Frequency-Domain Stability Conditions for Discrete-Time Switched Systems

We consider discrete-time switched systems with switching of linear time-invariant right-hand parts. The notion of a connected discrete switched system is introduced. For systems with the connectedness property, we propose necessary and sufficient frequency-domain conditions for the existence of a common quadratic Lyapunov function that provides the stability for a system under arbitrary switching. The set of connected switched systems contains discrete control systems with several time-varying nonlinearities from the finite sectors, considered in the theory of absolute stability. We consider the case of switching between three linear subsystems in more details and give an illustrative example.

Keywords

Discrete-time switched system, stability, Lyapunov functions, matrix inequalities

Publication date

30.09.2018

Number of characters

672

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GOST	Kamenetskiy V. Frequency-Domain Stability Conditions for Discrete-Time Switched Systems // Avtomatika i Telemekhanika – 2018. – Issue 8 C. 3-26 [Electronic resource]. URL: http://ras.jes.su/ait/s207987840000490-8-1-en (circulation date: 26.04.2024). DOI: 10.31857/S000523100001243-1
MLA	Kamenetskiy, Vladimir "Frequency-Domain Stability Conditions for Discrete-Time Switched Systems." Avtomatika i Telemekhanika 8 (2018):3-26. DOI: 10.31857/S000523100001243-1
APA	Kamenetskiy V. (2018). Frequency-Domain Stability Conditions for Discrete-Time Switched Systems. Avtomatika i Telemekhanika (8), pp.3-26 DOI: 10.31857/S000523100001243-1

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1. Shorten R., Wirth F., Mason O., et al. Stability Sriteria for Switched and Hybrid Systems // SIAM Rev. 2007. No. 4. P. 545–592.

2. Aleksandrov A.Yu., Platonov A.V. On Stability of Solutions for a Class of Nonlinear Difference Systems with Switching // Autom. Remote Control. 2016. V. 77. No. 5. P. 779–788.

3. Aleksandrov A.Yu., Martynyuk A.A., Platonov A.V. Dwell Time Stability Analysis for Nonlinear Switched Difference Systems // Nonlinear Dynam. Syst. Theory. 2016. V. 16. No. 3. P. 221–234.

4. Fiacchini M., Girard A., Jungers M. On the Stabilizability of Discrete-Time Switched Linear Systems: Novel Conditions and Comparisons // IEEE Trans. Autom. Control. 2016. V. 61. No. 5. P. 1181–1193.

5. Molchanov A.P. Lyapunov Functions for Nonlinear Discrete-Time Control Systems // Autom. Remote Control. 1987. V. 48. No. 6. P. 728–736.

6. Barabanov N.E. On the Lyapunov Exponent of Discrete Inclusions. I // Autom. Remote Control. 1988. V. 49. No. 2. P. 152–157.

7. Tsypkin Ya.3. Ob ustojchivosti v tselom nelinejnykh impul'snykh avtomaticheskikh sistem // Dokl. AN SSSR. 1962. T. 145. № 1. S. 52–65.

8. Yakubovich V.A. Absolute Stability of Pulsed Systems with Several Nonlinear or Linear but Nonstationary Blocks. I, II // Autom. Remote Control. 1967. No. 9. P. 1301–1313; 1968. No. 2. P. 244–263.

9. Shepelyavyj A.I. O kachestvennom issledovanii ustojchivosti v tselom i neustojchivosti dlya odnogo klassa amplitudno-impul'snykh sistem // Dokl. AN SSSR. 1970. T. 190. № 5. S. 49–56.

10. Shepel’yavi A.I. Absolute Instability of Nonlinear Pulse-Amplitude Control Systems. Frequency Criteria // Autom. Remote Control. 1972. V. 33. No. 6. P. 929–935.

11. Kamenetskij V.A. Absolyutnaya ustojchivost' diskretnykh sistem upravleniya s nestatsionarnymi nelinejnostyami // AiT. 1985. № 8. S. 172–176.

12. Barkin A.I. On Absolute Stability of Discrete Systems // Autom. Remote Control. 2008. V. 69. No. 10. P. 1647–1652.

13. Kamenetskiy V.A. Frequency-Domain Stability Conditions for Hybrid Systems // Autom. Remote Control. 2017. V. 78. No. 12. P. 2101–2119.

14. Mason O., Shorten R.S. On Common Quadratic Lyapunov Functions for Pairs of Stable Discrete-Time LTI Systems// IMA J. Appl. Math. 2004. V. 69. P. 271–283.

15. Barbashin E.A. Funktsii Lyapunova. M.: Fizmatlit, 1970.

16. Gelig A.X., Leonov G.A., Yakubovich V.A. Ustojchivost' nelinejnykh sistem s needinstvennym sostoyaniem ravnovesiya. M.: Nauka, 1978.

17. Kamenetskii V.A. Absolute Stability and Absolute Instability of Control Systems with Several Nonlinear Nonstationary Elements // Autom. Remote Control. 1983. V. 44. No. 12. P. 1543–1552.

18. Evtushenko Yu.G. Metody resheniya ehkstremal'nykh zadach i ikh primenenie v sistemakh optimizatsii. M.: Nauka, 1982.

19. Shorten R.S., Narendra K.S. On Common Quadratic Lyapunov Functions for Pairs of Stable LTI Systems whose System Matrices are in Companion Form // IEEE Trans. Autom. Control. 2003. V. 48. No. 4. P. 618–621.

20. King C., Nathanson M. On the Existence of a Common Quadratic Lyapunov Function for a Rank One Difference // Linear Algebra Appl. 2006. V. 419. No. 2–3. P. 400–416.

21. Chajkovskij M.M., Kurdyukov A.P. Algebraicheskie uravneniya Rikkati i linejnye matrichnye neravenstva dlya sistem diskretnogo vremeni. M.: IPU RAN, 2005.

22. Balandin D.V., Kogan M.M. Sintez zakonov upravleniya na osnove linejnykh matrichnykh neravenstv. M.: Fizmatlit, 2007.

23. Teoriya avtomaticheskogo upravleniya. Ch. I. Teoriya nelinejnykh i spetsial'nykhsistem avtomaticheskogo upravleniya / Pod red. A.A. Voronova. M.: Vyssh. shk., 1986.

24. Popov V.M. Giperustojchivost' avtomaticheskikh sistem. M.: Nauka, 1970.