Backstepping Research Articles

Towards Sustainable Transportation: Adaptive Trajectory Tracking Control Strategies of a Four-Wheel-Steering Autonomous Vehicle for Improved Stability and Efficacy

The objective of continuous increase in the evolution of autonomous and intelligent vehicles is to attain a trustworthy, economical, and safe transportation system. Four-wheel steering (4WS) vehicles are favored over traditional front-wheel steering (FWS) vehicles because they have excellent dynamic characteristics. This paper exhibits the trajectory tracking task of a two degree of freedom (2DOF) underactuated 4WS Autonomous Vehicle (AV). Because the system is underactuated, MIMO, and has a nontriangular form, the traditional adaptive backstepping control scheme cannot be utilized to control it. For the purpose of rectifying this issue, two-state feedback-based methods grounded on the hierarchical steps of the block backstepping controller are proposed and compared in this paper. In the first strategy, a modified block backstepping is applied for the entire dynamic system. Global stability of the overall system is manifested by Lyapunov theory and Barbalat’s Lemma. In the second strategy, a block backstepping controller has been applied after a reduction of the high-order model into various first-order subsystems, consisting of Lyapunov-based design and stability warranty. A trajectory tracking controller that can follow a double lane change path with high accuracy is designed, and then simulation experiments of the CarSim/Simulink connection are carried out against various vehicle longitudinal speeds and road surface roughness to demonstrate the effectiveness of the presented controllers. Furthermore, a PID driver model is introduced for comparison with the two proposed controllers. Simulation outcomes show that the proposed controllers can attain good response implementation and enhance the 4WS AV performance and stability. Indeed, enhancement of the stability and efficacy of 4WS autonomous vehicles would afford a sustainable transportation system by lessening fuel consumption and gas emissions.

Fixed-Time Backstepping Sliding-Mode Control for Interleaved Boost Converter in DC Microgrids

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.

Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell

Fuel cell electric vehicles (FCEVs) are among the devices that have emerged in recent years. To provide electricity to the electric motors, they use a proton-exchange membrane fuel cell (PEMFC) as the primary energy source and a secondary source consisting of an energy storage system (battery or supercapacitors). The addition of these sources to the motors and accessories of a vehicle requires the association of static converters to condition the different power sources. In addition, a high-efficiency and enhanced-reliability power converter is essential to connect the PEMFC to the vehicle’s DC bus. This paper proposes a robust feedback controller for a four-phase interleaved boost converter used with PEMFC. The proposed controller has double loops based on a state-feedback controller, and an inner loop which translates the differential equation of the system into a state representation by linearization around its operation points. The reference current is generated by state feedback in the outer loop; the state variable is defined by using a change variable. The strong robustness and highly dynamic characteristics of the proposed controller are demonstrated through its performance in terms of output voltage, source current, and settling time. The findings indicate that the proposed controller achieves a response time of 20 ms, resulting in an over 50% improvement compared to the controllers referenced in the literature. Additionally, it reduces both current and voltage ripple, keeping them each below 10%. Further, the controller gains synthesis is validated using the linear quadratic regulator (LQR) technique as well as boundary conditions, and its robustness is verified, taking into account the uncertainty of various operating conditions and discrepancies in circuit components. A double-loop super-twisting sliding mode controller, a backstepping control algorithm, and a PI controller are selected for comparison and discussion. Subsequently, the effectiveness of the proposed controller is evaluated through simulation with the parameters of a 500 W fuel cell system.

Fuzzy adaptive asynchronous sampled-data consensus control for nonlinear multi-agent systems under DoS attacks

This article investigates the problem of fuzzy adaptive resilient asynchronous sampled-data consensus control (SDCC) for a class of uncertain nonlinear strict-feedback multi-agent systems (MASs) under denial-of-service (DoS) attacks and with a directed graph. Firstly, the upper bounds on the sampling intervals and a distributed sampled-data observer are proposed to estimate the leader's output and its high-order derivatives under DoS attacks. Then, the sufficient conditions to ensure the asymptotic convergence of the designed distributed sampled-data observer are established. Based on the distributed sampled-data observer and backstepping control design technique, a distributed fuzzy adaptive resilient control algorithm is developed. It is proved that the proposed control scheme can guarantee that the controlled MASs are stable, and the consensus errors converge to a small neighbourhood of zero. Finally, a simulation example on heterogeneous vehicular platoons verifies the feasibility of the proposed control scheme.

Event-Triggered Backstepping Control of Fractional-Order Chaotic Systems with Dead Zone Via Disturbance Observer

This paper comes up with an adaptive event-triggered tracking control scheme for a kind of fractional-order strict feedback nonlinear systems with actuator dead-zone inputs and uncertain external disturbances. The presented strategy can overcome the defect in standard backstepping control scheme which leads to the “explosion of complexity” problem of virtual signals, and strengthen the transmission efficiency by an event-triggered scheme. An error compensation mechanism is presented for promoting the tracking accuracy and reducing the effect filter error. The designed controller guarantees that the tracking error converges into a small neighborhood near the origin, and all closed-loop signals are bounded. Moreover, Zeno behavior will be avoided. The numerical simulation is given to verify the accuracy and effectiveness of the designed technique.

Enhanced composite controller for PV/PMSG/PEMFC and BESS‐based DC microgrids voltage regulation: Integrating integral terminal sliding mode controller and recursive backstepping controller

AbstractThe variability of renewable energy sources due to weather patterns often leads to a mismatch between power generation and consumption within microgrids (MGs). This challenge is exacerbated when integrating bio‐renewable units, complicating stability maintenance in MGs. This research work introduces a novel solution to address this issue: a composite controller merging an integral terminal sliding mode controller with a recursive backstepping controller for direct current MGs (DCMGs). The proposed DCMG incorporates solar photovoltaic units, wind farms based on permanent magnet synchronous generators, proton exchange membrane fuel cells fuelled by hydrogen gas, an electrolyser, battery energy storage systems, and DC loads. First, a comprehensive mathematical model for the components within the DCMG is developed to design the proposed composite controller. This controller not only overcomes the inherent limitations and convergence issues of conventional SMCs but also ensures stable DC‐bus voltage and maintains power balance across various operational conditions. Moreover, a fuzzy logic‐based energy management system is introduced to regulate power flow, considering factors like battery state of charge and renewable energy sources' total power output. The control Lyapunov function confirms the DCMG system's asymptotic stability. Finally, the proposed controller's effectiveness is validated through simulations on both MATLAB/Simulink and Arduino Mega 2650 processor‐in‐the‐loop platforms under various operational conditions. In both platforms, the proposed controller surpasses an existing controller in terms of settling time, overshoot, and tracking error of the DC‐bus voltage.

A Novel Robust Hybrid Control Strategy for a Quadrotor Trajectory Tracking Aided with Bioinspired Neural Dynamics

This paper introduces a novel hybrid control strategy for quadrotor UAVs inspired by neural dynamics. Our approach effectively addresses two common issues: the velocity jump problem in traditional backstepping control and the control signal chattering in conventional sliding mode control. The proposed system combines an outer-loop bioinspired backstepping controller with an inner-loop bioinspired sliding mode controller, ensuring smooth trajectory tracking even under external disturbances. We rigorously analyzed the system’s stability using Lyapunov stability theory. To validate our algorithm’s effectiveness, we conducted trajectory tracking experiments in both disturbance-free and step-disturbance conditions, comparing it with the traditional backstepping control, conventional sliding mode control, and saturated sliding mode control. The results demonstrate that our algorithm not only tracks trajectories more effectively but also significantly outperforms these methods in suppressing velocity jumps and signal chattering.

A reference model-based backstepping anti-interference control method and its application to inverted pendulum

A reference model-based backstepping anti-interference control method is proposed, which performs well in the application of Inverted Pendulum system. Firstly, the backstepping controller of the single-stage straight-line inverted pendulum is designed and the reference model of the system is obtained. Input the actual state variables into the reference model. If the single-stage straight-line inverted pendulum is disturbed, it will deviate from the ideal value of the reference model. In order to improve the anti-interference performance of single-stage straight-line inverted pendulum system, the generalized state error is introduced, that is, the error between the output of the reference model and the actual output of the system. Then the controller is designed by Lyapunov stability theory based on generalized state error and backstepping control method. Finally, the reference model-based backstepping anti-interference controller is obtained. MATLAB/Simulation results show that the controller can effectively suppress the influence of external interference on the single-stage straight-line inverted pendulum. It has strong robustness and anti-interference ability.

Analysis of fly-around mission with spinning tether system for space station observation

This paper presents the mission scenarios of using spinning tether system to conduct space stations fly-around mission and validates its feasibility. The main challenge of fly-around mission lies in the difficulty of balancing low fuel consumption and long-term fly-around observation. To deal with this problem, a novel spinning tether system is proposed. Firstly, the fly-around process with spinning tether system is introduced, and the tether system is modeled based on Newton-Euler method with a novel description of spinning motion. Secondly, Given the unique structural limitations of space stations, two fly-around schemes and referenced fly-around trajectories are detailed. Thirdly, a backstepping controller is proposed for tracking the reference motion of fly-around satellites, and the fuel consumption among different fly-around schemes is compared and analyzed. In the end, numerical results validate that under the proposed control strategy, the spinning tether system can maintain a stable fly-around configuration in both the planar and vertical plane, the symmetrical formation configuration prevents the central space station from being affected by the motion of fly-around satellites. Moreover, energy consumption analysis indicates that tethered system can save 62.8 % of impulse compared to traditional schemes when flying in the planar plane, making it the most energy-efficient option.

Nonlinear integral backstepping control based on particle swarm optimization for a grid-connected variable wind energy conversion system during voltage dips

Double-fed induction generator (DFIG) wind turbines connected to the grid are particularly subject to grid problems such as voltage dips. Because of this, it might be challenging to maintain system stability and prevent system disconnections when using a Proportional-Integral (PI) Controller to operate this system. This paper applies the Integral Backstepping Control for the wind power plant system connected to the power grid. The Integral Backstepping is a nonlinear and recursive method that employs the Lyapunov theory to ensure the system's stability. The best selection of gain values for the Lyapunov function guarantees improved system control. These gains are frequently adjusted using the trial-and-error strategy, which is time-consuming and inefficient. This technique becomes more complex when many parameters need to be determined. It also limits the system's performance and restricts this nonlinear approach's advantages. The objective is to apply the particle swarm optimization for computing several constant values of the nonlinear approach by minimizing an integral absolute error criterion index. The weighted sum of errors is employed to solve the multiple objective problems. The proposed controller tracks the maximum power point, maintains the voltage of the DC-Link constant, and controls active and reactive power. The robustness of this method is verified in critical conditions, including system parameter variation and asymmetrical and symmetrical grid faults. The simulation findings highlight the effectiveness and robustness of the combination of Integral Backstepping with Particle Swarm Optimization in terms of reducing response time from 10.6 (ms) to 2 (ms), canceling static error, and improving overshoot compared to the vector control based on PI regulator. Besides, the DC-Link voltage ripples during the asymmetrical grid faults are reduced to ±1 (V) using the suggested controller. The latter can be implemented thanks to advancements in Central Processor Unit technology.

UAV Trajectory Tracking Using Proportional-Integral-Derivative-Type-2 Fuzzy Logic Controller with Genetic Algorithm Parameter Tuning.

Unmanned Aerial Vehicle (UAV)-type Quadrotors are highly nonlinear systems that are difficult to control and stabilize outdoors, especially in a windy environment. Many algorithms have been proposed to solve the problem of trajectory tracking using UAVs. However, current control systems face significant hurdles, such as parameter uncertainties, modeling errors, and challenges in windy environments. Sensitivity to parameter variations may lead to performance degradation or instability. Modeling errors arise from simplifications, causing disparities between assumed and actual behavior. Classical controls may lack adaptability to dynamic changes, necessitating adaptive strategies. Limited robustness in handling uncertainties can result in suboptimal performance. Windy environments introduce disturbances, impacting system dynamics and precision. The complexity of wind modeling demands advanced estimation and compensation strategies. Tuning challenges may necessitate frequent adjustments, posing practical limitations. Researchers have explored advanced control paradigms, including robust, adaptive, and predictive control, aiming to enhance system performance amidst uncertainties in a scientifically rigorous manner. Our approach does not require knowledge of UAVs and noise models. Furthermore, the use of the Type-2 controller makes our approach robust in the face of uncertainties. The effectiveness of the proposed approach is clear from the obtained results. In this paper, robust and optimal controllers are proposed, validated, and compared on a quadrotor navigating an outdoor environment. First, a Type-2 Fuzzy Logic Controller (FLC) combined with a PID is compared to a Type-1 FLC and Backstepping controller. Second, a Genetic Algorithm (GA) is proposed to provide the optimal PID-Type-2 FLC tuning. The Backstepping, PID-Type-1 FLC, and PID-Type-2 FLC with GA optimization are validated and evaluated with real scenarios in a windy environment. Deep robustness analysis, including error modeling, parameter uncertainties, and actuator faults, is considered. The obtained results clearly show the robustness of the optimal PID-Type-2 FLC compared to the Backstepping and PID-Type-1 FLC controllers. These results are confirmed by the numerical index of each controller compared to the PID-type-2 FLC, with 12% for the Backstepping controller and 51% for the PID-Type-1 FLC.

Reduced ultimate-bound of tracking error convergence for ETV system with unknown upper-bound mismatched perturbation

The electronic throttle valve (ETV) is widely used in automobile engines to obtain efficient fuel combustion, improved fuel economy, and reduced emissions. The control of the ETV system is a challenging problem due to the unmatched structure of perturbation and its unknown upper bound. As such, deep control concerns are required. For such a system, recent control approaches could partially solve the ultimate boundedness of convergence error, but they have failed to find a solution in the presence of unknown upper-bound perturbation. The contribution of this study is to develop a novel backstepping-based barrier function integral sliding mode controller (BS-BFISMC) to find a solution for the ETV system without the pre-knowledge of the upper bound of perturbation. The proposed control scheme gets the benefits of the backstepping algorithm (BS) and barrier function integral sliding mode control (BFISMC), and the combined scheme could lead to reduced ultimate-bound convergence of tracking error and result in chattering-free control action. A rigorous stability analysis has been conducted to prove the finding reached by the proposed control approach. The efficacy and effectiveness of the proposed controller have been shown by conducting a comparison study with other existing control methods; namely, conventional backstepping controller, robust continuous backstepping (CRBS), and backstepping-based quasi-integral sliding mode controller (BS-QISMC). The numerical results showed that the proposed BS-BFISMC yields the lowest level of ultimate bound of tracking errors and the least control efforts as compared to other controllers. Also, it has been shown that the proposed controller does not require the upper bound of perturbation, which is a pre-requisition of all compared controllers.

Adaptive neural network backstepping control for the magnetorheological semi-active air suspension system with uncertain mass and time-varying input delay

Aiming at rejecting external disturbance induced by the random road profile, this paper proposes an adaptive robust control strategy to address the control problem of semi-active air suspension system installed with magnetorheological dampers (MRD) for enhancing ride comfort and handling performance. To effectively depict the inherent nonlinear dynamics of the adjustable air spring, a radial basis function neural network (RBFNN) model is trained by the experimental data offline, and an adaptive learning algorithm is introduced to maintain the modeling accuracy. Moreover, to handle the prevalent sensitive parameter variation (such as sprung mass), a nonlinear estimator is designed to estimate the uncertain sprung mass. Furthermore, a delay compensator is integrated into the control law to mitigate the impact of the time-varying input delay caused by the actuator of MRD to restrain the chattering phenomena. Additionally, the stability of the closed-loop system is rigorously proven by employing a Lyapunov–Krasovskii functional, guaranteeing the boundedness of both tracking error and estimation error. Simulation results are displayed and analyzed, illustrating the feasibility and efficiency of the proposed control strategy in depressing the sprung mass acceleration and tire deflection, showing its significance in both control performance enhancement and chattering elimination.

Finite-time robust control of morphing aircraft based on time-varying observer

Morphable vehicles have strong nonlinear, strong coupling and strong time-varying characteristics in the process of variation, which brings great challenges to the design of their flight control systems. To address this problem, a time-varying gain extended state observer is proposed to accurately approximate the system coupling terms. Combining adaptive backstepping control technology and finite-time control theory, a finite-time bounded robust adaptive controller is designed. By designing the barrier Lyapunov function, the actual control law and the adaptive network update law are obtained step by step, ensuring that the system command tracking error can converge to a predetermined range within a finite time. Finally, the effectiveness of the designed control system is verified by attitude control simulation. The simulation results show that the morphable vehicle can track the command signal well in the process of variation and is basically not affected by the variation rate.

Compensation Function Observer-Based Backstepping Sliding-Mode Control of Uncertain Electro-Hydraulic Servo System

Observer-based control is the most commonly used method in the control of electro-hydraulic servo system (EHSS) with uncertainties, but it suffers from the drawback of low accuracy under the influence of large external load forces and disturbances. To address this problem, this paper proposes a novel compensation function observer-based backstepping sliding-mode control (BSMC) approach to achieve high-accuracy tracking control. In particular, the model uncertainties, including nonlinearities, parameter perturbations and external disturbances are analyzed and treated together as a lumped disturbance. Then, a fourth-order compensation function observer (CFO) is constructed, which fully utilizes the system state information to accurately estimate the lumped disturbance. On this basis, the estimate of the lumped disturbance is incorporated into the design of a backstepping sliding-mode controller, allowing the control system to compensate for the disturbance effect. The stability of the closed-loop control system under the CFO and BSMC is rigorously proven through the use of the Lyapunov theory, which guarantees that all the tracking error signals converge exponentially to the origin. Comparative simulations are carried out to show the effectiveness and efficiency of the proposed approach, i.e., compared with PID and ESO-based BSMC methods, the tracking accuracy is respectively improved by 94.86% and 88.19% under the influence of large external load forces and disturbances.

Enhanced control of grid-connected multi-machine wind power generation systems using fuzzy backstepping approaches

This research paper presents an approach for enhancing the performance of a multi-machine wind power generation system (WPGS) through the combination of nonlinear and intelligent control techniques. The suggested approach focuses on optimizing the WPGS, which utilizes permanent magnet synchronous generators (PMSGs) interconnected with the grid. To enable grid connection, a back-to-back configuration is employed for both the rectifier and inverter in a fifteen-switch configuration, with control achieved through pulse width modulation. The designed control algorithm is designed to effectively regulate the system and reduce the active and reactive power ripples. Initially, the WPGS model is provided. Subsequently, the fuzzy backstepping control (FBC) law is detailed, incorporating the Lyapunov stability technique. Fuzzy logic is used to adapt the gains in the backstepping approach, ensuring the control system can accommodate disturbances and system variations. As a result, this adaptive control approach enables optimal effusiveness of the entire system in both static and dynamic operational regimes. Experimental tests were conducted using MATLAB to compare the efficacy of the designed strategy against the traditional backstepping control approach. The obtained results demonstrated the robustness and effective reference tracking capabilities of the designed FBC approach, along with the complete elimination of speed overshoot across diverse wind conditions. These findings validate the designed control approach's effectiveness and superiority over the BC approach under varying conditions.

A Model Predictive Backstepping Control Approach for Angle Tracking of Steer-by-Wire System

A Model Predictive Backstepping Control Approach for Angle Tracking of Steer-by-Wire System

Artificial intelligent-based backstepping control of parallel-connected multi-machine wind generation system using a new fifteen-switch rectifier

This academic work offers a new variable-speed wind power generation system (NVS-WPGS) that employs two 5-phase permanent magnet synchronous generators (PMSGs) controlled through a fifteen-switch rectifier (FSR) architecture for a system that’s integrated to the power grid. The two 5-phase PMSGs are linked to the DC bus through an FSR, which, in turn, connects the DC link to the grid via a three-phase inverter. To ameliorate the operational efficiency of the wind system under investigation, the backstepping control (BSC) approach was employed. Building upon this technique, the Adaptive Neuro-Fuzzy Inference System (ANFIS) was trained and implemented for controlling the wind system. The goal is to boost system performance and control. While BSC is a strong control method for reference signal tracking, its dependence on system parameters is a downside. This dependence typically degrades performance. On the other hand, ANFIS exhibits robustness to variations in system parameters and uncertainties. The efficaciousness of the suggested control methodology was evaluated via experimental investigations conducted using the Matlab/Simulink program. The simulations were performed to verify the system’s capacity to maintain controlled variables in accordance with desired references, even in the presence of fluctuations in wind velocity. Simulations exhibit that the suggested ANFIS-BSC control system accurately tracks controlled intended references. Furthermore, system performance improved. Speed overshoot was decreased by 100%. Additionally, the system’s efficiency was enhanced, reaching a level of 96.5%, beating the BSC approach’s 96%. These findings underscore the remarkable effectiveness of the ANFIS-BSC approach compared to BSC methodologies.

Adaptive backstepping and sliding mode control of a quadrotor

This article proposes a simple robust adaptive control architecture with minimum parameter estimation laws for trajectory tracking of a quadrotor subjected to parametric variations and environmental disturbances. This simple control architecture aims to achieve highly accurate, robust, and fast trajectory tracking of the quadrotor in a short time with low computational cost. Firstly, the quadrotor model is divided into altitude, attitude, and position subsystems for which appropriate control methods are designed without prior knowledge of the upper bound of external disturbances. A simple adaptive fractional-order sliding mode control (AFSMC) is designed to enhance the tracking of the altitude subsystem and estimate the upper bound of the disturbances. Then, a simple adaptive backstepping control (ABC) is developed for the horizontal position to generate the required roll and pitch orientations. The adaptation laws not only estimate the upper bound of the disturbances but also adjust the controller gains thereby enhancing the robustness of the ABC. A nonsingular fast terminal sliding mode control (NFTSMC) is incorporated with a finite-time disturbance observer (FDO) to accurately suppress the disturbances, and follow the target rotation angles within a short finite-time. Simulation results showed that the compounded control structure ensures accurate, fast, and robust tracking. The AFSMC can achieve the desired altitude with a settling time of 0.31 s and root mean square error (RMSE) of 0.0454 m. The ABC can attain the target horizontal position coordinates with a settling time of (0.47 s, 0.47 s) and RMSE of (0.0370 m, 0.0518 s). The NFTSMC-FDO can achieve the desired attitude angles with a settling time of (0.11 s, 0.13 s, 0.52 s) and RMSE of (0.0098 rad, 0.0092 rad, 0.0935 rad). Performance comparisons with existing control methods in terms of settling time and RMSE demonstrated that the proposed control architecture is superior.

Adaptive predefined-time consensus for nonlinear multi-agent systems with input delay and performance constraints

This paper investigates the predefined-time tracking consensus issue for a class of non strict-feedback nonlinear multi-agent systems (MASs) under input delay and error performance constraints by adaptive backstepping control strategy. Firstly, the impact of input delay is eliminated by Pade approximation. With the help of an intermediate state variable, the considered MASs with input delay are revised as the independent-delay systems. Secondly, under constrained distributed error and corrected nth virtual error, the ln-type barrier Lyapunov function (BLF) is adopted to solve the error performance constrains and reduce the complexity of differentiation. Meanwhile, an adaptive predefined-time controller is designed to realize the tracking purpose by backstepping method and radial basis function neural networks (RBFNNs) approximation. Finally, a numerical simulation example and a practical application are presented to test the effectiveness for the proposed control strategy.