Backstepping Method Research Articles

Disturbance observer based fixed-time tracking control for unmanned aerial helicopters

In this paper, a fixed-time tracking control scheme based on the fixed-time disturbance observer is proposed for a 6-DOF unmanned aerial helicopter (UAH) with flight path constraints and composite disturbances. The composite disturbances are composed of external disturbances and system uncertainties. By using the hyperbolic tangent function, an improved disturbance observer is developed to estimate composite disturbances. Furthermore, a fixed-time back-stepping control method is employed for the position and attitude loops, which enables the UAH to track the expected path within the flight path constraints. Simulation results are given to demonstrate the effectiveness and advantages of the proposed scheme.

A Design of Model Predictive Control and Nonlinear Disturbance Observer-based Backstepping Sliding Mode Control for Terrain Following

In this study, we propose the terrain following algorithm using model predictive control and nonlinear disturbance observer-based backstepping sliding mode controller for an aircraft system. Terrain following is important for military missions because it helps the aircraft avoid detection by the enemy radar. The model predictive control is used to replace the generating trajectory and guidance with the flight path angle constraint. In addition, the aircraft is affected to the parameter uncertainty and unknown disturbance such as wind near the mountainous terrain. Therefore, we suggest the nonlinear disturbance-based backstepping sliding mode control method for the aircraft that has highly nonlinearity to enhance flight path angle tracking performance. Through the numerical simulation, the proposed method showed the better tracking performance than the traditional backstepping method. Furthermore, the proposed method presented the terrain following maneuver maintaining the desired altitude.

Discrete-Time Event-Triggered Type-2 fuzzy wavelet neural network control for Multi-Motor servo system

This paper investigates a discrete-time event-triggered adaptive neural network control scheme for a multi-motor servo system (MMSS) to realize a desired position tracking performance. First, a discrete-time model of the MMSS, including a dead-zone and a nonlinear friction, is established based on the Euler’s discretization. Then, a discrete-time adaptive neural network controller is designed by integrating a type-2 fuzzy wavelet neural network (T2FWNN) and the backstepping technology. The neural network is not only used to estimate uncertain nonlinearities, but also can handle the non-causal problem caused by the conventional backstepping method. Meanwhile, a fixed threshold event-triggered mechanism along with the incorporation of a dead-zone operator is superimposed into the actual controller thus saving communicational resources. Besides, stability analysis proves that all the signals in the closed MMSS are bounded, and the position tracking error converges to a small neighborhood of the origin. Finally, abundant simulation experience results demonstrate the effectiveness and robustness of the proposed scheme.

Tracking control via time-varying feedback for an uncertain robotic system with both output constraint and dead-zone input

This paper is devoted to the tracking control for an uncertain robotic system with both output constraint and dead-zone input. Remarkably, the distinctive characters of the system are reflected by system uncertainties and output constraint. First, more serious uncertainties are involved since unknown nonlinear dynamic matrices, external disturbance and the dead-zone input (see unknown slopes and break points therein) are simultaneously considered, but those of the related literature are not. Second, weaker conditions on the output constraint are allowed since the constraint functions considered are only first but not more order continuously differentiable while any their time derivatives are not necessarily available for feedback. This leads to the incapability of the traditional control schemes on this topic. To solve the control problem, a novel control framework is proposed based on time-varying feedback which overcomes the serious system uncertainties while relaxes the conditions on output constraints. Specifically, a state transformation with a time-varying gain is first introduced to derive a new system. Then, by using the traditional backstepping method with the introduction of the time-varying gain in the estimations of some uncertain terms, a time-varying feedback controller is explicitly designed, which ensures that all the states of the resulting closed-loop system are bounded while system output asymptotically tracks the reference signal without any violation of the output constraint. Finally, simulation results for two practical examples are provided to validate the effectiveness of the proposed theoretical results, and moreover, a comparison with PID method is given to show the superiority of the proposed method on tracking accuracy and robustness.

Optimal enhanced backstepping method for trajectory tracking control of the wheeled mobile robot

Optimal enhanced backstepping method for trajectory tracking control of the wheeled mobile robot

Safe tracking control for unmanned autonomous helicopter under actuator faults and outside disturbances

This paper considers the problem on tracking control for the medium-scale unmanned autonomous helicopter (UAH) system under external disturbances, actuator faults, and full state constraints. In comparison with existing results, this paper does not only consider asymmetric state constraints but also jointly take the faults and disturbances into account on the UAH system, estimating and compensating for them separately, which can effectively enhance the tracking performance. First, the dynamics of nonlinear UAH system is divided into the position subsystem and the attitude one. Second, an improved barrier Lyapunov function (BLF) is constructed to handle the problem of asymmetric state constraints. Third, in order to estimate the external disturbances and actuator faults, two coupled generalized proportional integral observers (GPIOs) and two fault estimators are designed, respectively. Fourth, based on above estimations, improved BLFs, and backstepping method, two tracking control laws for the position and attitude loops are presented and an overall closed-loop system is established. Then, a sufficient condition ensuring uniform boundedness of tracking and estimation errors is derived. Finally, some simulating results demonstrate the effectiveness and advantage of the proposed control scheme.

Adaptive disturbance observer-based finite-time command filtered control of nonlinear systems

This article examines the finite-time trajectory tracking control problem for a category of nonlinear perturbed systems. A novel observation mechanism is constructed to mitigate the effects of complex system disturbances. Then, a new adaptive finite-time backstepping nonlinear control algorithm is designed by integrating this disturbance observer with a command filter method. The synthesized control strategy not only effectively suppresses external disturbances and avoids the differential explosion problem of traditional backstepping methods but also ensures the convergence of the trajectory tracking error and filtering error compensation signal within a finite time. Finally, the validity of the suggested approach is confirmed via comparative simulations.

Design of Adaptive Finite-Time Backstepping Control for Shield Tunneling Systems with Constraints

This paper focuses on the finite-time tracking control problem of shield tunneling systems in the presence of constraints on the states and control input. By modeling the system based on the LuGre friction model, an effective method of tracking control in finite time is designed to overcome these actual constraints at the same time. First, the constraint on the system state is transformed into a symmetric constraint on the tracking error, and the constraint on control input is handled by designing an auxiliary differential equation. Then, radial basis function (RBF) neural networks are introduced to approximate the uncertainties. Next, using an adaptive finite-time backstepping method and choosing a logarithmic barrier Lyapunov function (BLF), a finite-time controller is designed to realize the finite-time stability of the closed-loop system. Finally, a simulation example is given to verify the correctness and validity of the theoretical results.

Robust Control Design of Under-Actuated Nonlinear Systems: Quadcopter Unmanned Aerial Vehicles with Integral Backstepping Integral Terminal Fractional-Order Sliding Mode

In this paper, a novel robust finite-time control scheme is specifically designed for a class of under-actuated nonlinear systems. The proposed scheme integrates a reaching phase-free integral backstepping method with an integral terminal fractional-order sliding mode to ensure finite-time stability at the desired equilibria. The core of the algorithm is built around proportional-integral-based nonlinear virtual control laws that are systematically designed in a backstepping manner. A fractional-order integral terminal sliding mode is introduced in the final step of the design, enhancing the robustness of the overall system. The robust nonlinear control algorithm developed in this study guarantees zero steady-state errors at each step while also providing robustness against matched uncertain disturbances. The stability of the control scheme at each step is rigorously proven using the Lyapunov candidate function to ensure theoretical soundness. To demonstrate the practicality and benefits of the proposed control strategy, simulation results are provided for two systems: a cart–pendulum system and quadcopter UAV. These simulations illustrate the effectiveness of the proposed control scheme in real-world scenarios. Additionally, the results are compared with those from the standard literature to highlight the superior performance and appealing nature of the proposed approach for underactuated nonlinear systems. This comparison underscores the advantages of the proposed method in terms of achieving robust and stable control in complex systems.

Local exponential stabilisation of the delayed Fisher's equation

Exponential stabilisation is undoubtedly an essential property to study for any control system. This property is often difficult to establish for nonlinear distributed systems and even more difficult if the state of such a system is subject to a time-delay. In this context, this paper is devoted to the study of the well-posedness and exponential stabilisation problems for the one-dimensional delayed Fisher's partial differential equation (PDE) posed in a bounded interval, with homogeneous Dirichlet condition at the left boundary and controlled at the right boundary by Dirichlet condition. Based on the infinite dimensional backstepping method for the linear and undelayed corresponding case, a feedback law is built. Under this feedback, we prove that the closed-loop system is well-posed in L 2 ( 0 , 1 ) and locally exponentially stable in the topology of the L 2 ( 0 , 1 ) -norm by the use of semigroup theory relating to differential difference equations and Lyapunov–Razumikhin analysis, respectively. A numerical example that reflects the effectiveness of the proposed approach is given.

Dynamic Modeling and Observer-Based Fixed-Time Backstepping Control for a Hypersonic Morphing Waverider

This paper proposes a fixed-time backstepping control method based on a disturbance observer for a hypersonic morphing waverider (HMW). Firstly, considering the disturbance of attitude channels, a dynamic model of a variable-span-wing HMW considering additional forces and moments is established, and an aerodynamic model of the aircraft is constructed using the polynomial fitting method. Secondly, the fixed-time stability theory and backstepping control method are combined to design an HMW fixed-time attitude controller. Based on the fixed-time convergence theory, a fixed-time disturbance observer is designed to achieve an accurate online estimation of disturbance and to compensate for the control law. In order to solve the problem of the “explosion of terms”, a nonlinear first-order filter is used instead of a traditional linear first-order filter to obtain the differential signal, ensuring the overall fixed-time stability of the system. The fixed-time stability of the closed-loop system is strictly proven via Lyapunov analysis. The simulation results show that the proposed method has good adaptability under different initial conditions, morphing speeds, and asymmetric morphing rates of the HMW.

Research on Magnetic Levitation Control Method under Elastic Track Conditions Based on Backstepping Method

The vehicle–guideway coupled self-excited vibration of maglev systems is a common control instability problem in maglev traffic while the train is suspended above flexible girders, and it seriously affects the suspension stability of maglev vehicles. In order to solve this problem, a nonlinear dynamic model of a single-point maglev system with elastic track is established in this paper, and a new and more stable adaptive backstepping control method combined with magnetic flux feedback is designed. In order to verify the control effect of the designed control method, a maglev vehicle–guideway coupled experimental platform with elastic track is built, and experimental verifications under rigid and elastic conditions are carried out. The results show that, compared with the state feedback controller based on the feedback linearization controller, the adaptive backstepping control law proposed in this paper can achieve stable suspension under extremely low track stiffness, and that it shows good stability and anti-interference abilities under elastic conditions. This work has an important meaning regarding its potential to benefit the advancement of commercial maglev lines, which may significantly enhance the stability of the maglev system and reduce the cost of guideway construction.

Lyapunov-based model predictive control for trajectory tracking of hovercraft with actuator constraints and external disturbances

In this study, a Lyapunov-based model predictive control (LMPC) method is developed to address the trajectory tracking problem of autonomous hovercrafts subjected to unknown complex disturbances from ocean environments and practical constraints of marine surface vessels, such as actuator increment limitations and saturation. Initially, the novel LMPC algorithm is applied to enhance tracking performance through rolling optimization and online optimization. Subsequently, a nonlinear disturbance observer is designed to estimate external disturbances caused by wind and dynamics uncertainties. Furthermore, to theoretically ensure control stability, contraction constraints are established using the nonlinear backstepping control method. This approach provides the essential conditions for the system's robustness and stability. Finally, the superiority and robustness of the designed LMPC trajectory tracking method are demonstrated through numerical simulations conducted in a marine environment.

Disturbance-Observer-Based Adaptive Prescribed Performance Formation Tracking Control for Multiple Underactuated Surface Vehicles

This study proposes a new disturbance-observer-based adaptive distributed formation control scheme for multiple underactuated surface vehicles (USVs) subject to unknown synthesized disturbances under prescribed performance constraints. A modified sliding mode differentiator (MSMD) is applied as a nonlinear disturbance observer to estimate unknown synthesized disturbances, which contain unknown environmental disturbances and system modelling uncertainties, thus enhancing the robustness of the system. Based on this, we impose the time-varying performance constraints on the position tracking error between the neighboring USVs. A novel differentiable error transformation equation is embedded in the prescribed performance control, and an adaptive prescribed performance controller is constructed by employing the backstepping method to ensure that the position tracking error remains within the prescribed transient and steady performance, and each USV realizes collision-free formation motion. Furthermore, a novel second-order nonlinear differentiator is introduced to extract the derivative information of the virtual control law. Finally, the numerical simulation results verify the effectiveness of the proposed control scheme.

Adaptive command filter control for switched time‐delay systems with dead‐zone based on an exact disturbance observer

AbstractThe focus of this article lies in the adaptive command filter tracking control problems for a class of switched time‐delay systems with an asymmetric dead‐zone and external disturbance under arbitrary switching. First, the external disturbance is estimated via an exact disturbance observer. Then, an echo state network (ESN) was adopted to approximate unknown function in system without tuning the weights between a reservoir and an input layer. Second, the time‐varying input of the dead‐zone model is considered as uncertain physical quantity of the system based on the dead‐zone slope boundary information. Third, an adaptive command filtered controller is employed to conquer the matter of “explosion of complexity” encountered in the traditional backstepping method. The filtering errors present in a dynamic surface control (DSC) approach are effectively eliminated by means of error compensation signals. A prescribed performance control (PPC) is resorted to confine the system output to meet prescribed steady‐state and transient tracking performances. The time delay terms are compensated by a Lyapunov–Krasovskii functional. The semi‐globally ultimately uniformly boundedness of all states in the closed‐loop system is ensured by the Liapunov's stability criterion. Finally, the validity of the developed control scheme is further verified through a simulation example.

Design of a Wind Tunnel Nozzle Section Trolley Control System Based on Backstepping Method

Abstract To accommodate different types of supersonic wind tunnel tests, frequent replacements of nozzle sections and test sections are required, involving multiple complex technical processes. To optimize operational procedures and improve efficiency, a control system for the wind tunnel nozzle section trolley based on the backstepping method was designed. The system uses radio frequency identification (RFID) signals for station positioning and intelligently adjusts the speed and direction of the control motor, thus driving the trolley to move back and forth automatically. Through the simulation of the motor motion control model, the system effectively reduces random disturbances in the motion system caused by load changes, achieves precise positioning of wind tunnel plugins and nozzle assemblies, enhances system operating accuracy, reduces production cycle times, and realizes a one-button positioning function for all components, playing a significant role in improving wind tunnel test efficiency and operational quality.

Design and analysis of UPQC in a microgrid using model reference adaptive control ensemble with back-stepping controller

In recent years, the power sector has shifted to decentralized power generation, exemplified by microgrids that combine renewable and traditional power sources. With the introduction of renewable energy resources and distributed generators, novel strategies are required to improve reliability and quality of power (PQ). In our proposed system, a model consisting of photovoltaics, wind energy, and fuel cells has been designed to share a network, bolstered by the integration of UPQC to rectify PQ issues. Notably, our model introduces a Back-stepping controller method featuring Model Reference Adaptive Control (MRAC) with online parameter tuning, offering superior adaptability and responsiveness. This approach not only ensures optimal grid management but also enhances efficiency and stability. Furthermore, the proposed model demands minimal additional infrastructure, leveraging existing resources to streamline implementation and maintenance, thereby promoting sustainability and cost-effectiveness. The research culminates in a comparative analysis between the MRAC-Back-stepping controller, Adaptive Neuro-Fuzzy Inference System (ANFIS), and Fuzzy controller, highlighting the efficacy and versatility of our proposed model in microgrid operations. A Matlab model has been designed along with a hardware setup to demonstrate the robustness of the model.

Distributed Adaptive Secure Consensus Tracking Control for Asynchronous Switching Nonlinear MASs Under Sensor Attacks and Actuator Faults

A distributed adaptive secure control algorithm is proposed for asynchronous switching nonlinear multi-agent systems (MASs) in pure-feedback form while considering the sensor attacks and actuator faults, which eliminates the effect of attack signals, actuator faults and uncertainties without knowing the sign of the control gain functions. Since the system states measured by the sensors are corrupted by the malicious attacker, only the post-attack system states can be obtained. Thus, an improved coordinate transformation is constructed by using the post-attack system states and a backstepping method based on the improved coordinate transformation is developed to solve the consensus tracking control problem of the researched systems under sensor attacks. The common Lyapunov function is designed to ensure the stability of all asynchronous switching subsystems of each agent. In the end, a simulation example is presented to illustrate the theoretical results’ validity. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper deal with the distributed adaptive secure consensus tracking control problem for asynchronous switching nonlinear MASs, whose models can describe many phenomena in nature such as biological processes, robot team formation and sensor networks. It is extremely challenging to achieve the tracking control problem of the researched asynchronous switching nonlinear system in an unsafe network environment, where the unknown sensor attacks signals, actuator faults coefficients, and randomly switched non-affine nonlinearities exist together for the followers. Furthermore, the control design and stability analysis of the researched system is based on the common Lyapunov approach, which makes the developed approach more engineering oriented.

Jerk Chaotic System: Analysis, Circuit Simulation, Control and Synchronization with Application to Secure Communication

Dynamical behaviors of 3D Jerk system were examined in terms of equilibrium, stability, dissipative, and phase space attractor in this study. The system’s practical applications were demonstrated through circuit realization and synchronization scheme via active backstepping control with its effectiveness demonstrated in secure communication. The viability of the theoretical model of 3D Jerk system was confirmed using electronic circuit workbench designed in MultiSIM enviroment. A nonlinear feedback controller was designed using the recursive backstepping technique to control and track a desire function. For secure communication application, active backstepping method was adopted to synchronize two identical chaotic systems evolving from different initial conditions. It was demonstrated that when the controller was activated, the systems synchronize successfully. The results of the active backstepping designed controllers were numerically applied in the area of secure communication, with the variable of the drive being encrypted information transmitted through a coupling channel. Using an additive encryption masking scheme, the encrypted signal was a superposition of sinusoidal information specified by period function and chaotic carrier generated from a variable of the Jerk system. The transmitted information signal was successfully retrieved from the chaotic response signal using an inverse function decryption algorithm, thereby confirming the effectiveness and robustness of the designed controller.

Unknown input observer based neuro-adaptive fault-tolerant control for vehicle platoons with sensor fault and output quantization

Sensor fault and output quantization are common issues acting on vehicle platoon, and they may lead to performance deterioration, instability and even insecurity of the platoon. Therefore, this paper investigates the fault-tolerant control (FTC) problem of vehicle platoons with regard to the above two issues. Considering the probabilistic sensor fault and quantized measurement signals, an unknown input observer (UIO) based fault detection algorithm with adaptive threshold is developed for sensor health status monitoring. Then, an augmented vehicle platoon model is constructed by introducing a low-pass output filter, and a robust UIO is established for state reconstruction. Based on the above results, a fault-tolerant control scheme is exploited by employing the back-stepping control method and adaptive radial basis function neural network (RBF NN) approximation technique, which is proved to be capable of achieving the time-domain string stability (TSS) of vehicle platoons in the presence of sensor fault and output quantization. Simulation results demonstrate the effectiveness of the proposed algorithms.