Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. (English) Zbl 0904.93036

Hybrid systems are systems of the form \[ \dot x=f_i(x), \quad x_{k+1} =f_i (x_k) \] with \(i\in Q\), \(Q\) being a finite set of indices. Even if each system defined by \(f_i(x)\) has the equilibrium at the origin stable, the switching might be such that for the “polysystem” the origin would not be stable. The stability is studied by using a set of switched Lyapunov functions (called multiple Lyapunov functions). Other properties such as the existence of a limit cycle and Bendixson-type theorems are discussed.


93D30 Lyapunov and storage functions
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Full Text: DOI Link


[1] Branicky M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Contr, 1998, 43: 475-482 · Zbl 0904.93036
[2] Liberzon D. Switching in Systems and Control. Boston: Birkhauser, 2003 · Zbl 1036.93001
[3] Sun Z D, Ge S S. Switched Linear Systems-Control and Design. New York: Springer-Verlag, 2004
[4] Zhao J, Spong M W. Hybrid control for global stabilization of the cart-pendulum system. Automatica, 2001, 37: 1941-1951 · Zbl 1005.93041
[5] Hespanha J P, Liberzon D, Angeli D, et al. Nonlinear norm-observability notions and stability of switched systems. IEEE Trans Autom Contr, 2005, 50: 154-168 · Zbl 1365.93349
[6] Zhai G S, Hu B, Yasuda K, et al. Disturbance attenuation properties of time-controlled switched systems. J Franklin Institute, 2001, 338: 765-779 · Zbl 1022.93017
[7] Lin H, Zhai G S, Antsaklis P J. Robust stability and disturbance attenuation analysis of a class of networked control systems. In: IEEE Conference on Decision and Control, Maui, Hawaii, 2003. 2: 1182-1187
[8] Gu K, Kharitonov V, Chen J. Stability of Time-Delay Systems. Boston, MA: Birkhauser, 2003 · Zbl 1039.34067
[9] Richard J. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 39: 1667-1694 · Zbl 1145.93302
[10] Fridman E, Shaked U. Delay-dependent stability and \(H_∞\) control: constant and time-varying delays. Int J Contr, 2003, 76: 48-60 · Zbl 1023.93032
[11] Han Q L, Gu K Q. On robust stability of time-delay systems with norm-bounded uncertainty. IEEE Trans Automat Contr, 2001, 46: 1426-1431 · Zbl 1006.93054
[12] Gao H J, Wang C H. Delay-dependent robust and filtering for a class of uncertain nonlinear time-delay systems. IEEE Trans Autom Contr, 2003, 48: 1661-1665 · Zbl 1364.93210
[13] He Y, Wu M, She J H, et al, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Sys Contr Lett, 2004, 51: 57-65 · Zbl 1157.93467
[14] He Y, Wu M, She J H, et al. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Trans Automat Contr, 2004, 49: 828-832 · Zbl 1365.93368
[15] Xie G M, Wang L. Quadratic stability and stabilization of discrete-time switched systems with state delay. In: Proc 43rd IEEE Conf Decision and Control, Atlantis, Paradise Island, Bahamas, 2004. 3235-3240
[16] Sun X M, Zhao J, Hill D J. Stability and \(L_2\)-gain analysis for switched delay systems: a delay-dependent method. Automatica, 2006, 42: 1769-1774 · Zbl 1114.93086
[17] Sun X M, Liu G P, Rees D, et al. Stability of systems with controller failure and time-varying delay. IEEE Trans Automat Contr, 2008, 53: 2391-2396 · Zbl 1367.93573
[18] Sun Y G,
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.