An improved path integration method for nonlinear systems under Poisson white noise excitation

In order to overcome some unsatisfactory trends and limitations of the traditional path integration (PI) method for Poisson white noise, a novel PI method is proposed in this paper, which includes two improved schemes. The first one is a new Transition Probability Density Function (TPDF) approximation which considers the randomness of the impulse happening time during each time intervals. The second one is a transformation of Chapman–Kolmogorov (CK) equation by a variable substitution instead of directly using it, whose numerical calculation is based on the back stepping Runge–Kutta scheme and the triangulation-based interpolation. Monte Carlo Simulations (MCS) are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems. The results show that compared with the traditional PI method, the improved PI method can give a more accurate description of the TPDF values, and provide more precise stationary Probability Density Function (PDF) results whenever the mean arrival rate is large or small. The improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary PDF results. Besides, it is discovered that cubic interpolation deserves to be applied in the improved PI method more than linear and natural interpolations.