High-level modeling and analysis of the traffic alert and collisionavoidance system (TCAS)

We demonstrate a high-level approach to modeling, analyzing, and verifying complex safety-critical systems through a case study on the traffic alert and collision avoidance system (TCAS); an avionics system that detects and resolves aircraft collision threats. Due to the complexity of the TCAS software and the hybrid nature of the closed-loop system, the traditional testing technique of exhaustive simulation does not constitute a viable verification approach. Moreover, the detailed specification of the system software employed to date as a means toward analysis and verification neither helps in intuitively understanding the behavior of the system nor enables the analysis of the closed-loop system behavior. We advocate defining high-level hybrid system models that capture the behavior not only of the software but also of the airplanes, sensors, pilots, etc. In particular, we show how the core components of TCAS can be captured by relatively simple hybrid I/O automata, which are amenable to format analysis. We then outline a methodology for establishing conditions under which TCAS guarantees sufficient separation in altitude for aircraft involved in collision threats. The contributions of the paper are the high-level models of the closed-loop TCAS system and the demonstration of the usefulness of high-level modeling, analysis, and verification techniques     

Moment estimation for chemically reacting systems by extended Kalman filtering

In stochastic models of chemically reacting systems that contain bimolecular reactions, the dynamics of the moments of order up to n of the species populations do not form a closed system, in the sense that their time-derivatives depend on moments of order n + 1. To close the dynamics, the moments of order n + 1 are generally approximated by nonlinear functions of the lower order moments. If the molecule counts of some of the species have a high probability of becoming zero, such approximations may lead to imprecise results and stochastic simulation is the only viable alternative for system analysis. Stochastic simulation can produce exact realizations of chemically reacting systems, but tends to become computationally expensive, especially for stiff systems that involve reactions at different time scales. Further, in some systems, important stochastic events can be very rare and many simulations are necessary to obtain accurate estimates. The computational cost of stochastic simulation can then be prohibitively large. In this paper, we propose a novel method for estimating the moments of chemically reacting systems. The method is based on closing the moment dynamics by replacing the moments of order n + 1 by estimates calculated from a small number of stochastic simulation runs. The resulting stochastic system is then used in an extended Kalman filter, where estimates of the moments of order up to n, obtained from the same simulation, serve as outputs of the system. While the initial motivation for the method was improving over the performance of stochastic simulation and moment closure methods, we also demonstrate that it can be used in an experimental setting to estimate moments of species that cannot be measured directly from time course measurements of the moments of other species.     

On the polymerisation of the epoxidised biodiesel: The importance of the epoxy rings position,the process and the products

Polyesters were prepared using epoxidised methyl esters of oleic acid (EP_OAME) and epoxidised biodiesel (mixture of methyl esters) from sunflower oil (EP_SOME) and linseed oil (EP_LOME) with cis-1,2-cyclohexanedicarboxylic anhydride and triethylamine. The kinetic of partial processes involved in the polymerisation were elucidated and related to epoxy rings position in the fatty acid chain. The activation energies (E_a) for the epoxy ring opening by the catalyst are 298, 216 and 136 kJ/mol for EP_OAME, EP_SOME and EP_LOME respectively. The reactions of the epoxy rings in the positions C9–C10, C12–C13 and C15–C16 with anhydride require average activation energies of 116, 32 and 22 kJ/mol, respectively. The compensation effect between activation energy and pre-exponential factor is observed. The polymerisation enthalpy, molecular weight, glass transition temperature and electrical properties were determined. The polyesters studied show promising properties for use in various technological applications.     

An interface between continuous and discrete-event controllers forvehicle automation

The work presented is part of a bigger effort to design an automated highway system (AHS) to improve the capacity and safety of current highways. Automation of highways and, in particular, platooning of vehicles raise a number of control issues. In the design proposed by Varaiya,(see IEEE Trans. Automat., Contr., vol.38, no.2, p.195-207, 1993), these issues are addressed by a hierarchical structure consisting of both discrete-event and continuous controllers. Our work is an attempt to construct a consistent interface between these two types of controllers. The proposed design is in the form of a set of finite-state machines (FSMs) that interact with the discrete controllers through discrete commands and with the continuous controllers by issuing commands that get translated to inputs for the vehicle actuators. The operation of the proposed design is verified using COSPAN and tested in simulation. The interface design provides insight into interesting problems related to the hybrid nature of the AHS as it touches on both the discrete and continuous worlds     

Maximizing the probability of attaining a target prior to extinction

We present a dynamic programming-based solution to the problem of maximizing the probability of attaining a target set before hitting a cemetery set for a discrete-time Markov control process. Under mild hypotheses we establish that there exists a deterministic stationary policy that achieves the maximum value of this probability. We demonstrate how the maximization of this probability can be computed through the maximization of an expected total reward until the first hitting time to either the target or the cemetery set. Martingale characterizations of thrifty, equalizing, and optimal policies in the context of our problem are also established.     

A game theoretic approach to controller design for hybrid systems

We present a method to design controllers for safety specifications in hybrid systems. The hybrid system combines discrete event dynamics with nonlinear continuous dynamics: the discrete event dynamics model linguistic and qualitative information and naturally accommodate mode switching logic, and the continuous dynamics model the physical processes themselves, such as the continuous response of an aircraft to the forces of aileron and throttle. Input variables model both continuous and discrete control and disturbance parameters. We translate safety specifications into restrictions on the system's reachable sets of states. Then, using analysis based on optimal control and game theory for automata and continuous dynamical systems, we derive Hamilton-Jacobi equations whose solutions describe the boundaries of reachable sets. These equations are the heart of our general controller synthesis technique for hybrid systems, in which we calculate feedback control laws for the continuous and discrete variables, which guarantee that the hybrid system remains in the “safe subset” of the reachable set. We discuss issues related to computing solutions to Hamilton-Jacobi equations. Throughout, we demonstrate out techniques on examples of hybrid automata modeling aircraft conflict resolution, autopilot flight mode switching, and vehicle collision avoidance     

Odd prime values of the Ramanujan tau function

We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(p ^n?1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime.     

Ten consecutive primes in arithmetic progression

In 1967 the first set of 6 consecutive primes in arithmetic progression was found. In 1995 the first set of 7 consecutive primes in arithmetic progression was found. Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression. This was made possible because of the increase in computer capability and availability, and the ability to obtain computational help via the Internet. Although it is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression, it is very likely that 10 primes will remain the record for a long time.