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Lectures in Feedback Design for Multivariable Systems pp 135–163Cite as

Coordination and Consensus of Linear Systems

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Part of the book series: Advanced Textbooks in Control and Signal Processing ((C&SP))

Abstract

In this chapter, we will see how the theory of asymptotic tracking can be fruitfully extended to address problems in which a large set of systems is controlled in such a way that certain variables of interest asymptotically coincide. The specific challenge addressed in this chapter resides in the fact that there is a limited exchange of information between individual systems, each one of which has access only to measurements of the outputs of limited number of neighbors.

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Notes

  1. 1.

    Note that all such systems have the same number of input and output components.

  2. 2.

    Further motivations for the interest of an information pattern such as the one described by (5.5) can be found in [1–6] and [20].

  3. 3.

    Background material on graphs can be found, e.g., in [7, 8]. All objects defined below are assumed to be independent of time. In this case the communication graph is said to be time-invariant. Extensions of definitions, properties and results to the case of time-varying graphs are not covered in this book and the reader is referred, e.g. to [9, 10].

  4. 4.

    Note that in a connected graph there may be more than one node with such property. See the example in Fig. 5.1.

  5. 5.

    In fact, if the graph \(\mathscr {G}\) possesses a root from which information can propagate to all other nodes along paths, the same is true for the graph \(\hat{\mathscr {G}}\), because the nonzero entries of \(\hat{A}\) coincide with the nonzero entries of A.

  6. 6.

    From now on, we drop the assumption—considered in the previous section—that \(p=m=1\).

  7. 7.

    For a couple of matrices \(A\in \mathbb R^{m\times n}\) and \(B\in \mathbb R^{p\times q}\), their Kroeneker product—denoted by \(A\otimes B\)—is the \(mp\times nq\) matrix defined as

    $$ A\otimes B = \left( \begin{matrix}a_{11}B &{} a_{12}B &{} \cdots &{} a_{1n}B \\ a_{21}B &{} a_{22}B &{} \cdots &{} a_{2n}B \\ \cdot &{} \cdot &{} \cdots &{} \cdot \\ a_{m1}B &{} a_{m2}B &{} \cdots &{} a_{mn}B \\ \end{matrix}\right) .$$
  8. 8.

    We use in what follows the property \((A\otimes B)(C\otimes D)=(AC\otimes BD)\), which—in particular—implies \((T^{-1}\otimes I_n)^{-1}=(T\otimes I_n)\).

  9. 9.

    See, e.g., [11].

  10. 10.

    Note that in this case the elements of R may be complex numbers.

  11. 11.

    The existence of such \(P>0\) is guaranteed by the assumption that the pair (A, C) is observable. See, e.g., [12].

  12. 12.

    See proof of Theorem A.2 in Appendix A.

  13. 13.

    See, e.g., [13]. See also [14, 15] and [21–23] for further reading.

  14. 14.

    The approach described hereafter is based on the work [16].

  15. 15.

    See, e.g., [17, p. 22].

  16. 16.

    That is, using \(x^*X^*Yy + y^*Y^*Xx \le d x^*X^*Xx + {1\over d} y^*Y^*Yy\), which holds for any \(d>0\).

  17. 17.

    The approach described in this section is motivated by the works of [18] and [19]. In particular, the work [18] shows that the approach outline above is in some sense necessary for the solution of the problem in question.

  18. 18.

    Note that the latter guarantees the existence a (unique) the solution pair of (5.47).

References

  1. Y. Kuramoto, Self-Entrainment of a Population of Coupled Non-Linear Oscillators, ed. by H. Araki, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol. 39 (Springer, Berlin, 1975), pp. 420–422

    Google Scholar 

  2. T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. |75, pp. 1226–1229 (1995)

    Google Scholar 

  3. J.K. Hale, Diffusive coupling, dissipation, and synchronization. J. Dyn. Diff. Eqns 9, 1–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Jadbabaie, J. Lin, A.S. Morse, Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules. IEEE Trans. Autom. Control 48, 988–1001 (2003)

    Article  MathSciNet  Google Scholar 

  5. R. Olfati-Saber, R.M. Murray, Consensus Problems in Networks of Agents With Switching Topology and Time-Delays. IEEE Trans. Autom. Control 49, 1520–1533 (2004)

    Article  MathSciNet  Google Scholar 

  6. S. Nair, N.E. Leonard, Stable synchronization of mechanical system networks. SIAM J. Control Optim. 47, 234–265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Fiedler, Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23, 298–305 (1973)

    MathSciNet  MATH  Google Scholar 

  8. R. Merris, A survey of graph Laplacians. Linear Multilinear Algebr. 39, 19–31 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Moreau, Stability of continuous-time distributed consensus algorithms (2004). arXiv:math/0409010v1 [math.OC]

  10. L. Moreau, Stability of multi-agent systems with time-dependent communication links. IEEE Trans. Autom. Control 50, 169–182 (2005)

    Article  MathSciNet  Google Scholar 

  11. J.A. Fax, R.M. Murray, Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49, 1465–1476 (2004)

    Article  MathSciNet  Google Scholar 

  12. J.C. Willems, Least square stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control 16, 621–634 (1971)

    Article  MathSciNet  Google Scholar 

  13. L. Scardovi, R. Sepulchre, Synchronization in networks of identical linear systems. Automatica 45, 2557–2562 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Wieland, From Static to Dynamic Couplings in Consensus and Synchronization among Identical and Non-Identical Systems, PhD Dissertation, Universität Stuttgart, 2010

    Google Scholar 

  15. Z. Li, Z. Duan, G. Chen, Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint. IEEE Trans. Circuits Syst.-I 57, 213–224 (2010)

    Article  MathSciNet  Google Scholar 

  16. J.H. Seo, H. Shim, J. Back, Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach. Automatica 45, 2659–2664 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Z. Lin, Low Gain Feedback (Springer, London, 1999)

    Google Scholar 

  18. P. Wieland, R. Sepulchre, F. Allgöwer, An internal model principle is necessary and sufficient for linear output synchronization. Automatica 47, 1068–1074 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Kim, H. Shim, J.H. Seo, Output consensus of heterogeneous uncertain linear multi-agent systems. IEEE Trans. Autom. Control 56, 200–206 (2011)

    Article  MathSciNet  Google Scholar 

  20. D.R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95, 215–233 (2007)

    Article  Google Scholar 

  21. H.L. Trentelman, K. Takaba, N. Monshizadeh, Robust Synchronization of Uncertain Linear Multi-Agent Systems. IEEE Trans. Autom. Control 58, 1511–1523 (2013)

    Article  MathSciNet  Google Scholar 

  22. A. Isidori, L. Marconi, G. Casadei, Robust Output Synchronization of a Network of Heterogeneous Nonlinear Agents Via Nonlinear Regulation Theory. IEEE Trans. Autom. Control 59, 2680–2691 (2014)

    Article  MathSciNet  Google Scholar 

  23. F. Delli Priscoli, A. Isidori, L. Marconi, A. Pietrabissa, Leader-Following Coordination of Nonlinear Agents Under Time-Varying Communication Topologies, IEEE Trans. Control Netw. Syst. Control, 2, 393–405 (2015)

    Google Scholar 

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  1. Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Università degli Studi di Roma “La Sapienza”, Rome, Italy

    Alberto Isidori

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Isidori, A. (2017). Coordination and Consensus of Linear Systems. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_5

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  • DOI: https://doi.org/10.1007/978-3-319-42031-8_5

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