Safety Margins for Flight Through Stochastic Gusts

W IND disturbances are a common factor in aviation accidents. Among 126 airplane accidents that occurred between 1979 and 2009 involving loss of control in flight, 14% listed windshear, turbulence, or thunderstorms as a cause or contributing factor. Moreover, 86% of the accidents initiated by atmospheric disturbances led to an upset flight condition, which usually entails an excursion of the aircraft state outside of the flight envelope. Sixtyfour percent of these accidents also involved inappropriate crew response [1]. The flight envelope, the set of airspeeds, altitudes, flight-path angles, and bank angles at which an airplane can maintain steady flight, is useful for identifying when an airplane is prone to loss of control. Airplanes are at a higher risk of loss of control when flying in unstable flight states. Most airplanes are designed to fly stably or stabilizably when flying steadily; therefore, flight states in the steady flight envelope are generally not conducive to loss of control. In prior work, the authors presented safety margins and adjusted flight envelopes called stationary flight envelopes for airplane flight through stochastic gusts [2]. Those margins and envelopes were primarily based on the instantaneous probability of exceeding the steady flight envelope, which also corresponds to the fraction of time spent outside the steady flight envelope. This Note connects several probabilistic safety margins for airplane flight in turbulence, including the instantaneous probability used in the authors’ prior work [2]. The safety margins, which are various measures related to the probability of a stochastic process deviating far from its meanvalue, include 1) the frequency of exceedance, which is the rate at which a stochastic process exceeds a given threshold; 2) the residence time, which is the average time at which a stochastic process exceeds a given threshold for the first time; 3) the logarithmic residence time, which is a dimensionless version of the residence time defined for thresholds far from the mean; 4) the instantaneous probability of exceedance, which is the probability that a stochastic process exceeds a given threshold at time t that, for an ergodic process, is also the fraction of time spent beyond the threshold; 5) the probability of exceedance at time t, which is the probability that a stochastic process has exceeded a given threshold at or before time t; and 6) the probability of exceedance per unit time, which is the probability of exceedance at time t divided by t. These safety margins are applied to the problem of airplane flight through stochastic wind gusts, where the exceedance thresholds are chosen to be the constraints that define the steady flight envelope. This Note expands upon work by Hoblit on frequency of exceedance [3] to show how to compute the probability per unit time of exceeding the flight envelope, the residence time within the flight envelope, and the logarithmic residence time within the flight envelope. This Note also shows that the logarithmic residence time derived from the frequency of exceedance equals the logarithmic residence time derived by Meerkov and Runolfsson [4] for linear time-invariant systems driven by white noise. The Note concludes that the dimensional and dimensionless measures of safety complement one another to understand the hazard posed by flight through stochastic gusts. Section II presents the dynamic model of airplane flight through stochastic gusts. Section III shows how to compute the probability per unit time of exceeding a threshold and the residence time within that threshold. Section IV introduces the logarithmic residence time and shows that two disparate methods to compute it give the same result. Finally, Sec. V presents a numerical example related to exceeding the steady flight envelope in turbulence that is validated in simulation in Sec. VI.