True proportional navigation with varying closing speed is called realistic true proportional navigation, which is implemented in practice. Our main goal is to derive the complete solutions of three-dimensional realistic true proportional navigation for nonmaneuvering and maneuvering targets. Three coupled nonlinear second-order state equations describing the relative motion are solved analytically without any linearization for performance and trajectory analysis. Properties of three-dimensional realistic true proportional navigation such as capture region, range-to-go, time-to-go, and two aspect angles within spherical coordinates are all obtained in closed form. Furthermore, the two-player game between three-dimensional realistic true proportional navigation and threedimensional ideal proportional navigation is investigated analytically in the pursuit-evasion scenario, where a realistic true proportional navigation guided missile is designed to pursue an ideal proportional navigation guided target. It is found that an ideal proportional navigation guided target is much harder to intercept than a realistic true proportional navigation guided target.

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The purpose of this study is to develop the adaptive terminal sliding mode scheme to control a MEMS resonator with a six-powered potential function for tracking a given reference signal in the presence of system uncertainties and external disturbances. The proposed adaptive controller includes the time-varying feedback gains can tackle the nonlinear dynamics without directly eliminating. Meanwhile, these time-varying feedback gains are adaptively updated according to the suitable updated rules without the known bounds of system uncertainties and external disturbances. Some sufficient conditions to guarantee the stability based on Lyapunov theory and numerical simulations are performed to demonstrate the effectiveness of the presented scheme.

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