Backstepping Method Research Articles

Occlusion Avoidance in Image-Based Control of Multiple Spacecraft

This study develops a novel image-based control scheme to address the on-orbit service mission that multiple spacecraft track a tumbling uncooperative target under a visibility constraint. The chaser spacecraft are commanded to hover above and track stably the feature points attached to the target as well as to avoid the mutual visual occlusion caused by interspacecraft motion. To this end, a new expression of image-based visual servoing dynamics for spacecraft is derived, where the physical meaning of each term is analyzed in detail so that the exact upper bounds of noncooperative terms can be estimated. After that, a novel occlusion description model is established by the projective theory of quadrics. The mathematical elements for describing the imaging elliptic curve and occlusion discrimination function are proposed, whose association with interspacecraft motion is given to guide occlusion avoidance controller design. Combining the artificial potential function with the backstepping control method, a robust control algorithm is designed. The proposed control scheme can achieve the six-degree-of-freedom control objective based on image space, and motion trajectory will not influence others’ visibility. Finally, the validity of the control scheme is verified by simulation for a given multispacecraft observation scenario.

Deep reinforcement learning based active surge control for aeroengine compressors

This study proposes an active surge control method based on deep reinforcement learning to ensure the stability of compressors when adhering to the pressure rise command across the wide operating range of an aeroengine. Initially, the study establishes the compressor dynamic model with uncertainties, disturbances, and Close-Coupled Valve (CCV) actuator delay. Building upon this foundation, a Partially Observable Markov Decision Process (POMDP) is defined to facilitate active surge control. To address the issue of unobservability, a nonlinear state observer is designed using a finite-time high-order sliding mode. Furthermore, an Improved Soft Actor-Critic (ISAC) algorithm is developed, incorporating prioritized experience replay and adaptive temperature parameter techniques, to strike a balance between exploration and convergence during training. In addition, reasonable observation variables, error-segmented reward functions, and random initialization of model parameters are employed to enhance the robustness and generalization capability. Finally, to assess the effectiveness of the proposed method, numerical simulations are conducted, and it is compared with the fuzzy adaptive backstepping method and Second-Order Sliding Mode Control (SOSMC) method. The simulation results demonstrate that the deep reinforcement learning based controller outperforms other methods in both tracking accuracy and robustness. Consequently, the proposed active surge controller can effectively ensure stable operation of compressors in the high-pressure-ratio and high-efficiency region.

Event-triggered trajectory tracking control for USV with prescribed performance and time delay based on differential flatness

This paper presents a trajectory tracking control scheme for underactuated surface vessels (USVs) with input delay. Firstly, the underactuated surface vessel system is transformed into a fully actuated system using differential flatness theory. To estimate the unknown nonlinear terms introduced in the transformation process, a fuzzy neural network (FNN) is employed. Secondly, to conserve control resources and communication bandwidth, the controller of the system under prescribed performance is designed using the backstepping method. This method updates the controller according to an event-triggered condition that is designed using a Lyapunov function. Finally, theoretical proof and simulation experiments are conducted to demonstrate the convergence and effectiveness of the proposed method.

Modeling and control design for half electric vehicle with wheel BLDC actuator and Pacejka's tire

We are considering the problem of controlling the longitudinal motion of half-vehicle electric vehicle (EV) or hybrid electric vehicle (HEV). A substantial part of the problem lies in the modeling of the controlled system dynamics, including those of: (i) the half-vehicle chassis motion; (ii) the actuator constituted of a brushless direct-current (BLDC), as an in-wheel motor, along with the associated voltage inverter; (iii) the tire. The dynamics of the latter are presently captured by Pacejka's model which is known to be more most accurate as it accounts for the various phenomena generated at the contact tire/road. The whole system model also takes into account other nonlinear effects, e.g., including the road load, the rolling resistance and the aerodynamic effects. Compared to those used in existing comparable works, the system model developed in this study is much more accurate. But it is also more complex, because it is of higher dimension and stronger nonlinearity, increasing substantially the complexity of the control design problem. The control objectives include tight regulation of the vehicle's speed in the various driving modes and safety of the on-board passengers. These objectives are technically reformulated in terms of reference trajectories, to be matched by the vehicle speeds despite unpredictable variations in aerodynamic forces and road parameters. This challenging control problem is dealt by developing a cascade-structure adaptive controller. The cascade structure is resorted to make the compensation for speed dynamics and disturbances decoupled from those of torque dynamics. The adaptive nature is realized by using parameter estimators, providing the controller with the capacity of compensating the effect of parameter uncertainty. The various controller components are designed making use of several different theoretical design tools including the backstepping control method and Lyapunov stability method. In addition to theoretical analysis, the performances of the developed controller are highlighted by simulation results in the presence of varying aerodynamic forces, road slope and various road surfaces. Comparison with simpler controller is also performed.

Nonsingular adaptive finite-time consensus control for uncertain nonlinear multi-agent systems with input quantization

This paperinvestigates the distributed consensus control for nonlinear multi-agent systems with input quantization in a directed communication topology. A nonsingular adaptive finite-time control (NAFTC) scheme is proposed by combining the filtered backstepping method with neural network control. Both chattering and singularity problems of the control signals are avoided thanks to its [Formula: see text] continuity, so that the contradiction between discontinuous- and [Formula: see text] continuous finite-time control with input quantization does not exist in our scheme. At the same time, a novel inequality for the input-output relationship of hysteresis quantizer is derived, which can simply handle the input quantization without requiring additional stability analysis. What is more, some common assumptions, such as Lipschitz assumption and Holder-continuity assumption, are not required in our scheme. Finally, stability analysis and numerical simulation are provided.

Formation fault-tolerant control for multiple UAVs with external disturbances

PurposeThe purpose of this paper is to construct an event-triggered finite-time fault-tolerant formation tracking controller, which can achieve a time-varying formation control for multiple unmanned aerial vehicles (UAVs) during actuator failures and external perturbations.Design/methodology/approachFirst, this study developed the formation tracking protocol for each follower using UAV formation members, defining the tracking inaccuracy of the UAV followers’ location. Subsequently, this study designed the multilayer event-triggered controller based on the backstepping method framework within finite time. Then, considering the actuator failures, and added self-adaptive thought for fault-tolerant control within finite time, the event-triggered closed-loop system is subsequently shown to be a finite-time stable system. Furthermore, the Zeno behavior is analyzed to prevent infinite triggering instances within a finite time. Finally, simulations are conducted with external disturbances and actuator failure conditions to demonstrate formation tracking controller performance.FindingsIt achieves improved performance in the presence of external disturbances and system failures. Combining limited-time adaptive control and event triggering improves system stability, increase robustness to disturbances and calculation efficiency. In addition, the designed formation tracking controller can effectively control the time-varying formation of the leader and followers to complete the task, and by adding a fixed-time observer, it can effectively compensate for external disturbances and improve formation control accuracy.Originality/valueA formation-following controller is designed, which can handle both external disturbances and internal actuator failures during formation flight, and the proposed method can be applied to a variety of formation control scenarios and does not rely on a specific type of UAV or communication network.

Nonlinear control of a hybrid pneumo‐hydraulic mock circuit of the cardiovascular system

AbstractHybrid cardiovascular mock circuits (HMC), designed for dynamic testing of Ventricular Assist Devices (VAD), offer physiologic accuracy by sequestering model complexity in silico and ease of construction by reducing number of model elements in vitro. Despite superior response time and precision, pneumatic actuation is avoided in HMCs due to nonlinear dynamics and noise. We tested the hypothesis that a HMC consisting of a variable elastance‐driven numerical cardiovascular circuit (CVS) coupled to a pneumo‐hydraulic physical circuit can be controlled without linearizing system dynamics. Reference left ventricular and aortic pressures were generated in silico in a seventh‐order electrical analogue of the CVS and transmitted via an electro‐hydraulic interface to the physical circuit. There, they were tracked in in vitro preload and afterload pneumo‐hydraulic reservoirs, respectively. The nonlinear pneumatic dynamics in the reservoirs was controlled using the Lyapunov stability criterion. A centrifugal pump, the speed (i.e., flow) of which was adjusted manually or using PID control, was interposed between the reservoirs and mimicked the VAD under evaluation. The flow of a recirculating gear pump was controlled by an integral backstepping method to equalize reservoir fluid volumes by rejecting pressure and flow disturbances. Sensor noise was reduced with discrete‐time Kalman filtering. Our results showed that normal, failing, and assisted cardiovascular physiologies simulated in silico were commensurate with clinical data obtained from subjects with similar pathologies. The numerical pressure references were tracked with high accuracy at the physical VAD terminals. Reservoir volumes remained stable at various combinations of heart rate, pressure, and VAD flow for prolonged durations with negligible steady‐state error. The HMC described here offers a stable performance testing platform for VAD prototypes.

Fixed-time stabilization for lower-triangular nonlinear systems under the p-normal form

This article investigates the fixed-time stabilization (FTS) problem for lower-triangular nonlinear systems under the [Formula: see text]-normal form. Under suitable assumptions condition of the system’s nonlinearities, a novel fixed-time partial-state feedback (PSF) control law is designed using backstepping design method and adding a power integrator (API) technique. With the help of one appropriate Lyapunov function, it is shown that the partial-state of the corresponding closed-loop system is fixed-time stable. Numerical simulation is given to illustrate the potency of the suggested continuous partial-sate feedback controller.

Cooperative Tracking Control for Nonlinear MASs Under Event-Triggered Communication.

The neural network-based adaptive backstepping method is an effective tool to solve the cooperative tracking problem for nonlinear multiagent systems (MASs). However, this method cannot be directly extended to the case without continuous communication. It is because the discontinuous communication results in discontinuous signals in this case, the standard backstepping method is inapplicable. To solve this problem, a hierarchical design scheme that involves distributed cooperative estimators and neural network-based decentralized tracking controllers is proposed. By introducing a dynamic event-triggered mechanism, cooperative intermediate parameter estimators are first designed to estimate the unknown parameters of the leader. By using the interpolation polynomial method, these estimators are extended to smooth estimators with high-order derivatives to guarantee that the backstepping method is applicable. Based on the state of the smooth estimators, a backstepping-based decentralized neural network tracking controller is designed. It is shown that the tracking errors are asymptotically convergent and all the signals in the closed-loop systems are bounded. Compared with the existing cooperative tracking results for nonlinear MASs with event-triggered communication, a more general class of MASs is considered in this article and a better performance in terms of asymptotic tracking is achieved. Finally, a simulation example is given to show the effectiveness of our developed method.

Time-Varying Optimal Formation Control for Second-Order Multiagent Systems Based on Neural Network Observer and Reinforcement Learning.

This article addresses a distributed time-varying optimal formation protocol for a class of second-order uncertain nonlinear dynamic multiagent systems (MASs) based on an adaptive neural network (NN) state observer through the backstepping method and simplified reinforcement learning (RL). Each follower agent is subjected to only local information and measurable partial states due to actual sensor limitations. In view of the distributed optimized formation strategic needs, the uncertain nonlinear dynamics and undetectable states may jointly affect the stability of the time-varying cooperative formation control. Furthermore, focusing on Hamilton-Jacobi-Bellman optimization, it is almost incapable of directly dealing with unknown equations. Above uncertainty and immeasurability processed by adaptive state observer and NN simplified RL are further designed to achieve desired second-order formation configuration at the least cost. The optimization protocol can not only solve the undetectable states and realize the prescribed time-varying formation performance on the premise that all the errors are SGUUB, but also prove the stability and update the critics and actors easily. Through the above-mentioned approaches offer an optimal control scheme to address time-varying formation control. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and digital simulation.

Adaptive multi-dimensional Taylor network tracking control of time-varying delay nonlinear systems subject to input saturation

For the tracking control problem of a class of nonlinear systems with input saturation constraint and time-varying delay, an adaptive tracking control scheme based on multi-dimensional Taylor network (MTN) is designed, and the proposed control scheme has the advantages of simple structure and small computation. A Gaussian error function is introduced to transform the input saturation into a linear model with bounded error. The Lyapunov-Krasovskii function is constructed, and the nonlinearity of the system is approximated using MTN, and an adaptive control strategy is constructed based on the backstepping method. The results show that the proposed control method can both ensure that all signals in the closed-loop system are bounded and achieve convergence of the tracking error to a small neighborhood near the origin. Numerical simulations verify the effectiveness of the method.

Toward the Intelligent, Safe Exploration of a Biomimetic Underwater Robot: Modeling, Planning, and Control.

Safe, underwater exploration in the ocean is a challenging task due to the complex environment, which often contains areas with dense coral reefs, uneven terrain, or many obstacles. To address this issue, an intelligent underwater exploration framework of a biomimetic robot is proposed in this paper, including an obstacle avoidance model, motion planner, and yaw controller. Firstly, with the aid of the onboard distance sensors in robotic fish, the obstacle detection model is established. On this basis, two types of obstacles, i.e., rectangular and circular, are considered, followed by the obstacle collision model's construction. Secondly, a deep reinforcement learning method is adopted to plan the plane motion, and the performances of different training setups are investigated. Thirdly, a backstepping method is applied to derive the yaw control law, in which a sigmoid function-based transition method is employed to smooth the planning output. Finally, a series of simulations are carried out to verify the effectiveness of the proposed method. The obtained results indicate that the biomimetic robot can not only achieve intelligent motion planning but also accomplish yaw control with obstacle avoidance, offering a valuable solution for underwater operation in the ocean.

A simplified model‐based nonlinear control with fast response and simple design flow for HB‐LLC resonant converter

SummaryLLC resonant converters have attracted much attention for its characteristics of high efficiency and high‐power density in recent years. Yet, with the diversity of application needs such as photovoltaic power generation and electric vehicles, traditional control solutions have gradually been unable to meet the increasing requirements, such as fast response to parameters perturbations. Besides, LLC converter is a high‐order nonlinear system, and it adds complexity to controller design. In order to solve these issues, taking the HB‐LLC converter as an example, this paper presents a simplified model‐based nonlinear control with fast response and simple design. The proposed controller is based on a second‐order model of LLC circuit, and the reverse design idea of the backstepping method is used to simplify the controller design. An experimental platform is built, and the results show that compared with PID control and linear active disturbance rejection control (LADRC), the proposed method has much smaller voltage undershoot (approximately 53.1% of PID and 68.4% of LADRC) and shorter settling time (approximately 27.8% of PID and 39.3% of LADRC) under input voltage change.

Tuning function‐based command filtered fault‐tolerant fixed‐time control for nonlinear systems with sensor/actuator faults

AbstractIn this article, the issue of fixed‐time fault‐tolerant control is studied for a class of nonlinear systems with sensor/actuator faults. The tuning functions and command filters are used to solve the problem of over‐parameterization and differential explosion in traditional backstepping method. The above methods simplify the derivation and reduce the calculation burden of the system respectively. Moreover, the introduction of fixed‐time control further improves the rapidity of the system. Another feature of this design is that the fixed‐time stability is finally realized by introducing a special Lyapunov function, rather than the practical fixed‐time stability. The tracking error can be guaranteed to converge to the origin in a fixed time. Simulation results show that the proposed control strategy is effective even if there are actuator/actuator faults in the system.

Fixed‐time adaptive fault‐tolerant tracking control for uncertain strict‐feedback nonlinear systems via command filtered backstepping

AbstractIn this article, the fault‐tolerant tracking control is addressed for uncertain strict‐feedback nonlinear systems with actuator faults. Neural networks are utilized to identify unknown dynamics in strict‐feedback nonlinear systems, and the adaptive technique is employed to estimate the parameter of actuator effectiveness. More importantly, a command filtered backstepping control method is improved by introducing a fixed‐time command filter and modifying virtual control laws with compensation mechanisms. By incorporating the adaptive neural networks into the command filtered backstepping design framework, a novel adaptive fault‐tolerant control law is constructed. Under the presented control law, the negative influence of the actuator fault and unknown dynamics is effectively compensated simultaneously. Besides, the “explosion of complexity” and “singularity” problems of backstepping is avoided. Moreover, the practical fixed‐time stability is guaranteed for the resulted closed‐loop system.

Observer‐based adaptive controller for a class of fractional‐order uncertain systems subject to unknown input and output quantizers and input nonlinearities

This work addresses design of an adaptive controller for nonlinear fractional‐order (FO) systems in the strict‐feedback form. The system is subject to unknown dynamics, unavailable state variables, quantized input and output, and input nonlinearity. The fuzzy logic system (FLS) is used to estimate the unknown dynamics, and instead of updating all the regressor weights, only the maximum of their norms is updated, which significantly reduces the computational load. Limitations imposed by data transmission bandwidth have been addressed by incorporating sector‐bounded quantizers, which include both logarithmic and hysteresis quantizers at the system input and output, and their errors have been considered in the controller design. It is assumed that system states are not measured. To estimate the system states, a linear observer has been used. Because of output quantization, the continuous output signal is not available, and therefore the observer has been designed based on the quantized output signal. In addition, the constraints of input nonlinearities, including symmetric and asymmetric saturation, backlash, hysteresis, and dead‐zone, have been considered in the controller design. Input nonlinearity and input quantization have been investigated simultaneously, which makes the analysis more challenging. Moreover, it is assumed that the type of input nonlinearity and its characteristics are not known, which makes the controller design more difficult. These problems have been resolved by using two novel adaptive laws. The adaptive backstepping method and the direct Lyapunov stability analysis have been used to construct the controller, and a nonlinear modified FO filter is used to avoid the explosion of complexity occurring in the traditional backstepping technique. Stability of the closed loop system has been established, and it has been shown that the tracking and state estimation errors converge to a small neighborhood of the origin. Finally, the proposed controller performance has been demonstrated via three simulation examples.

Adaptive finite-time incremental backstepping fault-tolerant control for flying-wing aircraft with state constraints

This paper proposes an incremental nonlinear fault-tolerant control scheme for flying-wing aircraft subject to state constraints under model uncertainties, external disturbances, and actuator faults. Firstly, the aircraft attitude dynamics system with state constraints is transformed into a free-constrained system by a nonlinear state-dependent function (NSDF), which simplifies the controller design in a direct and concise way. Based on the NSDF, a novel finite-time incremental backstepping (FT-IBS) method is proposed to achieve rapid attitude tracking and reduce the reliance on model knowledge, simultaneously. In this method, a command filter is designed to address the issues of complexity explosion and singularity, and particular stability compensators ensures the closed-loop system to achieve finite-time stability. Then, to overcome the lumped disturbances composed of uncertainties and faults, a new adaptive multivariable finite-time disturbance observer (AMFTDO) is designed for rapid estimation and compensation, and its adaptive strategy ensure high-precision tracking performance under complex time-varying lumped disturbances. Finally, comparative simulation results confirm that the proposed control scheme can achieve faster convergence, higher tracking accuracy, and stronger robustness.

Backstepping-based adaptive fuzzy tracking control for pure-feedback nonlinear multi-agent systems

This work developed an adaptive fuzzy tracking control strategy for pure-feedback nonlinear multi-agent systems via backstepping. The relevant unknown functions are addressed by fuzzy logic systems (FLSs), which take into account entirely unknown nonlinearities. The non-affine pure-feedback form is dealt with a Butterworth low-pass filter (BLF), and the converted systems can be used to construct a controller. The implementation of the first-order sliding-mode differentiator overcomes the difficulty of calculation explosion. The backstepping method is applied to design the virtual controller step by step, and the actual controller is designed successfully. Furthermore, through Lyapunov stability theory, the stability of systems is ensured, and that the tracking error is kept to a minimum. Finally, the simulation result shows the validity of the designed control strategy.

Stabilization of Fuzzy Prey-Predator Model Using Backstepping Method

Ecological system depends on prey predator interaction and therefore, as a result, diseases may spread among prey or predator or both of them. In this work, the fuzzy logic-based systems are used to elaborate a prey-predator model to study the effect of varying in the inflection rate. Formulation of prey predator model using fuzzy logic is more realistic depiction of the phenomena, since the initial population estimates may not be precisely known in the real-life situation, therefore the initial conditions may also be considered as fuzzy. The dynamical behaviour of the fuzzy exploited system is studied by using the backstepping method. Some references working on prey-predator model, in which they used classical control schemes with a very long and complicated steps. While the proposed method of this paper simplifies the work steps. Numerical simulation results are presented to validate the theoretical analysis.

Quadrotor control in complex manoeuvres using a hybrid backstepping and feedback linearisation control strategy

In this article, a novel control algorithm called the combined backstepping and feedback linearisation method, along with an uncertainty estimator, is presented for quadrotors. The objective is to develop a robust control algorithm capable of handling various flight conditions, compensating for uncertainties and disturbances, and effectively controlling high-speed manoeuvres. To accomplish this, the quadrotor dynamics model is first derived using the Newton–Euler method. Subsequently, a backstepping control algorithm is designed for the quadrotor's internal control layer, followed by the application of the feedback linearisation method to the external control layer. An estimator is also designed to mitigate the effects of disturbances and uncertainties. A comparison is made between the proposed combined backstepping and feedback linearisation algorithm and the backstepping method, revealing that the combined backstepping and feedback linearisation algorithm outperforms the backstepping method across different aspects. Notably, the combined backstepping and feedback linearisation algorithm achieves faster trajectory tracking and demonstrates fewer steady-state errors. Additionally, the integration of the uncertainty estimator into the combined backstepping and feedback linearisation algorithm effectively mitigates the detrimental effects of disturbances and uncertainties. Comparative results for tracking control are presented to evaluate the performance of the proposed algorithm across various scenarios and case studies.