The recent application of reliability analysis to controller synthesis has created the need for a
computationally efficient method for the estimation of the first excursion probabilities for linear dynamical
systems in higher dimensions. Simulation methods cannot provide an adequate solution to this specific application,
which involves numerical optimization of the system reliability with respect to the controller parameters,
because the total computational time needed is still prohibitive. Instead, an analytical approach is presented
in this paper. The problem reduces to the calculation of the conditional upcrossing rate at each surface
of the failure boundary. The correlation between upcrossings of the failure surface for the different failure
events may be addressed by the introduction of a multi-dimensional integral. An efficient algorithm is
adopted for the numerical calculation of this integral. Also, the problem of approximation of the conditional
upcrossing rate is discussed. For the latter there is no known theoretical solution. Three of the semi-empirical
corrections that have been proposed previously for scalar processes are compared and it is shown that the correction
should be based on the bandwidth characteristics of the system. Finally, examples that verify the validity
of the analytical approximations for systems in higher dimensions are discussed