Model-free adaptive consensus design for a class of unknown heterogeneous nonlinear multi-agent systems with packet dropouts

This paper studies the consensus problem for a class of unknown heterogeneous nonlinear multi-agent systems via a network with random packet dropouts. Based on the dynamic linearization technique, novel model-free adaptive consensus protocols with the data compensation mechanism are designed for both leaderless and leader-following cases. The advantage of this approach is that only neighborhood input and output data of the agents are required in the protocol design. For the stability analysis, a new Squeeze Theorem based method is developed to derive the theoretic results instead of the traditional contraction mapping principle used in model-free adaptive control. It is shown that the consensus can be achieved for both leaderless and leader-following cases if the communication topology is strongly connected. Finally, numerical simulations verifying the correctness of the theoretical results are given.

Distributed Estimator-Based Event-Triggered Neuro-Adaptive Control for Leader-Follower Consensus of Strict-Feedback Nonlinear Multiagent Systems

This article investigates the leader-follower consensus problem for strict-feedback nonlinear multiagent systems under a dual-terminal event-triggered mechanism. Compared with the existing event-triggered recursive consensus control design, the primary contribution of this article is the development of a distributed estimator-based event-triggered neuro-adaptive consensus control methodology. In particular, by introducing a dynamic event-triggered communication mechanism without continuous monitoring neighbors' information, a novel distributed event-triggered estimator in chain form is constructed to provide the leader's information to the followers. Subsequently, the distributed estimator is utilized to consensus control via backstepping design. To further decrease information transmission, a neuro-adaptive control and an event-triggered mechanism setting on the control channel are codesigned via the function approximate approach. A theoretical analysis shows that all the closed-loop signals are bounded under the developed control methodology, and the estimation of the tracking error asymptotically converges to zero, i.e., the leader-follower consensus is guaranteed. Finally, simulation studies and comparisons are conducted to verify the effectiveness of the proposed control method.

Adaptive Optimal Tracking Control of an Underactuated Surface Vessel Using Actor-Critic Reinforcement Learning

In this article, we present an adaptive reinforcement learning optimal tracking control (RLOTC) algorithm for an underactuated surface vessel subject to modeling uncertainties and time-varying external disturbances. By integrating backstepping technique with the optimized control design, we show that the desired optimal tracking performance of vessel control is guaranteed due to the fact that the virtual and actual control inputs are designed as optimized solutions of every subsystem. To enhance the robustness of vessel control systems, we employ neural network (NN) approximators to approximate uncertain vessel dynamics and present adaptive control technique to estimate the upper boundedness of external disturbances. Under the reinforcement learning framework, we construct actor-critic networks to solve the Hamilton-Jacobi-Bellman equations corresponding to subsystems of surface vessel to achieve the optimized control. The optimized control algorithm can synchronously train the adaptive parameters not only for actor-critic networks but also for NN approximators and adaptive control. By Lyapunov stability theorem, we show that the RLOTC algorithm can ensure the semiglobal uniform ultimate boundedness of the closed-loop systems. Compared with the existing reinforcement learning control results, the presented RLOTC algorithm can compensate for uncertain vessel dynamics and unknown disturbances, and obtain the optimized control performance by considering optimization in every backstepping design. Simulation studies on an underactuated surface vessel are given to illustrate the effectiveness of the RLOTC algorithm.

Fuzzy Adaptive Consensus Control for Nonlinear Multiagent Systems With Intermittent Actuator Faults

This article addresses the distributed adaptive fuzzy consensus fault-tolerant control (FTC) problem for a class of nonstrict-feedback nonlinear multiagent systems (NMASs) with intermittent actuator faults. The NMASs contain unknown nonlinear dynamics, and actuator faults are the type of intermittent faults. Unknown nonlinear functions have been handled based on fuzzy-logic systems (FLSs) approximation, and the distributed virtual controllers together with their parameter adaptive laws are first designed by combining the adaptive backstepping algorithm and the bounded estimation algorithm. To compensate for the intermittent actuator faults, the novel adaptive fuzzy consensus fault-tolerant controllers are then developed by co-designing the last virtual controllers. On the basis of the Lyapunov theory, the stability analysis of the closed-loop system are given, in which the tracking errors converge to zero asymptotically under the directed communication topologies theory. Finally, the proposed FTC scheme is carried on a group of one-link robotic manipulator systems, and its practicability and effectiveness are verified.

Distributed Fuzzy Optimal Consensus Control of State-Constrained Nonlinear Strict-Feedback Systems

This article investigates the distributed fuzzy optimal consensus control problem for state-constrained nonlinear strict-feedback systems under an identifier-actor-critic architecture. First, a fuzzy identifier is designed to approximate each agent's unknown nonlinear dynamics. Then, by defining multiple barrier-type local optimal performance indexes for each agent, the optimal virtual and actual control laws are obtained, where two fuzzy-logic systems working as the actor network and critic network are used to execute control behavior and evaluate control performance, respectively. It is proved that the proposed control protocol can drive all agents to reach consensus without violating state constraints, and make the local performance indexes reach the Nash equilibrium simultaneously. Simulation studies are given to verify the effectiveness of the developed fuzzy optimal consensus control approach.

Adaptive NN Optimal Consensus Fault-Tolerant Control for Stochastic Nonlinear Multiagent Systems

This article investigates the problem of adaptive neural network (NN) optimal consensus tracking control for nonlinear multiagent systems (MASs) with stochastic disturbances and actuator bias faults. In control design, NN is adopted to approximate the unknown nonlinear dynamic, and a state identifier is constructed. The fault estimator is designed to solve the problem raised by time-varying actuator bias fault. By utilizing adaptive dynamic programming (ADP) in identifier-critic-actor construction, an adaptive NN optimal consensus fault-tolerant control algorithm is presented. It is proven that all signals of the controlled system are uniformly ultimately bounded (UUB) in probability, and all states of the follower agents can remain consensus with the leader's state. Finally, simulation results are given to illustrate the effectiveness of the developed optimal consensus control scheme and theorem.

Leader-Follower Formation Learning Control of Discrete-Time Nonlinear Multiagent Systems

This article investigates the leader-follower formation learning control (FLC) problem for discrete-time strict-feedback multiagent systems (MASs). The objective is to acquire the experience knowledge from the stable leader-follower adaptive formation control process and improve the control performance by reusing the experiential knowledge. First, a two-layer control scheme is proposed to solve the leader-follower formation control problem. In the first layer, by combining adaptive distributed observers and constructed i_n -step predictors, the leader's future state is predicted by the followers in a distributed manner. In the second layer, the adaptive neural network (NN) controllers are constructed for the followers to ensure that all the followers track the predicted output of the leader. In the stable formation control process, the NN weights are verified to exponentially converge to their optimal values by developing an extended stability corollary of linear time-varying (LTV) system. Second, by constructing some specific "learning rules," the NN weights with convergent sequences are synthetically acquired and stored in the followers as experience knowledge. Then, the stored knowledge is reused to construct the FLC. The proposed FLC method not only solves the leader-follower formation problem but also improves the transient control performance. Finally, the validity of the presented FLC scheme is illustrated by simulations.

Consensus Tracking for High-Order Uncertain Nonlinear MASs via Adaptive Backstepping Approach

In this article, we focus on the problems of consensus control for nonlinear uncertain multiagent systems (MASs) with both unknown state delays and unknown external disturbances. First, a nonlinear function approximator is proposed for the system uncertainties deriving from unknown nonlinearity for each agent according to adaptive radial basis function neural networks (RBFNNs). By taking advantage of the Lyapunov-Krasovskii functionals (LKFs) approach, we develop a compensation control strategy to eliminate the effects of state delays. Considering the combination of adaptive RBFNNs, LKFs, and backstepping techniques, an adaptive output-feedback approach is raised to construct consensus tracking control protocols and adaptive laws. Then, the proposed consensus tracking scheme can steer the nonlinear MAS synchronizing to the predefined reference signal on account of the Lyapunov stability theory and inequality properties. Finally, simulation results are carried out to verify the validity of the presented theoretical approach.

Adaptive NN-Based Consensus for a Class of Nonlinear Multiagent Systems With Actuator Faults and Faulty Networks

This article addresses the problem of fault-tolerant consensus control of a general nonlinear multiagent system subject to actuator faults and disturbed and faulty networks. By using neural network (NN) and adaptive control techniques, estimations of unknown state-dependent boundaries of nonlinear dynamics and actuator faults, which can reflect the worst impacts on the system, are first developed. A novel NN-based adaptive observer is designed for the observation of faulty transformation signals in networks. On the basis of the NN-based observer and adaptive control strategies, fault-tolerant consensus control schemes are designed to guarantee the bounded consensus of the closed-loop multiagent system with disturbed and faulty networks and actuator faults. The validity of the proposed adaptively distributed consensus control schemes is demonstrated by a multiagent system composed of five nonlinear forced pendulums.

Resilient Delayed Impulsive Control for Consensus of Multiagent Networks Subject to Malicious Agents

Impulsive control is widely applied to achieve the consensus of multiagent networks (MANs). It is noticed that malicious agents may have adverse effects on the global behaviors, which, however, are not taken into account in the literature. In this study, a novel delayed impulsive control strategy based on sampled data is proposed to achieve the resilient consensus of MANs subject to malicious agents. It is worth pointing out that the proposed control strategy does not require any information on the number of malicious agents, which is usually required in the existing works on resilient consensus. Under appropriate control gains and sampling period, a necessary and sufficient graphic condition is derived to achieve the resilient consensus of the considered MAN. Finally, the effectiveness of the resilient delayed impulsive control is well demonstrated via simulation studies.

A Hierarchical Security Control Framework of Nonlinear CPSs Against DoS Attacks With Application to Power Sharing of AC Microgrids

In this article, we investigate the distributed resilient observers-based decentralized adaptive control problem for cyber-physical systems (CPSs) with time-varying reference trajectory under denial-of-service (DoS) attacks. The considered CPSs are modeled as a class of nonlinear multi-input uncertain multiagent systems, which can be used to model an AC microgrid system consisting of distributed generators. When the communication to a subsystem from one of its neighbors is attacked by a DoS attack, the transmitted information is unavailable and the existing distributed adaptive methods used to estimate the bound of the n th-order derivative of the reference trajectory become nonapplicable. To overcome this difficulty, we first design a new distributed estimator for each subsystem to ensure that the magnitude of the state of the estimator is larger than the bound of the n th-order derivative of the reference trajectory after a finite time. By employing the estimator state, a distributed observer with a switching mechanism is proposed. Then, a new block backstepping-based decentralized adaptive controller is developed. Based on the DoS communication duration property, convex design conditions of observer parameters are derived with the Lebesgue integral theory and the average dwell time method. It is proved that the output tracking errors will approach a compact set with the developed method. Finally, the design method is successfully applied to show the effectiveness of the proposed method to solve the power sharing problem for AC microgrids.

Control Design for Uncertain Higher-Order Networked Nonlinear Systems via an Arbitrary Order Finite-Time Sliding Mode Control Law

The authors proposed an arbitrary order finite-time sliding mode control (SMC) design for a networked of uncertain higher-order nonlinear systems. A network of n+1 nodes, connected via a directed graph (with fixed topology), is considered. The nodes are considered to be uncertain in nature. A consensus error-based canonical form of the error dynamics is developed and a new arbitrary order distributed control protocol design strategy is proposed, which not only ensures the sliding mode enforcement in finite time but also confirms the finite time error dynamics stability. Rigorous stability analysis, in closed-loop, is presented, and a simulation example is given, which demonstrates the results developed in this work.

Neuro-adaptive augmented distributed nonlinear dynamic inversion for consensus of nonlinear agents with unknown external disturbance

This paper presents a novel neuro-adaptive augmented distributed nonlinear dynamic inversion (N-DNDI) controller for consensus of nonlinear multi-agent systems in the presence of unknown external disturbance. N-DNDI is a blending of neural network and distributed nonlinear dynamic inversion (DNDI), a new consensus control technique that inherits the features of Nonlinear Dynamic Inversion (NDI) and is capable of handling the unknown external disturbance. The implementation of NDI based consensus control along with neural networks is unique in the context of multi-agent consensus. The mathematical details provided in this paper show the solid theoretical base, and simulation results prove the effectiveness of the proposed scheme.

Consensus Sliding-Mode Fault-Tolerant Control for Second-Order Multi-Agent Systems

This study investigates the consensus problem of second-order nonlinear multi-agent systems (MASs) with actuator faults via a sliding mode control approach. The consensus error dynamic is given based on the relative states of the neighbors. Then, a sliding mode surface based on consensus errors is proposed, and the asymptotic stability of the sliding mode is proved using the Lyapunov theory. Furthermore, a sliding-mode fault-tolerant consensus protocol is proposed to compensate for actuator faults. According to the sliding mode control theory, the proposed sliding-mode fault-tolerant controller ensures that the consensus of the MASs can be reached in a finite time. Finally, a simulation example of a second-order multi-robot system is presented to demonstrate the effectiveness of the proposed controller.

Neural-Network-Based Adaptive Event-Triggered Consensus Control of Nonstrict-Feedback Nonlinear Systems

The event-triggered consensus control problem is studied for nonstrict-feedback nonlinear systems with a dynamic leader. Neural networks (NNs) are utilized to approximate the unknown dynamics of each follower and its neighbors. A novel adaptive event-trigger condition is constructed, which depends on the relative output measurement, the NN weights estimations, and the states of each follower. Based on the designed event-trigger condition, an adaptive NN controller is developed by using the backstepping control design technique. In the control design process, the algebraic loop problem is overcome by utilizing the property of NN basis functions and by designing novel adaptive parameter laws of the NN weights. The proposed adaptive NN event-triggered controller does not need continuous communication among neighboring agents, and it can substantially reduce the data communication and the frequency of the controller updates. It is proven that ultimately bounded leader-following consensus is achieved without exhibiting the Zeno behavior. The effectiveness of the theoretical results is verified through simulation studies.

Observer-Based Event-Triggered Adaptive Fuzzy Control for Leader-Following Consensus of Nonlinear Strict-Feedback Systems

In this article, the leader-following consensus problem via the event-triggered control technique is studied for the nonlinear strict-feedback systems with unmeasurable states. The follower's nonlinear dynamics is approximated using the fuzzy-logic systems, and the fuzzy weights are updated in a nonperiodic manner. By introducing a fuzzy state observer to reconstruct the system states, an observer-based event-triggered adaptive fuzzy control and a novel event-triggered condition are designed, simultaneously. In addition, the nonzero positive lower bound on interevent intervals is presented to avoid the Zeno behavior. It is proved via an extension of the Lyapunov approach that ultimately bounded control is achieved for the leader-following consensus of the considered multiagent systems. One remarkable advantage of the proposed control protocol is that the control law and fuzzy weights are updated only when the event-triggered condition is violated, which can greatly decrease the data transmission and communication resource. The simulation results are provided to show the effectiveness of the proposed control strategy and the theoretical analysis.

Adaptive Consensus Tracking Control of Uncertain Nonlinear Multiagent Systems With Predefined Accuracy

In this article, we consider the leader-follower consensus control problem of uncertain multiagent systems, aiming to achieve the improvement of system steady state and transient performance. To this end, a new adaptive neural control approach is proposed with a novel design of the Lyapunov function, which is generated with a class of positive functions. Guided by this idea, a series of smooth functions is incorporated into backstepping design and Lyapunov analysis to develop a performance-oriented controller. It is proved that the proposed controller achieves a perfect asymptotic consensus performance and a tunable L₂ transient performance of synchronization errors, whereas most existing results can only ensure the stability. Simulation demonstrates the obtained results.

Multi-Lateral Teleoperation Based on Multi-Agent Framework: Application to Simultaneous Training and Therapy in Telerehabilitation

In this paper, a new scheme for multi-lateral remote rehabilitation is proposed. There exist one therapist, one patient, and several trainees, who are participating in the process of telerehabilitation (TR) in this scheme. This kind of strategy helps the therapist to facilitate the neurorehabilitation remotely. Thus, the patients can stay in their homes, resulting in safer and less expensive costs. Meanwhile, several trainees in medical education centers can be trained by participating partially in the rehabilitation process. The trainees participate in a "hands-on" manner; so, they feel like they are rehabilitating the patient directly. For implementing such a scheme, a novel theoretical method is proposed using the power of multi-agent systems (MAS) theory into the multi-lateral teleoperation, based on the self-intelligence in the MAS. In the previous related works, changing the number of participants in the multi-lateral teleoperation tasks required redesigning the controllers; while, in this paper using both of the decentralized control and the self-intelligence of the MAS, avoids the need for redesigning the controller in the proposed structure. Moreover, in this research, uncertainties in the operators' dynamics, as well as time-varying delays in the communication channels, are taken into account. It is shown that the proposed structure has two tuning matrices (L and D) that can be used for different scenarios of multi-lateral teleoperation. By choosing proper tuning matrices, many related works about the multi-lateral teleoperation/telerehabilitation process can be implemented. In the final section of the paper, several scenarios were introduced to achieve "Simultaneous Training and Therapy" in TR and are implemented with the proposed structure. The results confirmed the stability and performance of the proposed framework.

Fuzzy-Model-Based Output Feedback Reliable Control for Network-Based Semi-Markov Jump Nonlinear Systems Subject to Redundant Channels

This article investigates the reliable output feedback control problem for networked nonlinear semi-Markov jump systems, in which a control strategy with redundant channels is established to reduce the adverse effect caused by packet dropouts. The actuator faults are fully considered in the setup. On the basis of stochastic analysis theory and fuzzy-model-based method, some criteria are established to guarantee the σ -error mean-square stability for the considered systems. As a consequence, the reliable output feedback controller design method is proposed, which can be utilized to deal with the actuator failures problem effectively. Finally, two illustrative examples are employed to explain the availability of the presented design approach, where the single-link robot arm system model is contained.

Adaptive bounded neural network control for coordinated path-following of networked underactuated autonomous surface vehicles under time-varying state-dependent cyber-attack

This paper is concerned with the problem of coordinated path-following for networked underactuated autonomous surface vehicles in the presence of time-varying state-dependent cyber-attack. An adaptive bounded neural network controller is proposed to mitigate the malicious effect of the cyber-attack. At first, an individual path-following control law is designed for each vehicle by fusing a back-stepping technique, a line-of-sight guidance principle and a predictor-based neural network method. Second, a path update law is developed based on a synchronization approach together with an adaptive control method. The salient features of the proposed controller are presented as follows. First, an adaptive corrective signal is incorporated into the path update law design such that a desired formation can be achieved regardless of the time-varying state-dependent cyber-attack. Second, by using a saturation function and a projection operator, the proposed controller is bounded and the bound is known as a priori. It is proven that the closed-loop system is input-to-state practical stable in the face of time-varying state-dependent cyber-attack. Simulation results show the effectiveness of the proposed adaptive bounded neural network controller for coordinated path-following of networked underactuated autonomous surface vehicles subject to the cyber-attack.

Distributed Adaptive Fuzzy Event-Triggered Containment Control of Nonlinear Strict-Feedback Systems

In this paper, the adaptive fuzzy event-triggered containment control problem is addressed for uncertain nonlinear strict-feedback systems guided by multiple leaders. A novel distributed adaptive fuzzy event-triggered containment control is designed only using the information of the individual follower and its neighbors. Moreover, a distributed event-trigger condition with an adjustable threshold is developed simultaneously. The designed containment control law is updated in an aperiodic manner, only when event-triggered errors exceed tolerable thresholds. It is proved that the uniformly ultimately bounded containment control can be achieved, and there is no Zeno behavior exhibited by applying the proposed control scheme. Simulation studies are outlined to illustrate the effectiveness of the theoretical results and the advantages of the event-triggered containment control proposed in this paper.

Cooperative Path Following Ring-Networked Under-Actuated Autonomous Surface Vehicles: Algorithms and Experimental Results

This paper addresses the cooperative path following the problem of ring-networked under-actuated autonomous surface vehicles on a closed curve. A cooperative guidance law is proposed at the kinematic level such that a symmetric formation pattern is achieved. Specifically, individual guidance laws of surge speed and angular rate are developed by using a backstepping technique and a line-of-sight guidance method. Then, a coordination design is proposed to update the path variables under a ring-networked topology. The equilibrium point of the closed-loop system has been proven to be globally asymptotically stable. The result is extended to the cooperative path following the lack of sharing of a global reference velocity, and a distributed observer is designed to recover the reference velocity to each vehicle. Moreover, the cooperative path following the presence of an unknown sideslip is considered, and an extended state observer is developed to compensate for the effect of the unknown sideslip. Both simulation and experimental results are provided to illustrate the effectiveness of the proposed cooperative guidance law for the path following over a closed curve.

Distributed Adaptive Neural Network Control Applied to a Formation Tracking of a Group of Low-Cost Underwater Drones in Hazardous Environments

This paper addresses a formation tracking problem of multiple low-cost underwater drones by implementing distributed adaptive neural network control (DANNC). It is based on a leader-follower architecture to operate in hazardous environments. First, unknown parameters of underwater vehicle dynamics, which are important requirements for real-world applications, are approximated by a neural network using a radial basis function. More specifically, those parameters are only calculated by local information, which can be obtained by an on-board camera without using an external positioning system. Secondly, a potential function is employed to ensure there is no collision between the underwater drones. We then propose a desired configuration of a group of unmanned underwater vehicles (UUVs) as a time-variant function so that they can quickly change their shape between them to facilitate the crossing in a narrow area. Finally, three UUVs, based on a robot operating system (ROS) platform, are used to emphasize the realistic low-cost aspect of underwater drones. The proposed approach is validated by evaluating in different experimental scenarios.

Distributed Control of High-Order Nonlinear Input Constrained Multiagent Systems Using a Backstepping-Free Method

This paper presents novel cooperative tracking control for a class of input-constrained multiagent systems with a dynamic leader. Each follower agent is described by a high-order nonlinear dynamics in strict feedback form with input constraints. Our main contribution lies in presenting a system transformation method that can convert the input-constrained state feedback cooperative tracking control of agents into an unconstrained output feedback control of agents with dynamics in Brunovsky normal form. As a result, the original problem is simplified to be a simple stabilization of the transformed system for the agents. Thus, the use of the backstepping scheme is obviated, and the synthesis and computation are extremely simplified. It is strictly proved that all follower agents can synchronize to the leader with bounded synchronization errors, and all other signals in the closed-loop system are semi-global uniformly ultimately bounded. Finally, numerical analysis is carried out to validate the theoretical results and demonstrate the effectiveness of the proposed approach.

Distributed adaptive neural networks leader-following formation control for quadrotors with directed switching topologies

The leader-following formation problem is discussed for a team of quadrotors under directed switching topologies. To obtain a more general dynamic model, we describe the quadrotor system in a non-affine pure-feedback form with mismatched unknown nonlinearities. By employing an adaptive neural networks state observer to approximate the unknown nonlinear functions and to reconstruct the immeasurable inner states, we propose a novel distributed output feedback formation control protocol with the backstepping method combining with the dynamic surface control technique. From the Lyapunov stability theorem, all signals in the closed-loop formation system are proven to be cooperatively semiglobally uniformly ultimately bounded for any given bounded initial conditions. Finally, we proved that we verify the performance of the proposed formation control approach by a simulation study.

Consensus Maneuvering for a Class of Nonlinear Multivehicle Systems in Strict-Feedback Form

In this paper, a consensus maneuvering problem for nonlinear multivehicle systems in strict-feedback form is investigated. The consensus maneuvering problem includes a geometric task and a dynamic task. The geometric task means that all trajectories of follower vehicles converge to a parameterized path. The dynamic task is to drive the system to satisfy a desired dynamic assignment. A consensus maneuvering controller is developed for each vehicle based on a modular design approach. First, an estimator module is designed based on an echo state network, which is used to estimate uncertain nonlinearities. Then, a controller module is designed based on a modified dynamic surface control method through the use of a second-order nonlinear tracking differentiator. Finally, a path update law is designed based on a distributed maneuvering error feedback and a filtering scheme. The proposed controller is distributed in the sense that the path information is accessed by a small number of follower vehicles only. The stability of the closed-loop system cascaded by the estimator module and the controller module is analyzed based on input-to-state stability theory and cascade theory. Simulation results are provided to demonstrate the efficacy of the proposed consensus maneuvering controllers for uncertain nonlinear strict-feedback systems.

Adaptive Robust Tracking Control for Multiple Unknown Fractional-Order Nonlinear Systems

By applying the fractional Lyapunov direct method, we investigate the robust consensus tracking problem for a class of uncertain fractional-order multiagent systems with a leader whose input is unknown and bounded. More specifically, multiple fractional-order systems with heterogeneous unknown nonlinearities and external disturbances are considered in this paper, which include the second-order multiagent systems as its special cases. First, a discontinuous neural network-based (NN-based) distributed robust adaptive algorithm is designed to guarantee the consensus tracking error exponentially converges to zero under a fixed topology. Also the derived results are further extended to the case of switching topology by appropriately choosing multiple Lyapunov functions. Second, a continuous NN-based distributed robust adaptive algorithm is further proposed to eliminate the undesirable chattering phenomenon of the discontinuous controller, where the consensus tacking error is uniformly ultimately bounded and can be reduced as small as desired. It is worth noting that all the proposed NN-based robust adaptive algorithms are independent of any global information and thus are fully distributed. Finally, numerical simulations are provided to validate the correctness of the proposed algorithms.

Adaptive Fuzzy Containment Control of Nonlinear Systems With Unmeasurable States

The adaptive fuzzy containment control problem is discussed for high-order systems with unknown nonlinear dynamics and unmeasurable states guided by multiple dynamic leaders. A high gain observer is introduced to reconstruct the system states. Then, utilizing fuzzy logic systems to model followers' dynamics, an observer-based adaptive fuzzy containment control approach is presented using only the relative position of the neighbors. It is shown that the uniformly ultimately bounded containment control is realized under the condition that, each follower can obtain the information from at least one leader through a directed path. As an extension, an observer-based containment control with prescribed performance is developed, which guarantees the relative position error to be bounded by a specified bound. The obtained theoretical results are validated by simulation examples.

Two-Layer Distributed Formation-Containment Control of Multiple Euler-Lagrange Systems by Output Feedback

This paper addresses the distributed formation-containment (DFC) problem for multiple Euler-Lagrange systems with model uncertainties via output feedback in both constant and time-varying formation cases. First, a novel definition of the DFC problem is proposed using a two-layer framework. Since only parts of the followers can acquire the states of the dynamic leader, we design a distributed finite-time sliding-mode estimator to obtain accurate estimations of the desired position and velocity for each agent. Next, to deal with the absence of velocity sensors, we propose two DFC control laws combined with the high-gain observer for the leaders and the followers, respectively, while the time-varying formation in the first layer and the leader-based containment in the second layer can be achieved. Further, the adaptive neural networks are applied to deal with the model uncertainties due to their superior approximation capability. The uniform ultimate boundedness of all the state errors can be guaranteed by Lyapunov stability theory. In addition, a unified framework is given which can be transformed to four other basic distributed problems. Finally, simulation examples are presented to illustrate the feasibility of the theoretical results.

Connectivity-Preserving Consensus Tracking of Uncertain Nonlinear Strict-Feedback Multiagent Systems: An Error Transformation Approach

This brief addresses a distributed connectivity-preserving adaptive consensus tracking problem of uncertain nonlinear strict-feedback multiagent systems with limited communication ranges. Compared with existing consensus results for uncertain nonlinear lower triangular multiagent systems, the main contribution of this brief is to present an error-transformation-based design methodology to preserve initial connectivity patterns in the consensus tracking field, namely, both connectivity preservation and consensus tracking problems are considered for uncertain nonlinear lower triangular multiagent systems. A dynamic surface design based on nonlinearly transformed errors and neural network function approximators is established to construct the local controller of each follower. In addition, a technical lemma is derived to analyze the stability of the proposed connectivity-preserving consensus scheme in the Lyapunov sense.

Fully Distributed Adaptive Consensus Control of a Class of High-Order Nonlinear Systems With a Directed Topology and Unknown Control Directions

In this paper, we investigate the adaptive consensus control for a class of high-order nonlinear systems with different unknown control directions where communications among the agents are represented by a directed graph. Based on backstepping technique, a fully distributed adaptive control approach is proposed without using global information of the topology. Meanwhile, a novel Nussbaum-type function is proposed to address the consensus control with unknown control directions. It is proved that boundedness of all closed-loop signals and asymptotically consensus tracking for all the agents' outputs are ensured. In simulation studies, a numerical example is illustrated to show the effectiveness of the control scheme.

Leader-Follower Output Synchronization of Linear Heterogeneous Systems With Active Leader Using Reinforcement Learning

This paper develops optimal control protocols for the distributed output synchronization problem of leader-follower multiagent systems with an active leader. Agents are assumed to be heterogeneous with different dynamics and dimensions. The desired trajectory is assumed to be preplanned and is generated by the leader. Other follower agents autonomously synchronize to the leader by interacting with each other using a communication network. The leader is assumed to be active in the sense that it has a nonzero control input so that it can act independently and update its control to keep the followers away from possible danger. A distributed observer is first designed to estimate the leader's state and generate the reference signal for each follower. Then, the output synchronization of leader-follower systems with an active leader is formulated as a distributed optimal tracking problem, and inhomogeneous algebraic Riccati equations (AREs) are derived to solve it. The resulting distributed optimal control protocols not only minimize the steady-state error but also optimize the transient response of the agents. An off-policy reinforcement learning algorithm is developed to solve the inhomogeneous AREs online in real time and without requiring any knowledge of the agents' dynamics. Finally, two simulation examples are conducted to illustrate the effectiveness of the proposed algorithm.

Adaptive critic designs for optimal control of uncertain nonlinear systems with unmatched interconnections

In this paper, we develop a novel optimal control strategy for a class of uncertain nonlinear systems with unmatched interconnections. To begin with, we present a stabilizing feedback controller for the interconnected nonlinear systems by modifying an array of optimal control laws of auxiliary subsystems. We also prove that this feedback controller ensures a specified cost function to achieve optimality. Then, under the framework of adaptive critic designs, we use critic networks to solve the Hamilton-Jacobi-Bellman equations associated with auxiliary subsystem optimal control laws. The critic network weights are tuned through the gradient descent method combined with an additional stabilizing term. By using the newly established weight tuning rules, we no longer need the initial admissible control condition. In addition, we demonstrate that all signals in the closed-loop auxiliary subsystems are stable in the sense of uniform ultimate boundedness by using classic Lyapunov techniques. Finally, we provide an interconnected nonlinear plant to validate the present control scheme.