In this article, an adaptive boundary quantized fault-tolerant control scheme is developed for cascaded systems characterized by an axially moving belt and ordinary differential equations (ODEs). For the cascaded systems under investigation, the output of actuator dynamics serves as the input signal of the axially moving belt, such that the boundary control involving states of two subsystems is designed for vibration suppression. To mitigate the jittering during signal quantization, a sector-bounded quantizer is implemented. Moreover, the uncertainties in the actuator dynamics are approximated using fuzzy logic systems. The mixed effects of actuator failure and quantization are estimated via the bounded estimation approach, smoothing function, and adaptive technique. A dynamic event-triggered mechanism is introduced, incorporating an auxiliary dynamic variable to optimize communication resources. Based on this, a fuzzy event-based quantized fault-tolerant controller is developed for cascaded ODE-belt systems. Using the Lyapunov function method, the semi-global uniform boundedness of the closed-loop signals is guaranteed. Finally, simulation studies are conducted to validate the proposed control scheme.

In industrial scenarios, semantic segmentation of surface defects is vital for identifying, localizing, and delineating defects. However, new defect types constantly emerge with product iterations or process updates. Existing defect segmentation models lack incremental learning capabilities, and direct fine-tuning (FT) often leads to catastrophic forgetting. Furthermore, low contrast between defects and background, as well as among defect classes, exacerbates this issue. To address these challenges, we introduce a plug-and-play Transformer-based semantic complement module (TSCM). With only a few added parameters, it injects global contextual features from multi-head self-attention into shallow convolutional neural network (CNN) feature maps, compensating for convolutional receptive-field limits and fusing global and local information for better segmentation. For incremental updates, we propose multi-scale spatial pooling distillation (MSPD), which uses pseudo-labeling and multi-scale pooling to preserve both short- and long-range spatial relations and provides smooth feature alignment between teacher and student. Additionally, we adopt an adaptive weight fusion (AWF) strategy with a dynamic threshold that assigns higher weights to parameters with larger updates, achieving an optimal balance between stability and plasticity. The experimental results on two industrial surface defect datasets demonstrate that our method outperforms existing approaches in various incremental segmentation scenarios.

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This article pioneers the study of boundary-optimized fault-tolerant tracking control for flexible manipulators in a switching digraph with a heterogeneous linear leader. Compared with existing research, the proposed methods have several features. First, a distributed observer is designed to observe the leader's information in a general switching graph where communication can be interrupted. Second, a new partial differential equation (PDE)-based fault observer (FO) is designed to estimate unknown faults using only a few boundary states. Third, a novel long-term integral cost function is formulated to minimize angle-tracking errors, vibration deflections, and control energy in flexible manipulators. The ideal boundary optimal control laws are, then, derived and approximated using actor-critic neural networks (NNs) based on reinforcement learning (RL). Under the proposed fully distributed optimized fault-tolerant controllers, the closed-loop flexible manipulator's error states are proven uniformly ultimately bounded (UUB). Finally, the effectiveness of the proposed method is demonstrated through numerical simulation results.

Achieving visual localization with excellent interactive performance is challenging for mobile robots. Based on real-time photo-realistic view synthesis, 3D Gaussian splatting (3DGS) representation has demonstrated vast potential for robots engaging with the physical world. In this work, we propose a novel coarse-to-fine visual localization method named L3DGS based on the 3DGS radiance field representation. Particularly, during the coarse stage, we exploit novel views synthesized by the pretrained 3DGS map to create geometric feature correspondences to perform geometric alignment. Then, we integrate both geometric and photometric alignment to refine the camera pose. Unlike previous radiance field-based approaches, we leverage geometric feature correspondences and the innovative 3DGS map to improve the localization accuracy. In our experiments, we evaluate the proposed method across two real-world indoor and outdoor datasets. Consequently, compared to the baselines, the proposed method achieves competitive or superior experimental results.

This article mainly investigates the problem of vibration suppression and angle cooperative tracking control of a multiple flexible manipulators described by partial differential equations (PDEs) with input quantization, actuator failures, and unmodeled system dynamics. An intermediate control law is designed, and a smooth function with a positive integrable time-varying function is introduced. Besides, a new smooth function is constructed in the control law to handle the influence of quantization and actuator faults. Under the designed controller, the angles of all flexible manipulators can reach consensus through mutual communication, and the elastic deformation of each flexible manipulator can also be suppressed. Furthermore, the asymptotic stability of a closed-loop system is realized based on the Lyapunov function. Finally, numerical simulation validates the effectiveness of the method.

In this article, we investigate the event-triggered consensus control approaches for multiple single-link flexible manipulators formulated by partial differential equations with time-varying boundary disturbances. By designing an event triggering mechanism based on switching thresholds, the controller can only be updated after the trigger conditions are met, thereby avoiding the waste of network resources. Under local communication conditions, a finite-time distributed observer is constructed to estimate the leader state of a flexible manipulator. Fuzzy logic systems are applied to identify unmodeled dynamics. Disturbance observers are designed to estimate mixed perturbations consisting of disturbances and estimation errors. Moreover, the proposed control method can ensure that all signals in the closed-loop system are bounded, the elastic deflection of each flexible manipulator can be suppressed, the angle achieves consensus control, and Zeno behavior is avoided. Finally, the feasibility of the proposed theory is verified by numerical examples.

A novel binary channel fuzzy self-adjusted neural network (BCF-SANN) is proposed and researched for solving time-changing quadratic programming (QP) problems in this article. Unlike the fixed parameters of the typical zeroing neural network, the main parameters of the proposed BCF-SANN are time-changing, and its errors are adaptively quickly convergent. The biggest advantage of the novel neural network is that it combines a fuzzy self-adjusted controller, which takes the errors and derivatives of errors as fuzzy inputs and neural networks, further improving the convergence and robustness of the neural networks. To design the novel neural network, a time-changing QP problem is first established; then, using Lagrange's law, the time-changing QP problem is transformed into a time-changing matrix equation; and finally, based on the time-changing parameter neural dynamics method, a novel BCF-SANN is proposed. The detailed design process is given in this article, and the convergence and robustness of the proposed BCF-SANN are proved by theoretical analysis. Through comparative experiments, it is demonstrated that the proposed BCF-SANN has a faster convergence rate and stronger robustness than the traditional zeroing neural network and 1-D fuzzy recurrent neural network (RNN).

This study proposes a neural-network (NN)-based adaptive fixed-time control method for a two-degree-of-freedom (2-DOF) nonlinear helicopter system with input quantization and output constraints. First, a hysteresis quantizer is employed to mitigate chattering during signal quantization, and adaptive variables are utilized to eliminate errors in the quantization process. Subsequently, the system uncertainties are approximated using a radial basis function NN. Simultaneously, a logarithmic barrier Lyapunov function (BLF) is constructed to prevent the system outputs from violating the constraint boundaries. Based on a rigorous Lyapunov stability analysis and the fixed-time stability criterion, the signals of the closed-loop system are proven to be bounded within a fixed time. Finally, numerical simulations and experiments verified the feasibility of the proposed method.

This article explores the event-triggered asymptotic stabilization problem of an axially moving belt in the presence of output constraint, unknown system parameters, and high acceleration/deceleration (H-A/D). Instead of existing event-triggered control (ETC) schemes where only bounded stabilization can be obtained, adaptive event-triggered asymptotic stabilization control can be achieved in this article. For this, a new tangent barrier Lyapunov function (BLF) method is developed to ensure that the state satisfies its corresponding time-varying constraint conditions. Then, the unknown system parameter problems can be resolved by using the adaptive compensation technique. Considering the resource constraint of the communication channel and computation burden, a new ETC strategy is advanced to tackle the communication load and optimize system performance. Moreover, the asymptotic stabilization control can be realized through the developed boundary ETC. Lastly, simulation results are displayed to verify the designed boundary control algorithms.

Photoplethysmography (PPG) has been widely applied as a useful non-invasive sensing technology for the assessment of blood pressure and blood oxygen levels in commercial medical devices. In contrast with conventional amplifier-based analog front-ends (AFEs), a light-to-digital converter (LDC), which digitizes optical signals right after a photodiode, has received increasing attention as a promising architecture of PPG front-end for its superior power efficiency and compactness. The design challenge, however, remains in how to achieve high dynamic range (DR) in direct light-to-digital conversion within the rather constrained power budget. To tackle this problem, we propose a highly-programmable digital front-end IC in this paper to enable low-dark-current sensing in a PPG sensor with a novel MoSe2photodiode. A programmable second-order continuous-time delta-sigma ADC utilizing 1.5-bit quantizer serves as the core of this front-end IC. Its programmable architecture allows a minimally detectable input current of 0.4 μA, leading to a state-of-the-art dynamic range of 140 dB. The proposed LDC is implemented in TSMC 0.18 μm process and achieves an average power consumption of 14.87 μW. The simulated SNDR with the transient noise is 80.1 dB . Furthermore, an LED Driver is integrated to the proposed front-end IC to provide an output current ranging from 1 mA to 127 mA (7 bit). Compared to prior art, this work features outstanding performance with a smaller footprint.

Localization and pose estimation algorithms play an important role in intelligent transportation systems (ITSs), as ITS need to accurately sense and understand the traffic environment to support autonomous navigation, traffic flow management, and autonomous material handling. This article proposes a pose estimation method in the front end of lidar odometry with geometric constraints. The proposed method can accurately capture the geometric information in the environment and ensure the effectiveness of the point cloud participating in the registration to improve the accuracy of registration. In the back end, an enhanced pose estimation strategy combining rough registration and fine registration is adopted to further improve localization accuracy. Comprehensive experimental results show that the proposed method achieves higher localization accuracy against other baselines, which also demonstrates that the proposed method can cope with challenging scenes such as complex road conditions and dynamic objects.

This article investigates distributed adaptive fuzzy-based boundary vibration control and cooperative tracking for a directed flexible Timoshenko manipulator network with process and actuator faults, disturbances, and modeling uncertainties. In control design, fuzzy logic systems are adopted to estimate the process fault functions, and the adaptive method is used to compensate for the effects of actuator gain failures, disturbances, and modeling uncertainties. Then, by utilizing a combination of boundary consensus control, integral control, and adaptive fuzzy control, some novel distributed fault-tolerant controllers are provided to ensure the asymptotic convergence of vibration states and consensus tracking errors in the infinite-dimensional system. Different from the existing results, we prove the asymptotic convergence results directly from the perspective of definition based on some new analytical techniques. At last, some numerical simulation results are presented to illustrate the feasibility of the proposed control algorithms.

In this article, the event-triggered consensus tracking control problem is explored for a network of hyperbolic partial differential equations (PDEs) with boundary actuator dynamics described by ordinary differential equations. Control input appears in actuator dynamics rather than in the PDE subsystem, which poses an interesting open problem. Unknown nonlinear actuator dynamics and local interconnection information render existing boundary control algorithms no longer applicable. To handle this, based on the infinite and finite-dimension backstepping techniques, a novel distributed adaptive tracking controller is designed to realize the consensus tracking control, where the unknown nonlinearities are determined by using the fuzzy wavelet neural networks. Moreover, due to the resource constraint of the communication channel and computation burden, this article further develops a new event-triggered mechanism in the sensor-to-controller channel. Based on the proposed event-triggered distributed algorithm, it is amply illustrated that consensus tracking control can be realized. Finally, two examples are provided to demonstrate how effectively the designed tracking algorithm performs.

The leader-follower adaptive consensus control problem is addressed for partial differential equations (PDEs) multiagent systems (MASs), and these agents are composed of flexible manipulator systems with input nonlinearity, boundary uncertainties, and time-varying disturbances. Because of the spatial variables in the model, the design of adaptive protocols is more difficult than that of ordinary differential equation (ODE) MASs. By designing the Lyapunov function, a novel distributed boundary control (BC) protocol is constructed, which not only ensures the consensus of angular positions but also suppresses the boundary vibration of each agent. The hybrid effects of dead zones and input saturation on flexible manipulator systems are addressed using the approximation properties of neural networks (NNs). In addition, the disturbance adaptive laws are proposed to provide a control solution for bounded and time-varying disturbances. Furthermore, by applying the Lyapunov stability theory, the uniformly bounded stability of the multiflexible manipulator can be ensured. Finally, the feasibility of the presented control approach is verified using numerical examples.

For multiple flexible manipulator systems under an undirected switching topology graph with actuator faults, dead zones, and external disturbances, the proposed distributed control technique aims to attain joint angle consensus and vibration suppression. Therefore, in boundary controller design, the Nussbaum function is utilized to address the issue of uncertain control direction caused by fault tolerance and dead zones. Then, under the condition that the switching topology is connected, by using the backstepping technique, a new Lyapunov function is introduced so that the multiple flexible manipulators system can ensure angle cooperative control and vibration suppression while not knowing the control direction. Moreover, the adaptive law is designed to handle compound disturbances consisting of the unknown dead zones, additive faults, and boundary disturbances. Furthermore, we show that the flexible manipulator system is uniformly ultimately bounded and stable according to the Lyapunov stability theory. Ultimately, through the numerical simulation, the control strategy’s efficacy is confirmed.

Autonomous vehicles (AVs) play an important role in the next-generation intelligent transportation system (ITS), which requires AVs to have the ability of rapid decision-making and control. As a part of ITS, simultaneous localization and mapping (SLAM) technology is the basis for AVs to operate in a complex and unknown environment. In this article, a stereo vision odometry method in a dynamic environment is proposed, which can not only effectively overcome the influence of dynamic objects but also detect the position of dynamic objects. An optical flow filtering algorithm based on the quantitative histogram (QH) and the optical flow angle histogram (OFAH) of the feature points is proposed to obtain dynamic points. Furthermore, a multifeature fusion mechanism is used to perform binary segmentation and get the bounding box of dynamic object. Experiments show that the proposed method can improve the accuracy of pose estimation and detect moving objects in a dynamic environment.

This article investigates the cooperative vibration control problem for a flexible manipulator network with model uncertainties and boundary disturbances guided by multiple dynamic leaders. Different from the previous research on a single flexible manipulator, this article focuses on the containment control problem of multiple flexible manipulators. To this end, the boundary control algorithm is proposed for leader agents without disturbance and follower agents with disturbance and uncertainties. The fuzzy logic systems are applied to deal with uncertain and continuous functions in the system model. By constructing the appropriate Lyapunov function, not only the vibrations can be suppressed, but also the containment error between the convex hull spanned by the multiple leaders and all followers can be converged to zero by designing a distributed containment feedback control law. Furthermore, the control algorithm can ensure that all the signals in the closed-loop system are uniformity ultimately bounded; meanwhile, vibration amplitude and error converge to a small compact set. At last, the simulation results prove the effectiveness of the proposed control method.

This article focuses on vibration suppression of an Euler-Bernoulli beam which is subject to external disturbance. By integrating backstepping technique, an adaptive boundary iterative learning control (ABILC) is put forward to suppressing vibration. The adaptive law is proposed for handing the parameter uncertainty and the iterative learning term is designed to deal with periodic disturbance. An auxiliary system is utilized to compensate the effect of input nonlinearity. In addition, a barrier Lyapunov function is adopted to deal with asymmetric output constraint. With the proposed control strategy, the stability of the closed-loop system is proven based on rigorous Lyapunov analysis. In the end, the effectiveness of the proposed control is illustrated through numerical simulation results.

This article studies the problem of deformation reduction and attitude tracking for a rotated and extended flexible crane arm with input backlash-saturation and output asymmetrical constraint. By employing Halmilton's principle, the arm system model is formulated by a set of partial and ordinary differential equations (ODEs). Given the modeling inaccuracy, a radial neural network (RNN) is used to approximate system parameters. To better design the controllers, the backstepping technique is applied to the control design. For input nonlinearities with backlash and saturation, we reversely transform them as an asymmetric saturation constraint via a virtual input. A barrier Lyapunov function (BLF) containing logarithmic terms is constructed to guarantee the asymmetric output constraints and the uniformly ultimate boundedness and stability of the arm system are proved. Finally, to testify the effectiveness of the proposed controllers, numerical simulations are carried out, and responding simulation diagrams are displayed.

Most automatic product surface inspection methods in industry are data-hungry and task-specific. It is difficult to collect adequate labeled samples in practice due to factors including expensive data annotation cost, inadequate samples for some categories, and limitations on the initial production stage. In this article, a multiple guidance network (MGNet) is proposed to address these issues. In the network, the feature extraction machine (FEM) produces four feature maps of different functions to enhance the inspection ability of the algorithm. Also, the probability map generation (PMG) module is designed for coarse positioning of objects. Moreover, the structures of the mutual guidance and historical guidance (HG) guarantee that the network can fully utilize the information of the auxiliary dataset. Only one support sample containing the labeled objects is required for reference, and the network can determine whether the same labeled objects exist in the query images and locate them. For a comprehensive evaluation of MGNet, three experiments are carried out using three real-world datasets. Experiment results verify that the proposed method is promising for industrial product surface inspection with one labeled target sample.

This article focuses on the problem of vibration suppression and attitude tracking of a flexible rotatable manipulator. For the manipulator system suffering from parameter uncertainties, input saturations, time-varying output constraints, and periodic boundary disturbances, a new type of robust adaptive boundary control scheme is proposed. To cope with system parameter uncertainties and input saturations, radial basis neural network functions (RBNNFs) are introduced. To compensate for the periodic disturbance errors, the iterative learning control (ILC) is designed. In order to obtain a controller to guarantee the system stability, the backstepping technique is employed. Then, a modified Lyapunov function is constructed and system stability and uniform boundedness of output variables are proved. By conducting simulation experiment, the robustness and prescribed performance of the adaptive ILC-RBNNF-based controllers are testified.

The adaptive boundary vibration control and leader–follower angle tracking consensus problem of networked flexible Timoshenko manipulator systems in the case of parametric uncertainties and external disturbances are investigated. To mitigate the impacts of external disturbances and parameter uncertainties, we employ the adaptive tuning techniques to design some suitable adaptive terms. Then with network topology is connected, we propose some novel distributed adaptive continuous-time boundary control algorithms. Based on a novel combination of Lyapunov-based control and Barbalat’s lemma, the asymptotic control performances can be achieved. Besides, some novel distributed adaptive event-triggered boundary control algorithms are also proposed for the undisturbed Timoshenko manipulators. The advantage of the considered event-triggered algorithms is that they can avoid continuous updates of the controller and thereby can effectively save control resources. Moreover, Zeno behaviors are excluded for all agents to ensure the practicality of digital sampling. Finally, we illustrate the feasibility of the proposed control algorithms through numerical simulation examples.

In this article, neural network (NN)-based sliding mode control schemes are proposed for an n-link robotic manipulator with system uncertainties, input deadzone, and external perturbations. A novel error-shifting function is proposed to release initial conditions. NNs are employed to approximate the unknown parameters of both system uncertainties and input deadzone. To update the sliding mode scheme, two advanced sliding mode surfaces with error-shifting function and barrier function are proposed to reduce the dependency of prior information and to realize a finite time convergence result, collectively. It should be pointed out that the proposed methods do not require initial states to satisfy the prescribed constraint caused by the barrier function and can be applied under unknown initial conditions. Furthermore, finite-time convergence for both tracking errors and NN weights is guaranteed. The effectiveness of the proposed schemes is demonstrated by simulation and experiments on the KINOVA robot.

This article demonstrates the realization of angle tracking and deformation suppression by developing two boundary controllers for a flexible variable-length rotary crane arm with extraneous disturbances and asymmetric input-output constraints. The dynamic model description of this kind of crane arm system is several partial differential equations integrated into few ordinary differential equations. The S-curve acceleration and deceleration scheme is utilized to adjust the elongation rate of the arm. A kind of novel observer is put forward to tackle unknown extraneous disturbances. Auxiliary systems and barrier Lyapunov functions are introduced to meet the asymmetric input-output constraints. With the help of Lyapunov's theory, the global exponential stability and uniform boundedness are analyzed. The numerical simulations are finally provided to illuminate its availability of the designed control schemes.

In this article, vibration abatement problems of a riser system with system uncertainty, input deadzone, and output constraint are considered. For obtaining better control precision, a boundary control law is constructed by employing the backstepping method and Lyapunov's theory. The output constraint is guaranteed by utilizing a barrier Lyapunov function. Adaptive neural networks are designed to cope with the uncertainty of the riser and compensate for the effect caused by the asymmetric deadzone nonlinearity. With the designed controller, the output constraint is satisfied, and the system stability is guaranteed through Lyapunov synthesis. In the end, numerical simulation results are provided to display the performance of the developed adaptive neural network boundary control law.

Nonlinear and nonconvex optimization problems are vital and fundamental problems in science and engineering fields. In this article, a novel finite-time circadian rhythms learning network (called FT-CRLN) is proposed for solving nonlinear and nonconvex optimization problems with periodic noises. Different from the traditional recurrent neural networks, the proposed FT-CRLN can suppress the periodic noise notably and achieve excellent convergence performance in solving nonlinear and nonconvex problems. The theoretical analysis and rigorous mathematical proof verify the superior convergence, high accuracy, and strong robustness of the proposed FT-CRLN. The simulation results demonstrate the effectiveness and robustness of the proposed FT-CRLN in solving nonlinear and nonconvex problems compared with other state-of-art neural networks.

This article constructs two adaptive control laws to achieve deformation reduction and attitude tracking for a rotary variable-length crane arm with system parameter uncertainties and asymmetric input-output constraints. Two auxiliary systems are given to deal with the input constraints, an asymmetric-logarithm-barrier Lyapunov function is established for achieving the asymmetric output constrains, and five adaptive laws are constructed to handle system parameter uncertainties. Besides, the control design is based on a partial differential equation model, and the S-curve acceleration and deceleration method is used for regulating the arm extension speed. Both the system stability and uniform ultimate boundedness of the controlled crane arm are analyzed. Simulation results validate the effectiveness of our established control laws.

This article focuses on the vibration reducing and angle tracking problems of a flexible unmanned spacecraft system subject to input nonlinearity, asymmetric output constraint, and system parameter uncertainties. Using the backstepping technique, a boundary control scheme is designed to suppress the vibration and regulate the angle of the spacecraft. A modified asymmetric barrier Lyapunov function is utilized to ensure that the output constraint is never transgressed. Considering the system robustness, neural networks are used to handle the system parameter uncertainties and compensate for the effect of input nonlinearity. With the proposed adaptive neural network control law, the stability of the closed-loop system is proved based on the Lyapunov analysis, and numerical simulations are carried out to show the validity of the developed control scheme.

This article investigates the synchronization of fractional-order multi-weighted complex networks (FMWCNs) with order$\alpha \in (0,1)$. A useful fractional-order inequality${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V(x(t))$is extended to a more general form${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V^{\gamma }(x(t)),\gamma \in (0,1]$, which plays a pivotal role in studies of synchronization for FMWCNs. However, the inequality${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V^{\gamma }(x(t)),\gamma \in (0,1)$has been applied to achieve the finite-time synchronization for fractional-order systems in the absence of rigorous mathematical proofs. Based on reduction to absurdity in this article, we prove that it cannot be used to obtain finite-time synchronization results under bounded nonzero initial value conditions. Moreover, by using feedback control strategy and Lyapunov direct approach, some sufficient conditions are presented in the forms of linear matrix inequalities (LMIs) to ensure the synchronization for FMWCNs in the sense of a widely accepted definition of synchronization. Meanwhile, these proposed sufficient results cannot guarantee the finite-time synchronization of FMWCNs. Finally, two chaotic systems are given to verify the feasibility of the theoretical results.

For tackling the vibration suppression problem for a flexible riser system with external disturbance, input saturation, and output constraint, this article is constructed. A boundary iterative learning control is constructed together with Lyapunov’s theory and backstepping technology. The output constraint is largely solved by employing a barrier Lyapunov function. External disturbance is handled as the iteration goes on. Under the control of the designed method, the output is limited to a given region, and the system is proved to be closed-loop stable with the assistance of Lyapunov’s theorem of stability. Ultimately, the result of the simulation indicates that the method is effective and has a good control effect.

This article copes with vibration suppression and angular position tracking problems of a robotic manipulator system comprised of a rotating hub and a length-varying manipulator. To obtain precise dynamic response, the manipulator system is modeled in infinite-dimension with partial differential equations. S-curve acceleration/deceleration (S-CA/D) scheme is employed for speed regulation of the length-varying manipulator. Two novel observers are developed to estimate both the unknown disturbances and their time-derivatives, and two auxiliary systems are put forward to tackle input constraints. With assistance of the auxiliary systems and observers, two boundary control laws are put forward to manage vibration suppression and angular position tracking of the proposed manipulator system. Through Lyapunov’s theory, the closed-loop system is proved to be bounded. Numerical simulations have displayed the effectiveness of the observers and boundary control laws.

The vibration offsets reduction is studied for an industrial axially moving accelerated string subjected to the external disturbances and input saturation. A boundary control strategy is put forward to reduce the vibration offsets and an auxiliary term is introduced to handle the restriction of the input saturation. In addition, both infinite-dimensional observer and finite-dimensional observer are, respectively, designed to track unknown disturbances. Under the developed control strategy, the existence, uniqueness, and convergence of the solution of the closed-loop string system are discussed without resorting to model reduction methods. By choosing the proper control parameters, the simulations are given to verify the performance of the developed control strategy.