Consensus-based distributed cooperative learning control for a group of discrete-time nonlinear multi-agent systems using neural networks

This paper focuses on the cooperative learning capability of radial basis function neural networks in adaptive neural controllers for a group of uncertain discrete-time nonlinear systems where system structures are identical but reference signals are different. By constructing an interconnection topology among learning laws of NN weights in order to share their learned knowledge on-line, a novel adaptive NN control scheme, called distributed cooperative learning control scheme, is proposed. It is guaranteed that if the interconnection topology is undirected and connected, all closed-loop signals are uniform ultimate bounded and tracking errors of all systems can converge to a small neighborhood around the origin. Moreover, it is proved that all estimated NN weights converge to a small neighborhood of their common optimal value along the union of all state trajectories, which means that the estimated NN weights reach consensus with a small consensus error. Thus, all learned NN models have the better generalization capability than ones obtained by the deterministic learning method. The learned knowledge is also adopted to control a class of uncertain systems with the same structure but different reference signals. Finally, a simulation example is provided to verify the effectiveness and advantages of the distributed cooperative learning control scheme developed in this paper.     

Consensus control of leader-following nonlinear multi-agent systems with distributed adaptive iterative learning control

This paper addresses the consensus problem of leader-following nonlinear multi-agent systems with iterative learning control. The assumption that only a small portion of following agents can receive the information of leader agent is considered. To approximate the nonlinear dynamics of a given system, the radial basis function neural network is introduced. Then, a distributed adaptive iterative learning control protocol with an auxiliary control term is designed, where the estimates of nonlinear dynamics are applied in control protocol design and three adaptive laws are presented. Furthermore, the convergence of the proposed control protocol is analysed by Lyapunov stability theory. Finally, a simulation example is provided to demonstrate the validity of theoretical results.     

Adaptive neural control for multiagent systems with asymmetric time‐varying state constraints and input saturation

This article investigates the leader‐follower consensus problem of a class of non‐strict‐feedback nonlinear multiagent systems with asymmetric time‐varying state constraints (ATVSC) and input saturation, and an adaptive neural control scheme is developed. By introducing the distributed sliding‐mode estimator, each follower can obtain the estimation of leader's trajectory and track it directly. Then, with the help of time‐varying asymmetric barrier Lyapunov function and radial basis function neural networks, the controller is designed based on backstepping technique. Furthermore, the mean‐value theorem and Nussbaum function are utilized to address the problems of input saturation and unknown control direction. Moreover, the number of adaptive laws is equal to that of the followers, which reduces the computational complexity. It is proved that the leader‐follower consensus tracking control is achieved without violating the ATVSC, and all closed‐loop signals are semiglobally uniformly ultimately bounded. Finally, the simulation results are provided to verify the effectiveness of the control scheme.     

Adaptive Finite-time Consensus for Second-order Nonlinear Multi-agent Systems with Input Quantization

In this paper, the adaptive finite-time consensus (FTC) control problem of second-order nonlinear multi-agent systems (MASs) with input quantization and external disturbances is studied. With the help of finite time control technology, a novel distributed adaptive control protocol is constructed to achieve FTC performance for second-order nonlinear MASs by using the recursive method. The control input is quantized through a hysteresis quantizer, which reduces the communication rate of arbitrary two agents. The unknown functions are approximated by adopting the radial basis function neural networks. Under the consensus protocols and adaptive laws, it can be proved that velocity errors of arbitrary two agents reach a small region of zero in finite time as well as position errors. Finally, the effectiveness of the proposed method is illustrated via a simulation example.

     

Consensus in multi-agent systems with nonlinear uncertainties under a fixed undirected graph

This paper presents consensus algorithms by integrating cooperative control and adaptive control laws for multi-agent systems with unknown nonlinear uncertainties. An ideal multi-agent system without uncertainties is introduced first. The cooperative control law, based on an artificial potential function, is designed to make the ideal multi-agent system achieve consensus under a fixed and connected undirected graph. The presence of uncertainties will degenerate the performance, or even destabilize the whole multi-agent system. The L ₁ adaptive control law is therefore introduced to handle unknown nonlinear uncertainties. Two different consensus cases are considered: 1) normal consensus—where all agents reach an agreement on an initially undetermined position and velocity, and 2) consensus with a virtual leader—where all agents’ states converge to the virtual leader’s states. Under a fixed and connected undirected graph, the presented consensus algorithms enable the real multi-agent system to stay close to the ideal multi-agent system which achieves consensus with or without a virtual leader. Simulation results of 2-D consensus with nonlinear uncertainties are provided to demonstrate the presented algorithms.     

Distributed consensus control of nonlinear multi-agent systems with sensor uncertainty

This article focuses on the distributed consensus control problem for nonlinear multi-agent systems subject to sensor uncertainty. To be specific, we study nonlinear multi-agent systems of lower or upper triangular structure with unknown growth rate and sensor uncertainty. A new time-varying gain approach is proposed to construct observers as well as distributed output-feedback controllers. By selecting suitable design parameters, the leader-follower consensus of nonlinear multi-agent systems is achieved. Different from the existing results, a time-varying function in a logarithmic form is introduced to deal with unknown growth rate. Moreover, a monotonically increasing time-varying function is constructed to cope with uncertain sensor sensitivity. Two simulation examples are provided to demonstrate the effectiveness of the proposed distributed consensus control algorithms.     

Distributed cooperative learning algorithms using wavelet neural network

This paper investigates the distributed cooperative learning (DCL) problems over networks, where each node only has access to its own data generated by the unknown pattern (map or function) uniformly, and all nodes cooperatively learn the pattern by exchanging local information with their neighboring nodes. These problems cannot be solved by using traditional centralized algorithms. To solve these problems, two novel DCL algorithms using wavelet neural networks are proposed, including continuous-time DCL (CT-DCL) algorithm and discrete-time DCL (DT-DCL) algorithm. Combining the characteristics of neural networks with the properties of the wavelet approximation, the wavelet series are used to approximate the unknown pattern. The DCL algorithms are used to train the optimal weight coefficient matrix of wavelet series. Moreover, the convergence of the proposed algorithms is guaranteed by using the Lyapunov method. Compared with existing distributed optimization strategies such as distributed average consensus (DAC) and alternating direction method of multipliers (ADMM), our DT-DCL algorithm requires less information communications and training time than ADMM strategy. In addition, it achieves higher accuracy than DAC strategy when the network consists of large amounts of nodes. Moreover, the proposed CT-DCL algorithm using a proper step size is more accurate than the DT-DCL algorithm if the training time is not considered. Several illustrative examples are presented to show the efficiencies and advantages of the proposed algorithms.

     

切换拓扑下非线性多智能体系统自适应神经网络一致性

本文研究一类具有未知控制系数的非线性多智能体系统自适应神经网络分布式控制策略.首先,针对切换拓扑下具有未知控制系数的非线性多智能体系统一致性问题,提出一类自适应神经网络一致性控制算法.其中,采用神经网络函数逼近方法解决系统中的不确定性问题,并设计一项自适应光滑项处理有界扰动和神经网络函数逼近误差.随后,证明了切换拓扑下具有未知控制系数的非线性多智能体系统的一致性,并保证了闭环系统的有界性.此外,本文把相关的一致性算法扩展到了一般有向图含有一个有向生成树的情形.最后,通过仿真实例验证了本文所提算法的有效性.     

Two-loop reinforcement learning algorithm for finite-horizon optimal control of continuous-time affine nonlinear systems

This article proposes three novel time-varying policy iteration algorithms for finite-horizon optimal control problem of continuous-time affine nonlinear systems. We first propose a model-based time-varying policy iteration algorithm. The method considers time-varying solutions to the Hamiltonian–Jacobi–Bellman equation for finite-horizon optimal control. Based on this algorithm, value function approximation is applied to the Bellman equation by establishing neural networks with time-varying weights. A novel update law for time-varying weights is put forward based on the idea of iterative learning control, which obtains optimal solutions more efficiently compared to previous works. Considering that system models may be unknown in real applications, we propose a partially model-free time-varying policy iteration algorithm that applies integral reinforcement learning to acquiring the time-varying value function. Moreover, analysis of convergence, stability, and optimality is provided for every algorithm. Finally, simulations for different cases are given to verify the convenience and effectiveness of the proposed algorithms.     

Distributed control of uncertain multiagent systems for tracking a leader with unknown fractional‐order dynamics

This paper proposes distributed adaptive cooperative control algorithms for second‐order agents to track a leader with unknown dynamics. The models of the followers and the leader are composed of uncertain nonlinear components. The order of the leader's dynamics is unknown and can be fractional. Only the single output information is shared among neighbored agents. To simplify the control design, linearly parameterized neural networks are used to approximate the unknown functions. We first present an adaptive control for leaderless consensus and then extend the method to the tracking problem. Thorough theoretical proofs as well as numerical simulation are included to verify the results. Compared with relevant literature, the new approach applies to a larger variety of systems because (i) knowledge about the structure of leader's model is unnecessary; (ii) the unknown functions in different agents' dynamics can be diverse and arbitrary, in other words, the algorithms apply to heterogeneous agents; (iii) the results can be simply used without parameter calculations.     

Neural network-based adaptive consensus tracking control for multi-agent systems under actuator faults

In this paper, a distributed output feedback consensus tracking control scheme is proposed for second-order multi-agent systems in the presence of uncertain nonlinear dynamics, external disturbances, input constraints, and partial loss of control effectiveness. The proposed controllers incorporate reduced-order filters to account for the unmeasured states, and the neural networks technique is implemented to approximate the uncertain nonlinear dynamics in the synthesis of control algorithms. In order to compensate the partial loss of actuator effectiveness faults, fault-tolerant parts are included in controllers. Using the Lyapunov approach and graph theory, it is proved that the controllers guarantee a group of agents that simultaneously track a common time-varying state of leader, even when the state of leader is available only to a subset of the members of a group. Simulation results are provided to demonstrate the effectiveness of the proposed consensus tracking method.     

Distributed Adaptive Cooperative Tracking of Uncertain Nonlinear Fractional-order Multi-agent Systems

In this paper, the leader-following tracking problem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the interaction topology is undirected and connected and the unknown nonlinear uncertain dynamics can be parameterized by a neural network, an adaptive learning law is proposed to deal with unknown nonlinear dynamics, based on which a kind of cooperative tracking protocols are constructed. The feedback gain matrix is obtained to solve an algebraic Riccati equation. To construct the fully distributed cooperative tracking protocols, the adaptive law is also adopted to adjust the coupling weight. With the developed control laws, we can prove that all signals in the closed-loop systems are guaranteed to be uniformly ultimately bounded. Finally, a simple simulation example is provided to illustrate the established result.      

Relationships between the A priori and A posteriori errors in nonlinear adaptive neural filters

The lower bounds for the a posteriori prediction error of a nonlinear predictor realized as a neural network are provided. These are obtained for a priori adaptation and a posteriori error networks with sigmoid nonlinearities trained by gradient-descent learning algorithms. A contractivity condition is imposed on a nonlinear activation function of a neuron so that the a posteriori prediction error is smaller in magnitude than the corresponding a priori one. Furthermore, an upper bound is imposed on the learning rate eta so that the approach is feasible. The analysis is undertaken for both feedforward and recurrent nonlinear predictors realized as neural networks.     

状态时延和全状态约束下的多智能体系统自适应事件触发控制

研究有向通信图下非线性多智能体系统的一致控制问题.首先,通过引入性能函数,使输出误差满足预定性能;其次,采用障碍Lyapunov函数,保证所有状态满足约束条件,结合李雅普诺夫-克拉索夫斯基(Lyapunov-Krasovskii,LK)泛函和杨氏不等式消除状态时延的影响,利用径向基函数神经网络(radial basis...     

基于ELM的一类MIMO仿射非线性系统的鲁棒自适应控制

针对一类多输入多输出(MIMO) 仿射非线性动态系统, 提出一种基于极限学习机(ELM) 的鲁棒自适应神经控制方法. ELM随机确定单隐层前馈网络(SLFNs) 的隐含层参数, 仅需调整网络的输出权值, 能以极快的学习速度获得良好的推广性. 在所提出的控制方法中, 利用ELM逼近系统的未知非线性项, 针对ELM网络的权值、逼近误差及外界扰动的未知上界值分别设计参数自适应律, 通过Lyapunov 稳定性分析可以保证闭环系统所有信号半全局最终一致有界. 仿真结果表明了该控制方法的有效性.

     

Adaptive neural control and learning of affine nonlinear systems

This paper presents deterministic learning from adaptive neural network control of affine nonlinear systems with completely unknown system dynamics. Thanks to the learning capability of radial basis function, neural network (NN), stable adaptive NN controller is designed for the unknown affine nonlinear systems. The designed adaptive NN controller is rigorously shown that learning of the unknown closed-loop system dynamics can be achieved during the stable control process because partial persistent excitation condition of some internal signals in the closed-loop system is satisfied. Subsequently, neural learning controller using the knowledge obtained from deterministic learning is constructed to achieve closed-loop stability and improve control performance. Numerical simulation is provided to show the effectiveness of the proposed control scheme.     

RBF neural networks-based robust adaptive tracking control for switched uncertain nonlinear systems

In this paper, a robust adaptive H∞ control scheme is presented for a class of switched uncertain nonlinear systems. Radical basis function neural networks (RBF NNs) are employed to approximate unknown nonlinear functions and uncertain terms. A robust H∞ controller is designed to enhance robustness due to the existence of the compound disturbance which consists of approximation errors of the neural networks and external disturbance. Adaptive neural updated laws and switching signals are deducted from multiple Lyapunov function approach. It is proved that with the proposed control scheme, the resulting closed-loop switched system is robustly stable and uniformly ultimately bounded (UUB) such that good capabilities of tracking performance is attained and H∞ tracking error performance index is achieved. A practical example shows the effectiveness of the proposed control scheme.     

Adaptive tracking H-infinity control for switched nonlinear systems with unknown control gain sign

This paper addresses the adaptive tracking control scheme for switched nonlinear systems with unknown control gain sign. The approach relaxes the hypothesis that the upper bound of function control gain is known constant and the bounds of external disturbance and approximation errors of neural networks are known. RBF neural networks (NNs) are used to approximate unknown functions and an H-infinity controller is introduced to enhance robustness. The adaptive updating laws and the admissible switching signals have been derived from switched multiple Lyapunov function method. It’s proved that the resulting closed loop system is asymptotically Lyapunov stable such that the output tracking error performance and H-infinity disturbance attenuation level are well obtained. Finally, a simulation example of Forced Duffing systems is given to illustrate the effectiveness of the proposed control scheme and improve significantly the transient performance.     

Distributed adaptive fuzzy iterative learning control of coordination problems for higher order multi-agent systems

In this paper, the adaptive fuzzy iterative learning control scheme is proposed for coordination problems of Mth order (M ≥ 2) distributed multi-agent systems. Every follower agent has a higher order integrator with unknown nonlinear dynamics and input disturbance. The dynamics of the leader are a higher order nonlinear systems and only available to a portion of the follower agents. With distributed initial state learning, the unified distributed protocols combined time-domain and iteration-domain adaptive laws guarantee that the follower agents track the leader uniformly on [0, T]. Then, the proposed algorithm extends to achieve the formation control. A numerical example and a multiple robotic system are provided to demonstrate the performance of the proposed approach.     

Deadzone compensation in motion control systems using neuralnetworks

A compensation scheme is presented for general nonlinear actuator deadzones of unknown width. The compensator uses two neural networks (NNs), one to estimate the unknown deadzone and another to provide adaptive compensation in the feedforward path. The compensator NN has a special augmented form containing extra neurons whose activation functions provide a “jump function basis set” for approximating piecewise continuous functions. Rigorous proofs of closed-loop stability for the deadzone compensator are provided and yield tuning algorithms for the weights of the two NNs. The technique provides a general procedure for using NNs to determine the preinverse of an unknown right-invertible function