Linear Time-invariant Multi-agent Systems Research Articles

A Lyapunov Approach to Consensus of Linear Multi-Agent Systems with Infinite Distributed Communication Delays

This paper considers the leaderless consensus problem of linear time-invariant multi-agent systems with infinite distributed communication delays. A novel distributed low gain controller is proposed based on the solution to a parametric algebraic Riccati equation. It is shown via the newly developed Lyapunov-like method that not only the consensus of linear time-invariant multi-agent systems can be achieved exponentially under some mild assumptions but also an estimate of the exponential convergence rate of consensus is given in this work. The Lyapunov-like method is also extended to handle a special case of linear time-varying multi-agent systems. In addition, the obtained results include the results on the leaderless consensus of linear multi-agent systems with bounded distributed communication delays as special cases. To the best of our knowledge, this is the first work that develops the Lyapunov-like method for the leaderless consensus problems of both time-invariant and time-varying linear multi-agent systems with infinite distributed communication delays. Finally, a numerical example is presented to illustrate the effectiveness of the proposed controller.

Necessary and Sufficient Conditions of Formation-Containment Control of High-Order Multiagent Systems With Observer-Type Protocols

The analysis and design problems of formation-containment control for high-order linear time-invariant (LTI) multiagent systems (MASs) on directed graphs with observer-based output-feedback protocols are given in this work. To expand the feasibility of state formation configuration, two well-structured compensation signals are introduced for the leaders and followers in the protocols, respectively. Benefitting from the compensation signal of followers, the decoupling between formation control of leaders and containment control of followers is achieved. Thus, a necessary and sufficient condition is first established such that the formation-containment control for high-order LTI MASs can be achieved. Moreover, a heuristic iterative algorithm is developed to compute the controller gains, observer gains, as well as the compensation signals. Finally, two numerical examples are implemented to illustrate the time-varying formation-containment control of high-order MASs, which shows the validity and practicability of the theoretical results and algorithm.

Discrete Laguerre-based model predictive control for dynamic consensus of a vehicle platoon with time delay

Dynamic consensus of linear time-invariant multi-agent systems (MASs) using distributed model predictive control under the influence of input delay is addressed in this technical note. Model predictive control (MPC) is well suited to solve the consensus problem with its ability to handle multivariable systems and constraints. However, the increased computational complexity of MPC restricts its area of applications. An attempt to bridge this gap is made by proposing a Laguerre-based MPC design to bring down the computational load and make it viable for implementation. The effect of input delay on consensus is examined and the stability margin of a MAS with input delay is computed using the location of the closed-loop poles. A comparison of the proposed algorithm with conventional MPC establishes its superiority in convergence time, robustness to input delay and smoothness of the trajectories. The effectiveness of the proposed design is validated through the simulation of dynamic consensus on the benchmark problem of platoon configuration of vehicles. Abbreviations DLQR: discrete linear quadratic regulator; DMPC: distributed model predictive control; LMI: linear matrix inequality; LTI: linear time invariant; SISO: single-input single-output; MAS: multi-agent system; MIMO: multi-input multi-output; MPC: model predictive control

Distributed control of linear multiagent systems with global and local objectives

In this paper, we consider distributed control problems for high-order linear time-invariant multiagent systems with not only global but also local objectives over fixed directed communication graph topologies. The former is either leaderless synchronization or synchronization to a leader. Local objectives for a subset of agents, on the other hand, are tasks determined around the global objective by agent-specific exosystems. First, we construct reference models for all agents via two existing synchronization results, introduce two classes of distributed controllers, and define the considered problems. We then solve them by utilizing the converging-input converging-state property for a class of linear systems and the feedforward design methodology from the linear output regulation theory. Finally, numerical examples are presented to demonstrate the problems and the theoretical results.

An Optimal Control Approach for Consensus of General Linear Time-Invariant Multi-Agent Systems

Abstract In this paper, the consensus problem for general linear time-invariant (LTI) multi-agent systems (MASs) with a single input is studied in a new optimal control framework. The optimal cooperative control law is designed from a modified linear quadratic regulator (LQR) method and an inverse optimal control formulation. Three cost function terms are constructed to address the consensus, control effort, and cooperative tracking, respectively. Three distinct features of this approach can be achieved. First, the optimal feedback control law is derived analytically without involving any numerical solution. Second, this formulation guarantees both asymptotic stability and optimality. Third, the cooperative control law is distributed and only requires local information based on the communication topology to enable the agents to achieve consensus and track a desired trajectory. The performance of this optimal cooperative control method is demonstrated through an example of attitude synchronization of multiple satellites.

Consensus of Continuous-Time Multiagent Systems via Delayed Output Feedback: Delay Versus Connectivity

This article studies the consensus problem for continuous-time linear time-invariant multiagent systems, in which the agents exchange information through delayed communication channels in a fixed undirected topology. Our primary objective is to characterize explicitly the delay effect on consensus in the framework of distributed dynamic output feedback control and its implication on the graph network connectivity. For this purpose, we tackle and solve the consensus problem in the spirit of gain margin problem, using analytic interpolation techniques. We develop an explicit condition for consensus. The result provides a limit on network connectivity imposed by time delays as well as by the agent's unstable poles and nonminimum phase zeros.

A distributed control approach for heterogeneous linear multiagent systems

This paper considers an internal model-based distributed control approach to the cooperative output regulation problem of heterogeneous linear time-invariant multiagent systems over fixed directed communication graph topologies. First, a new definition of the linear cooperative output regulation problem is introduced in order to allow a broad class of functions to be tracked and rejected by a network of agents. Second, the solvability of this problem with three distributed control laws, namely dynamic state feedback, dynamic output feedback with local measurement and dynamic output feedback, is investigated by first considering a global condition and then providing an agent-wise local sufficient condition under standard assumptions. Finally, two numerical examples are provided to illustrate the selected contributions of this paper.

Distributed Containment Control of Multi-agent Systems with General Linear Dynamics and Time-delays

Containment control problems for high-order linear time-invariant multi-agent systems with fixed communication time-delays are investigated. Based on Lyapunov-Krasovskii functional method and the linear matrix inequality (LMI) method, sufcient conditions on the communication digraph, the feedback gains, and the allowed upper bound of the delays to ensure containment control of the multi-agent systems under the different containment control algorithms are given. Finally, numerical simulations are presented to demonstrate theoretical results.

Distributed Observer-based LQ Controller Design and Stabilization for Discrete-time Multi-agent Systems

This paper will investigate the distributed observer-based LQ controller design and stabilization problem for uncoupled identical linear time-invariant multi-agent systems with a performance index coupling the behavior between the multi-agents. A design method is proposed by applying the decomposition of the global discrete-time algebraic Riccati equations. A computationally tractable solution can be obtained by solving four local algebraic Riccati equations which have the same dimensions as a single agent. The stability condition is given in terms of the spectrum of two matrices representing the desired sparsity pattern of the distributed controller and distributed observer. A limited overall performance can also be guaranteed by the proposed distributed controller which is parameterized by two scalars. To illustrate the effectiveness of the algorithm, a numerical example is provided.

Disturbance observer‐based consensus control of input‐delayed LTI systems with matched disturbances: a predictor feedback approach

The study addresses the consensus problem of linear time invariant (LTI) multi-agent systems with constant input delay and matched external disturbances by using relative output information. First, distributed disturbance observers are constructed to estimate the disturbances generated by linear exosystems with unknown initial conditions. Next, a disturbance observer-based predictor feedback control law is developed for each agent to realise consensus disturbance rejection in the presence of input delay. Then, a linear matrix inequality-based control design algorithm is given to determine the satisfactory control parameters. It is proved that state consensus can be achieved for both leaderless and leader–follower networks in the presence of large input delay and matched disturbances under the control scheme. Finally, a simulation example is provided to validate the theoretical results.

On Stability and Inverse Optimality for a Class of Multi-agent Linear Consensus Protocols

This paper studies stability and inverse optimality for a class of linear consensus protocols applied to the identical linear time-invariant multi-agent systems, where communications among the agents are described by a fixed digraph, containing a spanning tree, whose scaled Laplacian is diagonalizable. The concept of the scaled Laplacian, normal Laplacian multiplied by a positive diagonal matrix, allows a more general graph topology to be handled. First, we show that partial stability and, even more, inverse optimality hold if either the scaling factors in each protocol are sufficiently large depending on graph properties, or the system matrix A satisfies the Lyapunov inequality. Then, duality principles between the agents’ and the consensus error dynamics are presented, which provide additional properties regarding consensus-related stability and inverse optimality. And the results are characterized in terms of the related symmetric Laplacian and its algebraic connectivity when the given graph is scaled undirected. Finally, through formation control simulation of a multi-agent mobile robot system for Г-scaled undirected and Г-scaled directed graphs, we have verified the theory of this paper on the partial stability and the inverse optimality conditions.

Consensus for heterogeneous networked multi-agent systems with switching topology and time-varying delays

This paper investigates consensus problem for heterogeneous discrete linear time-invariant (LTI) multi-agent systems subjected to time-varying network communication delays and switching topology. A new two-stage consensus protocol is proposed based on stochastic, indecomposable and aperiodic (SIA) matrix and pseudo predictive scheme. With pseudo predictive scheme the network delay is compromised. Consensus analysis based on seminorm is provided. Results give conditions for such systems with periodic switching topology and time-varying delays to reach consensus. Highlights of the paper include: the protocol can be implemented in a distributed manner; the pseudo predictive approach requires less computation and communication; the verification of consensus convergence does not require the global information about the communication topology; the protocol allows delay to be time-varying, topology to dynamically and asymmetrically switch and system mode to be unstable. Numerical and practical examples demonstrate the effectiveness of the theoretical results.

Distributed adaptive control and stability verification for linear multiagent systems with heterogeneous actuator dynamics and system uncertainties

ABSTRACTThe contribution of this paper is a control synthesis and stability verification framework for linear time-invariant multiagent systems with heterogeneous actuator dynamics and system uncertainties. In particular, we first propose a distributed adaptive control architecture in a leader-follower setting for this class of high-order multiagent systems. The proposed architecture uses a hedging method, which alters the ideal reference model dynamics of each agent in order to ensure correct adaptation in the presence of heterogeneous actuator dynamics of these agents. We then use Lyapunov stability theory and linear matrix inequalities to analyse the proposed architecture. This analysis reveals a stability condition, where evaluation of this condition with respect to a given graph topology allows stability verification of the controlled multiagent system. From a practical point of view, this condition also shows a fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in agent dynamics. Several illustrative numerical examples are also provided to demonstrate the efficacy of the proposed architecture.

Distributed adaptive control for time-varying formation of general linear multi-agent systems

ABSTRACTAn adaptive gain scheduling technique is investigated to discuss distributed time-varying formation (TVF) control problems for general linear time-invariant multi-agent systems (LTI-MASs), where two types of topologies are considered: (1) the interaction topology is undirected and connected, and (2) the interaction topology is directed. Two fully distributed adaptive TVF control protocols are, respectively proposed, which both assign a time-varying coupling weight to each node in the interaction topology. Two algorithms to design the constructed protocols are presented under the undirected and directed interaction topologies, respectively. A feasible TVF set is provided. The stabilities of two algorithms are, respectively proved based on the Lyapunov functional theory. For both undirected and directed interaction topologies, general LTI-MASs can achieve the given TVF using the designed fully distributed adaptive formation protocol without any global information about the interaction topology when the TVF satisfies the feasible set. Finally, theoretical results are illustrated with numerical simulation examples.

Distributed Decoupling of Partially-Unknown Interconnected Linear Multiagent Systems: State and Output Feedback Approaches

Distributed consensus algorithms have been widely designed for physically decoupled linear time-invariant multiagent systems operating at a fixed operating point. However, they do not guarantee the closed-loop stability and performance of multiagent systems in the presence of physical interconnections or changes in operating conditions of agents. In this paper, we propose a class of state- and output-coupled linear multiagent systems where each agent is characterized by its individual unknown operating point. Under the effect of these varying operating points of agents, we propose the distributed decoupling problem in order to mitigate the effect of interconnections, and ensure that each agent can operate at its individual operating point independent of its neighbors. We reformulate this decoupling task, and propose two linear quadratic regulator-based distributed leaderless consensus algorithms using coupled-state and -output measurements. In both scenarios, we establish sufficient conditions guaranteeing global exponential decoupling of agents.

Control design for synchronisation of identical linear multi-agent systems

ABSTRACTIn this paper, we study the state synchronisation problem of multi-agent systems subject to external additive perturbations. We consider high-order linear time-invariant multi-agent systems whose communication topology is encoded by an undirected and connected graph. We propose an control design technique based on a decentralised output feedback controller. We give sufficient conditions to ensure state synchronisation with bounded gain using a Lyapunov-based approach. These conditions are characterised in terms of matrix inequalities. Since these matrix inequality conditions are non-convex and cannot be solved straightforwardly, we propose a relaxation technique and an effective numerical procedure to design a suitable controller with guaranteed performance on the multi-agent distributed closed loop.

Distributed adaptive time-varying formation for multi-agent systems with general high-order linear time-invariant dynamics

This paper studies distributed time-varying formation control problems for general high-order linear time-invariant (LTI) multi-agent systems using an adaptive based approach. Firstly, a time-varying formation control protocol is constructed by the states of neighboring agents, which the adaptive gain scheduling technique is employed to estimate the coupling weights between neighboring agents. Different from the existing results on formation control, the formation can be specified by piecewise continuously differentiable vectors and no global information about the interaction topologies is required. Then an algorithm with two steps is presented to design the distributed adaptive formation control protocol, where a description of the feasible time-varying formation set is given. The stability of the algorithm is proved using the Lyapunov theory. It is shown that if the predefined time-varying formation belongs to the feasible formation set and each agent is stabilizable, then time-varying formation can be achieved by general high-order LTI multi-agent systems using the distributed adaptive formation protocol designed in the proposed algorithm. Finally, numerical examples are given to demonstrate the effectiveness of the theoretical results.

Output formation-containment analysis and design for general linear time-invariant multi-agent systems

Output formation-containment control problems for general linear time-invariant multi-agent systems with directed topologies are dealt with. Output formation-containment means that the outputs of leaders achieve the predefined formation, and at the same time the outputs of followers converge to the convex hull formed by the outputs of leaders. Firstly, static output protocols are presented for leaders and followers respectively. Then output formation-containment problems of multi-agent systems are transformed into asymptotic stability problems. Sufficient conditions with less computation complexity are proposed for multi-agent systems to achieve the output formation-containment. An explicit expression for the time-varying output formation reference function is derived to describe the macroscopic movement of the whole output formation-containment. Explicit expressions to describe the relationship among the outputs of followers, the time-varying output formation for the leaders and the output formation reference are derived. It is proven that the outputs of followers not only converge to the convex hull formed by those of leaders but also achieve certain time-varying formation specified by the convex combination of the desired output formation for the leaders. Moreover, an approach to determine the gain matrices in the protocols is given for multi-agent systems to achieve the output formation-containment. Finally, numerical simulations are provided to demonstrate the effectiveness of the theoretical results.

Actuator fault accommodation strategy for a team of multi-agent systems subject to switching topology

A cooperative actuator fault accommodation strategy for a team of linear time-invariant (LTI) multi-agent systems with a switching topology and directed communication network graph is studied in this paper. The faults can simultaneously occur in more than one agent. The proposed fault accommodation strategy is based on a consensus algorithm that weights the effort that each agent contributes to the consensus achievement of the team. The stability property of the proposed weighted consensus algorithm is investigated based on Lyapunov analysis. Moreover, a fault index is introduced to quantitatively measure the effects of inaccuracy in estimation of fault severities on consensus performance of the multi-agent team. This index is exploited to reconfigure the agents weights in such a manner that healthier agents contribute more control efforts to compensate for the faulty agents controller degradations. Both actuator loss-of-effectiveness and actuator saturation faults are investigated and the capabilities and effectiveness of our proposed approach is shown by numerical simulations and comparisons with centralized and decentralized fault recovery approaches that are available in the literature.

Containment analysis and design for general linear multi-agent systems with time-varying delays

Containment analysis and design problems for general high-order linear time-invariant multi-agent systems with time-varying delays are studied, where the interaction topology is directed. Using the state information of each agent and neighboring agents, a protocol with time-varying delays is constructed, where the motion modes of the leaders can be specified. Based on Lyapunov–Krasovskii stability theory, sufficient conditions for general linear multi-agent systems with time-varying delays to achieve containment are proposed which only include four linear matrix inequalities independent of the number of agents. Moreover, an approach to determine the gain matrices in the protocol is presented. Finally, a numerical example is given to demonstrate the effectiveness of the obtained theoretical results.