Backstepping Framework Research Articles

Event-Triggered Fuzzy Adaptive Predefined-Time Control for Fractional-Order Nonlinear Systems with Time-Varying Deferred Constraints and Its Application

This paper focuses on the fuzzy adaptive predefined-time control for fractional-order nonlinear systems with time-varying deferred constraints. First, a modified dynamic surface control technique is introduced to address the problem of computational complexity exposed in the backstepping framework, and the interval type-2 fuzzy logic systems are applied to model the unknown nonlinearities of the systems. Next, a shifting function and the barrier Lyapunov function with variational barrier bounds are formulated to deal with the constraints issue. Particularly, the constraint conditions can be satisfied within a predetermined time, even if they are transgressed initially. Furthermore, a switching threshold event-triggered controller is devised to balance the control energy and communication resources. With the help of the predefined-time stability criterion, it is proven that the presented predefined-time event-triggered controller can ensure that all the signals involved in the closed-loop system are bounded and the tracking error fluctuates to a small neighborhood of the origin in a predefined-time interval. Finally, two simulation examples are provided to confirm the effectiveness of the put-forward control algorithm.

Event-based adaptive formation and tracking control with predetermined performance for nonlinear multi-agent systems

The article focuses on the event-based predetermined performance formation tracking control for uncertain nonlinear multi-agent systems. Each agent is subject to actuator saturation constraint and full-state constraints. To meet the full-state constraints, a barrier function-based nonlinear mapping method is employed instead of the barrier Lyapunov function, such that the undesirable “feasibility conditions” are eradicated. To overcome actuator saturation, the auxiliary systems are utilized to let the controller be designed under the framework of backstepping. The approximation capacity of RBF NNs and adaptive methods are adopted, which can solve unknown uncertainties. An event-triggered predetermined performance formation tracking control strategy is designed, which can guarantee that the agents form the prescribed formation shape in a predetermined settling time and have a fine transient and steady-state tracking performance, while reducing the communication burden from the controller to the actuator. At last, an example of unmanned aerial vehicles is given to test the availability of the results.

Robust adaptive prescribed performance control for nonlinear systems with arbitrary initial states

The initial state of nonlinear MIMO strict feedback systems is often unpredictable and random, and existing prescribed performance control (PPC) techniques can only guarantee system output constraints within a fixed initial value performance boundary. In this paper, an adaptive PPC method tailored for nonlinear systems was introduced, enabling the system stable tracking regardless of the arbitrary initial states. To guarantee that the tracking error consistently meets the prescribed performance boundary(PPB) envelope, we introduce a nonlinear mapping between the initial value of the prescribed performance functions(PPFs) and the system tracking errors, resulting in an asymmetric time-varying performance boundary. Building upon this foundation, an adaptive PPC scheme is proposed under the backstepping framework, integrated with a nonlinear disturbance observer (NDO). This method ensures that the tracking error remains within the desired PPB, regardless of external disturbances or system initial states. To prevent ‘complexity explosion’, a dynamic surface control (DSC) technology is employed to filter the virtual control signals of each subsystem. Furthermore, the ultimate uniform boundedness of the closed-loop signal has been demonstrated, and a practical flight vehicles roll control case was introduced to validate the efficacy of the proposed method.

Fuzzy adaptive fixed-time control for output-constrained uncertain nonstrict-feedback time-delay systems

This paper deals with the problem of fuzzy adaptive fixed-time control for a class of uncertain nonlinear nonstrict feedback systems with time delays and output constraints. The negative effects of time delays on system performance and stability are overcome by introducing a new Barrier Lyapunov–Krasovskii (BLK) function and an innovative inequality. In addition, the singularity problem that arises in the process of deriving the control and adaptive laws, is solved by the proposed BLK function. The controller is designed in the backstepping framework which ensures that the system output does not violate predefined constraints. The proposed controller needs less information from the system and therefore has less implementation complexity than previous references, which leads to a more straightforward implementation. A fuzzy system approximation is utilized to overcome the uncertainties of the model. It is guaranteed that all closed-loop system signals converge to small neighborhoods around zero in a fixed time. Finally, an example is presented to verify that the proposed method achieves the control objectives.

Output feedback active disturbance rejection control of an electro-hydraulic servo system based on command filter

The output feedback active disturbance rejection control of a valve-controlled cylinder electro-hydraulic servo system is investigated in this paper. First, a comprehensive nonlinear mathematical model that encompasses both matched and mismatched disturbances is formulated. Due to the fact that only position information can be measured, a linear Extended State Observer (ESO) is introduced to estimate unknown states and matched disturbances, while a dedicated disturbance observer is constructed to estimate mismatched disturbances. Different from the traditional observer results, the design of the disturbance observer used in this study is carried out under the constraint of output feedback. Furthermore, an output feedback nonlinear controller is proposed leveraging the aforementioned observers to achieve accurate trajectory tracking. To mitigate the inherent differential explosion problem of the traditional backstepping framework, a finite-time stable command filter is incorporated. Simultaneously, considering transient filtering errors, a set of error compensation signals are designed to counter their negative impact effectively. Theoretical analysis affirms that the proposed control strategy ensures the boundedness of all signals within the closed-loop system. Additionally, under the specific condition of only time-invariant disturbances in the system, the conclusion of asymptotic stability is established. Finally, the algorithm’s efficacy is validated through comparative experiments.

Adaptive Fault-Tolerant Tracking Control for Discrete-Time Nonstrict-Feedback Nonlinear Systems With Stochastic Noises

This paper discusses the problem of fault-tolerant tracking control for discrete-time nonstrict-feedback nonlinear systems in the presence of stochastic noises and actuator faults. The system is characterized by a discrete-time nonstrict-feedback structure and multiplicative stochastic noise related to the states, which poses a challenge for the design and analysis of the fault-tolerant controller. Moreover, the considered actuator faults include multiplicative and additive faults without prior information. By integrating the properties of the backstepping framework and neural networks, and by introducing an adaptive fault compensation term, a novel fault-tolerant tracking control strategy is proposed. The effects of faults are compensated and the difficulties caused by the system structure are overcome, avoiding the algebraic loop problem and overcoming the causal contradiction. Given specific parameters, the designed fault-tolerant controller can ensure that the tracking error converges to an adjustable region regarding the origin and that all system signals are uniformly bounded concerning the mean-square sense. The simulation examples illustrate the effectiveness of the designed fault-tolerant control method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —As system complexity has been increasing, practical systems are often affected by faults when performing tracking control. Fault-tolerant control can provide acceptable robustness and improve system safety. On the one hand, many practical systems are nonlinear, difficult to model mathematically and accurately, and affected by stochastic noises. On the other hand, with digital control, the object is a discrete-time system. In practical applications, the upper bounds of the faults and the noises are often unknown. In this paper, a novel adaptive fault-tolerant control method is proposed for the more general uncertain discrete-time nonlinear systems, i.e., nonstrict-feedback nonlinear systems, with stochastic noises and actuator faults. Neural networks and an adaptive fault compensation term are integrated into the fault-tolerant controller. By adjusting the controller parameters properly, it is ensured that the tracking error converges to a small neighborhood of zero in the event of a fault occurring. In future research, sensor faults, input constraints, optimal control design, and applications in multi-agent systems will be considered.

Optimized backstepping‐based finite‐time containment control for nonlinear multi‐agent systems with prescribed performance

AbstractIn this article, a finite‐time optimal containment control method is proposed for nonlinear multi‐agent systems with prescribed performance. First, a neural network‐based reinforcement learning algorithm is developed under the optimized backstepping framework. The algorithm employs an identifier‐critic‐actor architecture, where the identifiers, critics and actors are used to estimate the unknown dynamics, evaluate the system performance, and optimize the system, respectively. Subsequently, in order to guarantee the transient performance of the tracking error, the original system is converted into an equivalent unconstrained system. Then, the tracking errors are allowed to converge to a prescribed set of residuals in finite time by combining prescribed performance control and finite‐time optimal control techniques. Furthermore, by using the Lyapunov stability theorem, it is verified that all signals are semi‐globally practical finite‐time stable, and all followers can converge to a convex region formed by multiple leaders. Finally, the effectiveness of the proposed scheme is demonstrated by a practical example.

Adaptive fixed-time fault-tolerant tracking control for rotary steerable drilling tool systems

In this paper, the problem of adaptive fixed-time fault-tolerant tracking control is investigated for rotary steerable drilling tool systems (RSDTSs). Markov jump system (MJS) is used to describe the RSDTS with varying parameters which are induced by the changing environment. By employing the smooth projection operator technique, adaptive laws are established to estimate the faults. Based on the fault compensation strategy, a new adaptive fixed-time fault-tolerant tracking control scheme is proposed to ensure that the RSDTS is globally stochastically practically fixed-time stable. In addition, to reduce the computational burden in the backstepping framework, the derivatives of the virtual control are directly derived using command filters. Finally, the experiment performed on the rotary steerable drilling tool systems prototype is exploited to demonstrate the feasibility and effectiveness of the proposed method.

Dynamic event-triggered adaptive neural control for MIMO nonlinear CPSs with time-varying parameters and deception attacks

This paper provides a dynamic event-triggered adaptive neural controller for the multi-input–multi-output (MIMO) non-strict feedback nonlinear cyber–physical systems (CPSs) with time-varying parameters encountering deception attacks (DAs). The single parameter learning method (SPLM) and the dynamic surface technology are combined under backstepping framework with adroitness, which not only resist the malicious sensor DA, but also reduce the complexity of the design scheme. Furthermore, a dynamic event-triggered strategy is added to update the threshold dynamically to further reduce the communication load. Theoretical analysis presents that the proposed control scheme can guarantee that all signals are bounded regardless encountering the DAs. Eventually, the validity of the developed methods is illustrated by a simulation case.

Adaptive predefined-time tracking control of uncertain permanent-magnet synchronous motor drive chaotic system

A predefined-time tracking control algorithm is formulated for the Permanent Magnet Synchronous Motor(PMSM) driver system subjected to chaotic oscillations and undeterministic parameters, which can determine the upper bound of settling-time by controller gains. Radial Basis Function Neural Networks(RBFNNs) are employed to approximate the unknown nonlinear functions as universal approximators. The innovative predefined-time tracking differentiators are constructed to estimate the derivatives of the virtual inputs, resolving “the explosion of complexity” in traditional backstepping method. Then, an adaptive predefined-time controller is proposed in the backstepping framework. Lyapunov’s theoretical analysis proves that all signals in the closed-loop systems are uniformly ultimately bounded and the output tracking errors can converge to a small neighborhood of the origin within the predefined-time. Finally, simulation studies are performed to demonstrate that the proposed control scheme can quash the chaotic behaviors of the PMSM drive system and ensure fast-tracking performance even under unknown parameters.

Adaptive Critic Learning-Based Optimal Bipartite Consensus for Multiagent Systems With Prescribed Performance.

Developing a distributed bipartite optimal consensus scheme while ensuring user-predefined performance is essential in practical applications. Existing approaches to this problem typically require a complex controller structure due to adopting an identifier-actor-critic framework and prescribed performance cannot be guaranteed. In this work, an adaptive critic learning (ACL)-based optimal bipartite consensus scheme is developed to bridge the gap. A newly designed error scaling function, which defines the user-predefined settling time and steady accuracy without relying on the initial conditions, is then integrated into a cost function. The backstepping framework combines the ACL and integral reinforcement learning (IRL) algorithm to develop the adaptive optimal bipartite consensus scheme, which contributes a critic-only controller structure by removing the identifier and actor networks in the existing methods. The adaptive law of the critic network is derived by the gradient descent algorithm and experience replay to minimize the IRL-based residual error. It is shown that a compute-saving learning mechanism can achieve the optimal consensus, and the error variables of the closed-loop system are uniformly ultimately bounded (UUB). Besides, in any bounded initial condition, the evolution of bipartite consensus is limited to a user-prescribed boundary under bounded initial conditions. The illustrative simulation results validate the efficacy of the approach.

Vision-based finite-time prescribed performance control for uncooperative UAV target-tracking subject to field of view constraints

This paper presents a vision-based finite-time prescribed performance controller for unmanned aerial vehicle (UAV) tracking of uncooperative aerial targets. The relative states between UAV and target are estimated by an onboard monocular camera. The inability of visual measurements to accurately determine the initial state of the target renders conventional prescribed performance controllers ineffective in such situations. As a result, it becomes essential to address the problem of prescribed performance control under conditions of uncertain initial values By utilizing an auxiliary transforming function, an Asymmetric Barrier Lyapunov Function (ABLF) and a finite-time prescribed performance function, a robust adaptive controller based on backstepping framework is proposed to deal with state constraints under unknown initial tracking conditions. It is proved that, the closed-loop relative position is capable of reaching the prescribed performance bound before the preset transforming time and converging to the prescribed steady-state error before a finite setting time. Simulation examples are provided to illustrated the effectiveness of the proposed tracking algorithm.

Distributed Event-Triggered Control for Manipulator with Fixed-Time Disturbance Observer

This article studies an event-triggered fixed-time trajectory tracking control problem of an n-joint manipulator system. Firstly, a fixed-time disturbance observer is proposed to reconstruct the total disturbance composed of external disturbances and model uncertainties, using the estimation as feedforward compensation to enhance the system robustness. Subsequently, based on the backstepping framework, a fixed-time controller with an event-triggering mechanism is designed for the manipulator to ensure the convergence of tracking errors to zero within a fixed time. Additionally, two event-triggering conditions are devised to reduce the transmission time of control input and the computation time of control output. Simultaneously, Zeno behavior is excluded through theoretical proof, validating the fixed-time stability of the closed-loop system. Finally, simulation verification is conducted on a two-joint manipulator, with results confirming the effectiveness of the control strategy.

Cooperative Control for Stochastic Multiagent Systems With Deferred Dynamic Constraints via a Novel Universal Barrier Function Approach.

This article investigates the cooperative control problem for stochastic multiagent systems (MASs) with dynamic constraints. A new universal barrier function is proposed, which is applicable to many systems with different types of constraint functions, even unconstrained systems. Several mapping functions are constructed to constrain the state variables directly without feasibility conditions, and the tracking control is achieved for stochastic MASs with deferred full-state constraints under the backstepping framework. In order to regulate the tracking error more precisely, the funnel error transformation is improved and the deferred funnel controller is developed by introducing a preassigned finite-time function. Based on the deferred funnel controller, the tracking error can be maintained within the predetermined funnel in the preassigned time. The convergence time can be defined according to the actual requirements, and it is independent of the design controller parameters and initial conditions. Finally, some simulation results are given to demonstrate the effectiveness of the proposed control algorithm.

VGESO-Based Finite-Time Fault-Tolerant Tracking Control for Quadrotor Unmanned Aerial Vehicle

Based on the variable gain extended state observer, a finite-time fault-tolerant control strategy is developed for the quadrotor unmanned aerial vehicle with actuator faults and external disturbances. Firstly, a novel variable gain extended state observer is designed to estimate the unknown external disturbances, which mitigates the initial peaking phenomenon existing in traditional extended state observer-based methods. Meanwhile, the neural networks are applied to accurately approximate unknown couplings online. Moreover, with the help of the projection operator technique, the unknown actuator faults are observed in real time. Combined with the backstepping framework, the finite-time robust fault-tolerant control scheme is constructed and the stability is strictly proved via Lyapunov’s theory. Finally, the validity of the developed control scheme is demonstrated through numerical simulations.

Observer-Based Adaptive Fuzzy Consensus Control of Nonlinear Multiagent Systems Encountering Deception Attacks

This paper investigates the fuzzy adaptive output- feedback consensus control problem for nonlinear multi-agent systems (MASs) with network communication encountering deception attacks. Fuzzy logic systems (FLSs) are used to approximate unknown nonlinear functions, and a novel fuzzy state observer is established to estimate the states of encountered deception attacks. For the case of unknown time-varying gains caused by deception attacks, a projection operator-based compensation mechanism is designed. Consequently, under the framework of adaptive backstepping, a novel adaptive fuzzy output- feedback consensus control method is proposed. The proposed control method can not only ensure that all signals remain bounded, but also make all followers maintain consensus with the trajectory of the leader. Finally, the simulation results are given to confirm the effectiveness of the proposed control scheme.

Composite Learning Adaptive Tracking Control for Full-State Constrained Multiagent Systems Without Using the Feasibility Condition.

This article proposes a distributed consensus tracking controller for a class of nonlinear multiagent systems under a directed graph, in which all agents are subject to time-varying asymmetric full-state constraints, internal uncertainties, and external disturbances. The feasibility condition generally required in the existing constrained control is removed by using the proposed nonlinear mapping function (NMF)-based state reconstruction technology, and the Lipschitz condition usually needed in the consensus tracking is also canceled based on the adaptive command-filtered backstepping framework. The composite learning of the neural network-based function approximator (NN-FAP) and the finite-time smooth disturbance observer (DOB) provides a novel scheme for handling internal and external uncertainties simultaneously. One advantage of this scheme is that the use of online historical data of the closed-loop system strengthens the excitation of NN's learning. Another advantage is that the DOB with NN-FAP embedding realizes that the finite-time observation for external disturbance in the case of the system dynamics is unknown. A complete controller design, sufficient stability analysis, and numerical simulation are provided.

Adaptive fuzzy finite-time PID backstepping control for chaotic systems with full states constraints and unmodeled dynamics

To solve the problem that the controlled object and system model cannot be fully mastered or the system parameters cannot be well obtained through conventional measurement methods, in this paper, a novel adaptive fuzzy finite-time PID backstepping control method is developed for a class of nonlinear chaotic systems with full state constraints and unmodeled dynamics. The unmodeled dynamics are handled by employing a dynamic signal. To avoid the violations of full state constraints (FSCs), a barrier Lyapunov function is used to implement a PID backstepping framework, whose merit consists in that three control gain constants KP, KI and KD with corresponding proportional relationships are designed to further improve control precision. Besides, the PID control method also embodies three dynamic gain components ΔKP, ΔKI and ΔKD, all of which are free functions that can be self-tuning online, and have a significant link with the adaptive law and modify accordingly. Finally, theoretical analysis confirms that all signals are ultimately uniformly bounded, the error signal converges to a nearby area near the origin in a finite-time, and the FSCs are not smashed. The practicality of the proposed method is confirmed by two simulation studies using the second-order Duffing chaotic system and third-order Chua's circuit.

Reinforcement learning-based consensus control for MASs with intermittent constraints

In this article, an adaptive optimal consensus control problem is studied for multiagent systems in the strict-feedback structure with intermittent constraints (the constraints appear intermittently). More specifically, by designing a novel switch-like function and an improved coordinate transformation, the constrained states are converted into unconstrained states, and the problem of intermittent constraints is resolved without requiring “feasibility conditions”. In addition, using the composite learning algorithm and neural networks to construct the identifier, a simplified identifier-actor-critic-based reinforcement learning strategy is proposed to obtain the approximate optimal controller under the framework of backstepping. Meanwhile, with the aid of the nonlinear dynamic surface control technique, the issue of “explosion of complexity” in backstepping is removed, and the requirements for filter parameters are loosened. Based on Lyapunov stability theory, it is demonstrated that all signals in the closed-loop system are bounded. Finally, two simulation examples are used to verify the effectiveness of the proposed method.

Pinning-Based Neural Control for Multiagent Systems With Self-Regulation Intermediate Event-Triggered Method.

A pinning-based self-regulation intermediate event-triggered (ET) funnel tracking control strategy is proposed for uncertain nonlinear multiagent systems (MASs). Based on the backstepping framework, a pinning control strategy is designed to achieve the tracking control objective, which only uses the communication weight between the agents without additional feedback parameters. Moreover, by designing a self-regulation triggered condition based on the tracking error, the intermediate triggered signal is calculated to replace the continuous signal in the controller, so as to achieve the goal of discontinuous update of the controller signal, and this mechanism does not need to add additional compensation function to the controller signal. At the same time, the funnel method is adopted to restrict the error of step n and avoid the possible negative impact caused by control signal. Furthermore, the nonlinear noncontinuous faults are compensated by the disturbance observer. Then, the Lyapunov stability theorem is used to prove that all signals of the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, some simulation results confirm the effectiveness of the proposed control scheme.