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A Practical Framework for the Analysis of Average Consensus Problem: Laplacian Generator and Analyzer Algorithm
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A Practical Framework for the Analysis of Average Consensus Problem: Laplacian IFAC-PapersOnLine 49-3 (2016) 341–346 and Analyzer Algorithm A Practical Framework Generator for the Analysis of Average Consensus Problem: Laplacian A Practical Framework for the Analysis of Average Consensus Problem: Laplacian Generator and Analyzer Algorithm Generator and Analyzer Algorithm Mustafa Saraoğlu* Aydın Polat* İlhan Mutlu* and Mehmet Turan Söylemez*
Mustafa Saraoğlu* Aydın Polat* İlhan Mutlu* and Mehmet Turan Söylemez* *ControlSaraoğlu* and Automation Department, Istanbul Technical University Mustafa AydınEngineering Polat* İlhan Mutlu* and Mehmet Turan Söylemez* Turkey ([email protected]; [email protected]; [email protected]; [email protected]). *Control and Automation Engineering Department, Istanbul Technical University and Automation Engineering Department, Istanbul Technical University Turkey*Control ([email protected]; [email protected]; [email protected]; [email protected]). Turkey ([email protected]; [email protected]; [email protected]; [email protected]). Abstract: This paper provides a topology based analysis for the average consensus problem regarding various configurations and possibilities. Consensus problem which can be defined as the agreement of the Abstract: paper provides a topology based problem analysis for the average problem regarding agents in aThis multi-agent topology is a common in various fieldsconsensus of science, technology and Abstract: This paper provides a topology based analysis for the average consensus problem regarding various configurations possibilities. Consensus problem which be is defined as the agreement of the engineering. The main and subject focused on within the scope of this can study, the selection of the topology various configurations and possibilities. Consensusproblem probleminwhich can fields be defined as the agreement of and the agents in a multi-agent topology is a for common various of science, regarding various performance criteria the agents to reach consensus. Basic conceptstechnology of multi-agent agents in a multi-agent topology is a on common problem inof various fields of selection science, technology and engineering. The main subject focused within the scope this study, is the of the topology systems are introduced along with an algebraic background. Graph Theory and its properties are also engineering. The main subject focused on within the scope of this study, isBasic the selection ofofthe topology regarding criteria the agents to reach consensus. concepts multi-agent covered asvarious a basis performance for understanding thefor behavior of multi-agent systems. A numerical approach in order regarding various performance criteria for the agents to reach consensus. Basic concepts of multi-agent systems are introduced an algebraic background. Graph andperformance its propertiescriteria, are also to determine the structurealong of thewith connection topology that satisfies the Theory predefined is systems are introduced along with an algebraic background. Graph Theory and its properties are also covered basis forcase understanding behavior of multi-agent systems. numerical proposed.asIna the end, studies are the included to verify the effectiveness ofAthe proposedapproach method. in order covered as a basis for understanding the behavior of multi-agent A numerical approach in order to determine the structure of the connection topology that satisfiessystems. the predefined performance criteria, is Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, to determine the structure of the connection topology that satisfies the predefined performance criteria, is © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. proposed. In the end, case studies are included to verify the effectiveness of the proposed method. Laplacian In Matrix, Numerical Topology Optimization proposed. the end, case studies are included to verify the effectiveness of the proposed method. Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, Laplacian Matrix, Numerical Topology Optimization Laplacian Matrix, Numerical Topology Optimization It is proposed within the scope of this study that the 1. INTRODUCTION boundaries for the numbers of connections to be maintained withinthethe scope of study thatusing the for is theproposed agents to reach consensus can this be determined Consensus problem’s1.origin may set back to ancient times of It INTRODUCTION It is proposed within the ofscope of thistostudy that the boundaries for the numbers connections be maintained 1. INTRODUCTION approaches. For this purpose, firstly, a set of humanity as it may be interpreted as social, political or a numerical fortothe numbers of connections to be maintained for the agents reach consensus can be determined using Consensus problem’s may set to ancient of boundaries performance criteria inthe terms of consensus location and the strategic problem. Theorigin very nature ofback the behavior of times animals for the agents to reach the consensus can be determined using Consensus problem’s origin may set back to ancient times of numerical approaches. For this purpose, firstly, consensus a set of humanity it may be been interpreted social, or a speed of convergence to that point in the average and humanasbeings have in scopeasfor manypolitical philosophers approaches. For this purpose, firstly, a and set the of humanity as it may bevery interpreted as social, political or a numerical performance criteria in terms of consensus location strategic problem. The nature of the behavior of animals problem should be defined for different topologies. Here, the as well as scientists. The consensus seen in the nature is often performance criteria intoterms of consensus location and the strategic problem. The very nature of the behavior of animals speed of convergence that point in the average consensus and human beings have been in scope for many philosophers topology refers to the relationship among agents in stunning and miraculous at some points. It is also referred to term speed of should convergence to that in the average consensus andwell human beings have been in scope forinmany philosophers problem be defined forpoint different topologies. Here, the as as scientists. Thecollective consensus seen the is often terms of information exchange. as swarm intelligence, behavior or nature herd behavior problem should be defined for different topologies. Here, the as well as scientists. The consensus seen in the nature is often term topology refers to the relationship among agents stunning and miraculous at some points. It is also referred to (Bouffanais, 2016). This alone was a great inspiration for term topology refers to the relationship among agents in in stunning and miraculous at some points. It is also referred to Average consensus problem, as we will be referring to as the as swarm intelligence, collective behavior model or herd behavior both engineers and scientist to understand, and realize terms of information exchange. terms of information exchange. as swarm intelligence, collective behavior or herd behavior consensus problem throughout this paper is one of the most (Bouffanais, 2016). This alone wasOne a great inspiration for the beauties of this deep concept. significant example consensus problem, we will be referring as the (Bouffanais, 2016). This alone was a great inspiration for Average fundamental problems of as consensus. In the toaverage both engineers and scientist to understand, model and realize Average consensus problem, as this we will beisreferring to asmost the about the modelling of consensus in nature is the biologically consensus problem throughout paper one of the both engineers scientist to understand, model andexample realize consensus problem the units which have the capability of the beauties ofand this deep concept. One significant problem throughout this paper is one of the most motivated work of Vicsek et al., (1995) which models the consensus fundamental problemsand of making consensus. the we average the beauties of this of deep concept.in One significant example evaluating information decisionsInwhich refer about themovement modelling nature is the biologically fundamental problems of consensus. In capability the average flocking andconsensus alignment problem. consensus problem the units which have the of about the modelling of consensus nature which is the biologically as agents must agree on a single point. This point will be motivated work of Vicsek et al.,in(1995) models the to consensus information problem theand units whichdecisions have thewhich capability of evaluating making we refer motivated work of Vicsek et al., (1995) which models the From themovement engineering point of problem. view, consensus problem referred to as consensus point which is indeed a saddle point flocking and alignment evaluating information and making decisions which we refer to asallagents on which a singlethese point. This should point will flocking movement and alignment agentsmust and agree a point agents go be in which can be informally defined asproblem. the problem of agreement for to as agents agree on a single This point will be asmust consensus point whichpoint. is indeed From of This view,problem consensus problem order toto reach consensus (DeGroot, 1974).a saddle It maypoint be of unitsthein engineering a multi-unit point system. consists of a referred to as consensus point which is indeed ashould saddle point From the be engineering point view, consensus problem referred for all agents a point which in which can informally definedofassystems. the problem of be agreement as and a general state thatthese each agents agent has andgo may series of challenges for multi-unit It must pointed considered for all to agents andconsensus a point which these agents should go be in which caninbea informally defined as the problem problem of agreement order reach (DeGroot, 1974). It may of units multi-unit system. This consists of a upon the agreed value. One of the simplest example out that the term multi-unit may refer to multi-agent systems converge order to reach consensus (DeGroot, 1974). It may be of units in a multi-unit system. This problem consists of a thatineach agenteach hasmeasuring and may series of challenges for systems multi-unit systems. It must be pointed for this case,asisaa general group ofstate sensors a room, and also to distributed which in fact is a problem of considered as athegeneral state that each agent has and may series ofthe challenges for multi-unit systems. It must be systems pointed considered converge upon agreed value. One of the simplest out that term multi-unit may refer to multi-agent 2011). cooperation among certain units and decision making an average of some local values (Avrachenkov et al.,example converge upon the agreed value. One of the simplest example out that the term multi-unit may refer to multi-agent systems group of sensors in a room, each problem measuring and also to distributed For this thatcase, kind isofa situation, the average consensus is mechanisms (DeGroot, systems 1974). which in fact is a problem of for this case,ofissome a group of values sensors(Avrachenkov in a room, each measuring and also to distributed systems units which and in fact is a problem of for an average local et al., 2011). cooperation among certain decision making more about measured values such as humidity or temperature of of some local values (Avrachenkov et problem al., 2011). cooperation among units this and cooperation decision making The problems(DeGroot, that maycertain occur during depend an Foraverage that kind situation, the average consensus is mechanisms 1974). rather than location or speed. that kind of situation, the average consensus problem is mechanisms (DeGroot, on various causes. For 1974). the sake of coordination, connection For more about measured values such as humidity or temperature The that may occur agents, during this about measured values such asand humidity or atemperature In literature, distributed networks reaching consensus and problems cooperation between thecooperation solution depend of the more rather than location or speed. The problems that may occur during this cooperation depend on various causes. For sake of coordination, connection location or speed. amongthan decision making units problem was firstly started by consensus problem is ofthegreat importance. Even though the rather on various causes. For the sake of coordination, connection distributed networks and reaching a consensus and cooperation between agents, the solution of the theliterature, work of Borkar and Varaiya (1982) and Tsitsiklis (1984). problem of maintaining a reliable connection is important as In literature, distributed networks and reaching a consensus and cooperation between agents, the solution of the the In among decision making units problem was firstly by consensus problem is of great importance. Even though The theoretical framework for consensus problems was it might seem, our focus in this paper is to satisfy the among decision making units problem was firstly started started by consensus is of great importance. Even though the the work of by Borkar Varaiya (1982) and Tsitsiklis (1984). problem ofproblem maintaining reliable is important as introduced the and studies of R.Olfati-Saber and Murray consensus, even in casesa that someconnection of these connections are the work of Borkar and Varaiya (1982) and Tsitsiklis (1984). problem ofseem, maintaining a reliable connection istoimportant as The theoretical framework for consensus problems was it might our focus in this paper is satisfy the lost. The theoretical framework consensus problems was it might seem, ourcases focus this of paper to satisfy are the introduced by the studies offorR.Olfati-Saber and Murray consensus, even in thatinsome theseisconnections consensus, even in cases that some of these connections are introduced by the studies of R.Olfati-Saber and Murray lost. lost. Copyright © 2016 IFAC 341 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 341Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 341 10.1016/j.ifacol.2016.07.057
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(2003) and (2004) based on the works of Fax and Murray (Fax, 2001), (Murray and Fax, 2004). One of the most useful areas of the consensus problem is the management in transportation systems. Platoon control is necessary for the intelligent and efficient coordination of vehicles. A group of heavy-duty vehicles can be put into formations, namely platoons, in order to increase the safety and fuel efficiency (Alam et al., 2015). This is achieved by forming a line of vehicles, keeping the intervehicular distances and reducing the aerodynamic drag force for the non-leading vehicles, just like the principle behind the flying bird formations. Driverless cars forming a platoon with a sensor based network topology is also a consensus problem which consists problems regarding collision avoidance, formation adjustment according to changing weather and road conditions, sustaining the formation under various disturbances (Wang et al., 2014).
Fig. 1. Graphical Representation and Determination of the Laplacian Matrix A graph consists of vertices and edges in graph theory. In a consensus problem, the relationship between the agents are included in the Laplacian matrix. A Laplacian matrix is a square matrix consisting of an adjacency matrix and a degree matrix and it is obtained by subtracting the adjacency matrix from the degree matrix. Further properties of the Laplacian matrix such as being positive semi-definite and diagonally dominant are also the important properties that make the analysis of a network possible (Cvetkovic et al., 1998).
2. CONSENSUS PROBLEM AND ITS FORMULATION Graph Theory is an important tool for the analysis of consensus problems. It can be directly described by the Laplacian Matrix which involves all the necessary properties that we need to model a consensus problem. Some important concepts related with the formulation of such problems are: • Agent: An individual robot or any unit that has a state along with the desire to reach consensus with other agents. • Topology: The topology of agents is related to their connections among each other. Most common way to express a topology is by using the graph theory. Each node in the graph indicates an agent.
3. MODELLING THE CONSENSUS PROBLEM It can be proposed that a graph representation alone is not enough to represent a complete average consensus problem. It is important to state that in order for agents to change their state values and solve an average consensus problem, a dynamical system which consists of integrators is essential (Olfati-Saber and Murray, 2004). A system representation formed by block diagrams was introduced by Olfati-Saber and Murray (Olfati Saber et al., 2007), as given in Figure 2.
• Consensus Time: The time that consensus is reached by all agents. • Consensus Position: The position of consensus which is related to the initial states of all agents and their information exchange relationships, namely their network topology. Algebraic graph theory has a set of properties and definitions that are directly applicable to consensus problems. It is essential to know these properties in order to determine the outcome of the consensus problem and how its outcome is influenced by the type of the graph. In areas involving networks such as the consensus problem, networks are usually defined by symmetrical matrices and the solutions for the problems lie within the fundamentals of the Graph Theory (Steven, 2010).
Fig. 2. Block diagram representation of an average consensus problem
An introductory example of forming a Laplacian Matrix is given in Figure 1.
The nature of the consensus problem can be modelled as an autonomous system with a state-space equation without an input. For the average consensus problem, convergence occurs around the average of the initial states of all agents if the communication graph is balanced (Olfati-Saber and Murray, 2004). Therefore the dynamics of the system can be written as (Olfati-Saber and Murray, 2004), (Lynch, 1997):
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𝑥𝑥𝑖𝑖̇ (𝑡𝑡) = ∑ 𝑥𝑥̇ = −𝐿𝐿𝐿𝐿
𝑗𝑗∈𝑁𝑁𝑖𝑖
Mustafa Saraoğlu et al. / IFAC-PapersOnLine 49-3 (2016) 341–346
𝑎𝑎𝑖𝑖𝑖𝑖 (𝑥𝑥𝑗𝑗 (𝑡𝑡) − 𝑥𝑥𝑖𝑖 (𝑡𝑡))
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algorithm handles the average consensus problem from a practical framework with some valuable insights. A flowchart of the algorithm, which is implemented in MATLAB™ with the aid of SIMULINK™, is shown in Figure 3.
(1) (2)
Equation (2) is a well-known dynamical linear system representation without any input. Thus, as it is known from the linear system theory, the eigenvalues of the Laplacian Matrix define the characteristic equation of the system and specify the behaviour of the system. Properties of the Laplacian Matrix can be exploited to determine the equilibrium point of the system which is when all agents agree on the same state value (Godsil and Royle, 2001); 𝑇𝑇
𝑥𝑥 ∗ = (∝, … , ∝) = ∝ 1
(3)
Fiedler stated (Fiedler, 1973) that the eigenvalues of Laplacian matrices can be sorted in ascending order starting from the first eigenvalue which is zero. From the outcome of this result, Fiedler defined the algebraic connectivity of a measure of connectivity (Merris, 1994). As it will be discussed further, the Graph Laplacian and their properties are the key elements in the analysis of the convergence of the consensus problem. Second smallest eigenvalue has a great significance as it is also related as Algebraic Connectivity (Chung, 1997) and also as Fiedler’s Eigenvalue (Fiedler, 1973). As it is mentioned earlier, the average consensus for balanced graphs occur around the average of all agents’ initial states. For a symmetrical adjacency matrix, all agents asymptotically converge to the value which is equal to the initial states of all agents and it is written as (Olfati-Saber and Murray, 2003): ∝ =
1 ∑ 𝑥𝑥𝑖𝑖 (0) 𝑛𝑛 𝑖𝑖
(4)
It is intuitively reasonable to state that (1) iteratively converges since the difference between states are always decaying and it is also stated that for any connected undirected graph, (1) solves the average consensus problem for any initial state (Tsitsiklis et al. 1986).
Fig. 3. The Flowchart of the Algorithm All the results obtained from the loop in the algorithm are reduced regarding the position and time criterion chosen. The consensus criterion used usually is the best possible values in terms of time and position. Shorter times with a consensus point around the average initial states are also considered as acceptable responses from the system.
4. PROPOSED APPROACH Theoretical solutions and analysis mostly concentrate on the second smallest eigenvalue as the key factor on the convergence rate. In this study, the importance of topologies has been examined along with the second smallest eigenvalue. In particular, it is shown that Different Laplacian Matrices with same eigenvalues may converge with different rates. Hence, an important element on the convergence of the system is that the topology of agents and the configurations of their connections.
In the next section, all possible configurations that a network topology may have for a fixed number of agents is considered as a case study. 5. CASE STUDY As a case study, a network topology consisting 3 agents is examined with the initial values: [5, 17, 37]. First step of the algorithm is to determine all the possible Laplacian Matrices and simulate each of them. Please note that 64 possible topologies (Laplacian matrices) exist for this case. Each topology is numbered from 1 to 64. Figure 4 below shows the consensus time and position for each combination representing a different topology. The cases where no suitable consensus occurs have been omitted.
The proposed approach is to consider a trade-off between the numbers of information channels that should be implemented and also to get a good performance in terms of consensus time and consensus location. The number of minimum connections between agents to reach consensus and also to get a relatively good result is also examined. The proposed 343
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The chart in the Figure 6 indicates the number of topologies which corresponds to the number of connections that a topology has. As it can be seen from the chart, consensus can be reached without establishing maximum number of connections, which leads to cost effective solutions. 5.1 Comparison of Same Consensus Times with Same Eigenvalues Out of 4 topologies where each have 5 connections, two topologies with identical consensus time were chosen to be examined as given in Figure 7.
Fig. 4. Graph of Consensus Time/Position versus Topology No According to the consensus criterion used the results are reduced, renumbered and presented below in Figure 5. The numbers in Figure 5 will be used from now on in order not to cause any confusion in comparison.
Fig. 7. Compared Topologies (No 15 and No 16)
Fig. 5. Reduced Graph of Consensus Time/Position versus Topology No
Fig. 8. Convergence of Topology No 15
Fig. 9. Convergence of Topology No 16
Fig. 6. Frequency of Topologies in terms of number of connections 344
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A graph representation of topology no 15 and topology no 16 are shown in Figure 8 and Figure 9 with their second smallest eigenvalues as λ. As a result, for the same eigenvalues, initial values, number of connections and consensus time, the consensus positions are different for two combinations due to different topologies. 5.2 Comparison of Same Consensus Position with Same Eigenvalues Out of 8 topologies where each have 4 connections, two topologies with identical consensus position were chosen to be examined in Figure 10.
Fig. 12. Convergence of Topology No 11 6. CONCLUSION As the comparison in the case study clearly indicates some valuable distinctions occur between properties of configurations. It can be seen that the network topology is quite important to get accurate results for the average consensus problem. The examples and cases can easily be extended to networks with larger scales of agents in the expense of larger computational times. The algorithm to use the proposed strategy is presented. To sum up, eigenvalues of the system are important for the system’s performance but they are not the only factors. The structure of the consensus connections has a significant role on the performance of the system.
Fig. 10. Compared Topologies (No 6 and No 11) A graph representation of topology no 6 and topology no 11 are shown in Figure 10 with their second smallest eigenvalues as λ. As a result, for the same eigenvalues, initial values, number of connections and consensus position, the consensus times are different for two combinations due to different topologies. The case study consists of comparisons of similar network topologies. Their comparisons in terms of time and their second smallest eigenvalues present us some valuable insight into the average consensus problem. As it is seen from the Figure 11 and Figure 12, second smallest eigenvalue alone does not completely determine the consensus time. Although both systems have the same system dynamics (because of the eigenvalues), topology had an effect on performance measures.
Further studies will concentrate on extending the number of nodes considered in this study together with some uncertainties in communications and connecting the results to the graph theory. It is obvious that the need for greater autonomy in transportation systems makes it necessary for scientists and engineers to understand the consensus problem from different perspectives in order to design more intelligent transportation systems. REFERENCES Alam, A., Besselink, B., Turri, V., and Johansson, K., H. (2015). Heavy-Duty Vehicle Platooning for Sustainable Freight Transportation: A Cooperative Method to Enhance Safety and Efficiency. IEEE CONTROL SYSTEMS 35(6), pp. 34-56. Avrachenkov, K., El Chamie, M., Neglia, G. (2011). A local average consensus algorithm for wireless sensor networks. Distributed Computing in Sensor Systems and Workshops (DCOSS), 2011 International Conference on, IEEE, Barcelona. Borkar, V., Varaiya, P. (1982), Asymptotic agreement in distributed estimation, IEEE Trans. Autom. Control, vol. AC-27, pp. 650–655. Bouffanais, R. (2016), Design and Control of Swarm Dynamics (First edition). Springer. ISBN 978-981-287750-5 Chung, F. R. K. (1997). Spectral Graph Theory. Providence, RI: Amer. Math. Soc.
Fig. 11. Convergence of Topology No 6 345
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Cvetković, D. M., Doob, M., and Sachs H. (1998), Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley. DeGroot, M. H. (1974), Reaching a consensus, J. Am. Statist. Assoc., vol. (69), pp. 118–121. Fax, J. A. (2001), “Optimal and cooperative control of vehicle formations,” Ph.D. dissertation, Control Dynamical Syst., California Inst.Technol., Pasadena, CA. Fax, J. A., Murray R. M. (2004), Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, vol. (49), pp. 1465–1476. Fiedler, M., (1973). Algebraic connectivity of graphs. Czechoslovak Math. J. 23(98), pp. 298 - 305. Godsil C., Royle G. (2001), Algebraic Graph Theory, ser. Graduate Texts in Mathematics. Berlin, Germany: Springer-Verlag, vol. 207. Lynch, N. A. (1997). Distributed Algorithms. San Francisco, CA: Morgan Kaufmann, 1997. Merris, R. (1994), Laplacian matrices of a graph: A survey, Linear Algebra its Appl., vol. (197), pp. 143–176. Olfati-Saber, R., Murray, R.M. (2003), Consensus protocols for networks of dynamic agents, Proc. 2003 Am. Control Conf., pp. 951–956. Olfati-Saber, R., Murray, R.M. (2004). Consensus Problems in Networks of Agents with Switching Topology and Time-Delays, Automatic Control, IEEE Transactions, vol. (49), pp. 1520 – 1533. Olfati-Saber, R., Fax, J.A., Murray, R.M. (2007). Consensus and Cooperation in Networked Multi-Agent Systems. Proceedings of the IEEE (Volume: 95), pp. 215 – 233. Steven, J. L. (2010), Linear Algebra with Applications (8th Edition), ISBN 978-0-13-600929-0 Tsitsiklis, J. N. (1984), Problems in decentralized decision making and computation, Ph.D. dissertation, Dept. Electr. Eng. Comput. Sci., Lab. Inf. Decision Syst., Massachusetts Inst. Technol., Cambridge, MA. Tsitsiklis, J. N., Bertsekas, D. P., Athans, M. (1986) Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Automatic Control, vol. (31), pp. 803–812. Vicsek, T., Cziroo´k, A., Ben-Jacob, E., Cohen, I., and Shochet, O., (1995). Novel type of phase transition in a system of self-deriven particles, Phys. Rev. Lett., vol. (75), pp. 1226–1229. Wang, L., Y., Syed, A., Yin, G., G., Pandya, A., and Zhang, H. (2014). Control of vehicle platoons for highway safety and efficient utility: Consensus with communications and vehicle dynamics, Journal of Systems Science and Complexity, vol. (27), pp. 605-631.
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A Practical Framework for the Analysis of Average Consensus Problem: Laplacian IFAC-PapersOnLine 49-3 (2016) 341–346 and Analyzer Algorithm A Practical Framework Generator for the Analysis of Average Consensus Problem: Laplacian A Practical Framework for the Analysis of Average Consensus Problem: Laplacian Generator and Analyzer Algorithm Generator and Analyzer Algorithm Mustafa Saraoğlu* Aydın Polat* İlhan Mutlu* and Mehmet Turan Söylemez*
Mustafa Saraoğlu* Aydın Polat* İlhan Mutlu* and Mehmet Turan Söylemez* *ControlSaraoğlu* and Automation Department, Istanbul Technical University Mustafa AydınEngineering Polat* İlhan Mutlu* and Mehmet Turan Söylemez* Turkey ([email protected]; [email protected]; [email protected]; [email protected]). *Control and Automation Engineering Department, Istanbul Technical University and Automation Engineering Department, Istanbul Technical University Turkey*Control ([email protected]; [email protected]; [email protected]; [email protected]). Turkey ([email protected]; [email protected]; [email protected]; [email protected]). Abstract: This paper provides a topology based analysis for the average consensus problem regarding various configurations and possibilities. Consensus problem which can be defined as the agreement of the Abstract: paper provides a topology based problem analysis for the average problem regarding agents in aThis multi-agent topology is a common in various fieldsconsensus of science, technology and Abstract: This paper provides a topology based analysis for the average consensus problem regarding various configurations possibilities. Consensus problem which be is defined as the agreement of the engineering. The main and subject focused on within the scope of this can study, the selection of the topology various configurations and possibilities. Consensusproblem probleminwhich can fields be defined as the agreement of and the agents in a multi-agent topology is a for common various of science, regarding various performance criteria the agents to reach consensus. Basic conceptstechnology of multi-agent agents in a multi-agent topology is a on common problem inof various fields of selection science, technology and engineering. The main subject focused within the scope this study, is the of the topology systems are introduced along with an algebraic background. Graph Theory and its properties are also engineering. The main subject focused on within the scope of this study, isBasic the selection ofofthe topology regarding criteria the agents to reach consensus. concepts multi-agent covered asvarious a basis performance for understanding thefor behavior of multi-agent systems. A numerical approach in order regarding various performance criteria for the agents to reach consensus. Basic concepts of multi-agent systems are introduced an algebraic background. Graph andperformance its propertiescriteria, are also to determine the structurealong of thewith connection topology that satisfies the Theory predefined is systems are introduced along with an algebraic background. Graph Theory and its properties are also covered basis forcase understanding behavior of multi-agent systems. numerical proposed.asIna the end, studies are the included to verify the effectiveness ofAthe proposedapproach method. in order covered as a basis for understanding the behavior of multi-agent A numerical approach in order to determine the structure of the connection topology that satisfiessystems. the predefined performance criteria, is Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, to determine the structure of the connection topology that satisfies the predefined performance criteria, is © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. proposed. In the end, case studies are included to verify the effectiveness of the proposed method. Laplacian In Matrix, Numerical Topology Optimization proposed. the end, case studies are included to verify the effectiveness of the proposed method. Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, Keywords: Consensus, Network Topology, Multi-Agent Systems, Average Consensus Problem, Laplacian Matrix, Numerical Topology Optimization Laplacian Matrix, Numerical Topology Optimization It is proposed within the scope of this study that the 1. INTRODUCTION boundaries for the numbers of connections to be maintained withinthethe scope of study thatusing the for is theproposed agents to reach consensus can this be determined Consensus problem’s1.origin may set back to ancient times of It INTRODUCTION It is proposed within the ofscope of thistostudy that the boundaries for the numbers connections be maintained 1. INTRODUCTION approaches. For this purpose, firstly, a set of humanity as it may be interpreted as social, political or a numerical fortothe numbers of connections to be maintained for the agents reach consensus can be determined using Consensus problem’s may set to ancient of boundaries performance criteria inthe terms of consensus location and the strategic problem. Theorigin very nature ofback the behavior of times animals for the agents to reach the consensus can be determined using Consensus problem’s origin may set back to ancient times of numerical approaches. For this purpose, firstly, consensus a set of humanity it may be been interpreted social, or a speed of convergence to that point in the average and humanasbeings have in scopeasfor manypolitical philosophers approaches. For this purpose, firstly, a and set the of humanity as it may bevery interpreted as social, political or a numerical performance criteria in terms of consensus location strategic problem. The nature of the behavior of animals problem should be defined for different topologies. Here, the as well as scientists. The consensus seen in the nature is often performance criteria intoterms of consensus location and the strategic problem. The very nature of the behavior of animals speed of convergence that point in the average consensus and human beings have been in scope for many philosophers topology refers to the relationship among agents in stunning and miraculous at some points. It is also referred to term speed of should convergence to that in the average consensus andwell human beings have been in scope forinmany philosophers problem be defined forpoint different topologies. Here, the as as scientists. Thecollective consensus seen the is often terms of information exchange. as swarm intelligence, behavior or nature herd behavior problem should be defined for different topologies. Here, the as well as scientists. The consensus seen in the nature is often term topology refers to the relationship among agents stunning and miraculous at some points. It is also referred to (Bouffanais, 2016). This alone was a great inspiration for term topology refers to the relationship among agents in in stunning and miraculous at some points. It is also referred to Average consensus problem, as we will be referring to as the as swarm intelligence, collective behavior model or herd behavior both engineers and scientist to understand, and realize terms of information exchange. terms of information exchange. as swarm intelligence, collective behavior or herd behavior consensus problem throughout this paper is one of the most (Bouffanais, 2016). This alone wasOne a great inspiration for the beauties of this deep concept. significant example consensus problem, we will be referring as the (Bouffanais, 2016). This alone was a great inspiration for Average fundamental problems of as consensus. 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It may of units multi-unit system. This consists of a upon the agreed value. One of the simplest example out that the term multi-unit may refer to multi-agent systems converge order to reach consensus (DeGroot, 1974). It may be of units in a multi-unit system. This problem consists of a thatineach agenteach hasmeasuring and may series of challenges for systems multi-unit systems. It must be pointed for this case,asisaa general group ofstate sensors a room, and also to distributed which in fact is a problem of considered as athegeneral state that each agent has and may series ofthe challenges for multi-unit systems. It must be systems pointed considered converge upon agreed value. One of the simplest out that term multi-unit may refer to multi-agent 2011). cooperation among certain units and decision making an average of some local values (Avrachenkov et al.,example converge upon the agreed value. One of the simplest example out that the term multi-unit may refer to multi-agent systems group of sensors in a room, each problem measuring and also to distributed For this thatcase, kind isofa situation, the average consensus is mechanisms (DeGroot, systems 1974). which in fact is a problem of for this case,ofissome a group of values sensors(Avrachenkov in a room, each measuring and also to distributed systems units which and in fact is a problem of for an average local et al., 2011). cooperation among certain decision making more about measured values such as humidity or temperature of of some local values (Avrachenkov et problem al., 2011). cooperation among units this and cooperation decision making The problems(DeGroot, that maycertain occur during depend an Foraverage that kind situation, the average consensus is mechanisms 1974). rather than location or speed. that kind of situation, the average consensus problem is mechanisms (DeGroot, on various causes. For 1974). the sake of coordination, connection For more about measured values such as humidity or temperature The that may occur agents, during this about measured values such asand humidity or atemperature In literature, distributed networks reaching consensus and problems cooperation between thecooperation solution depend of the more rather than location or speed. The problems that may occur during this cooperation depend on various causes. For sake of coordination, connection location or speed. amongthan decision making units problem was firstly started by consensus problem is ofthegreat importance. Even though the rather on various causes. For the sake of coordination, connection distributed networks and reaching a consensus and cooperation between agents, the solution of the theliterature, work of Borkar and Varaiya (1982) and Tsitsiklis (1984). problem of maintaining a reliable connection is important as In literature, distributed networks and reaching a consensus and cooperation between agents, the solution of the the In among decision making units problem was firstly by consensus problem is of great importance. Even though The theoretical framework for consensus problems was it might seem, our focus in this paper is to satisfy the among decision making units problem was firstly started started by consensus is of great importance. Even though the the work of by Borkar Varaiya (1982) and Tsitsiklis (1984). problem ofproblem maintaining reliable is important as introduced the and studies of R.Olfati-Saber and Murray consensus, even in casesa that someconnection of these connections are the work of Borkar and Varaiya (1982) and Tsitsiklis (1984). problem ofseem, maintaining a reliable connection istoimportant as The theoretical framework for consensus problems was it might our focus in this paper is satisfy the lost. The theoretical framework consensus problems was it might seem, ourcases focus this of paper to satisfy are the introduced by the studies offorR.Olfati-Saber and Murray consensus, even in thatinsome theseisconnections consensus, even in cases that some of these connections are introduced by the studies of R.Olfati-Saber and Murray lost. lost. Copyright © 2016 IFAC 341 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 341Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 341 10.1016/j.ifacol.2016.07.057
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(2003) and (2004) based on the works of Fax and Murray (Fax, 2001), (Murray and Fax, 2004). One of the most useful areas of the consensus problem is the management in transportation systems. Platoon control is necessary for the intelligent and efficient coordination of vehicles. A group of heavy-duty vehicles can be put into formations, namely platoons, in order to increase the safety and fuel efficiency (Alam et al., 2015). This is achieved by forming a line of vehicles, keeping the intervehicular distances and reducing the aerodynamic drag force for the non-leading vehicles, just like the principle behind the flying bird formations. Driverless cars forming a platoon with a sensor based network topology is also a consensus problem which consists problems regarding collision avoidance, formation adjustment according to changing weather and road conditions, sustaining the formation under various disturbances (Wang et al., 2014).
Fig. 1. Graphical Representation and Determination of the Laplacian Matrix A graph consists of vertices and edges in graph theory. In a consensus problem, the relationship between the agents are included in the Laplacian matrix. A Laplacian matrix is a square matrix consisting of an adjacency matrix and a degree matrix and it is obtained by subtracting the adjacency matrix from the degree matrix. Further properties of the Laplacian matrix such as being positive semi-definite and diagonally dominant are also the important properties that make the analysis of a network possible (Cvetkovic et al., 1998).
2. CONSENSUS PROBLEM AND ITS FORMULATION Graph Theory is an important tool for the analysis of consensus problems. It can be directly described by the Laplacian Matrix which involves all the necessary properties that we need to model a consensus problem. Some important concepts related with the formulation of such problems are: • Agent: An individual robot or any unit that has a state along with the desire to reach consensus with other agents. • Topology: The topology of agents is related to their connections among each other. Most common way to express a topology is by using the graph theory. Each node in the graph indicates an agent.
3. MODELLING THE CONSENSUS PROBLEM It can be proposed that a graph representation alone is not enough to represent a complete average consensus problem. It is important to state that in order for agents to change their state values and solve an average consensus problem, a dynamical system which consists of integrators is essential (Olfati-Saber and Murray, 2004). A system representation formed by block diagrams was introduced by Olfati-Saber and Murray (Olfati Saber et al., 2007), as given in Figure 2.
• Consensus Time: The time that consensus is reached by all agents. • Consensus Position: The position of consensus which is related to the initial states of all agents and their information exchange relationships, namely their network topology. Algebraic graph theory has a set of properties and definitions that are directly applicable to consensus problems. It is essential to know these properties in order to determine the outcome of the consensus problem and how its outcome is influenced by the type of the graph. In areas involving networks such as the consensus problem, networks are usually defined by symmetrical matrices and the solutions for the problems lie within the fundamentals of the Graph Theory (Steven, 2010).
Fig. 2. Block diagram representation of an average consensus problem
An introductory example of forming a Laplacian Matrix is given in Figure 1.
The nature of the consensus problem can be modelled as an autonomous system with a state-space equation without an input. For the average consensus problem, convergence occurs around the average of the initial states of all agents if the communication graph is balanced (Olfati-Saber and Murray, 2004). Therefore the dynamics of the system can be written as (Olfati-Saber and Murray, 2004), (Lynch, 1997):
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𝑥𝑥𝑖𝑖̇ (𝑡𝑡) = ∑ 𝑥𝑥̇ = −𝐿𝐿𝐿𝐿
𝑗𝑗∈𝑁𝑁𝑖𝑖
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𝑎𝑎𝑖𝑖𝑖𝑖 (𝑥𝑥𝑗𝑗 (𝑡𝑡) − 𝑥𝑥𝑖𝑖 (𝑡𝑡))
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algorithm handles the average consensus problem from a practical framework with some valuable insights. A flowchart of the algorithm, which is implemented in MATLAB™ with the aid of SIMULINK™, is shown in Figure 3.
(1) (2)
Equation (2) is a well-known dynamical linear system representation without any input. Thus, as it is known from the linear system theory, the eigenvalues of the Laplacian Matrix define the characteristic equation of the system and specify the behaviour of the system. Properties of the Laplacian Matrix can be exploited to determine the equilibrium point of the system which is when all agents agree on the same state value (Godsil and Royle, 2001); 𝑇𝑇
𝑥𝑥 ∗ = (∝, … , ∝) = ∝ 1
(3)
Fiedler stated (Fiedler, 1973) that the eigenvalues of Laplacian matrices can be sorted in ascending order starting from the first eigenvalue which is zero. From the outcome of this result, Fiedler defined the algebraic connectivity of a measure of connectivity (Merris, 1994). As it will be discussed further, the Graph Laplacian and their properties are the key elements in the analysis of the convergence of the consensus problem. Second smallest eigenvalue has a great significance as it is also related as Algebraic Connectivity (Chung, 1997) and also as Fiedler’s Eigenvalue (Fiedler, 1973). As it is mentioned earlier, the average consensus for balanced graphs occur around the average of all agents’ initial states. For a symmetrical adjacency matrix, all agents asymptotically converge to the value which is equal to the initial states of all agents and it is written as (Olfati-Saber and Murray, 2003): ∝ =
1 ∑ 𝑥𝑥𝑖𝑖 (0) 𝑛𝑛 𝑖𝑖
(4)
It is intuitively reasonable to state that (1) iteratively converges since the difference between states are always decaying and it is also stated that for any connected undirected graph, (1) solves the average consensus problem for any initial state (Tsitsiklis et al. 1986).
Fig. 3. The Flowchart of the Algorithm All the results obtained from the loop in the algorithm are reduced regarding the position and time criterion chosen. The consensus criterion used usually is the best possible values in terms of time and position. Shorter times with a consensus point around the average initial states are also considered as acceptable responses from the system.
4. PROPOSED APPROACH Theoretical solutions and analysis mostly concentrate on the second smallest eigenvalue as the key factor on the convergence rate. In this study, the importance of topologies has been examined along with the second smallest eigenvalue. In particular, it is shown that Different Laplacian Matrices with same eigenvalues may converge with different rates. Hence, an important element on the convergence of the system is that the topology of agents and the configurations of their connections.
In the next section, all possible configurations that a network topology may have for a fixed number of agents is considered as a case study. 5. CASE STUDY As a case study, a network topology consisting 3 agents is examined with the initial values: [5, 17, 37]. First step of the algorithm is to determine all the possible Laplacian Matrices and simulate each of them. Please note that 64 possible topologies (Laplacian matrices) exist for this case. Each topology is numbered from 1 to 64. Figure 4 below shows the consensus time and position for each combination representing a different topology. The cases where no suitable consensus occurs have been omitted.
The proposed approach is to consider a trade-off between the numbers of information channels that should be implemented and also to get a good performance in terms of consensus time and consensus location. The number of minimum connections between agents to reach consensus and also to get a relatively good result is also examined. The proposed 343
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The chart in the Figure 6 indicates the number of topologies which corresponds to the number of connections that a topology has. As it can be seen from the chart, consensus can be reached without establishing maximum number of connections, which leads to cost effective solutions. 5.1 Comparison of Same Consensus Times with Same Eigenvalues Out of 4 topologies where each have 5 connections, two topologies with identical consensus time were chosen to be examined as given in Figure 7.
Fig. 4. Graph of Consensus Time/Position versus Topology No According to the consensus criterion used the results are reduced, renumbered and presented below in Figure 5. The numbers in Figure 5 will be used from now on in order not to cause any confusion in comparison.
Fig. 7. Compared Topologies (No 15 and No 16)
Fig. 5. Reduced Graph of Consensus Time/Position versus Topology No
Fig. 8. Convergence of Topology No 15
Fig. 9. Convergence of Topology No 16
Fig. 6. Frequency of Topologies in terms of number of connections 344
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A graph representation of topology no 15 and topology no 16 are shown in Figure 8 and Figure 9 with their second smallest eigenvalues as λ. As a result, for the same eigenvalues, initial values, number of connections and consensus time, the consensus positions are different for two combinations due to different topologies. 5.2 Comparison of Same Consensus Position with Same Eigenvalues Out of 8 topologies where each have 4 connections, two topologies with identical consensus position were chosen to be examined in Figure 10.
Fig. 12. Convergence of Topology No 11 6. CONCLUSION As the comparison in the case study clearly indicates some valuable distinctions occur between properties of configurations. It can be seen that the network topology is quite important to get accurate results for the average consensus problem. The examples and cases can easily be extended to networks with larger scales of agents in the expense of larger computational times. The algorithm to use the proposed strategy is presented. To sum up, eigenvalues of the system are important for the system’s performance but they are not the only factors. The structure of the consensus connections has a significant role on the performance of the system.
Fig. 10. Compared Topologies (No 6 and No 11) A graph representation of topology no 6 and topology no 11 are shown in Figure 10 with their second smallest eigenvalues as λ. As a result, for the same eigenvalues, initial values, number of connections and consensus position, the consensus times are different for two combinations due to different topologies. The case study consists of comparisons of similar network topologies. Their comparisons in terms of time and their second smallest eigenvalues present us some valuable insight into the average consensus problem. As it is seen from the Figure 11 and Figure 12, second smallest eigenvalue alone does not completely determine the consensus time. Although both systems have the same system dynamics (because of the eigenvalues), topology had an effect on performance measures.
Further studies will concentrate on extending the number of nodes considered in this study together with some uncertainties in communications and connecting the results to the graph theory. It is obvious that the need for greater autonomy in transportation systems makes it necessary for scientists and engineers to understand the consensus problem from different perspectives in order to design more intelligent transportation systems. REFERENCES Alam, A., Besselink, B., Turri, V., and Johansson, K., H. (2015). Heavy-Duty Vehicle Platooning for Sustainable Freight Transportation: A Cooperative Method to Enhance Safety and Efficiency. IEEE CONTROL SYSTEMS 35(6), pp. 34-56. Avrachenkov, K., El Chamie, M., Neglia, G. (2011). A local average consensus algorithm for wireless sensor networks. Distributed Computing in Sensor Systems and Workshops (DCOSS), 2011 International Conference on, IEEE, Barcelona. Borkar, V., Varaiya, P. (1982), Asymptotic agreement in distributed estimation, IEEE Trans. Autom. Control, vol. AC-27, pp. 650–655. Bouffanais, R. (2016), Design and Control of Swarm Dynamics (First edition). Springer. ISBN 978-981-287750-5 Chung, F. R. K. (1997). Spectral Graph Theory. Providence, RI: Amer. Math. Soc.
Fig. 11. Convergence of Topology No 6 345
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