Aaron D Ames Alessandro Abate Shankar Sastry Sufficient
Aaron D. Ames Alessandro Abate Shankar Sastry Sufficient Conditions for the Existence of Zeno Behavior http: //chess. eecs. berkeley. edu zi +1 = = ªH (zi ); (k ¡ 1)2 zi : Abstract: We derive su±cient conditions for the existence of Zeno behavior in a class of hybrid systems. This is achieved by considering a class of hybrid systems termed diagonal ¯rst quadrant hybrid systems. To these hybrid system, we can associate a single discrete time dynamical system that describes its continuous evolution. We obtain su±cient conditions for the existence of Zeno Behavior by determining when this discrete time dynamical system is exponentially stable. Diagonal First Quadrant Hybrid Systems: k increasing A diagonal ¯rst quadrant hybrid system is a tuple H = (¡; D; G; R; F ); where ² ¡ = (Q; E) is an oriented graph that is a cycle. ² D = f. D g 2 is a set of domains of the Form q q Q Dq = fx 2 Rn : x 1 ¸ 0 and x 2 ¸ 0 g for every q 2 Q. Plots of the discrete time dynamical systems associated with the two tanks hybrid system with the following parameters: ² G = f. G g 2 is a set of guards of the form e e E Ge = fx 2 Rn : x 1 = 0 and x 2 ¸ 0 g for every e 2 E. ! R+ (here R+ = fx 2 R : x ¸ 0 g) given by For q 2 Q let ©q : R+ 0 0 Ã0 Ã Ã !! ! ² R = f. R g 2 is a set of reset maps such that e e E 1 = ©q(x) ¸ 2 q Re(x) = Re(x 1 ; x 2 ; : : : ; xn ) = (x 2 ; x 1 ; x 3 ; : : : ; xn )T for every e 2 E. exp ¸ 2 q log 1 ¸q a 1 q + ¸ 1 qx In°ow = k Out°ow of tank one = 1 Out°ow of tank two = 1. ¡ 1 a 2 : q For a diagonal ¯rst quadrant hybrid system H de¯ne ² F = f¤ x + a g 2 is a set of diagonal a±ne linear systems, i. e. , a 2 Rn q q q Q q £ 2 2 and ¤q Rn n is a diagonal matrix for every q Q. We denote ¤i; i q by ¸iq and refer to it as the ith eigenvalue. ªH(x) = ©q. K ± ©q. K ¡ 1 ± ¢ ¢ ¢ ± ©q 0 ! R+ , and Q = fq ; ; q g (as ordered by the edges). where ªH : R+ 1 k 0 0 The Discrete Time Dynamical System associated with a Hybrid System: The discrete time dynamical system associated to the diagonal ¯rst quadrant hybrid system H is given by zi +1 = ªH (zi ): Theorem: Main Theorem: If the linearization of the discrete time dynamical system associated to the diagonal ¯rst quadrant hybrid system H, zi +1 = DªH (0)zi ; Is stable at the origin, then H is Zeno Let H be a diagonal ¯rst quadrant hybrid system. Then if ¤qx + aq, q 2 Q, satis¯es the conditions: 9 > · > 1 ¸q 0 > > = a 1 q < 0 < a 2 q ) > H is Zeno: > ¯ ¯ > > ¯Q ¯ > > ¯ ¯ 2 a ¯ K qi ¯ < 1 ; i =0 a 1 qi May 11, 2005
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