Numerically stable square-root solutions in the family of adaptive Gaussian filters based on the general moment calculation principle for state estimation in continuous-discrete nonlinear stochastic systems

This paper solves the problem of accurate and robust state estimation in continuous-discrete stochastic systems whose process models are simulated by stochastic differential equations but their measurement equations enjoy a discrete-time nonlinear fashion. We show that any Gaussian filter designed for estimating the aforementioned systems can be presented in the form of some universal algorithm with its corresponding and easily-specified parameterization. In other words, such well-known state estimation schemes as the extended, unscented, cubature and quadrature Kalman filters are trivially implemented by adjusting the mean and covariance approximation weights and nodes in the algorithms devised below. The main features of state estimators under exploration are their adaptivity and numerical stability. The first one is grounded on ordinary differential equations (ODEs) developed and substantiated for evolution of deterministic samples utilized in calculations of the expectation and covariance in the process model of interest. These are effectively computed by variable-stepsize ODE solvers with automatic discretization error control and stepsize selection, including the widely known and commonly used built-in Matlab ODE solvers, which are available for integrating ODEs nowadays. The second characteristic is provided by employing square-root versions of the Gaussian filters designed, which are based on hyperbolic QR factorizations. Such square-root implementations are exceptionally robust to round-off and numerical integration error accumulations. Also, these preserve the symmetry and positivity of the covariances computed in exact arithmetic. Performances of our novel continuous-discrete square-root Gaussian filters are assessed and compared to their non-square-root counterparts in a simulated flight control scenario with ill-conditioned measurements.