Impulsive Consensus Tracking of Multiagent Systems with Quantization and Input Delays Using Position-Only Information

Zhou, Hong; Qin, Yu-Zhen; Feng, Kuo; Hu, Wen-Shan; Liu, Zhi-Wei

doi:https://doi.org/10.1155/2014/781048

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusions Acknowledgments References Copyright Related Articles

Special Issue

New Trends in Networked Control of Complex Dynamic Systems: Theories and Applications

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 781048 | https://doi.org/10.1155/2014/781048

Impulsive Consensus Tracking of Multiagent Systems with Quantization and Input Delays Using Position-Only Information

Hong Zhou,¹Yu-Zhen Qin,¹Kuo Feng,¹Wen-Shan Hu,¹and Zhi-Wei Liu¹

Academic Editor: Huaicheng Yan

Received02 Dec 2013

Accepted15 Jan 2014

Published20 Feb 2014

Abstract

This paper investigates the consensus tracking problem for second-order multi-agent systems without/with input delays. Randomized quantization scheme is considered in the communication channels, and impulsive consensus tracking algorithms using position-only information are proposed for the consensus tracking of multi-agent systems. Based on the algebraic graph theory and stability theory of impulsive systems, sufficient and necessary conditions for consensus tracking are studied. It is found that consensus tracking for second-order multi-agent systems without/with input delays can be achieved by appropriately choosing the sampling period and control gains which are determined by second/third degree polynomials. Simulations are performed to validate the theoretical results.

1. Introduction

During the last two decades, the consensus problem in multiagent systems has attracted considerable attention due to its broad applications including synchronization [1], formation control [2], flocking [3], and sensor networks [4]. The main objective of consensus problem in multiagent systems is to design distributed control that enables all agents in a network to reach an agreement with a certain characteristic.

Recently, the leader-following structure containing one leader and some followers has been introduced into the consensus. Particularly, consensus with a static leader is named consensus regulation problem, and consensus with an active leader is named consensus tracking problem [5]. In the consensus tracking problem, a portion of the followers can obtain the leader’s information while other followers can only obtain their neighbors’ information. The task of consensus tracking is to make all the followers track the leader. Many valuable results on consensus tracking have been obtained in the existing literatures with different features including variable topology [6, 7], coupling time delays [8], and network-based communication [9].

It is worthwhile to mention that most of the consensus tracking algorithms proposed in the above-mentioned literatures require both position and velocity information among neighbouring agents. Unfortunately, in order to reduce equipment costs and network traffic, the agents might not be equipped with velocity sensors in many applications such as some robots and air vehicles systems. Thus the velocity of each agent may not be obtained in these special multiagent systems. To address the limitation, consensus algorithms without any velocity information are proposed in [10–12].

On the other hand, in all of the aforementioned works, one major shortcoming of the proposed algorithms for the consensus tracking of multiagent systems is the reliance on the exchange of analog data. It demands quite a broad bandwidth and enough communication power in the information interaction. But in practical situations, communication channels are always with finite bandwidth and finite power so that only a finite number of bits can be transmitted [13–15]. When the constraints are considered, communication with unquantized data is impractical [16, 17]. The authors proposed distributed algorithms in which the agents utilize quantized communication information to communicate with each other [14, 18–20]. The probabilistic quantization scheme was adopted in the agents interaction [14, 18]. In [21], logarithmic quantizer was used to quantize the state of agents in the multiagent systems.

Inspired by the above discussion, randomized quantization scheme is considered in the communication channels in this paper. At the sampling instant, the states of agents are sampled and quantized utilizing this scheme before being transmitted to their neighbors. Since the impulsive control strategy is proved to be a very effective control strategy in many fields due to its various advantages such as smaller control effort (only operates at sampling times), less information required (only needs the information at sampling times), and simple implementation [22–25], it is taken into consideration in our work. In conclusion, the main contribution of this paper includes the following two aspects: the design of impulsive consensus tracking algorithms using the position-only information for the multiagent systems without/with input delays; the introduction of randomized quantization scheme which is applied to the quantization of agents’ information before communication, which is different from our previous works on impulsive algorithms [22, 24, 26]. It is worth pointing out that studying the effect of time delays on the consensus of multiagent systems is meaningful. Generally, there are two kinds of time delays in multiagent systems: communication delays and input delays. Communication delays are related to communication among agents while input delays are related to processing and connecting time for the packets arriving at each agent [27]. Input delay is studied in this paper. It is shown that the second-order multiagent systems without/with input delays can reach quantized consensus tracking if the sampling period and control gains are appropriately designed from polynomials with different orders.

The rest of the paper is organized as follows. In Section 2, preliminaries and problem formulation are given. The impulsive consensus tracking algorithms utilizing quantized communication for second-order multiagent systems without/with time delays are presented in Section 3 and the quantized consensus tracking is proved to be reached. In Section 4, numerical examples are given to illustrate the theoretical analysis. Conclusions are finally drawn in Section 5.

2. Preliminary and Problem Formulation

In this section, some basic concepts used throughout this paper are introduced. Let denote the set of real numbers and let . is the column vector. is the identity matrix of order . denotes the matrix with all elements equal to zero.

2.1. Preliminaries in Graph Theory

For a multiagent system with agents, let be a directed graph with the set of vertices , the set of edges , and a weighted adjacency matrix . An edge of is denoted by , where is called the parent vertex of and the child of . The adjacency elements associated with the edges are nonnegative. For , , and assume that . The set of neighbors of node is denoted by . The Laplacian matrix is defined as , for , and , for , where .

2.2. Definitions and Notations

For a multiagent system, an agent is called a leader if the agent has no neighbor and an agent is called a follower if the agent has neighbors. In this paper, we will consider a multiagent system consisting of one active leader and followers. Let denote the interaction topology of the followers and be Laplacian matrix of . Let denote the interaction topology of the multiagent system containing the followers and the leader. Let describe the connection between follower and the leader. if and only if there exists a direct path from the leader to the follower; otherwise, . The multiagent system with followers and one leader can be described by where and , , are the position and velocity of agent and is control input at time . , , and are the position, velocity, and the acceleration of the dynamic leader, respectively. The followers aim to track the leader in this paper. It is assumed that the communication among agents only occurs at sampling instants, and the sampling intervals are periodic with period . The sampling information needs to be quantized before being transmitted to the neighbor agents.

Randomized Quantization Scheme. In the following, a quantization scheme adopted in this paper is introduced. Suppose that the scalar value is quantized by a uniform quantizer with quantization interval . Obviously, there exists an integer such that . Inspired by [14, 28], the data can be quantized by the probabilistic manner: where . Define as the quantization operation. One has expectation and variance in the following form:

In this scheme, the randomized quantized value is equal to the original unquantized value in expectation. Let be the quantization error; the above formula can be rewritten as .

Definition 1. The quantized consensus tracking in the multiagent system (1) is said to be achieved if, for any initial state, where are monotonously increasing functions satisfying , .

Before presenting the consensus tracking analysis, we first give the following lemmas which will be used in our work.

Lemma 2 (see [29]). has a spanning tree if and only if has a simple eigenvalue together with the other eigenvalues whose real parts are all real ones. Therefore, contains a spanning tree if and only if all the eigenvalues of , , , have positive real parts.

Lemma 3 (see [30]). Consider a complex polynomial , where , and are real constants. Then is Hurwitz stable if and only if and .

Lemma 4 (see [31]). Matrix and are given. There exists a matrix norm such that

3. Consensus Tracking with Impulsive Algorithm

3.1. Impulsive Algorithm Utilizing Quantization Communication without Input Delays

Assume sampling period in the multiagent system is a positive constant , the sampling time sequence can be noted as a set , . On count of the velocities , , , cannot be obtained; we propose the following impulsive algorithm for system (1): where , , , . What is worth pointing out is that , where are the position information which are not quantized, and are denoted as the quantization error of agent at the time satisfying , where is the quantization interval of each agent in the system. Assume that is left-hand continuous at . , , and are the control gains that need to be designed.

Denote . From system (6), one can easily get where . Let , , for . Assume that is left-hand continuous at . Let . Then for Definition 1, the quantized consensus tracking is achieved if and only if

For , one has

Let , , and . Then controlled system (6) can be rewritten into a matrix form as where ,

Lemma 5. The controlled system (6) can achieve the quantized consensus tracking if and only if , where denotes the spectral radius of a matrix.

Proof. Define and thus the quantized consensus tracking of system algorithm (6) can be studied by analyzing since where denote the 1-norm and 2-norm, respectively. Then from (11), one has Necessity. If , the system (14) must be diverging and the system (6) cannot achieve the quantized consensus tracking since .
Sufficiency. If , then . According to Lemma 4, there exist a matrix norm and a constant satisfying provided that is small enough. And then from (14), one has where is the quantized interval. It is easy to know that since . Therefore, there exists a positive constant such that
Obviously, there exist constants and satisfying , ; that is,
According to Definition 1, it is easy to obtain that the quantized consensus tracking in the controlled system (6) can be achieved. Then this lemma is proved.

Theorem 6. The controlled system (6) can reach quantized consensus tracking if and only if the directed graph has a spanning tree and the polynomial is Hurwitz stable, where , , are the eigenvalues of .

Proof. Let where , , are the eigenvalues of . Assume that have different eigenvalues , . There exists a invertible matrix such that , where and where is the algebraic multiplicity of . Motivated by [30, 32], one has where , . According to Lemma 5, the quantized consensus tracking can be reached if and only if , .
Note that This implies that , if and only if , . As , the controlled system (6) can reach the quantized consensus tracking if and only if , . Note that
Let , . holds if and only if the polynomial is Schur stable.
Let
Obviously, the polynomial is Schur stable if and only if the polynomial is Hurwitz stable. And, then, the controlled system (6) can reach the quantized consensus tracking if and only if is Hurwitz stable. Then Theorem 6 is proved.

Remark 7. In Theorem 6, a necessary and sufficient condition for the quantized consensus tracking for the controlled system (6) without input delays is established as a polynomial with order two. For a given topology, whether directed or undirected, one could choose appropriate control gains and sampling period such that condition (19) is satisfied.

Theorem 8. Assume that is a directed graph. The controlled system (6) can achieve the quantized consensus tracking if and only if the directed graph contains a spanning tree and holds, where

Proof. According to Theorem 6, note that
From Lemma 2, if the directed graph has a spanning tree, then . According to Lemma 3, the complex polynomial is Hurwitz stable if and only if , and . It is easy to prove that is equivalent to . Then this proof is completed.

Corollary 9. Assume that is an undirected graph. The controlled system (6) can achieve the quantized consensus tracking if and only if the contains a spanning tree and holds, where is the maximum eigenvalue of .

Proof. According to Lemma 2, if the undirected graph has a spanning tree, then , . As this corollary is a special case of Theorem 8 where and , one can easily obtain the condition (29) by a simple calculation. Then Corollary 9 is proved.

3.2. Impulsive Algorithm Utilizing Quantization Communication with Input Delays

In some practical situations, the input time delays always exist, which cannot be ignored. When the time delays (assume that is time-invariant) are introduced into the protocol, one can consider the impulsive algorithm as where Similar to the above analysis, it is easy to obtain that, for , where Let where then the controlled system (30) can be rewritten as where ,

Lemma 10. When the input delays are considered, the controlled system (30) can achieve quantized consensus tracking if and only if .

Proof. It is easy to prove this lemma in a way that is similar to what is used in Lemma 5. Define . Analogously, one has , where . There exist positive constants such that , , so one has And then this proof is completed.

Theorem 11. The controlled (30) can achieve quantized consensus tracking if and only if the directed graph has a spanning tree and the polynomial is Hurwitz stable, where , , are the eigenvalues of , , , .

Proof. Let where , are the eigenvalues of . According to Lemma 10, it is easy to know that the controlled system (30) can achieve quantized consensus tracking if and only if , . Note that Let and then one has . Let It can be rewritten in the following form:
Obviously, polynomial is Schur stable if and only if polynomial is Hurwitz stable. It is easy to conclude that the controlled system (30) can achieve the quantized consensus tracking if and only if is Hurwitz stable. Then this theorem is proved.

Remark 12. Similar to Theorem 6, a necessary and sufficient condition for the quantized consensus tracking for the controlled system (30) is derived as a polynomial with order three in Theorem 11. For a given topology with determinate input delays , whether directed or undirected, one could also choose appropriate control gains , , and sampling period such that condition (39) is satisfied.

Corollary 13. Assume that the is an undirected graph. The controlled system (30) can achieve quantized consensus tracking if and only if the consists of a spanning tree and hold.

Proof. While the is the undirected graph, , are all real numbers. Thus , , are all real numbers too. According to Routh’s stability criterion, is stable if and only if , , , , and −. By solving the first four polynomials, one obtains condition (45). The fifth polynomial can be rewritten as condition (46). Therefore, quantized consensus tracking in controlled system (30) can be reached. This proof is completed.

4. Numerical Simulations

In this section, some numerical simulations are given to demonstrate the theorem analysis. Consider a multiagent system with 8 agents (a leader and 7 followers) which is shown in Figure 1. The Laplacian matrix is where , , and . Choose quantized interval . By simple calculation of , one has . Choose , , and . According to Theorem 8, the quantized consensus tracking can be reached if and only if . Let , as it is shown in Figure 2, the quantized tracking consensus can be achieved. While , it cannot be achieved, which is shown in Figure 3.

(a)

(b)

(a)

(b)

When input delays are taken into consideration, choose , , , , and , which satisfies Theorem 11. As it is shown in Figure 4, the quantized consensus tracking can be achieved. But when , , , , and , the system cannot achieve the consensus tracking, which is shown in Figure 5.

(a)

(b)

(a)

(b)

Remark 14. As a matter of fact, the quantized consensus tracking is clearly not a strict consensus; that is, all agents in the network do not have the same value as the leader. There are still some errors in quantized consensus tracking which can be found in the simulation results of Figures 4 and 6. According to Definition 1, this phenomenon is easily explained because the consensus quality is closely related to the quantized interval or even determined by it. Comparing Figure 4 with Figure 6, it can be found that the later errors () are bigger than the former () under the same control gains, sampling interval, and time delay. Therefore, in order to achieve a more accurate quantized consensus tracking, smaller quantized interval should be taken into account in practical applications.

(a)

(b)

5. Conclusions

In this paper, the consensus tracking problems for second-order multiagent systems without/with input delays are studied. We propose impulsive consensus tracking algorithms without velocity measurements. Considering the constraints of communication channels with finite bandwidth and finite power, randomized quantization scheme is introduced in our work. Information of agents in the multiagent systems is quantized using this scheme before being transmitted to their neighbors. Some sufficient and necessary conditions for consensus tracking are obtained. By appropriately choosing the sampling period and control gains such that this condition holds, the second-order multiagent systems without/with input delays can achieve quantized consensus tracking. Finally, numerical simulations are given to illustrate the theoretical analysis. Furthermore, future efforts will focus on adaptive and output feedback extensions of the developed controllers to reduce the required amount of state and model knowledge.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61304152, 61374064, and 61364017), National Key Technology Research and Development Program of the Ministry of Science and Technology of China (2013BAA01B01), and the China Postdoctoral Science Foundation funded project (2013T60738).

References

Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint,” IEEE Transactions on Circuits and Systems, vol. 57, no. 1, pp. 213–224, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Lin, B. Francis, and M. Maggiore, “Necessary and sufficient graphical conditions for formation control of unicycles,” IEEE Transactions on Automatic Control, vol. 50, no. 1, pp. 121–127, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory,” IEEE Transactions on Automatic Control, vol. 51, no. 3, pp. 401–420, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
R. Olfati-Saber and J. S. Shamma, “Consensus filters for sensor networks and distributed sensor fusion,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05), pp. 6698–6703, Seville, Spain, December 2005.
View at: Publisher Site | Google Scholar
Z. Meng, W. Ren, Y. Cao, and Z. You, “Leaderless and leader-following consensus with communication and input delays under a directed network topology,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 1, pp. 75–88, 2011.
View at: Publisher Site | Google Scholar
Y. Hong, J. Hu, and L. Gao, “Tracking control for multi-agent consensus with an active leader and variable topology,” Automatica, vol. 42, no. 7, pp. 1177–1182, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Ni and D. Cheng, “Leader-following consensus of multi-agent systems under fixed and switching topologies,” Systems & Control Letters, vol. 59, no. 3-4, pp. 209–217, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Hu and Y. Hong, “Leader-following coordination of multi-agent systems with coupling time delays,” Physica A, vol. 374, no. 2, pp. 853–863, 2007.
View at: Publisher Site | Google Scholar
L. Ding, Q.-L. Han, and G. Guo, “Network-based leader-following consensus for distributed multi-agent systems,” Automatica, vol. 49, no. 7, pp. 2281–2286, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
H. Su, X. Wang, and G. Chen, “A connectivity-preserving flocking algorithm for multi-agent systems based only on position measurements,” International Journal of Control, vol. 82, no. 7, pp. 1334–1343, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Mei, W. Ren, and G. Ma, “Distributed coordination for second-order multi-agent systems with nonlinear dynamics using only relative position measurements,” Automatica, vol. 49, no. 5, pp. 1419–1427, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-W. Liu, Z.-H. Guan, X. Shen, and G. Feng, “Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements,” IEEE Transactions on Automatic Control, vol. 57, no. 10, pp. 2639–2643, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
A. Kashyap, T. Başar, and R. Srikant, “Quantized consensus,” Automatica, vol. 43, no. 7, pp. 1192–1203, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. C. Aysal, M. Coates, and M. Rabbat, “Distributed average consensus using probabilistic quantization,” in Proceedings of the 14th IEEE/SP WorkShop on Statistical Signal Processing (SSP '07), pp. 640–644, Madison, Wis, USA, August 2007.
View at: Publisher Site | Google Scholar
H. Yan, Z. Su, H. Zhang, and F. Yang, “Observer-based $H_{\infty}$ control for discrete-time stochastic systems with quantisation and random communication delays,” IET Control Theory & Applications, vol. 7, no. 3, pp. 372–379, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
H. Zhang, H. Yan, F. Yang, and Q. Chen, “Quantized control design for impulsive fuzzy networked systems,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 6, pp. 1153–1162, 2011.
View at: Publisher Site | Google Scholar
Z.-H. Guan, C. Meng, R.-Q. Liao, and D.-X. Zhang, “Consensus of second-order multi-agent dynamic systems with quantized data,” Physics Letters A, vol. 376, no. 4, pp. 387–393, 2012.
View at: Publisher Site | Google Scholar
T. C. Aysal, M. J. Coates, and M. G. Rabbat, “Rates of convergence for distributed average consensus with probabilistic quantization,” in Proceedings of the Allerton Conference on Communication, Control, and Computing, Urbana, Ill, USA, September 2007.
View at: Google Scholar
Z.-W. Liu, Z.-H. Guan, T. Li, X.-H. Zhang, and J.-W. Xiao, “Quantized consensus of multi-agent systems via broadcast gossip algorithms,” Asian Journal of Control, vol. 14, no. 6, pp. 1634–1642, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
T. C. Aysal, M. J. Coates, and M. G. Rabbat, “Distributed average consensus with dithered quantization,” IEEE Transactions on Signal Processing, vol. 56, no. 10, pp. 4905–4918, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
D. V. Dimarogonas and K. H. Johansson, “Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control,” Automatica, vol. 46, no. 4, pp. 695–700, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z.-H. Guan, Z.-W. Liu, G. Feng, and M. Jian, “Impulsive consensus algorithms for second-order multi-agent networks with sampled information,” Automatica, vol. 48, no. 7, pp. 1397–1404, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Zhang, H. Yan, T. Liu, and Q. Chen, “Fuzzy controller design for nonlinear impulsive fuzzy systems with time delay,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 5, pp. 844–856, 2011.
View at: Publisher Site | Google Scholar
Z.-W. Liu, H. Zhou, Z.-H. Guan, W.-S. Hu, L. Ding, and W. Wang, “Distributed impulsive consensus of the multiagent system without velocity measurement,” Abstract and Applied Analysis, vol. 2013, Article ID 825307, 8 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
H. Zhang, Z.-H. Guan, and G. Feng, “Reliable dissipative control for stochastic impulsive systems,” Automatica, vol. 44, no. 4, pp. 1004–1010, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-H. Guan, Z.-W. Liu, G. Feng, and Y.-W. Wang, “Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control,” IEEE Transactions on Circuits and Systems, vol. 57, no. 8, pp. 2182–2195, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-P. Tian and C.-L. Liu, “Consensus of multi-agent systems with diverse input and communication delays,” IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2122–2128, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Zhang and Y.-P. Tian, “Consensus of data-sampled multi-agent systems with random communication delay and packet loss,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 939–943, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms, Princeton University Press, Princeton, NJ, USA, 2009.
View at: MathSciNet
W. Yu, W. X. Zheng, G. Chen, W. Ren, and J. Cao, “Second-order consensus in multi-agent dynamical systems with sampled position data,” Automatica, vol. 47, no. 7, pp. 1496–1503, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 2012.
W. Yu, L. Zhou, X. Yu, J. Lu, and R. Lu, “Consensus in multi-agent systems with second-order dynamics and sampled data,” IEEE Transactions on Industrial Informatics, vol. 9, no. 4, pp. 2137–2146, 2012.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2014 Hong Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

858

Downloads

909

Citations