisons of PNG and pure NMPC-IM (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>15</mn> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>1.5</mn> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> g): (<b>a</b>) Trajectories. (<b>b</b>) Control input. Green represents PNG and purple represents pure NMPC-IM.</p>
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<p>Different <math display="inline"><semantics> <msub> <mi>k</mi> <mi>t</mi> </msub> </semantics></math> in Maximum Input Determination: (<b>a</b>) Trajectories. (<b>b</b>) Control input.</p>
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<p>Comparisons of PNG, NMPC, NMPC-IM (sin): (<b>a</b>) Trajectories. (<b>b</b>) Control input. (<b>c</b>) Prediction horizon of NMPC-IM.</p>
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<p>Individually considerate of IM: (<b>a</b>) Trajectories. (<b>b</b>) Control input. (<b>c</b>) Prediction horizon.</p>
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<p>Comparisons of PNG, NMPC, NMPC-IM (exp): (<b>a</b>) Trajectories. (<b>b</b>) Control input. (<b>c</b>) Prediction horizon of NMPC-IM.</p>
Full article ">Figure 13
<p>Different initial heading angles of <span class="html-italic">M</span>: (<b>a</b>) Trajectories. (<b>b</b>) Control input.</p>
Full article ">Figure 14
<p>Addition of disturbance observer: (<b>a</b>) Trajectories. (<b>b</b>) Control input. <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>s</mi> <mi>u</mi> <mo>)</mo> </mrow> </semantics></math> = (36.64 s, 1.4346 g). (<b>c</b>) Prediction horizon of NMPC-IM. (<b>d</b>) Estimated and true values of <math display="inline"><semantics> <msub> <mi>a</mi> <mrow> <mi>T</mi> <mi>r</mi> </mrow> </msub> </semantics></math>. (<b>e</b>) Estimated and true values of <math display="inline"><semantics> <msub> <mi>a</mi> <mrow> <mi>T</mi> <mi>θ</mi> </mrow> </msub> </semantics></math>.</p>
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