Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems

In this work a procedure for obtaining polytopic lambda-contractive sets for Takagi Sugeno fuzzy systems is presented, adapting well-known algorithms from literature on discrete-time linear difference inclusions (LDI) to multi-dimensional summations. As a complexity parameter increases, these sets tend to the maximal invariant set of the system when no information on the shape of the membership functions is available. lambda-contractive sets are naturally associated to level sets of polyhedral Lyapunov functions proving a decay-rate of lambda. The paper proves that the proposed algorithm obtains better results than a class of Lyapunov methods for the same complexity degree: if such a Lyapunov function exists, the proposed algorithm converges in a finite number of steps and proves a larger lambda-contractive set.This work has been supported by Projects DPI2011-27845-C02-01 and DPI2011-27845-C02-02, both from Spanish Government.Arino, C.; Perez, E.; Sala Piqueras, A.; Bedate, F. (2014). Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems. Journal of The Franklin Institute. 351(7):3559-3576. https://doi.org/10.1016/j.jfranklin.2014.03.014S35593576351

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. This implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.Comment: 33 pages, 4 figure