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Some stability tests for switched descriptor systems. (English) Zbl 1429.93309

Summary: In this paper we first derive some general Lyapunov based stability conditions for switched linear descriptor systems and illustrate them with a simple mechanical system. Next we develop simple conditions for specific classes of systems. In particular, we consider the important case when switching between a standard system and an index one descriptor system, and systems where switching occurs between an index one and an index two descriptor system. Examples are given to illustrate these results.

MSC:

93D23 Exponential stability
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C05 Linear systems in control theory
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[1] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory (1994), SIAM · Zbl 0816.93004
[2] Dai, L., Singular control systems, (Lecture notes in control and information sciences (1989), Springer Verlag: Springer Verlag Berlin)
[3] Liberzon, D.; Hespanha, J. P.; Morse, A. S., Stability of switched systems: a lie-algebraic condition, Systems & Control Letters, 37, 3, 117-122 (1999) · Zbl 0948.93048
[4] Liberzon, D.; Trenn, S., On stability of linear switched differential algebraic equations, (Decision and control, 2009 held jointly with the 2009 28th chinese control conference. CDC/CCC 2009. proceedings of the 48th IEEE conference on (2009), IEEE), 2156-2161
[5] Liberzon, D.; Trenn, S.; Wirth, F., Commutativity and asymptotic stability for linear switched daes, (Decision and control and european control conference (CDC-ECC), 2011 50th IEEE conference on (2011), IEEE), 417-422
[6] Mironchenko, A.; Wirth, F.; Wulff, K., Stabilization of switched linear differential algebraic equations and periodic switching, IEEE Transactions on Automatic Control, 60, 8, 2102-2113 (2015) · Zbl 1360.93609
[7] Mori, Y.; Mori, T.; Kuroe, Y., A solution to the common lyapunov function problem for continuous-time systems, (Decision and control, 1997, Proceedings of the 36th IEEE conference on, Vol. 4 (1997), IEEE), 3530-3531
[8] Narendra, K. S.; Balakrishnan, J., A common lyapunov function for stable lti systems with commuting a-matrices, IEEE Transactions on Automatic Control, 39, 12, 2469-2471 (1994) · Zbl 0825.93668
[9] Owens, D. H.; Debeljkovic, D. L., Consistency and liapunov stability of linear descriptor systems: A geometric analysis, IMA Journal of Mathematical Control and Information, 2, 2, 139-151 (1985) · Zbl 0637.93051
[10] Sajja, S. (2016). Matlab code for example 2. http://smarttransport.ucd.ie/wordpress/wp-content/uploads/DescriptorSystemsExample.zip; Sajja, S. (2016). Matlab code for example 2. http://smarttransport.ucd.ie/wordpress/wp-content/uploads/DescriptorSystemsExample.zip
[11] Sajja, S.; Corless, M.; Zeheb, E.; Shorten, R., On dimensionality reduction and the stability of a class of switched descriptor systems, Automatica, 49, 6, 1855-1860 (2013) · Zbl 1360.93145
[12] Sajja, S.; Corless, M.; Zeheb, E.; Shorten, R., Stability of a class of switched descriptor systems, (American control conference (ACC), 2013 (2013), IEEE), 54-58 · Zbl 1360.93145
[13] Shorten, R.; Cairbre, F. O., A proof of global attractivity for a class of switching systems using a non-quadratic lyapunov approach, IMA Journal of Mathematical Control and Information, 18, 3, 341-353 (2001) · Zbl 0992.93084
[14] Shorten, R.; Corless, M.; Wulff, K.; Klinge, S.; Middleton, R., Quadratic stability and singular siso switching systems, IEEE Transactions on Automatic Control, 54, 11, 2714-2718 (2009) · Zbl 1367.93444
[15] Stykel, T., Analysis and numerical solution of generalized Lyapunov equations (2002), Institut für Mathematik, Technische Universität: Institut für Mathematik, Technische Universität Berlin · Zbl 1014.34037
[16] Tanwani, A.; Trenn, S., On observability of switched differential-algebraic equationss, (49th IEEE conference on decision and control (2010), IEEE), 5656-5661
[17] Trenn, S., Distributional differential algebraic equations (2009), Zugl.: Ilmenau, Techn. Univ., (Ph.D. thesis) · Zbl 1360.34002
[18] Trenn, S., Switched differential algebraic equations, (Dynamics and control of switched electronic systems (2012), Springer), 189-216
[19] Trenn, S.; Wirth, F., Linear switched daes: Lyapunov exponents, a converse lyapunov theorem, and barabanov norms, (51st IEEE conference on decision and control (2012), IEEE), 2666-2671
[20] Zhai, G.; Xu, X., A commutation condition for stability analysis of switched linear descriptor systems, Nonlinear Analysis. Hybrid Systems, 5, 3, 383-393 (2011) · Zbl 1238.93078
[21] Zhou, L.; Ho, D. W.C.; Zhai, G., Stability analysis of switched linear singular systems, Automatica, 49, 5, 1481-1487 (2013) · Zbl 1319.93069
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