More specifically, I study the stability of switched systems (a special case of hybrid systems) under arbitrary switching. Switched systems have numerous applications in diverse fields (see [4][5]). In the particular case of linear switched systems, finding a necessary and sufficient condition for stability is equivalent to solving one of the oldest open problems in control theory, the "Absolute Stability Problem" [6]. I am primarily interested in stability analysis using tools from the theory of optimal and geometric control (see the survey paper [7]).
The optimal control approach provides a complete solution to the absolute stability problem in the planar case [8] (see also [9]). When combined with Lie-algebraic techniques, it also leads to stability conditions for nonlinear switched systems [10][11]. This approach provided the first partial solutions to an open problem on the stability of nonlinear switched systems with a nilpotent Lie-algebra (see Problem 6.4 in Unsolved Problems in Mathematical Systems and Control Theory).
Another research topic I am pursuing is fuzzy modeling and control. My research focus is on (1) systematic design and analysis of fuzzy models and controllers [12]; and (2) using fuzzy modeling to develop mathematical models describing animal behavior and biological systems (see e.g. [13][14][15]). Fuzzy tools are also useful for knowledge-based computing in artificial neural networks [16].
Recently, I am also working on analyzing switched power converters using both time- and frequency-domain approaches [17][18].
References
[1] van der Schaft, A. and Schumacher, H. An Introduction to Hybrid Dynamical Systems, Springer, 2000.
[2] Alur, R. et al. "Modeling and analyzing biomolecular networks", Computing in Science and Engineering, 4(1):20-31, 2002.
[3] Branicky, M. "Introduction to hybrid systems", in D. Hristu-Varsakelis and W.S. Levine (eds.), Handbook of Networked and Embedded Control Systems, pp. 91-116. Birkhauser, 2005.
[4] Liberzon, D. Switching in Systems and Control, Birkhauser, 2003.
[5] Sun, Z. and Ge, S. S. Switched Linear Systems: Control and Design, Springer, 2005.
[6] Vidyasagar, M. Nonlinear Systems Analysis, Prentice Hall, 1993.
[7] Margaliot, M. "Stability analysis of switched systems using variational principles: an introduction", Automatica, 42:2059-2077, 2006.
[8] Margaliot, M. and Langholz, G. "Necessary and sufficient conditions for absolute stability: The case of second-order systems", IEEE Trans. Circuits and Systems-I, 50(2):227-234, 2003.
[9] Holcman, D. and Margaliot, M. "Stability analysis of switched homogeneous systems in the plane", SIAM J. Control and Optimization, 41(5):1609-1625, 2003.
[10] Margaliot, M. and Liberzon, D. "Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions", Systems and Control Letters, 55:8-16, 2006.
[11] Sharon, Y. and Margaliot, M. "Third-order nilpotency, finite switchings and asymptotic stability", J. of Differential Equations, 233:136-150, 2007.
[12] Margaliot, M. and Langholz, G. New Approaches to Fuzzy Modeling and Control - Design and Analysis, World Scientific, 2000.
[13] Tron, E. and Margaliot, M. "Mathematical modeling of observed natural behavior: a fuzzy logic approach", Fuzzy Sets Systems, 146(3):437-450, 2004.
[14] Tron, E. and Margaliot, M. "How does the Dendrocoleum lacteum orient to light? a fuzzy modeling approach", Fuzzy Sets Systems, 155(2):236-251, 2005.
[15] Laschov, D. and Margaliot, M. "Mathematical modeling of the Lambda switch: a fuzzy logic approach", J. of Theoretical Biology, 260:475-489, 2009.
[16] Kolman, E. and Margaliot, M. Knowledge-Based Neurocomputing: A Fuzzy Logic Approach, Springer-Verlag, 2009.
[17] Margaliot, M. and Weiss, G. "The low-frequency distortion in D-class amplifiers", IEEE Trans. Circuits Systems—II, 57:772-776, 2010.
[18] Ruderman, A. and Reznikov, B. and Margaliot, M. "Analysis of a flying capacitor converter: a sampled-data modeling approach", submitted.