Numerical methods for Engineers and Scientists

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Introduction To Perturbation Techniques

In this paper, we investigate robust resource allocation algorithm design for multiuser downlink multiple-input single-output (MISO) unmanned aerial vehicle (UAV) communication systems, where we account for the various uncertainties that are unavoidable in such systems and, if left unattended, may severely degrade system performance. We jointly optimize the two-dimensional (2-D) trajectory and the transmit beamforming vector of the UAV for minimization of the total power consumption. The algorithm design is formulated as a non-convex optimization problem taking into account the imperfect knowledge of the angle of departure (AoD) caused by UAV jittering, user location uncertainty, wind speed uncertainty, and polygonal no-fly zones (NFZs). Despite the non-convexity of the optimization problem, we solve it optimally by employing monotonic optimization theory and semidefinite programming relaxation which yields the optimal 2-D trajectory and beamforming policy. Since the developed optimal resource allocation algorithm entails a high computational complexity, we also propose a suboptimal iterative low-complexity scheme based on successive convex approximation to strike a balance between optimality and computational complexity. Our simulation results reveal not only the significant power savings enabled by the proposed algorithms compared to two baseline schemes, but also confirm their robustness with respect to UAV jittering, wind speed uncertainty, and user location uncertainty. Moreover, our results unveil that the joint presence of wind speed uncertainty and NFZs has a considerable impact on the UAV trajectory. Nevertheless, by counteracting the wind speed uncertainty with the proposed robust design, we can simultaneously minimize the total UAV power consumption and ensure a secure trajectory that does not trespass any NFZ.

Multiuser MISO UAV Communications in Uncertain Environments With No-Fly Zones: Robust Trajectory and Resource Allocation Design

This paper presents the unified cooperative strategies for the salvo attack of multiple missiles on the basis of the traditional proportional navigation (PN) algorithm. The cooperative guidance laws are developed in a quite simple formulation consisting of a PN component for target capture and a coordination component for simultaneous arrival. The centralized coordination algorithms have better performance when the global information of time-to-go can be obtained by each missile, whereas the decentralized coordination algorithms come into effect when the group member is only able to collect information from its neighbors. The cooperative strategies can achieve a simultaneous attack against both stationary and maneuvering targets. Numerical simulations are used to demonstrate that the proposed approaches are feasible and flexible for the salvo attack of multiple missiles. The capturability analyses of the cooperative guidance laws are also performed by discussing the selection of gain parameters.

Unified approach to cooperative guidance laws against stationary and maneuvering targets

Three-dimensional cooperative guidance laws against stationary and maneuvering targets

Analytical entry guidance for coordinated flight with multiple no-fly-zone constraints

The proposed guidance law is used for guiding a missile against a maneuvering target while satisfying a circular no-fly zone (NFZ) constraint. It consists of two parts: virtual-target guidance (VTG) and boundary-constraint handling scheme (BCHS). In order that the missile avoids the NFZ, VTG first maps the actual target to a virtual one, then obtains the relative motion between the virtual target and missile, and finally uses proportional navigation to steer the missile to the virtual target. The missile also hits the actual target when it hits the virtual target because the virtual and actual targets are coincident at this moment. In some cases, especially when the initial velocity vector of the missile points toward the center of the NFZ, if the evasive action taken by VTG is found to be insufficient, then BCHS will be enabled to keep the missile from entering the NFZ unless the target enters the NFZ.

Guidance law with circular no-fly zone constraint

A composite guidance law is proposed for intercepting moving target while strictly satisfying the constraints on multiple No-Fly Zones (NFZs) distributed arbitrarily. The research has two major steps. In the first step, by considering only one NFZ, a guidance law is developed with three parts: Orientation Adjustment Scheme (OAS), Boundary-Constraint Handling Scheme (BCHS), and Proportional Navigation (PN). OAS determines the major flight direction by predicting the collision point of the missile and target. BCHS controls the missile to approach and then fly along the boundary of the NFZ smoothly so as to bypass the NFZ through a short path. PN is used to intercept the target in the endgame phase. In the second step, we use the multi-step decision process to set up a series of appropriate waypoints in order to avoid multiple NFZs. The superior performance of the proposed guidance law has been demonstrated by trajectory simulations.

Guidance law for intercepting target with multiple no-fly zone constraints

Analytical solutions to three-dimensional hypersonic gliding trajectory over rotating Earth

The analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory are proposed by perturbation theory. First, the analytical solution of downrange distance is approximated by the outer solutions of a two-timescale singular perturbation system where the solution error is less than 2%. Then, the zeroth-order solutions and first-order corrections of altitude and flight-path angle are developed using regular perturbation techniques which could decompose the coupled and nonlinear system into a sequence of linear problems to solve in order. Because the first-order system for altitude and flight-path angle is a linear time-varying system that cannot be solved by traditional method such as Laplace transform, two methods are put forward to simplify the system: spectral decomposition method (SDM) and linear transformation method (LTM). SDM assumes the system to be time invariant and then uses spectral decomposition to solve it. LTM introduces a small compensation into the system such that the system can be decoupled into two independent first-order differential equations by linear transformation. Then, by combining the advantages of the two kinds of solutions, more accurate integrated analytical solutions are proposed. The simulation results show that the analytical solutions are in good agreement with the numerical solutions. In addition, the computation time of the analytical solutions is about two orders of magnitude less than that of the numerical solutions.

Wenbin Yu

Papers

Analytical entry guidance for coordinated flight with multiple no-fly-zone constraints

Guidance law with circular no-fly zone constraint

Guidance law for intercepting target with multiple no-fly zone constraints

Analytical solutions to three-dimensional hypersonic gliding trajectory over rotating Earth

Analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory