Riccati Design for Synchronization of Discrete-Time Systems

  • Frank L. LewisEmail author
  • Hongwei Zhang
  • Kristian Hengster-Movric
  • Abhijit Das
Part of the Communications and Control Engineering book series (CCE)


In this chapter, design methods are given for synchronization control of discrete-time multi-agent systems on directed communication graphs. The graph is assumed to have fixed topology and contain a spanning tree. The graph properties complicate the design of synchronization controllers due to the interplay between the eigenvalues of the graph Laplacian matrix and the required stabilizing gains. A method is given that decouples the design of the synchronizing feedback gains from the detailed graph properties. It is based on computation of the agent feedback gains using a local Riccati equation design. Conditions are given for synchronization based on the relation of the graph eigenvalues to a bounded circular region in the complex plane that depends on the agent dynamics and the Riccati solution. The notion of ‘synchronization region’ is used. Convergence to consensus and robustness properties are investigated. This chapter also investigates the design of distributed observers for identical agents using a local Riccati design. A cooperative observer design guaranteeing convergence of the estimates of all agents to their actual states is proposed. The notion of a convergence region for distributed observers on graphs is introduced.


Tracking Error Riccati Equation Feedback Gain Communication Graph Control Node 
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  1. 1.
    Duan Z, Chen G, Huang L (2009) Disconnected synchronized regions of complex dynamical Networks. IEEE Trans Automat Contr 54(4):845–849CrossRefMathSciNetGoogle Scholar
  2. 2.
    Elia N, Mitter SK (2001) Stabilization of linear systems with limited information. IEEE Trans Automat Contr 46(9):1384–1400CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Fu M, Xie L (2005) The sector bound approach to quantized feedback control. IEEE Trans Automat Contr 50(11):1698–1711CrossRefMathSciNetGoogle Scholar
  4. 4.
    Hengster-Movric K, You K, Lewis FL, Xie L (2013) Synchronization of discrete-time multi-agent systems on graphs using riccati design. Automatica, vol. 49(2):414–423Google Scholar
  5. 5.
    Lewis FL, Vrabie D, Syrmos VL (2012) Optimal Control, 3rd Edn. Wiley, New YorkCrossRefGoogle Scholar
  6. 6.
    Li Z, Duan Z, Chen G, Huang L (2010) Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans Circuits Syst I, Reg papers, 57(1):213–224Google Scholar
  7. 7.
    Li Z, Duan Z, Chen G (2011) Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems, Series B 16(2):489–505Google Scholar
  8. 8.
    Pecora LM, Carroll TL (1998) Master stability functions for synchronized coupled systems. Phys Rev Lett 80(10):2109–2112CrossRefGoogle Scholar
  9. 9.
    Sinopoli B, Schenato L, Franceschetti M, Poolla K, Sastry S (2004) Kalman filtering with intermittent observations. IEEE Trans Automat Contr 49(9):1453–1464CrossRefMathSciNetGoogle Scholar
  10. 10.
    Wang XF, Chen G (2002) Pinning control of scale free dynamical networks. Physica A 310(3–4):521–531CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    You K, Xie L (2011) Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Trans Automat Contr 56(10):2262–2275CrossRefMathSciNetGoogle Scholar
  12. 12.
    Zhang H, Lewis FL, Das A (2011) Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback. IEEE Trans Automat Contr 56(8):1948–1952CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Frank L. Lewis
    • 1
    Email author
  • Hongwei Zhang
    • 2
  • Kristian Hengster-Movric
    • 3
  • Abhijit Das
    • 4
  1. 1.UTA Research InstituteUniversity of Texas at ArlingtonFort WorthUSA
  2. 2.School of Electrical EngineeringSouthwest Jiaotong UniversityChengduChina, People’s Republic
  3. 3.UTA Research InstituteUniversity of Texas at ArlingtonFort WorthUSA
  4. 4.Advanced Systems Engineering​Danfoss Power Solutions (US) Company​AmesUSA

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