Citation Context

...t1) and ˆµ∞(t1), respectively. � Let di µ(t) dti and diν(t) dti represent the ith time derivative of µ(t) and ν(t) along the trajectories of system (7) and (9) respectively. The following result from =-=[17]-=- shows that if a sufficiently large but finite number of these derivatives match point-wise, then, the difference between solutions to (8) and (9), i.e. µ(t) and ν(t) converges to an arbitrarily small...

Citation Context

...an cover all such ingredients [16,6]. While such models have a vast potential of application, results related to their analysis and verification are limited, and often based on Monte-Carlo simulation =-=[4,15]-=-. For certain subclasses of piecewise deterministic Markov processes, of probabilistic hybrid automata, and of stochastic hybrid systems, reachability probabilities can be approximated (e.g. [20,7,17]...

Citation Context

...ith a probabilistic notion of safety for stochastic systems. Several other methods have been developed for analysis of stochastic hybrid systems, for example, based on statistical moments computation =-=[11]-=-, discretization into Markov chains [12], [13], and construction of barrier functions [14]. The approach that we propose in this paper is very different from the previous approaches, in the sense that...

Citation Context

.... 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.1 Introduction Roughly speaking, hybrid systems are dynamical systems with both continuous and discrete dynamics. The study of hybrid systems has received considerable attention in recent years due to their applications in a diverse range of scientific and engineering problems. Examples include transportation systems such as air traffic management systems [21], [23] and automated highway systems [24], robotics [8], computer and communication networks [14], and automotive systems [3]. Besides engineering applications, hybrid systems are also found useful in modeling biological systems [2]. While impressive progress has been made so far in the study of hybrid systems, a majority of the efforts focus exclusively on deterministic hybrid systems, which are not suitable for modeling practical systems with inherent uncertainty. For example, the trajectory of an aircraft is subject to the perturbations of wind [6], the traffic in a computer network may fluctuate and components may break down at random intervals, and stochastic noises exist in the gene...

Citation Context

...ments will be necessary–all of which can be determined as above. For third or higher order propensity functions, the resulting higher order moments can be also be easily expressed in terms of the first two using moment using generating functions as described in the example section below. 2. Lognormal Distribution. Based on a lognormal distribution for X, one obtains the following equation: E[XiXjXk] = E[XiXj ]E[XjXk]E[XiXk] E[Xi]E[Xj ]E[Xk] . As before, this leads to a closed-system when substituted in the moment Eqn. (1), provided that the reactions in the system are at most bimolecular. In ([13]) it was shown that this moment closure results without any a priori assumptions on the shape of the distribution for X by matching all (or a large number of) the time derivatives of the exact solution for Eqn. (1) with the corresponding time derivatives of the approximate solution for Eqn. (2), for a given set of initial conditions. However, for systems with third or higher order terms in the reaction rates, it is more difficult to find expressions for the higher moments necessary to close the system. When the population standard deviations are not much smaller than the means, choosing ϕ(·) b...

Citation Context

... this section, we present a framework of stochastic hybrid systems originally proposed in [11] and subsequently extended in [5] by incorporating the concept of switching diffusions [8]. See [1], [2], =-=[9]-=-, [13], [14] for other relevant works on stochastic hybrid systems. In the extended framework, the state (Xt,mt) of a stochastic hybrid system H consists of two parts: Xt is the continuous state and m...

Citation Context

...een used as a modelling framework in various applications, such as air traffic management [29], manufacturing systems [27], communication networks [31], and stochastic modelling of chemical reactions =-=[32]-=-. When the system is stochastic, answering the safety verification question in a worst-case non-stochastic manner (i.e., to verify whether or not a trajectory of the system can reach the unsafe set) a...