Consensus in time-delayed multi-agent systems with quantized dwell times

doi:10.1016/j.sysconle.2017.03.012

Systems & Control Letters

Volume 104, June 2017, Pages 59-65

https://doi.org/10.1016/j.sysconle.2017.03.012 Get rights and content

Abstract

This paper investigates the effects of quantized dwell times in solving consensus problems in time-delayed multi-agent systems with the asynchronous property of unpredictable quantization jumps in inter-agent communications. The quantized dwell times can be introduced on purpose or be inherent from quantizing devices. They beneficially avoid undesirable fast chattering of quantizers’ outputs and mathematical difficulties in convergence analysis. The main result shows that the largest eigenvalue of the interaction graph Laplacian well characterizes the dependence of maximum allowable time delay on quantized dwell times and quantizer parameters under the assumption of heterogeneous time-varying transmission delays. Simulations are presented to show the effectiveness of the main result.

Introduction

In multi-agent systems, numerous studies have shown the significant effects of information flows on the methodology for coordinating inter-connected autonomous agents [1], and particular effort has been put into the networks with finite capacity or bandwidth constraints [2], [3], [4]. One popular topic concerning this issue is the consensus protocol design with only quantized information. A quantizer digitizes incoming information and usually gives piecewise constant output. So at any instant, only finite bits of channel capacity are required for the transmission of quantized data [5], [6], [7], [8]. Typical quantizers include uniform, logarithmic [9], [10], [11], and probabilistic quantizers [12], [13]. Several fundamental and interesting quantization principles have been presented for the gossip consensus of discrete-time systems [5], [12], [14], [15], [16], [17], [18]. More general settings of time-varying topologies in discrete-time systems were studied and tight polynomial bounds on convergence times were provided in [19]. However, for continuous-time systems, quantized networks also give rise to the possibility of infinite switchings of quantizer outputs within a finite period of time [6]. This chattering behavior increases the burden of data transmission and it theoretically leads to the non-existence of solutions in the traditional framework of differential equations. Certainly, the concept of differential inclusions can be used to tackle the encountered mathematical difficulties [10], [11], [20], [21]. To prevent infinitely frequent chattering, in [22], the mechanism of hysteretic quantizers was successfully presented for quantized average consensus. This paper explores the feasibility of quantized consensus of multi-agent systems when quantized dwell times are direct introduced to avoid chattering problems.

Due to decentralized interactions, quantized multi-agent systems also exhibit inherently the feature of asynchronous encoding of quantizers and asynchronous information transmission. Asynchronous consensus was formally addressed for continuous-time and discrete-time models in [23], [24], [25]. Examples of asynchronous problems include independent way-point updates of agents [23], asynchronous state updates [25], [26], asynchronous data-sampling schedules [27], and asynchronous computation [28]. For analysis purpose, asynchronous events were modeled by an iteration graph with infinite vertices and were dealt with by nonlinear paracontractions theory in [25]. They can also be simply modeled by a sequence of interaction graphs, together with additional joint or periodical connectivity conditions. In such a setup, the concept of “analytic synchronization” and Lyapunov methods were used in the consensus analysis in [23], [24] and [26], [27], respectively.

This paper modifies the popular quantizers, such as uniform and logarithmic quantizers, by the introduction of quantized dwell times, and studies the effectiveness of these quantizers in the distributed consensus control of multi-agent systems. Specifically, a dwell time is given after each output change of quantizers; during dwell times, the involved quantizers have no response to the processed data. Consequently, any two consecutive output jumps of any quantizer are separated by a time interval of at least the dwell-time length, and the output chattering of quantizers is successfully avoided. This approach is different from the one based on hysteretic quantizers proposed in [22], which avoids output chattering by constructing overlapping level sets of quantization. Technically, by our approach, the minimum event interval is obviously not less than the dwell times; whereas, with hysteretic quantizers, the minimum event interval depends on both agent states and switching sequences of quantization events.

This paper tries to consider a general setting of data transmission delays, which can be fixed, time-varying, or random, and are only required to be upper bounded. However, because of the dynamic complexity induced by asynchronous quantization as well as delays, we only present the complete and interesting results on the average consensus problem in a bi-directional network of single-integrators. In the consensus analysis, we devise a new method with the help of incidence matrices and two groups of time-dependent diagonal matrices to describe the unstructured delay dependence of states and the asynchronous relationship between each pair of quantizers. The evolution of states is considered on a joint sequence of quantization times, which is similar to the idea from the method of “analytic synchronization” [23], [24]. Under the assumption of heterogeneous time-varying delays on information channels, we show that the largest eigenvalue of the graph Laplacian well measures the robustness of the network with respect to asynchronous quantization and time delays. This conclusion is consistent with the classic result given in [29] for time-delayed continuous-time systems without quantization.

This paper is organized as follows. In Section 2, the model is presented and the motivation for studying quantized dwell times is given; In Section 3, we perform the convergence analysis and discuss the delay robustness of the studied protocol with quantized dwell times; In Section 4, simulations are given to show the correctness of the main result. The paper is concluded in Section 5.

Section snippets

Problem formulation

This section first gives the individual models and then defines the quantizers with quantized dwell times. Finally, the problem is formulated equivalently in the form of edge dynamics.

Consensus analysis

First, we summarize the main result in the following theorem.

Theorem 1

Suppose that Assumption 1 is satisfied and the time delays $τ^{i} (t)$ , $i = 1, 2, \dots, m$ , are bounded above. Denote $τ_{\max} = sup {τ^{i} (t) : t \geq 0, i = 1, 2, \dots, m}$ . If the interaction topology $G$ is connected and dwell time $h$ satisfies $2 h + τ_{\max} < \frac{1}{λ_{\max}},$ then, for any agent $i$ , following protocol (2) , $\{\begin{matrix} \underset{t \to \infty}{lim sup} | x_{i} (t) - x_{j} (t) | \leq σ, j \in N_{i}, & if Q^{h} (\cdot) = Q_{u}^{h} (\cdot); \\ lim_{t \to \infty} x_{i} (t) = κ, & if Q^{h} (\cdot) = Q_{log}^{h} (\cdot), \end{matrix}$ where $λ_{\max}$ is the largest eigenvalue of the graph Laplacian $D D^{T}$

Simulations

In this section, we consider a group of $5$ agents with the topology depicted in Fig. 2. It can be obtained that the largest eigenvalue of the graph Laplacian $λ_{\max} = 4.3028$ . In the simulations, suppose that $h = 0.05$ and $τ_{\max} = 0.13$ , which satisfies the condition of Theorem 1. We choose the quantization parameter $σ = 0.05$ in $Q_{u}^{h} (\cdot)$ and $σ = 2$ in $Q_{log}^{h} (\cdot)$ and the initial states $x (0) = {[1 2 3 4 5]}^{T}$ and $τ^{i} (0) = 0$ , $i = 1, 2, \dots, 5$ . The derivatives of time delays are random, piece-wise constant and belong to $[- 0.25, 0.25]$ . The

Conclusions

This paper presented an approach to eliminate the Zeno behavior of quantizer outputs in asynchronous multi-agent systems by introducing quantized dwell times. A sufficient condition in terms of maximum time delay and dwell time was given for the average state consensus. The main result shows that the delay robustness is mainly determined by the largest eigenvalue of the graph Laplacian of the interaction topology. We are also interested in the extension of the work to more general settings,

References (31)

CaoY. et al.
Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication
Automatica
(2009)
LiH. et al.
Distributed model predictive control of constrained nonlinear systems with communication delays
Systems Control Lett.
(2013)
KashyapA. et al.
Quantized consensus
Automatica
(2007)
DimarogonasD.V. et al.
Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control
Automatica
(2010)
LiuH. et al.
Quantization effects on synchronized motion of teams of mobile agents with second-order dynamics
Systems Control Lett.
(2012)
GuoM. et al.
Consensus with quantized relative state measurements
Automatica
(2013)
YuanD. et al.
Distributed average consensus via gossip algorithm with real-valued and quantized data for $0 < q < 1$
Systems Control Lett.
(2010)
YuanD. et al.
Distributed dual averaging method for multi-agent optimization with quantized communication
Systems Control Lett.
(2012)
CarliR. et al.
Gossip consensus algorithms via quantized communication
Automatica
(2010)
FranceschelliM. et al.
Quantized consensus in Hamiltonian graphs
Automatica
(2011)

CaiK. et al.

Convergence time analysis of quantized gossip consensus on digraphs

Automatica

(2012)

FrascaP.

Continuous-time quantized consensus: convergence of Krasovskii solutions

Systems Control Lett.

(2012)

CeragioliF. et al.

Discontinuities and hysteresis in quantized average consensus

Automatica

(2011)

FaxJ.A. et al.

Information flow and cooperative control of vehicle formations

IEEE Trans. Automat. Control

(2004)

LiK. et al.

Robust and efficient quantization and coding for control of multidimensional linear systems under data rate constraints

Internat. J. Robust Nonlinear Control

(2007)

Cited by (33)

Semi-global asymptotic state agreement of nonlinear multi-agent systems with communication delays under directed switching topologies
2024, Nonlinear Analysis: Hybrid Systems
This paper investigates the semi-global state agreement problem for continuous-time nonlinear multi-agent systems with communication delays under direct and switching topologies. A systematic solution to the semi-global state agreement is presented and based on which a sufficient and necessary condition on the control laws is obtained, as long as the direct and switching communication topologies satisfy a mild joint connectivity assumption. Illustrative simulations are provided to validate the efficacy of the proposed theoretical results.
Monotone decreasing LKF method for secure consensus of second-order mass with delay and switching topology
2023, Systems and Control Letters
Citation Excerpt :
Considering that random effects often cause the change of communication topology among agents, it is more practical to investigate second-order MASs with switching topology. There have been many interesting results concerning consensus of MASs with different dwell time (DT) [11–14]. Note that a common idea in studying the dynamical performance of switched systems with delay is that a multiple LKF has to jump high at switching instants, which is one of the main sources of conservative results and seems to be insurmountable.
This letter investigates secure global asymptotic consensus (SGAC) with non-weighted $L_{2}$ gain of second-order multi-agent systems (MASs) under deception attacks by designing time-delayed state feedback control with switching topology. Compared with the existing methods in which the Lyapunov–Krasovskii functional (LKF) has to jump high at switching instants, the most important breakthrough is that a discretized LKF is designed such that it is monotone decreasing at switching instants, which not only can obtain low conservative results but also is convenient to study consensus with non-weighted $L_{2}$ gain without any additional inequality transformation. The Results are universal since they can be applied with no difficulty to any other switched time-delay systems. A numerical simulation illustrates the less conservative SGAC.
Event-triggered consensus control and fault estimation for time-delayed multi-agent systems with Markov switching topologies
2021, Neurocomputing
Citation Excerpt :
Time-delay phenomenon is very common in complex networked systems due to the frequent information exchange between various digital devices. The consensus problem of MASs with time-delay has also attracted the interest of many scholars from different academic backgrounds [6–8]. In practice, the agents are often subject to stochastic perturbations, which may lead to the changes of network topology structure.
This paper focuses on the consensus control and fault estimation problems for a class of time-delayed multi-agent systems with Markov switching topologies. Two different event-triggered mechanisms are adopted with hope to reduce burden of shared network and improve energy efficiency. Under Markov process, by establishing the consensus control protocol and designing a novel adaptive fault estimation observer, the consensus control and fault estimation problems are transformed into two stochastic stability problems in different forms. Then, according to the switching Lyapunov function method and free-weighting matrix technique, two delay-dependent stability criteria on the consensus control and fault estimation are derived, respectively. However, the two criteria containing nonlinear coupling terms are not standard linear matrix inequalities (LMIs) and cannot be solved directly with the LMI toolbox. In order to eliminate the coupling terms, two improved path-following algorithms are presented. These algorithms depend on the initial conditions, so it is very crucial to choose the appropriate preset parameters. The computational complexity is increasing with the number of iterations, system size and matrix dimension, which is a fully new challenge for the study of consensus control and fault estimation of multi-agent systems. Based on the algorithms, the switching consensus controller gains and model gain matrices of fault estimation can be efficiently solved out. Finally, a simulation example of tailless fighter airplanes is given to illustrate the practicality and validity of the theoretical results.
Distributed constrained optimization problem of heterogeneous linear multi-agent systems with communication delays
2021, Systems and Control Letters
Citation Excerpt :
With the development of multi-agent systems [1–9], distributed optimization problems have attracted considerable attention due to various applications.
This paper addresses the constrained distributed optimization problem of heterogeneous linear multi-agent systems, where the agents with linear dynamics are subject to local set constraints, global nonlinear inequality constraints and heterogeneous communication delays. Agents collaborate to minimize a global objective function by communicating with their neighbors in a graph. Each agent’s decision variable is constrained in a local set. The decision variables of all agents are coupled by global inequality constraints which are modeled by nonlinear functions. To handle the heterogeneous constant communication delays, the scattering transformation between neighbors is employed. We design a new distributed control law to investigate the passivity of systems of individual agents in the presence of constraints and communication delays. Integrating the proposed control law with the scattering transformation, we prove that systems converge to the optimal solution which minimizes the global objective functions. Simulations of heterogeneous linear multi-agent systems are presented to illustrate the effectiveness of the distributed control law.
Event-triggered formation tracking control of nonholonomic mobile robots without velocity measurements
2020, Automatica
Citation Excerpt :
The authors in Borgers and Heemels (2014) have pointed out that, many existing results in ETM would exhibit Zeno behavior when there is disturbance in the system. This problem was tackled by a time-event hybrid mechanism in Xiao and Chen (2016) and Xiao et al. (2016), and by the introduction of dwell times in Xiao et al. (2017) and Xiao et al. (2018). Inspired by Borgers and Heemels (2014), a mixed ETM is adopted in this paper.
This paper investigates an event-triggered formation tracking problem of nonholonomic mobile robots in a leader’s coordinate frame. The proposed control protocol can be implemented in directed sensor networks based on only relative information detected by onboard sensors of mobile robots. A novel observer is proposed, and by virtue of it, neither velocity measurements nor communication among robots is required. Furthermore, by introducing a super-twisting sliding mode differentiator, the convergence of estimation errors is no longer dependent on the convergence of system states, which makes it possible to adopt an event-triggered mechanism, by which the continuous update of control command is avoided and Zeno behavior is excluded. Finally, simulations illustrate the effectiveness of our control protocol.
Winner-take-all competition with heterogeneous dynamic agents
2020, Neurocomputing
Citation Excerpt :
Inspired by the swarm behavior of biosystem, more and more researchers are paying attention to swarm behaviour and trying to apply it in industry systems. The classical topics of swarm behaviour mainly focus on consensus [1], formation [2], flocking [3], containment [4], winner-take-all [5], etc. Winner-take-all competition [6] is one of the foundational Lotka-Volterra models, it is motivated by the competition observed and lateral inhibition among neurons in the brain [7].
Winner-take-all competition exists widely in nature and has been applied in many engineering fields. This paper mainly investigates a group of heterogeneous dynamic agents, which produce the winner-take-all competition. For the heterogeneous system consisting of first- and second-order dynamic agents, we propose two different kinds of protocols with and without velocity measurements, respectively. Firstly, we employ the Lasalle’s invariant principle to solve the equilibrium points of the proposed system. Secondly, we prove that the proposed protocols can solve the winner-take-all problems for the heterogeneous systems. The results reveal that winner is independent from the dynamics of agents, but is determined by inputs. Finally, some examples are also gave to verify the validity of the proposed protocols.

View all citing articles on Scopus

^☆: This work was supported in part by the National Natural Science Foundation of China (NSFC, Grant Nos. 61422302, 61273030) and the Program for New Century Excellent Talents in University (Grant No. NCET-13-0178), and in part by the Natural Sciences and Engineering Research Council of Canada .

View full text

Consensus in time-delayed multi-agent systems with quantized dwell times☆

Abstract

Introduction

Section snippets

Problem formulation

Consensus analysis

Simulations

Conclusions

Automatica

Systems Control Lett.

Automatica

Automatica

Systems Control Lett.

Automatica

Systems Control Lett.

Systems Control Lett.

Automatica

Automatica

Automatica

Systems Control Lett.

Automatica

Information flow and cooperative control of vehicle formations

IEEE Trans. Automat. Control

Robust and efficient quantization and coding for control of multidimensional linear systems under data rate constraints

Internat. J. Robust Nonlinear Control