×

zbMATH — the first resource for mathematics

Stability of switched positive linear systems with average dwell time switching. (English) Zbl 1244.93129
Summary: In this paper, the stability analysis problem for a class of Switched Positive Linear Systems (SPLSs) with average dwell time switching is investigated. A Multiple Linear Copositive Lyapunov Function (MLCLF) is first introduced, by which the sufficient stability criteria in terms of a set of linear matrix inequalities, are given for the underlying systems in both continuous-time and discrete-time contexts. The stability results for the SPLSs under arbitrary switching, which have been previously studied in the literature, can be easily obtained by reducing MLCLF to the common linear copositive Lyapunov function used for the system under arbitrary switching those systems. Finally, a numerical example is given to show the effectiveness and advantages of the proposed techniques.

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
15B48 Positive matrices and their generalizations; cones of matrices
93C05 Linear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benvenuti, L.; Farina, L., A tutorial on the positive realization problem, IEEE transactions on automatic control, 49, 5, 651-664, (2004) · Zbl 1365.93001
[2] Cui, C.; Long, F.; Li, C., Disturbance attenuation for switched system with continuous-time and discrete-time subsystems: state feedback case, ICIC express letters, 4, 1, 205-212, (2010)
[3] De Leenheer, P.; Aeyels, D., Stabilization of positive linear systems, Systems & control letters, 44, 4, 259-271, (2001) · Zbl 0986.93059
[4] Ding, X.; Shu, L.; Liu, X., On linear copositive Lyapunov functions for switched positive systems, Journal of the franklin institute, 348, 2099-2107, (2011) · Zbl 1231.93058
[5] Fainshil, L.; Margaliot, M.; Chigansky, P., On the stability of positive linear switched systems under arbitrary switching laws, IEEE transactions on automatic control, 54, 4, 897-899, (2009) · Zbl 1367.93431
[6] Farina, L.; Rinaldi, S., Positive linear systems, (2000), Wiley Interscience Series New York
[7] Feng, J.; Lam, J.; Li, P.; Shu, Z., Decay rate constrained stabilization of positive systems using static output feedback, International journal of robust and nonlinear control, 21, 44-54, (2011) · Zbl 1207.93080
[8] Fornasini, E.; Valcher, M.E., Linear copositive Lyapunov functions for continuous-time positive switched systems, IEEE transactions on automatic control, 55, 8, 1933-1937, (2010) · Zbl 1368.93593
[9] Gurvits, L.; Shorten, R.; Mason, O., On the stability of switched positive linear systems, IEEE transactions on automatic control, 52, 6, 1099-1103, (2007) · Zbl 1366.93436
[10] Jadbabaie, A.; Lin, J.; Morse, A.S., Co-ordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE transactions on automatic control, 48, 6, 988-1001, (2003) · Zbl 1364.93514
[11] Kaczorek, T., Positive 1D and 2D systems, (2002), Springer Verlag London · Zbl 1005.68175
[12] Knorn, F.; Mason, O.; Shorten, R.N., On linear co-positive Lyapunov functions for sets of linear positive systems, Automatica, 45, 8, 1943-1947, (2009) · Zbl 1185.93122
[13] Liberzon, D., Switching in systems and control, (2003), Birkhauser Berlin · Zbl 1036.93001
[14] Li, P.; Lam, J.; Wang, Z.; Date, P., Positivity-preserving \(H_\infty\) model reduction for positive systems, Automatica, 47, 7, 1504-1511, (2011) · Zbl 1220.93036
[15] Lin, H.; Antsaklis, P.J., Stability and stabilizability of switched linear systems: a survey of recent results, IEEE transactions on automatic control, 54, 2, 308-322, (2009) · Zbl 1367.93440
[16] Liu, X.; Dang, C., Stability analysis of positive switched linear systems with delays, IEEE transactions on automatic control, 56, 7, 1684-1690, (2011) · Zbl 1368.93599
[17] Margaliot, M.; Branicky, M.S., Nice reachability for planar bilinear control systems with applications to planar linear switched systems, IEEE transactions on automatic control, 54, 4, 900-905, (2009) · Zbl 1367.93061
[18] Mason, O., & Shorten, R. (2003). A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems. In Proc. ameri. contr. conf. (pp. 4469-4470).
[19] Mason, O., & Shorten, R. (2004). Some results on the stability of positive switched linear systems. In Proc. of the 43rd IEEE conference on decision and control (pp. 4601-4606). Atlantis, Bahamas.
[20] Mason, O.; Shorten, R., On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE transactions on automatic control, 52, 7, 1346-1349, (2007) · Zbl 1366.34077
[21] Rami, M.A., Solvability of static output-feedback stabilization for LTI positive systems, Systems & control letters, 60, 704-708, (2011) · Zbl 1226.93116
[22] Shi, P.; Xia, Y.; Liu, G.; Rees, D., On designing of sliding mode control for stochastic jump systems, IEEE transactions on automatic control, 51, 1, 97-103, (2006) · Zbl 1366.93682
[23] Sun, X.; Wang, W.; Liu, G.; Zhao, J., Stability analysis for linear switched systems with time-varying delay, IEEE transactions on circuits and systems II: analog and digital signal processing, 38, 2, 528-533, (2008)
[24] Sun, X.; Zhao, J.; Hill, D., Stability and \(L_2\)-gain analysis for switched delay systems: a delay-dependent method, Automatica, 42, 10, 1769-1774, (2006) · Zbl 1114.93086
[25] Wang, D.; Wang, W.; Shi, P., Delay-dependent exponential stability for switched delay systems, Optimal control applications and methods, 30, 4, 383-397, (2009)
[26] Wang, D.; Wang, W.; Shi, P., Exponential \(H_\infty\) filtering for switched linear systems with interval time-varying delay, International journal of robust and nonlinear control, 19, 5, 532-551, (2009) · Zbl 1160.93328
[27] Xu, S.; Chen, T., Robust \(H_\infty\) control for uncertain discrete-time stochastic bilinear systems with Markovian switching, International journal of robust and nonlinear control, 15, 5, 201-217, (2005) · Zbl 1078.93025
[28] Xue, X.; Li, Z., Asymptotic stability analysis of a kind of switched positive linear discrete systems, IEEE transactions on automatic control, 55, 9, 2198-2203, (2010) · Zbl 1368.93615
[29] Zhai, G.; Matsune, I.; Imae, J.; Kobayashi, T., A note on multiple Lyapunov functions and stability condition for switched and hybrid systems, Int. J. innovative computing, information and control, 5, 5, 1189-1200, (2009)
[30] Zhang, L.; Gao, H., Asynchronously switched control of switched linear systems with average Dwell time, Automatica, 46, 5, 953-958, (2010) · Zbl 1191.93068
[31] Zhang, L.; Jiang, B., Stability of a class of switched linear systems with uncertainties and average Dwell time switching, Int. J. innovative computing, information and control, 6, 2, 667-676, (2010)
[32] Zhang, L.; Shi, P., Stability, \(l_2\)-gain and asynchronous \(H_\infty\) control of discrete-time switched systems with average Dwell time, IEEE transactions on automatic control, 54, 9, 2193-2200, (2009)
[33] Zhao, J.; Hill, D., On stability, \(L_2\)-gain and \(H_\infty\) control for switched systems, Automatica, 44, 5, 1220-1232, (2008) · Zbl 1283.93147
[34] Zhao, X.; Zeng, Q., Delay-dependent \(H_\infty\) performance analysis and filtering for Markovian jump systems with interval time-varying-delays, International journal of adaptive control and signal processing, 24, 8, 633-642, (2010) · Zbl 1204.93123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.