Stabilisation of discrete-time switched positive linear systems via time- and state-dependent switching laws
Stabilisation of discrete-time switched positive linear systems via time- and state-dependent switching laws
- Author(s): Y. Tong ; C. Wang ; L. Zhang
- DOI: 10.1049/iet-cta.2011.0293
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- Author(s): Y. Tong 1 ; C. Wang 1 ; L. Zhang 1
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View affiliations
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Affiliations:
1: Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, People's Republic of China
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Affiliations:
1: Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, People's Republic of China
- Source:
Volume 6, Issue 11,
19 July 2012,
p.
1603 – 1609
DOI: 10.1049/iet-cta.2011.0293 , Print ISSN 1751-8644, Online ISSN 1751-8652
This study investigates the stabilisation problem of discrete-time switched positive linear systems by means of piecewise linear copositive Lyapunov functions. Two stabilisation strategies are designed under time- and state-dependent switching cases, respectively. The former case aims at determining an upper bound of the minimum dwell time to guarantee that the underlying system is stable for any switching signal with dwell time greater than this bound. The latter case is focused on deriving a state-dependent switching law stabilising the underlying system from the solution of a family of so-called linear copositive Lyapunov–Metzler inequalities. In each case, a sufficient stabilisation condition is given first, then based on which an associated guaranteed cost is further proposed. A practical system derived from the distributed power control in communication networks is given to illustrate the effectiveness and applicability of the theoretical results.
Inspec keywords: time-varying systems; piecewise linear techniques; linear systems; distributed control; Lyapunov methods; stability; power control; discrete time systems
Other keywords:
Subjects: Control system analysis and synthesis methods; Discrete control systems; Power and energy control; Stability in control theory; Time-varying control systems
References
-
-
1)
- E.H. Vargas , P. Colaneri , R. Middleton , F. Blanchini . Discrete-time control for switched positive systems with application to mitigating viral escape. Int. J. Robust Nonlinear Control , 10 , 1093 - 1111
-
2)
- J.C. Geromel , P. Colaner . Stability and stabilization of continuous-time switched linear systems. SIAM J. Control Optim. , 5 , 1915 - 1930
-
3)
- O. Mason , R. Shorten . On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Trans. Autom. Control , 7 , 1346 - 1349
-
4)
- S. Bundfuss , M. Dür . Copositive Lyapunov functions for switched systems over cones. Syst. Control Lett. , 5 , 342 - 345
-
5)
- M.A. Rami . Solvability of static output-feedback stabilization for LTI positive systems. Syst. Control Lett. , 704 - 708
-
6)
- O. Mason , V.S. Bokharaie , R. Shorten . (2009) Stability and D-stability for switched positive systems, Positive systems.
-
7)
- T. Kaczorek . (2002) Positive 1D and 2D systems.
-
8)
- X. Liu . Stability analysis of switched positive systems: a switched linear copositive Lyapunov function method. IEEE Trans. Circuits Syst. II, Express Briefs , 5 , 414 - 418
-
9)
- L. Fainshil , M. Margaliot , P. Chigansky . On the stability of positive linear switched systems under arbitrary switching laws. IEEE Trans. Autom. Control , 4 , 897 - 899
-
10)
- A. Jadbabaie , J. Lin , A.S. Morse . Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control , 6 , 988 - 1001
-
11)
- M.A. Rami , F. Tadeo . Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II , 2 , 151 - 155
-
12)
- D. Liberzon , A.S. Morse . Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. , 5 , 59 - 70
-
13)
- F. Knorn , O. Mason , R.N. Shorten . On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica , 8 , 1943 - 1947
-
14)
- G. James , V. Rumchev . Stability of positive linear discrete-time systems. Bull. Pol. Acad. Sci. Tech. Sci. , 1 , 1 - 8
-
15)
- X. Liu , L. Wang , W. Yu , S. Zhong . Constrained control of positive discrete-time systems with delays. IEEE Trans. Circuits Syst. II , 2 , 193 - 197
-
16)
- G.J. Foschini , Z. Miljanic . A simple distributed autonomous power control algorithm and its convergence. IEEE Trans. Veh. Technol. , 4 , 641 - 646
-
17)
- X. Liu , W. Yu , L. Wang . Necessary and sufficient asymptotic stability criterion for 2-D positive systems with time-varying state delays described by Roesser model. IET Control Theory Appl. , 5 , 663 - 668
-
18)
- J. Daafouz , J. Bernussou . Parameter dependent Lyapunov functions for discrete-time systems with time varying parameter uncertainties. Syst. Control Lett. , 5 , 355 - 359
-
19)
- J.C. Geromel , P. Colaner . Stability and stabilization of discrete-time switched systems. Int. J. Control , 7 , 719 - 728
-
20)
- Zappavigna, A., Colaneri, P., Geromel, J., Middleton, R.: `Stabilization of continuous-time switched linear positive systems', Proc. America Control Conference, 2010, Baltimore, America, p. 3275–3280.
-
21)
- H. Lin , P.J. Anstaklis . Stability and stabilisability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control , 308 - 322
-
22)
- R.A. Decarlo , M.S. Branicky , S. Pettersson , B. Lennartson . Perspective and results on the stability and stabilizability of hybrid systems. Proc. IEEE , 1069 - 1081
-
23)
- E. Fornasini , M.E. Valcher . Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Trans. Autom. Control , 8 , 1933 - 1937
-
24)
- Zappavigna, A., Colaneri, P., Geromel, J., Shorten, R.: `Dwell time analysis for continuous-time switched linear positive systems', Proc. the America Control Conference, 2010, Baltimore, America, p. 6256–6261.
-
25)
- R. Shorten , F. Wirth , O. Mason , K. Wulff , C. King . Stability criteria for switched and hybrid systems. Siam Rev. , 4 , 545 - 592
-
26)
- D.U. Campos-Delgado , J.M. Luna-Rivera , F.J. Martnez-Lopez . Distributed power control algorithms in the uplink of wireless code-division multiple-access systems. IET Proc. Control Theory , 5 , 795 - 805
-
27)
- J.P. Hespanha , A.S. Morse . Switching between stabilizing Controllers. Automatica , 11 , 1905 - 1917
-
28)
- D. Liberzon . (2003) Switching in systems and control.
-
29)
- J. Daafouz , P. Riedinger , C. Iun . Stability analysis and control synthesis for switched systems: a switched lyapunov function approach. IEEE Trans. Autom. Control , 11 , 1883 - 1887
-
30)
- X. Liu , W. Yu , L. Wang . Stability analysis of positive systems with bounded time-varying delays. IEEE Trans. Autom. Control , 7 , 600 - 604
-
31)
- R. Shorten , F. Wirth , D. Leith . A positive systems model of TCP-like congestion control: asymptotic results. IEEE Trans. Autom. Control , 3 , 616 - 629
-
32)
- M. Twardy . On the alternative stability criterion for positive systems. Bull. Pol. Acad. Sci. Tech. Sci. , 4 , 379 - 383
-
33)
- X. Liu , C. Dang . Stability analysis of positive switched linear systems with delays. IEEE Trans. Autom. Control , 7 , 1684 - 1690
-
34)
- L. Gurvits , R. Shorten , O. Mason . On the stability of switched positive linear systems. IEEE Trans. Autom. Control , 6 , 1099 - 1103
-
35)
- Paul, A., Akar, M., Mitra, U., Safonov, M.G.: `A switched system model for stability analysis of distributed power control algorithms for cellular communications', Proc. 2004 American Control Conference, 2004, Boston, Massachusetts.
-
36)
- M.S. Branicky . Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control , 4 , 475 - 482
-
37)
- A. Berman , R.J. Plemmons . (1994) Nonnegative matrices in the mathematical sciences.
-
38)
- L. Farina , S. Rinaldi . (2000) Positive linear systems: theory and applications.
-
39)
- W. Mitkowski . Dynamical properties of Metzler systems. Bull. Pol. Acad. Sci. Tech. Sci. , 4 , 309 - 312
-
1)