Performance analysis of switched linear systems under arbitrary switching via generalized coordinate transformations


For a continuous-time switched linear system, the spectral abscissa is defined as the worst-case divergence rate under arbitrary switching, which is critical for characterizing the asymptotic performance of the switched system. In this study, based on the generalized coordinate transformations approach, we develop a computational scheme that iteratively produces sequences of minimums of matrix set μ1 measures, where the limits of the sequences are upper bound estimates of the spectral abscissa. A simulation example is presented to illustrate the effectiveness of the proposed scheme.

This is a preview of subscription content, access via your institution.


  1. 1

    Williams S M, Hoft R G. Adaptive frequency domain control of PWM switched power line conditioner. IEEE Trans Power Electron, 1991, 6: 665–670

    Google Scholar 

  2. 2

    Zhao J, Spong M W. Hybrid control for global stabilization of the cart-pendulum system. Automatica, 2001, 37: 1941–1951

    MathSciNet  MATH  Google Scholar 

  3. 3

    Zhang W, Hu J H. Dynamic buffer management using optimal control of hybrid systems. Automatica, 2008, 44: 1831–1840

    MathSciNet  MATH  Google Scholar 

  4. 4

    Branicky M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control, 1998, 43: 475–482

    MathSciNet  MATH  Google Scholar 

  5. 5

    Hespanha J P, Morse A S. Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, 2002. 2655–2660

    Google Scholar 

  6. 6

    Liberzon D, Morse A S. Basic problems in stability and design of switched systems. IEEE Control Syst Mag, 1999, 19: 59–70

    MATH  Google Scholar 

  7. 7

    Liberzon D, Hespanha J P, Morse A S. Stability of switched systems: a Lie-algebraic condition. Syst Control Lett, 1999, 37: 117–122

    MathSciNet  MATH  Google Scholar 

  8. 8

    Wang Y, Wang W, Liu G P. Stability of linear discrete switched systems with delays based on average dwell time method. Sci China Inf Sci, 2010, 53: 1216–1223

    MathSciNet  Google Scholar 

  9. 9

    Shorten R, Wirth F, Mason O, et al. Stability criteria for switched and hybrid systems. SIAM Rev, 2007, 49: 545–592

    MathSciNet  MATH  Google Scholar 

  10. 10

    Lin H, Antsaklis P J. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Autom Control, 2009, 54: 308–322

    MathSciNet  MATH  Google Scholar 

  11. 11

    Sun Z D, Ge S S. Stability Theory of Switched Dynamical Systems. Berlin: Springer, 2011

    Google Scholar 

  12. 12

    Zhu Y Z, Zhang L X, Lin W Y, et al. Benefits of redundant channels in observer-based H∞ control for discrete-time switched linear systems. Sci China Tech Sci, 2016, 59: 55–62

    Google Scholar 

  13. 13

    Blanchini F. Nonquadratic Lyapunov functions for robust control. Automatica, 1995, 31: 451–461

    MathSciNet  MATH  Google Scholar 

  14. 14

    Blanchini F, Miani S. A new class of universal Lyapunov functions for the control of uncertain linear systems. IEEE Trans Autom Control, 1996, 44: 641–647

    MathSciNet  MATH  Google Scholar 

  15. 15

    Sun Z D. Recent advances on analysis and design of switched linear systems. Control Theory Technol, 2017, 15: 242–244

    MATH  Google Scholar 

  16. 16

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

    MathSciNet  MATH  Google Scholar 

  17. 17

    Mason P, Boscain U, Chitour Y. Common polynomial Lyapunov functions for linear switched systems. SIAM J Control Optim, 2006, 45: 226–245

    MathSciNet  MATH  Google Scholar 

  18. 18

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

    MathSciNet  MATH  Google Scholar 

  19. 19

    Molchanov A P, Pyatnitskiy Y S. Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst Control Lett, 1989, 13: 59–64

    MathSciNet  Google Scholar 

  20. 20

    Mancilla-Aguilar J L, Garc´ia R A. A converse Lyapunov theorem for nonlinear switched systems. Syst Control Lett, 2000, 41: 67–71

    MathSciNet  MATH  Google Scholar 

  21. 21

    Peleties P, Decarlo R. Asymptotic stability of m-switched systems using Lyapunov-like functions. In: Proceedings of the American Control Conference, Boston, 1991. 1679–1684

    Google Scholar 

  22. 22

    Pettersson S, Lennartson B. Stability and robustness for hybrid systems. In: Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, 1996. 1202–1207

    Google Scholar 

  23. 23

    Elsner L. The generalized spectral-radius theorem: an analytic-geometric proof. Linear Algebra Appl, 1995, 220: 151–159

    MathSciNet  MATH  Google Scholar 

  24. 24

    Blondel V D, Nesterov Y. Computationally efficient approximations of the joint spectral radius. SIAM J Matrix Anal Appl, 2005, 27: 256–272

    MathSciNet  MATH  Google Scholar 

  25. 25

    Sun Z D. A note on marginal stability of switched systems. IEEE Trans Autom Control, 2008, 53: 625–631

    MathSciNet  MATH  Google Scholar 

  26. 26

    Chitour Y, Mason P, Sigalotti M. On the marginal instability of linear switched systems. Syst Control Lett, 2011, 61: 7322–7327

    MathSciNet  MATH  Google Scholar 

  27. 27

    Barabanov N E. Ways to compute the Lyapunov index for differential nesting. Avtomatika I Telemekhanika, 1989, 50: 475–479

    Google Scholar 

  28. 28

    Sun Z D. Matrix measure approach for stability of switched linear systems. In: Proceedings of the 7th IFAC Symposium Nonlinear Control System, Pretoria, 2007. 557–560

    Google Scholar 

  29. 29

    Xiong J D, Sun Z D. Approximation of extreme measure for switched linear systems. In: Proceedings of the 9th IEEE International Conference on Control and Automation, Santiago, 2011. 722–725

    Google Scholar 

  30. 30

    Lin M L, Sun Z D. Approximating the spectral abscissa for switched linear systems via coordinate transformations. J Syst Sci Complex, 2016, 29: 350–366

    MathSciNet  MATH  Google Scholar 

  31. 31

    Lin M L, Sun Z D. Approximation of the spectral abscissa for switched linear systems via generalized coordinate transformations. In: Proceedings of the 11th World Congress on Intelligent Control and Automation, Shenyang, 2014. 2208–2212

    Google Scholar 

  32. 32

    Protasov V Y, Jungers R M. Analysing the stability of linear systems via exponential Chebyshev polynomials. IEEE Trans Autom Control, 2016, 61: 795–798

    MathSciNet  MATH  Google Scholar 

  33. 33

    Gurvits L. Stability of discrete linear inclusion. Linear Algebra Appl, 1995, 231: 47–85

    MathSciNet  MATH  Google Scholar 

  34. 34

    Shih M H, Wu J W, Pang C T. Asymptotic stability and generalized Gelfand spectral radius formula. Linear Algebra Appl, 1997, 252: 61–70

    MathSciNet  MATH  Google Scholar 

  35. 35

    Parrilo P A, Jadbabaie A. Approximation of the joint spectral radius using sum of squares. Linear Algebra Appl, 2008, 428: 2385–2402

    MathSciNet  MATH  Google Scholar 

  36. 36

    Sun Z D, Shorten R N. On convergence rates of simultaneously triangularizable switched linear systems. IEEE Trans Autom Control, 2005, 50: 1224–1228

    MathSciNet  MATH  Google Scholar 

  37. 37

    Vidyasagar M. Nonlinear Systems Analysis. Englewood Cliffs: Prentice-Hall, 1993

    Google Scholar 

  38. 38

    Blanchini F. The gain scheduling and the robust state feedback stabilization problems. IEEE Trans Autom Control, 2000, 45: 2061–2070

    MathSciNet  MATH  Google Scholar 

  39. 39

    Macduffee C. The Theory of Matrices. New York: Chelsea, 1946

    Google Scholar 

  40. 40

    Hartwig R E. The resultant and the matrix equation AX = XB. SIAM J Appl Math, 1972, 22: 538–544

    MathSciNet  MATH  Google Scholar 

  41. 41

    Zahreddine Z. Matrix measure and application to stability of matrices and interval dynamical systems. Int J Math Math Sci, 2003, 2003: 75–85

    MathSciNet  MATH  Google Scholar 

Download references


This study was supported by National Key Basic Research Program (973 Program) (Grant No. 2014CB845302), National Natural Science Foundation of China (Grant Nos. 91546203, 61273121), and Young and Middle-aged Foundation of Fujian Education Research (Grant No. JAT160294).

Author information



Corresponding author

Correspondence to Zhendong Sun.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, M., Sun, Z. Performance analysis of switched linear systems under arbitrary switching via generalized coordinate transformations. Sci. China Inf. Sci. 62, 12203 (2019).

Download citation


  • generalized coordinate transformation
  • matrix set measure
  • spectral abscissa
  • switched linear system