A Gaussian Sum Method to Analyze Bounded Acceleration Guidance Systems

To analyze a highly nonlinear dynamical system, it is necessary to evaluate the time evolution of the probability density function (pdf) of the states. In this paper, a novel Gaussian sum method is proposed to approximate the non-Gaussian pdf of a dynamical system whose nonlinear elements depend only on a single state variable. Using the Chapman-Kolmogorov equation, a set of recursive equations is derived to propagate the mean, covariance, and weight of the Gaussian sum components. There are several multiple integrals in the recursive equations which are analytically reduced to single integrals. As an application, the proposed approach is applied to statistically analyze the engagement of two pursuers with bounded acceleration against a randomly step evader maneuver. Pursuers are assumed to use the zero-lag version of the recently developed optimal cooperative guidance law (OCGL) or the well-known proportional navigation (PN). The results show that the accuracy of the proposed method is equivalent to Monte Carlo simulation which requires extensive computational effort. Utilizing the proposed analytical Gaussian sum method, OCGL and PN are statistically investigated and compared with each other. The results demonstrate that OCGL is superior to PN, especially when the guidance system is highly in saturation.