Distributed finite‐time tracking of multiple Euler–Lagrange systems without velocity measurements

This paper investigates the distributed finite‐time tracking problem of networked agents with multiple Euler–Lagrange dynamics. To achieve finite‐time tracking, a distributed finite‐time protocol is first proposed on the basis of both relative position and relative velocity measurements. By using tools from homogeneous theory, it is theoretically shown that the proposed protocol can guarantee finite‐time tracking in the presence of control input constraints. On the basis of the state feedback analysis and with the aid of second‐order sliding‐mode observer approach, a new class of finite‐time tracking protocols based only on the relative position measurements is developed and employed. It is proved that the multiple agents equipped with the designed protocols can track the target location in finite time. Furthermore, a decentralized finite‐time protocol based on a distributed estimator is proposed to solve the finite‐time tracking problems with a dynamic leader. The effectiveness of the theoretical results is finally illustrated by numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Leader‐following consensus with connectivity preservation of uncertain Euler–Lagrange multi‐agent systems

The leader‐following consensus problem for multiple Euler–Lagrange systems has been extensively studied for various scenarios. Under the assumption that the communication graph is jointly connected, one of our recent papers gave the solution for the case where the leader system can generate a combination of arbitrary step signal, arbitrary ramp signal, and arbitrary sinusoidal signals. In practice, it is desirable to enable the control law the capability of maintaining the connectivity of the communication graph, thus achieving the leader‐following consensus without assuming the connectivity of the communication graph. We call such a problem as leader‐following consensus with connectivity preservation. By combining the adaptive control technique and potential function technique, we will show that such a problem is solvable. By employing different potential functions, our approach may also lead to the solution of such problems as rendezvous, flocking and swarming. Copyright © 2017 John Wiley & Sons, Ltd.     

Model‐independent consensus control of Euler‐Lagrange agents with interconnecting delays

In this paper, we introduce, for the first time, a proportional‐integral‐derivative controller to solve the consensus problem for networked Euler‐Lagrange agents. The proposed control scheme solves both the leaderless and the leader‐follower consensus problems without requiring the exact knowledge of any part of the agent model. Contrary to the previous studies in the literature, we do not require either to cancel any part of the agent model, which renders robust our approach, nor employ discontinuous controllers. It is proven that both consensus problems are solved globally and asymptotically and the presence of bounded time‐varying communicating delays is considered.     

Distributed event‐triggered tracking control with a dynamic leader for multiple Euler‐Lagrange systems under directed networks

The distributed tracking control for multiple Euler‐Lagrange systems with a dynamic leader is investigated in this article via the event‐triggered approach. Only a portion of followers have access to the leader, and the communication topology among all agents is directed that contains a directed spanning tree rooted at the leader. The case that the leader's generalized velocity is constant is first considered, and a distributed event‐based control law is developed by using a velocity estimator. When the leader's generalized velocity is time‐varying, novel distributed continuous estimators are proposed to avoid the undesirable chattering effect while guaranteeing that the estimate errors converge to zeros. With the designed distributed estimators, another distributed event‐based control protocol is provided. Controller update frequency and resource consumption in our work can be reduced by applying the aforementioned two distributed control laws, and the tracking errors can converge to zeros. In addition, it is rigorously proved that no agent exhibits Zeno behavior. Finally, the effectiveness of the proposed distributed event‐based control laws is elucidated by a number of simulation examples.     

Leader‐following consensus of multiple uncertain Euler‐Lagrange systems

A new adaptive distributed controller is developed for the leader‐following consensus problem of multiple uncertain Euler‐Lagrange systems. A distinct feature of our proposed approach as opposed to the existing ones is that it does not need the exchange of controller's state among the communication network. As a consequence, it not only makes the implementation of the controller much easier but also reduces the communication cost. The effectiveness of the main result is demonstrated by some exemplary applications to cooperative control of multiple two‐link robot arms.     

Finite‐time consensus of Euler‐Lagrange agents without velocity measurements via energy shaping

Motivated by the energy‐shaping framework and the properties of homogeneous systems, this paper deals with the problem of achieving consensus of multiple Euler‐Lagrange (EL) systems using the energy shaping plus damping injection principles of passivity‐based control. We propose a method to derive a novel family of decentralized controllers that is capable of solving the leaderless and the leader‐follower consensus problems in finite‐time in networks of fully actuated EL systems without employing velocity measurements. As in the energy‐shaping methodology, the controller is another EL system and the plant‐controller interconnection is the gradient of a suitable defined potential function. The potential energy and dissipation functions, of the controller, are provided with some homogeneous properties in order to achieve finite‐time convergence. This paper provides several simulations that corroborate the performance of different controllers.     

Tracking control of uncertain Euler–Lagrange systems with finite‐time convergence

A unified solution is presented to the tracking control problem of Euler–Lagrange systems with finite‐time convergence. A reconstruction module is designed to estimate the overall of unmodeled dynamics, disturbance, actuator misalignment, and multiple actuator faults. That reconstruction is accomplished in finite time with zero error. A nonsingular terminal sliding mode controller is then synthesized, and the resultant closed‐loop system is also shown to be finite‐time stable with the reference trajectory followed in finite time. Unlike most sliding mode control methods to handle system uncertainties, the designed control has less conservativeness and stronger fault tolerant capability. A rigid spacecraft system is used to demonstrate the effectiveness and potential of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Leader‐following consensus with disturbance rejection for uncertain Euler‐Lagrange systems over switching networks

In this paper, the leader‐following consensus with disturbance rejection problem of uncertain Euler‐Lagrange systems is studied by the adaptive distributed observer approach. We first present a key lemma that guarantees the existence of an exponentially convergent adaptive distributed observer for linear leader systems without exponential growing modes over jointly connected switching and directed communication networks. This lemma also provides a specific Lyapunov function for the error dynamics of the adaptive distributed observer, which will play a crucial role in establishing one of the main results. A special case of this result where there is no disturbance will extend the existing result for a neutrally stable leader system and undirected communication networks to the case where the communication networks are directed and the leader's positional signal includes the class of ramp signals. Two examples will be given to demonstrate the effectiveness of the new results.     

Pose consensus in networks of heterogeneous robots with variable time delays

This paper proposes a novel pose (position and orientation) consensus controller for networks of heterogeneous robots modeled in the operational space. The proposed controller is a distributed proportional plus damping scheme that, with a slight modification, solves both the leader–follower and leaderless consensus problems. A singularity‐free representation, unit quaternion, is used to describe the robots orientation, and the network is represented by an undirected and connected interconnection graph. Furthermore, it is shown that the controller is robust to interconnection variable time delays. Experiments with a network of two 6‐degrees‐of‐freedom robots are presented to illustrate the performance of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Periodic motion planning for virtually constrained Euler–Lagrange systems

The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler–Lagrange systems subject to (n-1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler–Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart–pendulum system and how then to make them orbitally exponentially stable following the machinery developed in [A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164–1176]. The extension here also considers time-varying virtual constraints.     

Distributed consensus filter on directed switching graphs

We consider distributed estimation on a directed graph with switching topologies. Motivated by a recent PI consensus filter, we modify the protocol and remove the requirement of bidirectional exchange of neighboring gains for fixed topologies. We then extend the protocol to switching topologies and propose a new hybrid consensus filter design. Convergence results under both balanced directed, and general directed graphs are given for switching graphs. Consensus error bounds are analytically derived in the case of time‐varying inputs. Satisfactory simulation results are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

Impulsive consensus of multi‐agent directed networks with nonlinear perturbations

This paper investigates the consensus problem of a multi‐agent directed network with nonlinear perturbations (MNNP). Our main objective is to study how the nonlinear perturbations affect a consensus of a multi‐agent network. Based on a reduced‐order transformation, it is shown that an MNNP cannot reach a consensus under a hypothesis even though the discussed network has a spanning tree. The hypothesis shows a detailed relationship between the nonlinear perturbations and the structure of the multi‐agent network. An impulsive approach is then introduced, and a simple criterion is presented to guarantee the consensus of all agents for an MNNP. Finally, a numerical example is provided to illustrate the effectiveness of the obtained criterion. Copyright © 2011 John Wiley & Sons, Ltd.     

Group consensus of multi-agent systems in directed networks with noises and time delays

In this paper, group consensus problems in fixed directed networks of dynamic agents are investigated. Group consensus means that the agents in each group share a consistent value while there is no agreement between any two groups. Based on algebraic graph theory, sufficient conditions guaranteeing group consensus under the proposed control protocol in the presence of random noises and communication delays are derived. The analysis uses a stability result of Mao for stochastic differential delay equations, which ensures the consensus can be achieved almost surely and exponentially fast. Numerical examples are provided to demonstrate the availability of the obtained results as well as the effect of time delay/noise intensity.     

Distributed event‐triggered consensus control with fully continuous communication free for general linear multi‐agent systems under directed graph

This paper considers the distributed event‐triggered consensus problem for multi‐agent systems with general linear dynamics under a directed graph. We propose a novel distributed event‐triggered consensus controller with state‐dependent threshold for each agent to achieve consensus. In this strategy, continuous communication in both controller update and triggering condition monitoring is not required, which means the proposed strategy is fully continuous communication free. Each agent only needs to monitor its own state continuously to determine if the event is triggered. Additionally, the approach shown here provides consensus with guaranteed positive inter‐event time intervals. Therefore, there is no Zeno behavior under the proposed consensus control algorithm. Finally, numerical simulations are given to illustrate the theoretical results.     

Reaching global leaderless consensus in directed networks via smooth and bounded control

This short communication revisits the leaderless consensus problem in directed networks. The network topology is described by a general directed graph which is only needed to contain at least a directed spanning tree, and is not necessarily strongly connected. A bounded and smooth control law using the hyperbolic tangent function is developed as the consensus protocol. Explicit stability analysis based on Lyapunov's second method and the concept of condensation graph is given. The proposed approach transforms the leaderless consensus problem into two subproblems: reaching leader-following consensus under a graph containing a directed spanning tree with the leader being the root node and reaching leaderless consensus under a strongly connected graph. Remarkably, a perturbed system is formulated as the bridge between these two subproblems which are solved by Lyapunov's second method. The global leaderless consensus is achieved based on the stability of the perturbed system. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed approach.     

Distributed formation‐containment control with Euler‐Lagrange systems subject to input saturation and communication delays

This paper is concerned with the problem of formation‐containment on networked systems, with interconnected systems modeled by the Euler‐Lagrange equation with bounded inputs and time‐varying delays on the communication channels. The main results are the design of control algorithms and sufficient conditions to ensure the convergence of the network. The control algorithms are designed as distributed dynamic controllers, in such a way that the number of neighbors of each agent is decoupled from the bound of the control inputs. That is, in the proposed approach the amplitude of the input signal does not directly increase with the number of neighbors of each agent. The proposed sufficient conditions for the asymptotic convergence follow from the Lyapunov‐Krasovskii theory and are formulated in the linear matrix inequalities framework. The conditions rely only on the upper bound of delays and on a subset of the controller parameters, but they do not depend on the model of each agent, which makes it suitable for networks with agents governed by distinct dynamics. In order to illustrate the effectiveness of the proposed method we present numerical examples and compare with similar approaches existing in the literature.     

Distributed finite‐time containment control for multiple Euler‐Lagrange systems with communication delays

In this paper, distributed finite‐time containment control for multiple Euler‐Lagrange systems with communication delays and general disturbances is investigated under directed topology by using sliding‐mode control technique. We consider that the information of dynamic leaders can be obtained by only a portion of the followers. Firstly, a nonsingular fast terminal sliding surface is selected to achieve the finite‐time convergence for the error variables. Then, a distributed finite‐time containment control algorithm is proposed where the neural network is utilized to approximate the model uncertainties and external disturbances of the systems. Furthermore, considering that error constraint method can improve the performance of the systems, a distributed finite‐time containment control algorithm is developed by transforming the error variable into another form. It is demonstrated that the containment errors are bounded in finite time by using Lyapunov theory, graph theory, and finite‐time stability theory. Numerical simulations are provided to show the effectiveness of the proposed methods.     

Reaching a consensus in networks of high-order integral agents under switching directed topologies

Consensus problem of high-order integral multi-agent systems under switching directed topology is considered in this study. Depending on whether the agent’s full state is available or not, two distributed protocols are proposed to ensure that states of all agents can be convergent to a same stationary value. In the proposed protocols, the gain vector associated with the agent’s (estimated) state and the gain vector associated with the relative (estimated) states between agents are designed in a sophisticated way. By this particular design, the high-order integral multi-agent system can be transformed into a first-order integral multi-agent system. Also, the convergence of the transformed first-order integral agent’s state indicates the convergence of the original high-order integral agent’s state, if and only if all roots of the polynomial, whose coefficients are the entries of the gain vector associated with the relative (estimated) states between agents, are in the open left-half complex plane. Therefore, many analysis techniques in the first-order integral multi-agent system can be directly borrowed to solve the problems in the high-order integral multi-agent system. Due to this property, it is proved that to reach a consensus, the switching directed topology of multi-agent system is only required to be ‘uniformly jointly quasi-strongly connected’, which seems the mildest connectivity condition in the literature. In addition, the consensus problem of discrete-time high-order integral multi-agent systems is studied. The corresponding consensus protocol and performance analysis are presented. Finally, three simulation examples are provided to show the effectiveness of the proposed approach.     

Cooperative controller design for synchronization of networked uncertain Euler–Lagrange systems

This paper addresses the synchronization problems with/without a dynamic leader for a team of distributed Lagrange systems on digraph. A systematic way to design and analyze the distributed control algorithms is presented. The contributions of the paper are twofold. First, the adaptive coordination control protocols are proposed for synchronization of networked uncertain Lagrange systems with/without tracking. This protocol can guarantee synchronization in finite time. Second, the design of the distributed tracking controller for the networked dynamic systems is proposed by using Lyapunov methods. The development is suitable for the general digraph communication topologies. Simulation examples are included to demonstrate the effectiveness of the proposed algorithms. Copyright © 2014 John Wiley & Sons, Ltd.     

Integral sliding mode control for Euler‐Lagrange systems with input saturation

This paper investigates the robust control for the Euler‐Lagrange (EL) system with input saturation by using the integral sliding mode control and adaptive control. An integral sliding mode surface that is suitable for solving the problem of the input constraint is given based on the saturation function. By using the integral sliding mode surface, two robust antisaturation controllers are designed for the EL system with external disturbances. The first controller can deal with the external disturbances with known bounds, whereas the second one can compensate the external disturbances with unknown bounds by using the adaptive control. Finally, the effectiveness of the proposed controllers is demonstrated by strict theoretical analysis and numerical simulations.