Chi-Ching Yang

A unified approach to proportional navigation

In this paper, the two major classes of proportional navigation (PN), namely, true proportional navigation (TPN) and pure proportional navigation (PPN) are analyzed and solved by a unified approach. The analytical tools used in the line-of-sight (LOS) referenced systems such as TPN, realistic true proportional navigation (RTPN), generalized true proportional navigation (GTPN) and ideal proportional navigation (IPN), are extended here to handle the interceptor velocity referenced systems such as PPN and its variants. It is found that the above two branches of guidance systems belong to a more general PN scheme which defines the acceleration of the interceptor as being proportional to the LOS rate with direction normal to an arbitrarily assigned vector L/spl I.oarr/. For example, L/spl I.oarr/ of TPN is LOS, and L/spl I.oarr/ of PPN is the interceptors velocity. Every PN scheme associates with a specific form of L/spl I.oarr/. The optimal PN (OPN) problem which concerns the determination of the optimal direction L/spl I.oarr/ is also addressed. Under the proposed general PN scheme, its six special cases, i.e., TPN, RTPN, GTPN, IPN, PPN, and OPN are solved in a unified way from which many new relations among them can be revealed, and their performances can be compared on a common basis.

Analytical Solution of Three-Dimensional Realistic True Proportional Navigation

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. I. Introduction ROPORTIONAL navigation (PN) for short-range tactical misP siles is the optimal interceptive law in the sense of producing zero miss distance with the least integral square control effort. PN has been studied since the 1940s. During the four decades that followed, proportional navigation has been explored in many different ways, such as true proportional navigation, pure proportional navigation (PPN), generalized proportional navigation, realistic true proportional navigation (RTPN), and ideal proportional navigation It has long been recognized that there exists a significant difference in the way in which PN guidance law is analyzed and in the way in which it is implemented. Most analytical studies of PN assume that the closing velocity in the PN guidance law is constant, whereas in realistic situations, the instantaneous closing speed may be continuously estimated or measured using devices such as homing seekers with Doppler radar. To remove this difference, RTPN, which adapts to varying closing speed, has recently been proposed. Performance and trajectory analysis of two-dimensional RTPN was studied by

Analytical solution of 3D true proportional navigation

The commanded acceleration applying in the direction normal to the line of sight (LOS) between the interceptor and its target is called true proportional navigation (TPN). The main goal of this work is to derive the complete analytical solutions of three-dimensional (3D) TPN 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 3D TPN 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 comparison between 3D TPN and 3D realistic true proportional navigation (RTPN) is investigated analytically in the two-player game where a TPN-guided missile is commanded to pursue a RTPN-guided target, and vice versa.

Optimal pure proportional navigation for maneuvering targets

The optimal pure proportional navigation (PPN) guidance law with time-varying navigation gains is considered. Unlike the conventional optimal PPN approach where linearized model was assumed in the optimization process, this work exploits the exact nonlinear formulation of PPN to derive analytically the optimal time trajectory of the navigation gain to minimize a performance index which is a weighted sum of the final time and the integral of the squared acceleration. It is verified that the PPN scheme with constant navigation gain is not only optimal in the vicinity of the interception point, but also optimal for the whole trajectory, if the navigation constant is designed by the methodology proposed here. Based on the optimization results for nonmaneuvering targets, a recursive optimal PPN scheme is proposed for maneuvering targets, wherein the optimal navigation gain and time-to-go are predicted recursively during the interception, and trajectory and performance of the interceptor guided by optimal recursive PPN scheme are evaluated analytically.

Analytical solution of generalized 3D proportional navigation

The main goal of this paper is to establish a systematic analysis on three-dimensional (3D) guidance laws. The concept of generalized three-dimensional proportional navigation is introduced and its analytical solution is derived. The three second-order nonlinear differential equations describing the 3D pursuit-evasion scenario are obtained in terms of the unit angular momentum which is an useful index measuring the departure tendency of the relative motion from a fixed plane. The existing difficulties in solving this set of coupled nonlinear equations are conquered and the solution shows that many past studies on the two-dimensional proportional navigation are merely special cases of the present framework.

Analytical Solution of 3D Realistic True Proportional Navigation

Abstract The main goal of this paper is to derive the complete solutions of three-dimensional realistic true proportional navigation (3D RTPN) for nonmaneuvering 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 3D RTPN such as capture region, range-to-go, time-to-go, and two aspect angles within spherical coordinates are all obtained in closed form.

Experiment-Aided Controller Design of Rotor Systems With a Magnetic Bearing

A new design methodology for the vibration control of rotor systems with a magnetic bearing is developed in this paper. The methodology combines the experimental design method in quality control engineering and the conventional PD control technique such that their advantages in implementation feasibility and performance-robustness can be integrated together. A quality loss index defined by the summation of the infinity norm of unbalanced vibration is used to characterize the system dynamics. By using the location of the magnetic bearing and PD feedback gains as design parameters, the controller of experiment-aided design achieves the best system performance. In addition, it is robust to operating speed variations. A rotor system consisting of 4 rigid disks, 3 isotropic bearings, and 1 magnetic bearing is applied to illustrate the feasibility and effectiveness of the experiment-aided controller design.© 1995 ASME

Guidance-Law Synthesis for Interception and Rendezvous

In this paper, we consider guidance-law synthesis problems where the required conditions for interception or rendezvous are specified first, and a class of guidance laws is then synthesized to meet the given conditions. This approach is different from the existing analysis oriented methods where guidance laws are proposed first and then their performances are analyzed to check whether the interception or rendezvous conditions are satisfied. The synthesized guidance laws can be parametrized in terms of two free functions which provide additional design freedoms to satisify some strengthened interception or rendezvous conditions.

Optimal Design of Sensor/Actuator Location and Feedback Gain by Taguchi Method

Abstract This paper presents a controller design methodology for optimal of sensor/actuator location and feedback gain. The methodology combines Taguchi method in quality control engineering and classical PID control such that their advantages in implementation feasibility and performance robustness can be integrated together. Instead of searching for all possible combinations of design parameters, a reduced set of experimental design-the orthogonal array in Taguchis method-is applied to search for a near optimal design. Several orders of magnitude in the effort of controller design can thus be reduced. The controller is shown to achieve the best possible system performance while maintaining the closed loop system stability. A rotor system is applied to illustrate the feasibility and effectiveness of the controller.

Sarason Approach to Optimal Two-Block and Four-Block H∞ Norm

Abstract This paper introduces a new method for computing optimal 2-block and 4-block H ∞ norm. It is shown that the spectral radii of 2-block and 4-block operators can be found by calculating the largest singular value of a finite dimensional matrix formed from the matrix representation of the Sarason operator. This novel approach reveals the intimate relations among the Hankel, Toeplitz and Sarason operators and provides an efficient way for computing optimal H ∞ norm.