A new look at bounding integrity risk in the presence of time-correlated errors

A new approach is developed to upper bound integrity risk for safety-critical GNSS applications when the autocorrelation functions of measurement and process noise are uncertain. The algorithm is recursive and significantly expands the applicable range of existing methods. In Kalman filtering, models must be specified for autocorrelated measurement and process noise. This paper assumes that the noise processes are the output of known linear systems driven by white noise. However, the system parameters are only known to lie in specified intervals. Following the approach in [6], an augmented state model is first developed to propagate the estimate error covariance matrix when there is parameter uncertainty. It is shown that the error variance for a specified state is a polynomial in the unknown parameters whose order grows linearly with time. An exact upper bound on integrity risk is determined by maximizing the variance. Next, a recursive algorithm is derived to update the variance's Taylor series expansion. This enables an approximate bound to be established by maximizing a polynomial whose order is low and remains fixed over time. The Taylor remainder theorem is used to introduce conservatism and provide justification that the approximate bound is at least as large as the exact bound. For a one-dimensional navigation problem, we show that the number of series terms needed to obtain a tight upper bound on integrity risk is less than 10.