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Stability Theory of Switched Dynamical Systems pp 197–238Cite as

Connections and Implications

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Part of the book series: Communications and Control Engineering ((CCE))

Abstract

Chapter 5 establishes the connections and implications among the switched stability/stabilization problems and several fundamental control problems. For absolute stability of Lur’e systems, an elegant connection to the guaranteed stability of switched linear systems is established. Utilizing this connection, computational algorithms are presented to verify absolute stability for planar Lur’e systems. Another implication of the guaranteed stability criteria is the consensus analysis of multi-agent systems with dynamic neighbors, and exponential agreement is reached if the graph is always strongly connected. For an intelligent system with linear local controllers and a fuzzy rule, it is naturally converted into a piecewise linear system, and hence the stability analysis can be conducted by means of the stability criteria presented in Chapter 3. This brings a new design method and a fresh observation to the fuzzy control problem. For a SISO linear process with unknown parameters, an adaptive control framework is established based on appropriate partitions of the parameter space and proper stabilizing switching strategy among the local controllers that are designed to stabilize the system in a local sense. Finally, for controllable switched linear systems, a multilinear feedback design approach is proposed to tackle the stabilization problem. The main idea is to associate with each subsystem a set of candidate linear controllers such that the extended switched system is stabilizable. By utilizing the pathwise state-feedback switching design diagram, the problem of stabilization is solved in a constructive manner.

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References

  1. Anderson BDO, Brinsmead TS, De Bruyne F, Hespanha JP, Liberzon D, Morse AS. Multiple model adaptive control, part 1: finite controller coverings. Int J Robust Nonlinear Control. 2000;10(11–12):909–29.

    Article  MATH  Google Scholar 

  2. Angeli D, Mosca E. Lyapunov-based switching supervisory control of nonlinear uncertain systems. IEEE Trans Autom Control. 2002;47(3):500–5.

    Article  MathSciNet  Google Scholar 

  3. Astrom KJ, Wittenmark B. Adaptive control. 2nd ed. Eaglewood Cliffs: Prentice Hall; 1995.

    Google Scholar 

  4. Balde M, Boscain U. Stability of planar switched systems: the nondiagonalizable case. Commun Pure Appl Anal. 2008;7:1–21.

    MathSciNet  MATH  Google Scholar 

  5. Balde M, Boscain U, Mason P. A note on stability conditions for planar switched systems. Int J Control. 2009;82(10):1882–8.

    Article  MathSciNet  MATH  Google Scholar 

  6. Baldi S, Battistelli G, Mosca E, Tesi P. Multi-model unfalsified adaptive switching supervisory control. Automatica. 2010;46(2):249–59.

    Article  MATH  Google Scholar 

  7. Biggs N. Algebraic graph theory. Cambridge: Cambridge University Press; 1993.

    Google Scholar 

  8. Boscain U. Stability of planar switched systems: the linear single input case. SIAM J Control Optim. 2002;41:89–112.

    Article  MathSciNet  MATH  Google Scholar 

  9. Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in systems and control theory. Philadelphia: SIAM; 1994.

    Google Scholar 

  10. Cao SG, Rees NW, Feng G. Universal fuzzy controllers for a class of nonlinear systems. Fuzzy Sets Syst. 2001;122(1):117–23.

    Article  MathSciNet  MATH  Google Scholar 

  11. Fax JA, Murray RM. Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control. 2004;49(9):1465–76.

    Article  MathSciNet  Google Scholar 

  12. Feng G. Stability analysis of discrete time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Trans Fuzzy Syst. 2004;12(1):22–8.

    Article  Google Scholar 

  13. Feng G. A survey on analysis and design of model-based fuzzy control systems. IEEE Trans Fuzzy Syst. 2006;14(5):676–97.

    Article  Google Scholar 

  14. Fu M, Barmish B. Adaptive stabilization of linear systems via switching control. IEEE Trans Autom Control. 1986;31(12):1097–103.

    Article  MathSciNet  MATH  Google Scholar 

  15. Godsil C, Royle GF. Algebraic graph theory. Berlin: Springer; 2001.

    MATH  Google Scholar 

  16. Hespanha JP, Liberzon D, Morse AS. Overcoming the limitations of adaptive control by means of logic-based switching. Syst Control Lett. 2003;49(1):49–65.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hespanha JP, Liberzon D, Morse AS. Hysteresis-based switching algorithms for supervisory control of uncertain systems. Automatica. 2003;39:263–72.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hespanha JP, Liberzon D, Morse AS, Anderson BDO, Brinsmead TS, De Bruyne F. Multiple model adaptive control, part 2: Switching. Int J Robust Nonlinear Control. 2001;11(5):479–96.

    Article  MATH  Google Scholar 

  19. Hockerman-Frommer J, Kulkarni SR, Ramadge PJ. Controller switching based on output prediction errors. IEEE Trans Autom Control. 1998;43(5):596–607.

    Article  Google Scholar 

  20. Holcman D, Margaliot M. Stability analysis of switched homogeneous systems in the plane. SIAM J Control Optim. 2003;41(5):1609–25.

    Article  MathSciNet  MATH  Google Scholar 

  21. Hong YG, Gao LX, Cheng D, Hu JP. Lyapunov-based approach to multiagent systems with switching jointly connected interconnection. IEEE Trans Autom Control. 2007;52(5):943–8.

    Article  MathSciNet  Google Scholar 

  22. Jadbabaie A, Lin J, Morse AS. Coordination of groups of mobile agents using nearest neighbor rules. IEEE Trans Autom Control. 2003;48(6):988–1001.

    Article  MathSciNet  Google Scholar 

  23. Ji M, Egerstedt M. Distributed coordination control of multiagent systems while preserving connectedness. IEEE Trans Robot. 2007;23(4):693–703.

    Article  Google Scholar 

  24. Johansson M, Rantzer A, Arzen K-E. Piecewise quadratic stability of fuzzy systems. IEEE Trans Fuzzy Syst. 1999;7(6):713–22.

    Article  Google Scholar 

  25. Lin Z, Broucke M, Francis B. Local control strategies for groups of mobile autonomous agents. IEEE Trans Autom Control. 2004;49(4):622–9.

    Article  MathSciNet  Google Scholar 

  26. Loewy R. On ranges of real Lyapunov transformations. Linear Algebra Appl. 1976;13(1):79–89.

    Article  MathSciNet  MATH  Google Scholar 

  27. Margaliot M, Gitizadeh R. The problem of absolute stability: a dynamic programming approach. Automatica. 2004;40(7):1247–52.

    Article  MathSciNet  MATH  Google Scholar 

  28. Margaliot M, Langholz G. Necessary and sufficient conditions for absolute stability: the case of second-order systems. IEEE Trans Circuits Syst I, Fundam Theory Appl. 2003;50(2):227–34.

    Article  MathSciNet  Google Scholar 

  29. Margaliot M, Yfoulis C. Absolute stability of third-order systems: a numerical algorithm. Automatica. 2006;42(10):1705–11.

    Article  MathSciNet  MATH  Google Scholar 

  30. Moreau L. Stability of continuous-time distributed consensus algorithms. In: Proc IEEE CDC; 2004. p. 3998–4003.

    Google Scholar 

  31. Morse AS. Logic-based switching and control. In: Francis BA, Tannenbaum AR, editors. Feedback control, nonlinear systems, and complexity. New York: Springer; 1995. p. 173–95.

    Chapter  Google Scholar 

  32. Morse AS. Supervisory control of families of linear set-point controllers, part 1: Exact matching. IEEE Trans Autom Control. 1996;41(10):1413–31.

    Article  MathSciNet  MATH  Google Scholar 

  33. Morse AS. Supervisory control of families of linear set-point controllers, part 2: Robustness. IEEE Trans Autom Control. 1997;42(11):1500–15.

    Article  MathSciNet  MATH  Google Scholar 

  34. Narendra KS, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans Autom Control. 1994;39(12):2469–71.

    Article  MathSciNet  MATH  Google Scholar 

  35. Narendra KS, Balakrishnan J. Adaptive control using multiple models. IEEE Trans Autom Control. 1997;42(2):171–87.

    Article  MathSciNet  MATH  Google Scholar 

  36. Olfati-Saber R, Murray RM. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control. 2004;49(9):1520–33.

    Article  MathSciNet  Google Scholar 

  37. Pyatnitskiy ES, Rapoport LB. Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems. IEEE Trans Circuits Syst I, Fundam Theory Appl. 1996;43(3):219–29.

    Article  MathSciNet  Google Scholar 

  38. Rapoport LB. Asymptotic stability and periodic motions of selector-linear differential inclusions. In: Garofalo F, Glielmo L, editors. Robust control via variable structure and Lyapunov techniques. New York: Springer; 1996. p. 269–85.

    Chapter  Google Scholar 

  39. Shorten RN, Narendra KS. Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems. Int J Adapt Control Signal Process. 2003;16:709–28.

    Article  Google Scholar 

  40. Shorten RN, Narendra KS, Mason O. A result on common quadratic Lyapunov functions. IEEE Trans Autom Control. 2003;48(1):110–3.

    Article  MathSciNet  Google Scholar 

  41. Stefanovic M, Safonov M. Safe adaptive switching control: stability and convergence. IEEE Trans Autom Control. 2008;53(9):2012–21.

    Article  MathSciNet  Google Scholar 

  42. Sugeno M. On stability of fuzzy systems expressed by fuzzy rules with singleton consequents. IEEE Trans Fuzzy Syst. 1999;7(2):201–24.

    Article  MathSciNet  Google Scholar 

  43. Sun Z. Robust switching of switched linear systems. In: Proc IFAC NOLCOS; 2010. p. 256–9.

    Google Scholar 

  44. Sun Z, Ge SS. Switched linear systems: control and design. London: Springer; 2005.

    MATH  Google Scholar 

  45. Sun Z, Huang J. A note on connectivity of multi-agent systems with proximity graphs and linear feedback protocol. Automatica. 2009;45(9):1953–6.

    Article  MATH  Google Scholar 

  46. Sun Z, Peng Y. Stabilizing design for switched linear control systems: a constructive approach. Trans Instrum Meas. 2010;32(6):706–35.

    Article  Google Scholar 

  47. Szabo Z, Bokor J, Balas G. Generalized piecewise linear feedback stabilizability of controlled linear switched systems. In: Proc IEEE CDC; 2008. p. 3410–4.

    Google Scholar 

  48. Szabo Z, Bokor J, Balas G. Controllability and stabilizability of linear switched systems. In: Proc ECC; 2009.

    Google Scholar 

  49. Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern. 1985;15(1):116–32.

    MATH  Google Scholar 

  50. Tanaka K, Sugeno M. Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 1992;12(2):135–56.

    Article  MathSciNet  Google Scholar 

  51. Tanaka K, Wang HO. Fuzzy control systems design and analysis: a linear matrix inequality approach. New York: Wiley; 2001.

    Google Scholar 

  52. Veres SM. The geometric bounding toolbox, user’s manual & reference. UK: SysBrain; 2001.

    Google Scholar 

  53. Vinnicombe G. Frequency domain uncertainty and the graph topology. IEEE Trans Autom Control. 1993;38:1371–83.

    Article  MathSciNet  MATH  Google Scholar 

  54. Xie G, Wang L. Controllability implies stabilizability for discrete-time switched linear systems. In: Hybrid systems: computation and control. Berlin: Springer; 2005. p. 667–82.

    Chapter  Google Scholar 

  55. Xie G, Wang L. Periodic stabilizability of switched linear control systems. Automatica. 2009;45(9):2141–8.

    Article  MATH  Google Scholar 

  56. Zadeh LA. Fuzzy sets. Inf Control. 1965;8(3):338–53.

    Article  MathSciNet  MATH  Google Scholar 

  57. Zadeh LA. Fuzzy algorithm. Inf Control. 1968;12:94–102.

    Article  MathSciNet  MATH  Google Scholar 

  58. Zavlanos MM, Jadbabaie A, Pappas GJ. Flocking while preserving network connectivity. In: Proc IEEE CDC; 2007. p. 2919–24.

    Google Scholar 

  59. Zhang W, Abate A, Hu JH, Vitus MP. Exponential stabilization of discrete-time switched linear systems. Automatica. 2009;45(11):2526–36.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. College Automation Science & Engineering, Center for Control and Optimization, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Zhendong Sun

  2. Department of Electrical and Computer Engineering, The National University of Singapore, Singapore, Singapore

    Shuzhi Sam Ge

  3. Robotics Institue and Institute of Intelligent Systems and Information Technology, University of Electronic Science and Technology of China, Chengdu, People’s Republic of China

    Shuzhi Sam Ge

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  1. Zhendong Sun
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  2. Shuzhi Sam Ge
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Correspondence to Zhendong Sun .

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© 2011 Springer-Verlag London Limited

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Sun, Z., Ge, S.S. (2011). Connections and Implications. In: Stability Theory of Switched Dynamical Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-256-8_5

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  • DOI: https://doi.org/10.1007/978-0-85729-256-8_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-255-1

  • Online ISBN: 978-0-85729-256-8

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