1 Aircraft Conflict Detection in Presence of Spatially Correlated Wind Field

In this research we study the probabilistic safety of a pair of aircraft flying in the vicinity of each other in a random wind field. A central part of our effort is the modeling of the wind field as a spatially correlated random field, so that the closer the two points in space, the more similar the values of the random field at those two points. This is in line with experimental measurements of the atmospheric wind in, e.g., [1]. We assume kinetic model for the aircraft dynamics, which has a nominal velocity profile that is typically piecewise constant, and a state feedback term modeling the correctional effort of the pilot for the cross-track deviation from the ideal flight path. To compute the probability of conflict, namely, the probability of two aircraft coming closer than a prescribed 5 nmi distance, we discretize the space into grids, and construct Markov chains on the grids that approximate the stochastic processes modeling aircraft motions. The probability of conflict can then be computed to arbitrary precision by using the Markov chains, with a fine enough grid size. In [3], we study the problem in two dimensional airspace, and in [2], we study a simplified version of the problem in three dimensional airspace. A stochastic hybrid system approach is applied in [5] to model the genetic network regulating the biosynthesis of an antibiotic called subtilin in Bacillus subtilis. Each B. subtilis cell is modeled as a stochastic hybrid system, whose continuous variables are the concentrations of various proteins inside the cell, and whose discrete variables are the ON/OFF states of random switches for protein production. The dynamics of the continuous variables follow ordinary differential equations with parameters dependent on the discrete variables, namely, a protein is being produced at a higher rate if the corresponding switch in the genetic network is ON. On the other hand, the discrete variables (switches) change values randomly according to Markov chains whose transition probabilities are functions of the concentrations of modulating proteins, modeling the activations and deactivations of the production of these protein species. While the dynamics of the continuous variables are deterministic, the randomness of the system arises from the probabilistic switchings of the discrete variables, which indirectly makes the evolution of the continuous variables random as well. In our model, the interaction of a B. subtilis cell with the environment is modeled through the input and output …