Backstepping Procedure Research Articles

Adaptive Saturated Obstacle Avoidance Trajectory Tracking Control for Euler–Lagrange Systems With Velocity Constraints

ABSTRACTThis article pays attention to the obstacle avoidance trajectory tracking control problem for uncertain Euler–Lagrange (EL) systems subject to velocity constraints and input saturation. The existing obstacle avoidance results do not consider velocity constraints under input saturation, which means that an EL system may not be able to obtain sufficient control inputs to avoid a collision with an obstacle if it has a high speed when approaching the obstacle. Therefore, the velocity constraints in the obstacle avoidance tracking control are considered in this paper. A novel velocity constraint function that depends on the distance between the system and the obstacle is proposed. Integral‐multiplicative Lyapunov‐barrier functions (LBFs) are constructed and incorporated into the backstepping procedure to design an adaptive fuzzy obstacle avoidance tracking control scheme. Moreover, an auxiliary dynamic system is designed by constructing a bounded nonlinear vector related to an auxiliary variable to compensate for the effects of saturation. Through the Lyapunov method and boundedness analysis for the barrier function, it is shown that the protocol achieves obstacle avoidance for the EL system without violating the velocity constraints inside the obstacle detection region, while also guaranteeing the ultimate uniform boundedness of all the closed‐loop signals. Numerical simulations are presented to demonstrate the efficacy of the proposed control strategy.

Barrier-function based adaptive trajectory tracking control for high-order nonlinear systems with collision avoidance

This paper considers the problem of trajectory tracking and collision avoidance for a class of high-order nonlinear strict feedback systems with unknown nonlinearities. The main issue is how to ensure collision avoidance and tracking performance simultaneously in the presence of unknown nonlinear functions. To address the issue, an integral-multiplicative barrier Lyapunov function (BLF) is integrated into the backstepping procedure to remove the dynamic mismatching issue of the existing SUM-type BLF. It has been proven that the proposed adaptive approach ensures both collision avoidance and tracking performance of high-order nonlinear systems in multi-obstacle environments, and all the signals in the closed-loop system are uniformly ultimately bounded (UUB). Simulation results confirm the effectiveness of the proposed method.

Finite-time adaptive NN dynamic surface control for nonstrict nonlinear FOSs subject to input dead-zone and full-states constraints

This article focuses on a kind of nonstrict nonlinear fractional-order systems (FOSs) suffering from state constraints and dead-zone input. Meanwhile, a finite-time adaptive dynamic surface control (DSC) approach based on backstepping technology and approximation principle of radial basis function neural network (RBFNN) is developed. To overcome the problem of inherent computational complexity, a fractional-order filter is applied to approach the virtual controller and its fractional-order derivative in each step of the backstepping procedure. The barrier Lyapunov function (BLF) is employed to handle the state constraints, and finite-time stability criteria on the basis of fractional-order Lyapunov method are introduced to prove the finite-time convergence of the tracking error into a small region around the origin. It is shown that all the solutions of the closed-loop system are bounded, while the state constraints are satisfied within a predetermined finite time. Finally, two examples are provided to demonstrate the effectiveness of the presented control scheme.

Adaptive stabilization for a class of wave equations with uncertain nonlinear boundary actuator dynamics

This paper investigates the stabilization of a wave equation with uncertain boundary actuator dynamics. The remarkable characters of the system under investigation are revealed by the essential nonlinearities involved in the boundary actuator and the serious parametric unknowns coming from anti-damping coefficient at uncontrolled boundary and those in the actuator, which result into the incapability of the traditional control methods on this topic. For this, a novel adaptive control strategy is proposed in this paper. Specifically, a Riemann variables based state transformation is first introduced to change the original system into a new one whose stabilization implies that of the original system. Then, a state-feedback controller joint with some adaptive laws which are smartly chosen for the compensation of unknown parameters is explicitly designed by a finite-dimensional backstepping procedure. Owing to the compensation of boundary actuator dynamics, the performance analysis of the closed-loop system is more complicated than that of the related literature. For this, a rigorous recursive step is given to show the stability of the resulting closed-loop system. Finally, a numerical example is provided to verify the effectiveness of the proposed theoretical results.

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 fast finite-time secondary control of islanded microgrids: A disturbance observer-based approach

For islanded microgrids with inverter-based distributed generators (DGs), this article focuses on the fast finite-time secondary voltage and frequency control problem considering system uncertainties. The distributed controller is designed based on the command filtered backstepping procedure, and a disturbance observer is constructed to estimate the lumped system uncertainties (including parameter perturbations, unmeasurable variables, etc.). In addition, the singularity issue derived from differentiating the virtual control law is avoided by using a smooth switch function. The proposed controller guarantees that the tracking errors of voltage and frequency converge to a small region around the origin in a finite time, while achieving desired active power sharing ratio. Case studies are conducted on a microgrid system to demonstrate the effectiveness of the proposed strategy in the presence of load changes and parameter perturbations.

Neural-Network-Based Adaptive Event-Triggered Asymptotically Consensus Tracking Control for Nonlinear Nonstrict-Feedback MASs: An Improved Dynamic Surface Approach.

In this article, the asymptotic tracking control problem for a class of nonlinear multi-agent systems (MASs) is researched by the combination of radial basis function neural networks (RBF NNs) and an improved dynamic surface control (DSC) technology. It's important to emphasize that the MASs studied in this article are nonlinear and nonstrict-feedback systems, where the nonlinear functions are unknown. In order to satisfy the requirement that all items in the controller must be available, the unknown nonlinearities in the system are flexibly approximated by utilizing RBF NNs technique. Moreover, the issue of ``complexity explosion'' in the backstepping procedure is handled by improving the traditional DSC technology, and meanwhile, the influences of the boundary layers caused by the filters in the DSC procedure are eliminated skillfully through the compensation terms. In addition, the relative threshold event-triggered strategy is developed for the designed controllers to reduce the waste of communication resources, where Zeno phenomenon is successfully avoided. It is observed that the new presented control strategy ensures that all the closed-loop systems variables are uniformly ultimately bounded (UUB), and furthermore all the outputs of followers are able to track the output of the leader with zero tracking errors. Finally, the simulation results are presented to show the effectiveness of the obtained design scheme.

Neighbor event-triggered adaptive distributed control for multiagent systems with dead-zone inputs

<abstract><p>The paper focused on the distributed tracking problem for a specific class of multi-agent systems, characterized by bandwidth constraint and dead zone actuators, where the bandwidth limitations exist in neighbor agents and the dead zone nonlinearity refers to a generalized mathematical model. Initially, a series of event-triggered mechanisms with relative thresholds were established for neighbor agents, ensuring that control signals were transmitted only when necessary. Next, the generalized dead zone models were decomposed into two parts: indefinite terms with control coefficients and disturbance-like terms, resulting in unpredictability and damaging effects. Subsequently, based on the backstepping procedure, final consensus controllers with multiple polynomial compensators were constructed. These controllers offset the coupling coefficients caused by event-triggered mechanisms and dead zone non-smooth. Stability analysis was given to substantiate the theoretical correctness of this method and support the claim of Zeno behavior avoidance. Finally, simulation studies were performed for the feasibility of our proposed methodology.</p></abstract>

Adaptive Fuzzy Tracking Control for Switched Nonlinear Systems Under FDI Attacks and Input Saturation: A Flexible Transient Performance Approach.

The present study proposes an adaptive fuzzy tracking control strategy for switched nonlinear systems, capable of effectively addressing false data injection (FDI) attacks and input saturation, while achieving flexible prescribed performance control as well as semi-global uniform ultimate boundness for the resultant system. Compared to the previous work, the provided control strategy exhibits two notable strengths: 1) it introduces a novel modified fixed-time pregiven performance function to effectively balance input saturation and output constraint and 2) the detrimental impacts resulting from FDI attacks are successfully mitigated by implementing the fuzzy logic systems approximation technique in the backstepping procedure. A set of switching fuzzy observers are established to estimate the unobservable states while a first-order differential filter is utilized to handle the complexity explosion problem. Finally, the mass-spring-damper system is given to substantiate the developed approach.

Neural Network Adaptive Iterative Learning Control for Strict-Feedback Unknown Delay Systems Against Input Saturation.

Neural network adaptive iterative learning control (ILC) is developed in this article to treat strict-feedback nonlinear systems with unknown state delays and input saturation. These delays are treated by constructing the Lyapunov-Krasovskii (L-K) functions for each subsystem. A command filter is employed to avoid the derivative explosion caused by continuous differentiation of the virtual controller. Corresponding auxiliary systems are designed and integrated into the backstepping procedure to compensate input saturation and the unimplemented part of the filter. Hyperbolic tangent functions and radial basis function neural networks (RBF NNs) are employed to treat singularity and related unknown terms, respectively. The convergence of the resultant strict-feedback systems is ensured in the framework of composite energy function (CEF). Finally, a simulation example is adopted to substantiate the validity of the proposed algorithm.

Adaptive NN Controller of Nonlinear State-Dependent Constrained Systems With Unknown Control Direction.

Various constraints commonly exist in most physical systems; however, traditional constraint control methods consider the constraint boundaries only relying on constant or time variable, which greatly restricts applying constraint control to practical systems. To avoid such conservatism, this study develops a new adaptive neural controller for the nonlinear strict-feedback systems subject to state-dependent constraint boundaries. The nonlinear state-dependent mapping is employed in each step of backstepping procedure, and the prescribed transient performance on tracking error and the constraints on system states are ensured without repeatedly verifying the feasibility conditions on virtual controllers. The radial basis function neural network (NN) with less parameters approach is introduced as an identifier to estimate the unknown system dynamics and reduce computation burden. For removing the effect of unknown control direction, the Nussbaum gain technique is integrated into controller design. Based on the Lyapunov analysis, the developed control strategy can ensure that all the closed-loop signals are bounded, and the constraints on full system states and tracking error are achieved. The simulation examples are used to illustrate the effectiveness of the developed control strategy.

Prescribed performance control of a robotic manipulator with unknown control gain and assigned settling time

This paper presents a control method for trajectory tracking of a robotic manipulator, subject to practical constraints and uncertainties. The proposed method is established upon an adaptive backstepping procedure incorporating a tangent-type barrier Lyapunov function and it preserves some important metrics of trajectory tracking such as fast and user-defined settling time response and robustness against actuation faults and unknown control gain. The proposed design maintains the system trajectory within a prescribed performance bound and relaxes the assumption of the bounded initial condition. These salient features preserve the system within a safety bound and, consequently, guarantee the system stability and safety. The performance of the proposed control method is validated on a 3-DOF PUMA 560 robotic manipulator benchmark model, with different operation scenarios. The simulation results confirm the effectiveness and robustness of the proposed control method.

Adaptive robust output tracking of uncertain strict-feedback nonlinear systems with full state constraints via control schemes with simple structure

In this paper, the problem of output tracking control is considered for a class of uncertain strict-feedback nonlinear systems with virtual control coefficients and full state constraints. It is assumed only that the nonlinear uncertainties of the systems are some smooth functions enough to guarantee the existence of solutions to the differential equations which describe uncertain systems, and that the virtual control coefficients are unknown. Based on backstepping algorithm, by combining the barrier Lyapunov function with an integral inequality reported in the current control literature, a design method is presented whereby a simple adaptive robust output tracking control scheme can be synthesised. The presented design method can evade the repeated differentiation problem appearing in using backstepping algorithm, and need not know all the nonlinear upper bounds of uncertainties, which are repeatedly employed at each step of backstepping procedure. It is also shown that the output tracking error of the uncertain nonlinear systems with both unknown virtual control coefficients and full state constraints can converge uniformly exponentially towards a ball which can be as small as desired. Finally, a numerical example is given, and the corresponding simulations are also implemented to demonstrate the simplicity of the proposed method and the validity of the theoretical results.

Discrete-Time Prescribed Performance Control of an Air-Vehicle’s Seeker Stabilized Platform

Most of the existing prescribed performance control (PPC) methodologies are developed in the continuous-time domain. In this article, a discrete-time PPC (DPPC) strategy is investigated for an air-vehicle’s seeker stabilized platform. Unlike the existing DPPC whose convergence time drifts with the sampling time, the proposed controller is able to guarantee tracking errors with fixed convergence time via devising a new discrete-time performance function, which improves the engineering practicability. Moreover, a new disturbance observer is constructed to estimate both system uncertainties and external disturbances. In addition, the back-stepping procedure is used to design a DPPC approach for the sake of stabilizing transformed errors. This endows tracking errors with desired fixed-time prescribed performance. Finally, the efficiency of design is verified via compared simulations.

Adaptive Iterative Learning Control for a Class of Nonlinear Strict-Feedback Systems With Unknown State Delays.

In this article, an adaptive iterative learning control scheme is presented for a class of nonlinear parametric strict-feedback systems with unknown state delays, aiming to achieve the point-wise tracking of desired trajectory in a finite interval. The appropriate Lyapunov-Krasovskii functions are established to compensate the influence of time-delay uncertainties on the control systems. As the main features, the proposed approach integrates the command filter into the backstepping procedure to avoid the differential explosion problem that may occur with the increase of system order, and introduces the hyperbolic tangent functions into the learning controller to handle the singularity problem thus maintaining the continuity of input signal. The results of theoretical analysis and numerical simulation demonstrate that the tracking errors at the entire period will converge to a compact set along the iteration axis. Compared with the existing works, the proposed control scheme is promising to manifest the better performance and practicability owing to the learning mechanism, the dynamic model, as well as the implementation of controller.

Bumpless transfer distributed adaptive backstepping control of nonlinear multi-agent systems with circular filtering under DoS attacks

This paper studies the problem of resilient distributed cooperative control for a class of uncertain nonlinear multi-agent systems (MASs) under Denial-of-Service (DoS) attacks. To avoid repeatedly differentiating the discontinuous communication signals caused by the DoS attacks in the backstepping procedure, a new circular switching filter is constructed. Compared with the existing results, the circular switching filter can sufficiently improve the tolerance capacity for the DoS attacks in the sense that the DoS attacks can be tolerated as long as the network under attacks is intermittently connected. To address the issue of discontinuity of the switching controller, a bumpless transfer controller is further proposed by smoothing the discontinuous communication signals based on an artificial time delay mechanism. Simulation results illustrate the effectiveness of the proposed control schemes.

Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems

This paper studies the asymptotic control problem of a class of large-scale high-order nonlinear systems (LSHONSs), and an asymptotic fuzzy adaptive dynamic surface controller is developed. Unknown nonlinear terms are learned online by fuzzy logic systems (FLSs) such that the accurate nonlinear model is released in the controller design procedure, where the parameters of FLSs are updated by developing adaptive laws. To compensate for the “boundary error” caused by the dynamic surface control method where a linear filter is added in the backstepping procedure to handle the “explosion of complexity” problem, a nonlinear filter is proposed to eliminate the boundary layer error. Some simulations are given to demonstrate the effectiveness of the proposed algorithm.

Fuzzy adaptive tracking control of fractional-order multi-agent systems with partial state constraints and input saturation via event-triggered strategy

This paper addresses the problem of adaptive tracking control for fractional-order multi-agent systems subject to input saturation and partial state constraints, utilizing an event-triggered strategy with output feedback. The interconnected graph of multiple agents is assumed to be directed and contain a spanning tree. Since only the system output is available, a fractional-order state observer is designed to obtain the unmeasurable states. The nonlinear dynamics of the system are assumed to be unknown and are approximated by a fuzzy logic system. By utilizing the barrier Lyapunov function, backstepping procedure and dynamic surface control technique, an event-triggered adaptive control scheme is proposed to ensure that the constrained states remain within their bounds, all system signals are bounded, tracking error converges to a bounded set containing the origin, and no Zeno behavior occurs. Finally, the validity of the presented method is verified by simulation.

Fixed-time adaptive fuzzy event-triggered control for uncertain nonlinear systems with output constraint and actuator failures

For a category of uncertain nonlinear systems with output constraint and actuator failures, the fixed-time event-triggered control method is investigated. Output constraint and actuator failures are common in practical systems, which may affect the performance of the system. Most of the existing schemes to deal with these problems only guarantee the system's asymptotic stability. It is a challenge to design event-triggered based controllers to achieve fixed-time stability of a system with the above constraints. To handle the above problems, barrier Lyapunov functions (BLFs) and extended fuzzy logic systems (EFLSs) are introduced to the backstepping procedure. Besides, to reduce redundant data transmissions, an event-triggered mechanism (ETM) is designed. On this foundation, a fixed-time event-triggered control method is proposed. Technically, by using the proposed method, the controlled system is practically fixed-time stable, and the system output abides by the corresponding constrained requirements. Meanwhile, the communication resources are saved. Finally, some simulation examples are used to demonstrate the effectiveness of the control method.

Adaptive Learning Gain-Based Control for Nonlinear Systems With External Disturbances: Application to PMSM

The performance of backstepping control with an external disturbance guarantees the upper bound of the tracking error, which is determined by an unknown upper bound of the disturbance and design parameters. Therefore, tuning the design parameters is necessary to obtain a satisfactory control performance. Generally, a high gain is used to reduce the tracking error under the unknown disturbance in the previous methods. However, the use of an unnecessarily high gain may amplify the high-frequency measurement noise and peaking phenomenon. In this study, we propose an adaptive learning gain (ALG)-based nonlinear control for a strict-feedback nonlinear system with an external disturbance to achieve the desired control performance without any information about the disturbance and gain tuning. An ALG-based nonlinear controller using a backstepping procedure is designed to track the desired trajectory. The ALG update law is established to suppress the tracking error within the desired bound. The amplification of the measurement noise in the control input is reduced, as the ALG is updated to avoid the use of an unnecessarily high gain when the controller achieves more than the desired performance.