Conventional Backstepping Research Articles

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.

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.

Neuro-adaptive path following control of autonomous ground vehicles with input deadzone

This article investigates the path-following control problem of an autonomous ground vehicle (AGV) with unknown external disturbances and input deadzones. Neural networks are used to estimate unknown external disturbances, dead zones, and nonlinear functions. The minimum learning parameter scheme is employed to adjust the neural network to reduce the computational load. A backstepping control is proposed to facilitate the tracking of the target path. The steady-state path-following error is decreased by adding an integral error term to the backstepping controller. Command filtering is employed to address the explosion of the complexity issue of the conventional backstepping approach, and the filtering error is compensated via an auxiliary signal. Lyapunov stability study indicates that the AGV closed-loop system is bounded by the proposed control with reasonable accuracy. At last, simulations are given to demonstrate the potential of the proposed scheme in path-following control.

A novel approach to the FPIBSC strategy for an electric power steering system

The electric power steering system is used to improve comfort and steering feel for the user. In this article, we propose to use an integrated nonlinear control strategy for the electric steering system, which is described based on a complicated dynamic model. This work provides two new contributions. First, this control algorithm combines backstepping and proportional–integral (PI) techniques with parameters adjusted by a fuzzy algorithm, so it is called Fuzzy proportional–integral backstepping control (FPIBSC). The output of the PI algorithm is the input of the backstepping technique, while system stability is evaluated based on the Lyapunov function with virtual control variables. Second, road reaction torque is calculated based on a spatial dynamic model, which considers the influence of many other factors. Numerical simulation methods are used to evaluate the performance of the system. According to research findings, the value of the steering motor angle (controlled object) continuously tracks the desired value with negligible error. Under some specific conditions, the error between signals can be reduced to zero. In addition, other outputs that are obtained from the FPIBSC algorithm also tend to follow the reference signal with high accuracy. The phase difference phenomenon only occurs when using the conventional backstepping algorithm instead of FPIBSC. The assisted torque increases as speed decreases or the driver torque increases. In general, the system’s stability is always guaranteed under many different simulation conditions.

Fast Finite‐Time Dynamic Surface Synchronization Control for Telerobotics System with Prescribed Performance

AbstractThis study focuses on the fast finite‐time synchronization control problem for nonlinear telerobotics system with asymmetric time‐vary delays and uncertainties. A finite‐time prescribed performance function is incorporated into the backstepping control framework such that the synchronization error will remain within a predefined funnel boundary. To increase the rate of convergence, the selected error transformation function is a arctangent function. The utilization of dynamic surface control method aims to decrease computational complexity by mitigating the need for repetitive differentiation of virtual signals in the conventional backstepping algorithm. In the meantime, the non‐power approximate signals of fuzzy neural network algorithms are utilized to replace the system uncertainties, effectively addressing the passivity issue associated with time‐delayed channels. Both theoretical analysis and experimental results are present to verify that the synchronization error can converge to a small neighborhood around zero in the fast finite time and the closed‐loop system remains stable.

Adaptive disturbance rejection neural output feedback control of hydraulic manipulator systems

This paper proposes an adaptive disturbance rejection neural output feedback control (ADRNC) scheme for multi-degree-of-freedom (n-DOF) hydraulic manipulator systems, subjected to unknown nonlinearities, external disturbances and unmeasured system states. The controller design is formulated by integrating Radial Basis Function Neural Networks (RBFNNs) with state and disturbance observers using the backstepping method. The RBFNNs are synthesized to handle unknown nonlinear functions and the residual estimate error, coupled with external disturbances, is estimated through the combination of state observer and disturbance observer. The unique features of the proposed controller lies in its capability to estimate both matched and unmatched lumped disturbances. The auxiliary disturbance estimation law is guided by the neural learning weights and estimated system states provided by state observers. By effectively utilizing neural networks to approximate and mitigate most nonlinear uncertainties, the workload of the disturbance observer is substantially reduced. High-gain feedback is therefore avoided and improved tracking performance can be expected. Moreover, to avoid the tedious analysis and the problem of “explosion of complexity” in the conventional backstepping method, we employ a first-order sliding-mode differentiator. Rigorous analysis via Lyapunov methods establishes the stability of the entire closed-loop system, ensuring guaranteed and satisfactory tracking performance under the integrated influence of unknown nonlinearities, unmeasured states, and external disturbances. Extensive simulations are conducted to verify the effectiveness of the nested control strategy.

Time-varying gain extended state observer-based adaptive optimal control for disturbed unmanned helicopter

In this paper, the robust adaptive optimal tracking control problem is addressed for the disturbed unmanned helicopter based on the time-varying gain extended state observer (TVGESO) and adaptive dynamic programming (ADP) methods. Firstly, a novel TVGESO is developed to tackle the unknown disturbance, which can overcome the drawback of initial peaking phenomenon in the traditional linear ESO method. Meanwhile, compared with the nonlinear ESO, the proposed TVGESO possesses easier and rigorous stability analysis process. Subsequently, the optimal tracking control issue for the original unmanned helicopter system is transformed into an optimization stabilization problem. By means of the ADP and neural network techniques, the feedforward controller and optimal feedback controller are skillfully designed. Compared with the conventional backstepping approach, the designed anti-disturbance optimal controller can make the unmanned helicopter accomplish the tracking task with less energy. Finally, simulation comparisons demonstrate the validity of the developed control scheme.

Design of Backstepping Control Based on a Softsign Linear–Nonlinear Tracking Differentiator for an Electro-Optical Tracking System

To address the problems of a low tracking accuracy and slow error convergence in high-order single-input, single-output electro-optical tracking systems, a backstepping control method based on a Softsign linear–nonlinear tracking differentiator is proposed. First, a linear–nonlinear tracking differentiator is designed in conjunction with the Softsign excitation function, using its output as an approximate replacement for the conventional differentiation process. Then, this is combined with backstepping control to eliminate the “explosion of complexity” problem in conventional backstepping procedures due to repeated derivation of virtual control quantities. This reduces the workload of parameter tuning, takes into account the rapidity and stability of signal convergence, and improves the trajectory tracking performance. This method can ensure the boundedness of the system signal. The effectiveness and superiority of this control method are verified through simulations and experiments.

Distributed Leader Follower Formation Control of Mobile Robots Based on Bioinspired Neural Dynamics and Adaptive Sliding Innovation Filter

This paper investigated the distributed leader follower formation control problem for multiple differentially driven mobile robots. A distributed estimator is first introduced and it only requires the state information from each follower itself and its neighbors. Then, we propose a bioinspired neural dynamic based backstepping and sliding mode control hybrid formation control method with proof of its stability. The proposed control strategy resolves the impractical speed jump issue that exists in the conventional backstepping design. Additionally, considering the system and measurement noises, the proposed control strategy not only removes the chattering issue existing in the conventional sliding mode control but also provides smooth control input with extra robustness. After that, an adaptive sliding innovation filter is integrated with the proposed control to provide accurate state estimates that are robust to modeling uncertainties. Finally, we performed multiple simulations to demonstrate the efficiency and effectiveness of the proposed formation control strategy.

Decentralised adaptive neural finite-time prescribed performance control for nonlinear large-scale systems based on command filtering

In this research, the issue of decentralised adaptive neural finite-time prescribed performance control is discussed for nonstrict-feedback large-scale nonlinear interconnected systems subject to dead zones input and unknown control direction. The obstacle of ‘explosion of complex’ occurred in conventional backstepping design can be surmounted by adopting the command filter technique and nonlinearities are approximated by introducing an adaptive neural control approach. To handle the obstacles due to unknown directions and unknown interconnections, Nussbaum-type functions and two smooth functions are used and designed. Meanwhile, error compensation signals are introduced to deal with the problem associated with the dynamic surface method. To constraint the output tracking error within a predefined boundary in finite time, an improved performance function, i.e. finite-time performance function is introduced. Different from existing control results, the developed control methodology does not require any information on the boundedness of dead-zone parameters. It is further proved that the constructed controller not only assures the semi-global boundedness of all the controlled system signals, but also makes the output tracking errors reach within a predefined small set. Finally, both numerical and practical examples are supplied to further validate the effectiveness of the presented theoretic result.

Prescribed performance event-triggered fuzzy optimal tracking control for strict-feedback nonlinear systems

The prescribed performance event-triggered optimal tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with external disturbances. First, the disturbance observers are constructed to estimate external disturbances. Then, the controller contains an adaptive fuzzy controller and an optimal compensation term, in which the unknown nonlinearities and cost function are approximated by the fuzzy logic systems. An adaptive fuzzy controller is established by the dynamic surface control (DSC) technique, which is addressed to handle “computation complexity” issue occurred in conventional backstepping approach. Based on adaptive dynamic programming (ADP) method, an optimal compensation term is developed by minimizing the cost function. Furthermore, communication load is reduced via embedding event-triggered mechanism. In addition, the tracking error can be restricted in the prescribed region with the aid of the prescribed performance control. Thus, the whole control scheme can not only ensure that the tracking error converges to a boundary but also achieve optimization, reduce the communication burden and avoid “computation complexity” issue. The boundedness of all signals in the closed-loop is proved. Finally, the simulation examples illustrate the validity of the designed control scheme.

Adaptive fractional backstepping intelligent controller for maximum power extraction of a wind turbine system

Controlling wind power plants is a challenging issue, however. This is due to its highly nonlinear dynamics, unknown disturbances, parameter uncertainties, and quick variations in the wind speed profiles. So robust controllers are needed to overcome these challenges. This paper suggests two novel control approaches for doubly fed induction generator-based wind turbines. Its key objective is to regulate the generator speed and rotor currents. A radial basis function (RBF) neural network disturbance observer based fractional order backstepping sliding mode control (SMC) is presented to control the rotor currents. This RBF neural network-based disturbance observer estimates unknown disturbances. Also, a new adaptive fractional order terminal SMC is suggested for the control of the generator speed. This robust chattering-free controller that does not require any information about the bound of uncertainties fractional calculus is adopted in the SMC design to eliminate undesired chattering phenomena. The controller parameters are optimally tuned utilizing the ant colony optimization algorithm. The proposed approach was validated using a simulation study entailing various conditions. Its performance was also compared to that of the conventional backstepping and conventional backstepping sliding mode controller. The simulations results verified the approach's ability to maximize power extraction from the wind and properly regulate the rotor currents. The proposed method has about 20% less tracking error than the other two methods, which means 20% higher efficiency.

Adaptive Finite-Time Fuzzy Control for Uncertain Nonlinear Systems with Asymmetric Full-State Constraints

This paper studies the adaptive finite-time fuzzy control issue associated with uncertain nonlinear systems that exhibit asymmetric constraints on the full state. A distinct function, constrained by nonlinear states, is designed to mitigate the excessive breach of these full-state boundaries. Unlike the standard barrier Lyapunov function (BLF) method, this approach solves symmetric and asymmetric full-state constraints without modifying the controller structure, and it does not require any additional assumptions about virtual control to be met. Simultaneously employing approximating functions using fuzzy logic systems and incorporating dynamic surface control technology integrated with a first-order filter, the unknown nonlinear functions emanating from the suggested controller strategy are estimated. Additionally, this approach addresses the prevalent problem of complexity explosion observed in conventional backstepping techniques. An adaptive finite-time fuzzy tracking control strategy is introduced, ensuring that all signals and tracking errors of the controlled system remain bounded in finite time. Finally, two simulation examples are given to illustrate the effectiveness of the proposed control scheme, confirming that all states remain within the predefined regions.

Distributed Adaptive-Neural Finite-Time Consensus Control for Stochastic Nonlinear Multiagent Systems Subject to Saturated Inputs.

In this article, the problem of distributed finite-time consensus control for a class of stochastic nonlinear multiagent systems (MASs) (with directed graph communication) in the presence of unknown dynamics of agents, stochastic perturbations, external disturbances (mismatched and matched), and input saturation nonlinearities is addressed and studied. By combining the backstepping control method, the command filter technique, a finite-time auxiliary system, and artificial neural networks, innovative control inputs are designed and proposed such that outputs of follower agents converge to the output of the leader agent within a finite time. Radial-basis function neural networks (RBFNNs) are employed to approximate unknown dynamics, stochastic perturbations, and external disturbances. To overcome the complexity explosion problem of the conventional backstepping method, a novel finite-time command filter approach is proposed. Then, to deal with the destructive effects of input saturation nonlinearities, the finite-time auxiliary system is designed and developed. By mathematical analysis, it is proven that the mentioned MAS (injected by the proposed control inputs) is semiglobally finite-time stable in probability (SGFSP) and all consensus tracking errors converge to a small neighborhood of the zero during a finite time. Finally, a numerical simulation onto a group of four single-link robot manipulators is carried out to illustrate the effectiveness of the suggested control scheme.

Compound learning adaptive neural network optimal backstepping control of uncertain fractional-order predator–prey systems

Reinforcement learning as an effective strategy is widely utilized in optimal control. However, when updating critic–actor weight vectors based on the square of Bellman residual, it often leads to substantial computational complexity. This paper formulates a compound learning optimal backstepping control programme that can efficaciously reduce the computational burden for fractional-order predator–prey systems (FOPPS) with uncertainties. To economize resource, a reinforcement learning technology is adopted to realize the optimal control in view of neural networks under identifier–critic–actor structure. To address the computational complexity issue raised above, a simple positive definite function is proposed to update critic–actor weight vectors. Fractional-order filters are utilized to estimate virtual signals and their fractional-order derivatives for tackling the “explosion of complexity” problem existing in the conventional backstepping technology. Simultaneously, to enhance the approximation accuracy of uncertainties in FOPPS, a compound learning updating law is built by using tracking error and prediction error. In accordance with the stability analysis, the formulated scheme ensures that the output of FOPPS can track the reference signal with the expected accuracy and all signals are bounded. Eventually, a numerical simulation is presented to validate the effectiveness of the proposed control strategy.

Adaptive neural dynamic surface control of load/grid connected voltage source inverters with LC/LCL filters

Utilizing passive filters such as L, LC and LCL is preferred to cancel out high-frequency harmonics caused by pulse width modulation of voltage source inverters. However, the LC and LCL filters have shown better harmonic attenuation than the conventional L filter. Nevertheless, the control process of LC and LCL filters is more complicated due to their higher-order dynamics. The problem gets more challenging in the presence of uncertainties such as load and grid impedance variations. To overcome these challenges, two novel adaptive neural dynamic surface controllers are proposed for LC and LCL filters in the load and grid-connected modes, respectively. Meanwhile, the issue of computational complexity inherent in the conventional backstepping method is avoided here by utilizing the dynamic surface control technique. Furthermore, the matched and unmatched uncertainties of LC/LCL filters are approximated via multi-input multi-output radial basis function neural networks. Stability of the closed-loop systems is guaranteed by converging the tracking errors to a small neighborhood of the origin. Simulations are given to illustrate the effectiveness and potential of the proposed adaptive neural dynamic surface control methods under the load and grid impedance changes.

Hierarchical sliding mode-based adaptive fuzzy control for uncertain switched under-actuated nonlinear systems with input saturation and dead-zone

PurposeThis paper aims to study the issues of adaptive fuzzy control for a category of switched under-actuated systems with input nonlinearities and external disturbances.Design/methodology/approachA control scheme based on sliding mode surface with a hierarchical structure is introduced to enhance the responsiveness and robustness of the studied systems. An equivalent control and switching control rules are co-designed in a hierarchical sliding mode control (HSMC) framework to ensure that the system state reaches a given sliding surface and remains sliding on the surface, finally stabilizing at the equilibrium point. Besides, the input nonlinearities consist of non-symmetric saturation and dead-zone, which are estimated by an unknown bounded function and a known affine function.FindingsBased on fuzzy logic systems and the hierarchical sliding mode control method, an adaptive fuzzy control method for uncertain switched under-actuated systems is put forward.Originality/valueThe “cause and effect” problems often existing in conventional backstepping designs can be prevented. Furthermore, the presented adaptive laws can eliminate the influence of external disturbances and approximation errors. Besides, in contrast to arbitrary switching strategies, the authors consider a switching rule with average dwell time, which resolves control problems that cannot be resolved with arbitrary switching signals and reduces conservatism.

Modeling, analysis, and adaptive neural modified-backstepping control of an uncertain horizontal pendubot with double flexible joints

Position control of an underactuated uncertain horizontal pendubot with double flexible joints (PDFJ) that has only one actuator is a new and challenging topic. This study establishes the dynamics model of the PDFJ, analyzes its characteristics, and presents an adaptive neural modified-backstepping method to achieve its position control. Through theoretical analysis, we obtain the relationship between underactuated system dynamics and the kinetic energy derivative of the two links. On this basis, an adaptive neural modified-backstepping method is proposed, which deals with the effects of parameter perturbation, model uncertainty, and external disturbance and makes the PDFJ reach the target position. This method relaxes the strict restriction of conventional backstepping methods and dynamic surface control methods that demands that the relative degree of each virtual subsystem must be one, and it can still alleviate a differential explosion problem. Finally, we demonstrate its effectiveness through simulation and experimental results.

Adaptive backstepping fuzzy synchronization control of fractional-order chaotic systems with input saturation and external disturbances

In this paper, an adaptive backstepping command filtered controller is proposed for a class of uncertain strict feedback fractional-order chaotic systems with input saturation and external disturbances. A command filter is designed to avoid the “explosion of complexity” problem in the conventional backstepping technique. To tackle with filter error and improve synchronization accuracy, a compensation mechanism is provided. Meanwhile, fuzzy logic systems are utilized to approximate unknown functions, and disturbance observers are constructed to reduce the impact of unknown disturbances. In particular, to reduce the chattering phenomenon, a smooth function rather than the sign function is used in the controller design, and the stability of the closed-loop system can be guaranteed by the proposed synchronization controller. A simulation study is provided to confirm the practicality and validity of the proposed method.

Distributed Event-Triggered Fixed-Time Leader–Follower Formation Tracking Control of Multiple Underwater Vehicles Based on an Adaptive Fixed-Time Observer

This paper focuses on the fixed-time leader–follower formation control of multiple underwater vehicles (MUVs) in the presence of external disturbances. First, an adaptive fixed-time disturbance observer (AFxDO) is developed to deal with unknown time-varying environmental disturbances. The developed AFxDO guarantees the fixed-time convergence property of the disturbance observation error and no prior information on the external disturbances or their derivatives is required. Then, with the aid of the developed AFxDO, a distributed event-triggered fixed-time backstepping controller was developed to achieve the leader–follower formation tracking control of MUVs. To solve the “explosion of complexity” problem inherent in the conventional backstepping, a nonlinear filter is introduced to obtain the derivative of the virtual control law. Furthermore, to reduce the communication burden, the event-triggered mechanism is integrated into the formation tracking controller. The stability analysis shows that the closed-loop MUV system is practical fixed-time stable. Finally, simulation results demonstrate the effectiveness of the proposed scheme.