Nonlinear Observer Research Articles

The nursing robot, equipped with a 6-degree-of-freedom (6-DOF) humanoid manipulator, has been applied in elderly and disabled care to execute complex and random nursing tasks with its advantages in automation and intelligence. Especially, when the nursing robot performs daily care tasks such as serving tea and pouring water, the good trajectory tracking performance of its manipulator is a crucial capability. However, nonlinear coupling, model uncertainty, joint friction, unknown external disturbances, and particularly the fact that manipulator does not satisfy Pieper criterion-are the main challenges, which degrade control performance. Few existing studies have simultaneously addressed all these issues to improve the control accuracy of the manipulator. Therefore, to achieve the good tracking performance for manipulator, a robust control method combining sliding mode control (SMC), radial basis function neural network (RBFNN), and nonlinear disturbance observer (NDO) is proposed. An improved Jacobian-based gradient descent method solves inverse kinematics, with the improved gradient descent driven inverse kinematics (IGDIK) module ensuring accuracy; RBFNN compensates for model uncertainty; NDO handles disturbances and friction. Simulations and experiments demonstrate enhanced trajectory tracking accuracy and stability, validating its effectiveness for the target manipulator.

Determining Lorentzian manifold from non-linear wave observation at a single point

Non-invasive tidal volume estimation with wearable sensors using a high-gain observer and deep learning.

Adaptive nonlinear control and observation for grid-connected wind-BESS systems with unknown demand

Abstract Resilience to disturbances has been a major focus of research in recent years regarding unmanned aerial vehicle (UAV) applications. This study presents a trajectory‐tracking control system that integrates a nonlinear disturbance observer (NDO) and a model predictive control (MPC). The control architecture consists of inner‐loop attitude control and outer‐loop position control, and the adaptive control primarily addresses uncertainties in the rotational terms of wind disturbances. The goal of NDO is to estimate the total disturbances and translate them into attitude commands to reduce their impact. When the adaptive control law is incorporated into the inner control loop, the influence of wind disturbances on quadrotor UAVs could be reduced. Different types of disturbances that may occur during flights are simulated to validate the performance of the proposed control system. Feasibility and control performance on real‐time embedded hardware have been validated through flight experiments. Both simulation results and the outcomes from these experiments show that the proposed approach, termed NDOMPC control system, could improve robustness and enhance control performance.

Abstract This paper presents a high‐frequency sliding mode control (HFSMC) approach that utilizes a nonlinear disturbance observer (NDO) and an improved tracking differentiator (TD) for achieving robust trajectory tracking control of a quadrotor unmanned aerial vehicle (UAV) in the presence of lumped disturbances and parameter uncertainties. Firstly, the quadrotor control system is decoupled into an inner‐loop subsystem focused on attitude adjustment and an outer‐loop subsystem dedicated to position control. The hierarchical control mechanism of the inner–outer loop solves the under‐actuation problem. Secondly, NDO is utilized to estimate and counteract lumped disturbances in real time, enhancing the system disturbance rejection capability. Additionally, a high‐frequency switching function is introduced into the reaching law to improve the reaching speed and handle parameter uncertainties, while the improved TD is used to smooth the desired attitude signals and their derivatives, reducing the chattering inherent in sliding mode control. This scheme effectively alleviates control input chattering while enhancing controller robustness, offering a simpler design and stronger disturbance rejection compared to nonsingular fast terminal sliding mode control (NFTSMC). Finally, the stability of the system is proven using globally uniformly ultimately bounded (GUUB) and Lyapunov theory. The effectiveness of the control strategy was validated through simulation experiments.

Active control and nonlinear vibration isolation are effective methods for achieving optimal vibration isolation performance. However, the excessive energy consumption and potential instability of these approaches substantially limit their development and widespread application. This study proposes an active bionic quasi-zero stiffness stretcher vibration isolation (ABQZSSVI) system, inspired by the efficient cushioning and vibration damping mechanisms of birds’ legs. The ABQZSSVI system reduces the demand for energy consumption of active control through a nonlinear quasi-zero stiffness (QZS) design. The polarization sway instability caused by the asymmetric stiffness is further mitigated using a nonlinear robust sliding mode controller (SMC) with a nonlinear disturbance observer. Asymmetric stiffness characteristics and vibration damping performance were experimentally tested. Results demonstrate that the proposed ABQZSSVI system significantly alleviates the polarization sway instability, and enhances stretcher stability, and improves both the low-frequency vibration isolation performance and energy efficiency.

Considering the complex offshore environment, where small robotic fish are exposed to surface disturbances from wave forces, suspended particles, and model uncertainties, as well as deep-water disturbances from bottom currents, this paper proposes an inverse sliding mode control strategy based on a disturbance observer to enhance trajectory tracking robustness. The research focuses on a tail-fin-driven bionic robotic fish. Utilizing its kinematic and dynamic models, a virtual control law based on an agent dynamics model is proposed, with the tail fin integrated as a real control input. A nonlinear disturbance observer with controllable convergence characteristics is developed to estimate and compensate for disturbances, including wave forces and model uncertainties. Additionally, a velocity error correction function is introduced to mitigate the impact of strong disturbances. Based on Lyapunov theory, an adaptive sliding mode control law is derived to ensure system stability. The control law for the caudal fin swing angle and bias is obtained by inverting the virtual control inputs, applied to the robotic fish’s accurate model. Numerical simulations show that the disturbance observer’s tracking error remains below 5%, and the trajectory tracking error is within 0.1 meters, representing only 2.2% of the robotic fish’s body length. Compared to mainstream control methods, the proposed approach significantly enhances robustness in contrast to the conventional sliding mode control with observers, and exhibits substantially smaller tracking errors, especially during trajectory transitions.

This paper proposes a novel command-filtered backstepping control method integrated with nonlinear disturbance observers (NDO-CFBS) to improve early-stage rehabilitation training for patients with lower-limb dysfunction across multiple postures, including sitting, lying, and standing. A backstepping control strategy is first developed for the human–robot system, considering the characteristics of passive rehabilitation training. Nonlinear disturbance observers (NDOs) are designed to compensate for model uncertainties and external disturbances. A command filter is then introduced to efficiently compute the derivatives of the virtual control variables within the backstepping framework, leading to the formulation of the NDO-CFBS method. Finally, simulation results of trajectory tracking performance are presented for the multi-posture lower limb rehabilitation robot, demonstrating the effectiveness of the proposed control strategy.

ABSTRACTThis paper presents an integrated framework for time‐sequential trajectory planning and adaptive control of a transfer manipulator operating in compact spaces, with the aim of ensuring the safety of operation, reducing total load transfer time, and improving efficiency. Unlike conventional approaches that treat trajectory planning and control as separate processes, this work proposes a tightly coupled framework that simultaneously optimizes both trajectory generation and motion control for enhanced performance. First, a radial basis function neural network (RBFNN) is employed to online approximate state‐dependent unknown nonlinearities in the system. Second, a composite nonlinear disturbance observer (NDO) incorporating RBFNN approximation results is designed to compensate for both external disturbances and approximation errors, thereby significantly reducing the system uncertainties. Third, an adaptive finite‐time prescribed performance (AFPP) control method is developed to guarantee output constraints, establishing fundamental safety conditions for time‐overlapped trajectory planning. Based on this high‐precision controller with guaranteed output constraints, time‐sequential trajectories with overlapping phases are optimally designed considering physical space constraints, which not only ensures operational safety in compact spaces but also substantially improves load transfer efficiency. The simulation results validate the robustness and effectiveness of the proposed approach, demonstrating significant improvements in both safety and control precision in highly constrained environments.

This paper proposes a predefined time transient coordinated control strategy based on an adaptive nonlinear extended state observer (ANLESO) to address the adaptability challenges of mode transition control in power-split hybrid electric vehicles (PS-HEVs). Firstly, building upon a conventional dynamic coordinated control framework, the influence of varying acceleration conditions and external disturbances on mode transition performance is analyzed. To enhance disturbance estimation under both positive and negative as well as large and small errors, an ANLESO is developed, which not only improves the speed and accuracy of disturbance observation but also guarantees symmetric convergence performance with respect to estimation errors. Subsequently, a predefined time feedback controller is developed based on the theory of predefined time control. Theoretical stability analysis demonstrates that the convergence time of the system is independent of the initial state and can be guaranteed within a predefined time. Finally, the feasibility and superiority of the proposed control strategy are validated through Hardware-in-the-Loop (HIL) testing and vehicle experimentation. The results show that, compared with PID control based on a linear expansion state observer, the proposed strategy reduces the mode transition time by 45.7% and mitigates drivability shock by 59.2%.

In ocean engineering, path following serves as a fundamental capability for autonomous underwater vehicles (AUVs), enabling essential operations such as environmental exploration and inspection. However, for robotic dolphins employing dorsoventral undulatory propulsion, the periodic pitching induces strong coupling between propulsion and attitude, posing significant challenges for precise path following in disturbed environments. In this paper, a real-time robust path-following control framework is proposed for robotic dolphins to address these challenges. First, a novel robotic dolphin platform is presented by integrating a dorsoventral propulsion mechanism with a passive peduncle joint, followed by the systematic formulation of a full-state dynamic model. Then, a minimum-snap-based path optimizer is constructed to generate smooth and dynamically feasible trajectories, improving path quality and motion safety. Subsequently, a robust model predictive controller is developed, which incorporates control surface dynamics, a nonlinear disturbance observer, and a Sigmoid-based disturbance-grading mechanism to ensure fast attitude response and precise tracking performance. Finally, extensive simulations under various environmental disturbances validate the effectiveness of the proposed approach in both trajectory optimization and robust path following. The proposed framework not only demonstrates strong robustness in path following and disturbance rejection, but also provides practical guidance for future underwater missions such as long-term environmental monitoring, inspection, and rescue.

Horizontal vibration of car is a critical issue affecting the stability and ride comfort of high-speed elevators. This phenomenon primarily caused by the uncertain nonlinear disturbances induced by guide rails, guide shoes, and other various internal and external perturbations. To mitigate these vibrations, a novel output feedback stabilization control strategy based on nonlinear extension state observer (NLESO) is proposed. First, an eight-degree-of-freedom dynamics model considering the uncertain nonlinear disturbances was established for the horizontal vibrations. Second, NLESOs were developed to observe and compensate for the internal and external perturbations of elevator car system. Subsequently, a stabilization controller with NLESOs was designed based on the output feedback of the dynamics model using a finite-time stabilization control law. The finite-time stability of NLESOs was proven using Lyapunov’s theorem. Finally, numerical simulations for the proposed method were conducted under two typical rail excitations using MATLAB/Simulink and compared with higher-order sliding mode control (HOSMC), linear extension state observer (LESO), and passive control schemes. The results indicate that the proposed method significantly reduces the displacement acceleration and angular deflection acceleration of horizontal vibrations, with root mean square values decreasing by more than 68% and 67%, respectively, compared to passive control. This demonstrates the superior stability and finite-time convergence of the proposed method, effectively suppressing horizontal vibrations during high-speed elevator operation.

As global demand for clean and renewable energy continues to rise, wind power has become a critical component of the sustainable energy transition. However, the increasingly complex operating conditions and structural configurations of modern wind turbines pose significant challenges for system reliability and control. Specifically, accurate load torque estimation is crucial for supporting the long-term stable operation of the wind power system. This paper presents a novel load torque estimation approach based on a nonlinear extended state observer (NLESO) for wind turbines with permanent magnet synchronous generators. In this method, the load torque is estimated using current measurements and observer-derived acceleration, thereby eliminating the need for torque sensors. This not only reduces hardware complexity but also improves system robustness, particularly in harsh or fault-prone environments. Furthermore, the stability of the observer is rigorously proven through Lyapunov theory using the variable gradient method. Finally, simulation results under different wind speed conditions validate the method’s accuracy, robustness, and adaptability.

Abstract Aiming at the problem of unknown model accuracy, high sample representativeness requirements, and the loss of time series features in traditional model-based and data-driven fault diagnosis for intelligent ship sensors, this paper proposes a hybrid fault diagnosis method that combines both approaches. A nonlinear passive observer is constructed to generate system residual signals at first. Then, a convolutional neural network and a gated recurrent unit neural network are used to extract local and time series features, respectively. Moreover, the self-attention mechanism is introduced to further distinguish the important relationship between different time points of the signal. Finally, the fault diagnosis is realized through the classifier. Experimental results based on an intelligent ship model show that the diagnosis rate increased by 7.4% compared to models without an observer. Compared to traditional machine learning and deep learning methods, the proposed model achieves a diagnostic accuracy of over 99%, demonstrating superior performance.

Abstract For the hypersonic vehicle interception problem, this paper proposes a cooperative guidance law identification and trajectory prediction method based on bearing-only measurement information. Firstly, under the condition that dual interceptor vehicles can obtain bearing-only measurement information, a nonlinear state and observation model based on the cooperative observation of the dual interceptor vehicles is constructed. Then, the cubature Kalman filter is used for state estimation and guidance law identification. Finally, trajectory prediction is performed based on the estimation results, and interceptor vehicles adopt proportional guidance for interception. The results show that the method can realize the accurate estimation of the guidance law parameters and state information of the hypersonic vehicle without relying on the distance measurement information, and the trajectory prediction based on the estimation results meets certain accuracy and speed requirements.

Adaptive sliding mode load-following control of a small modular reactor via reinforcement learning, nonlinear extended state observer, and neural network

We consider the problem of global finite-time stability for a class of nonlinear systems. The novelty in this paper is to consider a nonlinear finite time observer design, which introduces a finite-time observer for nonlinear systems that can be put into a nonlinear canonical form up to an output injection. The proof is based on the Lyapunov theory for Finite-Time Stability and the observer design method.

The reliability of the doubly-fed induction generator (DFIG) in wind turbines is essential for safe and efficient operation. To this end, this article proposes a fractional-order sliding-mode adaptive control scheme for a DFIG subjected to actuator faults. In this scheme, a nonlinear fault observer is designed to estimate actuator faults and the system uncertainties. Then, the observer estimates are used to design the adaptive fractional-order sliding mode controller for the faulty DFIG. The developed control method assumes no knowledge of the faults, uncertainties, or bounds. Moreover, the Lyapunov stability criterion is used to demonstrate the system's closed-loop stability. In the simulation section, simulation tests are conducted on a 7.5 KW DFIG-based subject to different actuator fault profiles and uncertainties, and the proposed controller is compared to some existing control strategies. The results validate the superior performance and robustness of the proposed controller compared with existing methods.

This paper presents the implementation of a real-time nonlinear state observer applied to an erbium-doped fiber laser system. The observer is designed to estimate population inversion, a state variable that cannot be measured directly due to the physical limitations of measurement devices. Taking advantage of the fact that the laser intensity can be measured in real time, an observer was developed to reconstruct the dynamics of population inversion from this measurable variable. To validate and strengthen the estimate obtained by the observer, a Recurrent Wavelet First-Order Neural Network (RWFONN) was implemented and trained to identify both state variables: the laser intensity and the population inversion. This network efficiently captures the system’s nonlinear dynamic properties and complements the observer’s performance. Two metrics were applied to evaluate the accuracy and reliability of the results: the Euclidean distance and the mean square error (MSE), both of which confirm the consistency between the estimated and expected values. The ultimate goal of this research is to develop a neural control architecture that combines the estimation capabilities of state observers with the generalization and modeling power of artificial neural networks. This hybrid approach opens up the possibility of developing more robust and adaptive control systems for highly dynamic, complex laser systems.