Advertisement

Designed Switching

  • Zhendong Sun
  • Shuzhi Sam Ge
Part of the Communications and Control Engineering book series (CCE)

Abstract

Chapter 4 is devoted to the stabilizing switching design for switched dynamical systems under controlled switching. It is proven that a switched Lyapunov function exists if the system is globally asymptotically stabilizable. However, counterexamples exhibit that even stabilizable planar switched linear systems may not admit any convex switched Lyapunov function. To overcome the intrinsic difficulty, we introduce a class of nonconvex functions known as min functions that are piecewise quadratic and prove that each stabilizable switched linear system admits a min function as a switched Lyapunov function. To further address the stabilizability and robustness of switched linear system, we propose a pathwise state-feedback switching strategy, which accounts to concatenating a finite number of switching paths based on appropriate partitions of the state space. By aggregating the overall system into a discrete-time piecewise linear system, we are able to prove that the switching strategy exponentially stabilizes the original switched linear system whenever it is asymptotically stabilizable. We develop a computational procedure to calculate a stabilizing pathwise state-feedback switching law for an asymptotically stabilizable switched linear system. To further investigate the robustness of the pathwise state-feedback switching strategy, we define a (relative) distance between two switching signals and prove that the closed-loop system is robust against structural/unstructural/switching perturbations.

Keywords

Lyapunov Function Switching Signal Quadratic Lyapunov Function Switch Linear System State Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 15.
    Bacciotti A. Stabilization by means of state space depending switching rules. Syst Control Lett. 2004;53(3–4):195–201. MATHCrossRefMathSciNetGoogle Scholar
  2. 16.
    Bacciotti A, Mazzi L. A discussion about stabilizing periodic and near-periodic switching signals. In: Proc IFAC NOLCOS; 2010. p. 250–5. Google Scholar
  3. 26.
    Bertsekas DP. Dynamic programming and optimal control, vol II. 3rd ed. Nashua: Athena Scientific; 2010. Google Scholar
  4. 35.
    Blanchini F, Savorgnan C. Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions. In: Proc IEEE CDC; 2006. p. 119–24. Google Scholar
  5. 36.
    Blanchini F, Savorgnan C. Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions. Automatica. 2008;44(4):1166–70. CrossRefMathSciNetGoogle Scholar
  6. 58.
    Clarke FH, Ledyaev YS, Sontag ED, Subbotin AI. Asymptotic controllability implies feedback stabilization. IEEE Trans Autom Control. 1997;42(10):1394–407. MATHCrossRefMathSciNetGoogle Scholar
  7. 60.
    Daafouz J, Riedinger P, Iung C. Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans Autom Control. 2002;47(11):1883–7. CrossRefMathSciNetGoogle Scholar
  8. 78.
    Feng G, Ma J. Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems. IEEE Trans Circuits Syst I, Fundam Theory Appl. 2001;48(11):1337–44. CrossRefGoogle Scholar
  9. 80.
    Feron E. Quadratic stabilizability of switched systems via state and output feedback. Massachusetts Inst Tech, Tech Rep CICS-P-468; 1996. Google Scholar
  10. 81.
    Feuer A, Goodwin GC, Salgado M. Potential benefits of hybrid control for linear time invariant plants. In: Proc ACC; 1997. p. 2790–4. Google Scholar
  11. 87.
    Geromel JC, Colaneri P. Stability and stabilization of continuous-time switched linear systems. SIAM J Control Optim. 2006;45(5):1915–30. MATHCrossRefMathSciNetGoogle Scholar
  12. 106.
    Hespanha JP, Liberzon D, Morse AS. Logic-based switching control of a nonholonomic system with parametric modeling uncertainty. Syst Control Lett. 1999;38(3):167–77. MATHCrossRefMathSciNetGoogle Scholar
  13. 107.
    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. MATHCrossRefMathSciNetGoogle Scholar
  14. 114.
    Hu TS, Lin ZL. Composite quadratic Lyapunov functions for constrained control systems. IEEE Trans Autom Control. 2003;48(3):440–50. CrossRefMathSciNetGoogle Scholar
  15. 115.
    Hu TS, Ma LQ, Lin ZL. Stabilization of switched systems via composite quadratic functions. IEEE Trans Autom Control. 2008;53(11):2571–85. CrossRefMathSciNetGoogle Scholar
  16. 118.
    Ishii H, Francis BA. Stabilizing a linear system by switching control with dwell time. IEEE Trans Autom Control. 2002;47(12):1962–73. CrossRefMathSciNetGoogle Scholar
  17. 129.
    Kellet CM, Teel A. Weak converse Lyapunov theorems and control-Lyapunov functions. SIAM J Control Optim. 2004;42:1934–59. CrossRefMathSciNetGoogle Scholar
  18. 130.
    Khalil HK. Nonlinear systems. 3rd ed. Upper Saddle River: Prentice Hall; 2002. MATHGoogle Scholar
  19. 132.
    Kolmanovsky I, McClamroch NH. Developments in nonholonomic control problems. IEEE Control Syst Mag. 1995;15(6):20–36. CrossRefGoogle Scholar
  20. 150.
    Lin H, Antsaklis PJ. Switching stabilizability for continuous-time uncertain switched linear systems. IEEE Trans Autom Control. 2007;52(4):633–46. CrossRefMathSciNetGoogle Scholar
  21. 151.
    Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Autom Control. 2009;54(2):308–22. CrossRefMathSciNetGoogle Scholar
  22. 152.
    Lin Y, Sontag ED, Wang Y. A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim. 1996;34(1):124–60. MATHCrossRefMathSciNetGoogle Scholar
  23. 169.
    McClamroch NH, Kolmanovsky I. Performance benefits of hybrid control design for linear and nonlinear systems. Proc IEEE. 2000;88(7):1083–96. CrossRefGoogle Scholar
  24. 181.
    Mount DM. Bioinformatics: sequence and genome analysis. 2nd ed. New York: Cold Spring Harbor Laboratory Press; 2004. Google Scholar
  25. 183.
    Narendra KS, Balakrishnan J. Adaptive control using multiple models. IEEE Trans Autom Control. 1997;42(2):171–87. MATHCrossRefMathSciNetGoogle Scholar
  26. 185.
    Notredame C. Recent progresses in multiple sequence alignment: a survey. Pharmacogenomics. 2002;3(1):131–44. CrossRefGoogle Scholar
  27. 191.
    Pearson WR, Lipman DJ. Improved tools for biological sequence comparison. Proc Natl Acad Sci USA. 1988;85:2444–8. CrossRefGoogle Scholar
  28. 192.
    Peng YP. Feedback stabilization and performance optimization of switched systems. PhD dissertation, South China Univ Tech; 2010. Google Scholar
  29. 198.
    Rifford L. Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J Control Optim. 2000;39(4):1043–64. MATHCrossRefMathSciNetGoogle Scholar
  30. 211.
    Sontag ED. Smooth stabilization implies coprime factorization. IEEE Trans Autom Control. 1989;34(4):435–43. MATHCrossRefMathSciNetGoogle Scholar
  31. 213.
    Stanford DP, Urbano JM. Some convergence properties of matrix sets. SIAM J Matrix Anal Appl. 1994;14(4):1132–40. CrossRefMathSciNetGoogle Scholar
  32. 216.
    Sun Z. Stabilizability and insensitiveness of switched systems. IEEE Trans Autom Control. 2004;49(7):1133–7. CrossRefGoogle Scholar
  33. 217.
    Sun Z. A modified stabilizing law for switched linear systems. Int J Control. 2004;77(4):389–98. MATHCrossRefGoogle Scholar
  34. 218.
    Sun Z. A general robustness theorem for switched linear systems. In: Proc IEEE ISIC; 2005. p. 8–11. Google Scholar
  35. 219.
    Sun Z. Combined stabilizing strategies for switched linear systems. IEEE Trans Autom Control. 2006;51(4):666–74. CrossRefGoogle Scholar
  36. 220.
    Sun Z. Stabilization and optimal switching of switched linear systems. Automatica. 2006;42(5):783–8. MATHCrossRefMathSciNetGoogle Scholar
  37. 222.
    Sun Z. Converse Lyapunov theorem for switched stability of switched linear systems. In: Proc Chinese contr conf; 2007. p. 678–80. Google Scholar
  38. 225.
    Sun Z. Stabilizing switching design for switched linear systems: a state-feedback path-wise switching approach. Automatica. 2009;45(7):1708–14. MATHCrossRefGoogle Scholar
  39. 231.
    Sun Z. Switching distance and robust switching for switched linear systems. Automatica. 2010; submitted. Google Scholar
  40. 235.
    Sun Z, Ge SS. On stability of switched linear systems with perturbed switching paths. J Control Theory Appl. 2006;4(1):18–25. MATHCrossRefMathSciNetGoogle Scholar
  41. 250.
    Tokarzewski J. Stability of periodically switched linear systems and the switching frequency. Int J Syst Sci. 1987;18(4):697–726. MATHCrossRefMathSciNetGoogle Scholar
  42. 252.
    Veres SM. The geometric bounding toolbox, user’s manual & reference. UK: SysBrain; 2001. Google Scholar
  43. 257.
    Wang W, Nesic D. Input-to-state stability and averaging of linear fast switching systems. IEEE Trans Autom Control. 2010;55(5):1274–9. MathSciNetGoogle Scholar
  44. 258.
    Wicks MA, Peleties V, DeCarlo RA. Construction of piecewise Lyapunov functions for stabilizing switched systems. In: Proc IEEE CDC; 1994. p. 3492–7. Google Scholar
  45. 259.
    Wicks MA, Peleties P, DeCarlo RA. Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. Eur J Control. 1998;4(2):140–7. MATHGoogle Scholar
  46. 263.
    Wu AG, Feng G, Duan GR, Gao HJ. A stabilizing slow-switching law for switched discrete-time linear systems. In: Proc IEEE MSC; 2010. p. 2099–104. Google Scholar
  47. 264.
    Wu J, Sun Z. New slow-switching laws for switched linear systems. In: Proc IEEE ICCA; 2010. p. 2274–7. CrossRefGoogle Scholar
  48. 267.
    Xie G, Wang L. Periodical stabilization of switched linear systems. J Comput Appl Math. 2005;18(1):176–87. CrossRefMathSciNetGoogle Scholar
  49. 276.
    Zhai G, Lin H, Antsaklis PJ. Quadratic stabilizability of switched linear systems with polytopic uncertainties. Int J Control. 2003;76(7):747–53. MATHCrossRefMathSciNetGoogle Scholar
  50. 277.
    Zhai G, Yasuda K. Stability analysis for a class of switched systems. Trans Soc Instrum Control Eng. 2000;36(5):409–15. Google Scholar
  51. 279.
    Zhang W, Abate A, Hu JH, Vitus MP. Exponential stabilization of discrete-time switched linear systems. Automatica. 2009;45(11):2526–36. MATHCrossRefGoogle Scholar
  52. 280.
    Zhang W, Hu JH. On optimal quadratic regulation for discrete-time switched linear systems. In: Egerstedt M, Mishra B, editors. Hybrid systems: computation and control. Berlin: Springer; 2008. p. 584–97. CrossRefGoogle Scholar
  53. 281.
    Zhao J, Hill DJ. Dissipativity theory for switched systems. IEEE Trans Autom Control. 2008;53(4):941–53. CrossRefMathSciNetGoogle Scholar
  54. 282.
    Zhao J, Hill DJ. Passivity and stability of switched systems: a multiple storage function method. Syst Control Lett. 2008;57(2):158–64. MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.College Automation Science & Engineering, Center for Control and OptimizationSouth China University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of Electrical and Computer EngineeringThe National University of SingaporeSingaporeSingapore
  3. 3.Robotics Institue and Institute of Intelligent Systems and Information TechnologyUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

Personalised recommendations