in more practical models of Mutli-agent systems like ant colony or particle swarm, there are some criteria for measuring the convergence of agents. in these models when agents start to interact with each other, after a while, they find an optimal solution for a given problem (they converge on a solution). this solution could be referred as a consensus which agents agree upon.
Dear Bahram Zaeri,
Thank you for giving me a head start.
Are "consensus" and "stability" the same thing? and what about the sensitivity of the consensus? is it not possible to find some condition(s) under which an agent of a multi-agent system may deviate from he consensus?
In the control and design of a Multi-Agent System, what is physical meaning of consensus? Is it an equation or condition or set of equations?
Please help me and provide some insight into these queries?
yes. consensus in physical conditions is viable.(as I said in my comment). since every agent behaves in a random manner, so there are some situations in which an agent might deviate(like swarm based methods). consensus can be defined as a way of interaction between agents(direct or indirect) which agents can communicate and reach an agreement.
a theoretical framework for consensus in multi-agent systems has been introduced here:
http://www.eecs.harvard.edu/~rad/courses/cs266/papers/reza-tac07.pdf
See my article:
N. Cai, J. Xi, Y. Zhong, “Swarm stability of high order linear time-invariant swarm systems”, IET Control Theory Appl., 2011, 5(2), pp. 402-408.
It contains basic and complete analysis for the consensus of general dynamic linear multi-agent systems.
Reza's discussions only consider first-order cases.
As an alternative, you could interpret consensus as some form of "correctness" (cf. [1]). Then, you may distinguish partial correctness from total correctness. The former asserts that if the system yields a solution, then this solution will satisfy a given specification, e.g. it is a valid solution for the given problem. The latter regards termination, i.e. asserting that the system terminates within a finite amount of time after it has been started, such that the final state of the system represents a valid solution.
Finally, if this behavior additionally is independent of the system’s starting conditions, the heuristic is said to be self-stabilizing (see [2], here you can find some feasible proof methods, too).
Also, see [3] for a framework to formally describe your system, which you have to do in order to be able to reason about it.
[1] K. R. Apt, “Ten Years of Hoare’s Logic: A Survey—Part I,” ACM Trans. Program. Lang. Syst., vol. 3, no. 4, pp. 431–483, 1981.
[2] S. Dolev, Self-Stabilization. Cambridge, MA, USA: MIT Press, 2000.
[3] N. Lynch, Distributed Algorithms. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1996.
As another alternative you can consider that consensus can be maintaining connectivity between agents, in the sense that even perturbation happens in their environment, agents still communicate to accomplish their goal. The latter is the finality for which agents exist.
Recommended links of a i) evolutionary game theory ii) sociobiology, iii) biomimicry nature:
Evolutionary game theory: Axelrod (article is linked, book is a more detailed treatment)
Sociobiology: Passino ,Seeley, Isscher
Biomimicry: Feng Tan, Jean-Jacques Slotine- concensus as "quorum sensing"?
Last but not least- something on the "softer side" in the form of an analogy/fable
Bernard Mendelville's Fable of the Bees
http://www-ee.stanford.edu/~hellman/Breakthrough/book/pdfs/axelrod.pdf
http://webcache.googleusercontent.com/search?q=cache:6CgvPEWoT1kJ:arxiv.org/abs/1303.3934+&cd=1&hl=en&ct=clnk&gl=my
http://www2.ece.ohio-state.edu/~passino/PapersToPost/SwarmCognition-BES.pdf
http://oll.libertyfund.org/titles/846
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