1. Motivations In a deterministic setting: Difficult to assess global properties (stability, reachability) Model glitches: Zenoness De-abstaction is the solution? Not always! No a-priori check Ill-posedness: Grazing events Problematic to simulate them Event detection can be hard May 11 th, 2005 Stochastic Approximations of Deterministic Hybrid systems Alessandro Abate Aaron Ames Shankar Sasty http://chess.eecs.berkeley.edu Setting Define a (deterministic) Hybrid System as a 6-tuple HS = (Q,E,D,G,R,F): Q = {1,...,m} ½ Z: set of discrete states-finite, subset of the integers. E ½ Q £ Q: set of edges which define relations between the domains. For e = (i,j) 2 E denote its source by s(e) = i and its target by t(e) = j; the edges in E can be indexed, E = {e 1,…,e |E| }. D = {D i } i 2 Q : set of domains where D i µ R n. G = {G e } e 2 E : set of guards, where G e µ D s(e). R = {R e } e 2 E : set of reset maps, continuous from G e µ D s(e) to R e (G e ) µ D t(e) and Lipschitz. F = {f i } i 2 Q : set of vector fields or ordinary differential equations (ODEs), such that f i is Lipschitz on R n. The solution to the ODE f i with i.c. x 0 2 D i is x i (t) where x i (t_0) = x_0. The guards are spacial, given by the zero level sets of smooth functions: {g e } e 2 E such that G e = {x: g e (x) = 0}; we also assume that g e (x) ¸ 0 for all x 2 D s(e) \ G e. In this work we shall introduce stochasticity on the Reset Maps. Conjecture. Given a hybrid system HS, there exists a non-trivial stochastic hybrid system SHS whose probabilistic behavior encompasses the deterministic behavior of the hybrid system HS: B (HS) µ B (SHS), where B (HS) is the behavior of the hybrid system HS. Moreover, SHS yields itself more easily to analysis. References: oA. Abate and A. D. Ames and S. Sastry: ``Stochastic Approximations of Hybrid Systems". Proceedings of the ACC 2005. oA. Abate and A. D. Ames and S. Sastry: ``Characterizing the Behavior of Deterministic Hybrid Systems through Stochastic Approximations". In preparation. oA. Abate: Analysis of Stochastic Hybrid Systems. MS Thesis, EECS Department, UC Berkeley, May 2004. oA.Abate, L. Shi, S. Simic, S. Sastry: ``A Stability Criterion for Stochastic Hybrid Systems". Proceedings of the MTNS 2004. Leuven, BG, July 2004. Contact: aabate@eecs.berkeley.edu Simulations Theorem: Given a hybrid system HS, the non trivial stochastic hybrid system SHS h, dependent on a parameter h (the integration step), verifies the following lim h -> 0 SHS h = HS. Theorem: Given a hybrid system HS, the non trivial stochastic hybrid system SHS h admits no Zeno behavior, 8 h > 0. Consider the ODE dx/dt = f(x). Examine the IVP I = (f,[t 0,t F ],x 0 ) on [t 0,t F ] with x t 0 =x 0. Usually solved by Numerical Integration: approximated solution x n (t) on [t 0,t F ], such that x n (t 0 )=x 0. Step size is h. Assume a numerical scheme produces a solution that is accurate of order M(t,h); M(t,0)=M(0,h)=0. Global bound on the error: 9 a constant C I such that ||x(t) - x n (t)|| · C I M(t-t 0,h). After proper “guard linearization”, assume that: At every point in time, the probability that the actual solution switches from the current domain to the one identified by a guard is given by the proportion of the volume sphere centered around the numerical solution that lies beyond the guard, by the volume of this sphere. Given an initial condition for the execution, we have the following knowledge: P ij (t) = P {q(t)=j|q(t 0 )=i}, 8 t ¸ t 0. This way, build a transition probability matrix P(t). Define time-dependent jump intensities: G(t) = dP(t)/dt, 8 t ¸ t 0. The trajectories of the original Deterministic Hybrid System HS can be handled as continuous- time Markov processes, solutions of the approximated Stochastic Hybrid System SHS h. decreasing step-size h Event Detection Two-tanks system: Eliminating Zeno The original trajectory is Zeno; the approximated isn’t. Propagation of the error cones. HS are hard! 2 Examples: Idea