Reciprocally Convex Approach for the Stability of Networked Control Systems

Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 110)


This chapter deals with the problem of stability analysis for networked control systems via the time-delayed system approach. The network-induced delays are modeled as two additive time-varying delays in the closed-loop system. To check the stability of such particular featured systems, an appropriate Lyapunov–Krasovskii functional is proposed and the Jensen inequality lemma is applied to the integral terms that are derived from the derivative of the Lyapunov–Krasovskii functional. Here, the cascaded structure of the delays in the system enables one to partition the domain of the integral terms into three parts, which produces a linear combination of positive functions weighted by inverses of convex parameters. This is handled efficiently by the authors’ lower bounds lemma that handles the so-called reciprocally convex combination.


Reciprocally convex combination Delay systems Stability Networked control systems 



This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009743).

Poogyeon Park also gratefully acknowledges the LG YONAM Foundation for its financial support through the Professors’ overseas research program for sabbatical research leave at University of Maryland.


  1. 1.
    Antsaklis P, Baillieul J (2004) Guest editorial special issue on networked control systems. IEEE Trans Autom Contr 31(9):1421–1423CrossRefMathSciNetGoogle Scholar
  2. 2.
    Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  3. 3.
    Du B, Lam J, Shu Z, Wang Z (2009) A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components. IET Contr Theor Appl 3(4):383–390CrossRefMathSciNetGoogle Scholar
  4. 4.
    Gao H, Chen T, Lam J (2008) A new delay system approach to network-based control. Automatica 44(1):39–52CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems, 1st edn. Birkhäuser BostonGoogle Scholar
  6. 6.
    Ko JW, Lee WI, Park PG (2011) Delayed system approach to the stability of networked control systems. In: Proceedings of the international multiconference of engineers and computer scientists 2011 (IMECS 2011), Hong Kong. Lecture notes in engineering and computer science, pp 772–774Google Scholar
  7. 7.
    Lam J, Gao H, Wang C (2007) Stability analysis for continuous systems with two additive time-varying delay components. Syst Contr Lett 56(1):16–24CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Li H, Chow MY, Sun Z (2009a). State feedback stabilisation of networked control systems. IET Contr Theor Appl 3(7):929–940CrossRefMathSciNetGoogle Scholar
  9. 9.
    Li T, Guo L, Wu L (2009b). Simplified approach to the asymptotical stability of linear systems with interval time-varying delay. IET Contr Theor Appl 3(2):252–260CrossRefMathSciNetGoogle Scholar
  10. 10.
    Park PG, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47:235–238CrossRefMATHGoogle Scholar
  11. 11.
    Peng C, Tian YC (2008) Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay. IET Contr Theor Appl 2(9):752–761CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Electrical EngineeringPohang University of Science and TechnologyPohangRepublic of Korea
  2. 2.Division of IT Convergence EngineeringPohang University of Science and TechnologyPohangRepublic of Korea

Personalised recommendations