Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 716547, 10 pages

http://dx.doi.org/10.1155/2015/716547

## Velocity Control for Coning Motion Missile System Using Direct Discretization Method

Nanjing University of Science and Technology, Nanjing 210094, China

Received 3 September 2015; Accepted 19 November 2015

Academic Editor: Jing Na

Copyright © 2015 Rui-sheng Sun and Chao Ming. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new coning motion control methodology, which takes into account the terminal speed constraint to design the velocity control system for a missile. By using a direct discretization method to transform the optimal control problem into a nonlinear dynamic programming problem, the optimal trajectory and velocity profile are obtained to satisfy the design index requirement. In order to perform the velocity control, a virtual moving target is proposed for the missile to chase along the optimized trajectory. Consequently, after building velocity control model, a velocity control law and control parameters of the coning motion are completed through the dynamic inversion theory. The simulation results suggest that the proposed control law has a good performance and could be applied to the guidance for the missile with terminal speed control constraint.

#### 1. Introduction

Modern warfare demands increasingly high performance requirements for missile, for example, reentry warhead, cluster bombs, and rocket-assisted torpedo. It needs not only high guidance precision, but also desired velocity in terminal phase. The reason for this is threefold. The first one is the conceptual requirement in terminal phase. For instance, a rocket-assisted torpedo should fly not too fast to guarantee that the matrix and the subtorpedo can separate securely. The second is the requirement of the large launch window and large envelope. For a missile system, it should not only get as large range as possible, but also hit the enough short-range targets, which requires the missile to consume superfluous kinetic energy in a timely manner. The third is to ensure the performance of control system. For example, the radar signals can fail in transfer due to the high-speed motion of radar-guided missile, which makes reentry warhead surface be surrounded by a serious aerodynamic heating of plasma. Consequently, it is necessary to control the deceleration of the missile flight speed in a proper time, which has become a focus for researchers.

Generally speaking, there are three kinds of approaches to control the velocity of missile when flying in the air. Firstly, we can control the velocity by changing the thrust force magnitude. Enomoto et al. [1] introduced a velocity control system for a leader-following UAV through changing the thrust force magnitude and using the dynamic inversion with the two-time scale approach. However, this control approach can only be suitable for the missile with propulsion system still on.

Secondly, according to the flight status and the terminal velocity constraint, the missile velocity can be governed by using the trajectory optimization method. By using this approach, the flight velocity is controlled by changing the flight height profile of the missile, which can change its drag force due to its changing dynamic pressure. In [2], a defined problem of optimal control was transformed into a two-point boundary problem, through applying the Pontryagin minimum principle. In this work, the optimal control and the optimal trajectory corresponding to this control at a skip three-dimensional entry of space vehicle into the planetary atmosphere were determined related to the obtaining of the maximum terminal velocity. Saraf et al. [3] presented an entry control algorithm for future space transportation vehicles through tracking the reference drag acceleration and heading angle profiles so as to satisfy all entry constraints including flight velocity control. Bruyère et al. [4, 5] designed a sideslip velocity autopilot for a model of tactical missile in order to meet the requirements of sideslip velocity over full flight envelope.

Finally, the missile velocity can be controlled by means of the coning motion method. It is a type of control technique that the missile axis movement is tapered at a fixed angle of rotation around the velocity vector, which will produce a velocity-centered conical surface, a coning motion [6] that can make the missile increase induced drag in order to adjust the flight velocity through producing induced angle of attack. Obviously, the coning motion control is an effective approach to control the flight velocity of missile [7–9]. Song [10] presented a new velocity control method of coning motion through simulating the reentry to improve flight control accuracy and control robustness of the terminal speed of the reentry vehicle when antidesigning the speed control method for Pershing II. Reference [11] put forward a velocity magnitude control program which separates deceleration motion into two different forms in the capability range of warhead guidance and control systems, in order to meet the requirement of fall velocity of a reentry maneuvering warhead and to avoid designing a complicated ideal velocity curve. In the field of noncontrolled rocket, Mao et al. did some work for the coning motion on the producing mechanism [12], motion characteristics and stability analysis methods [8, 13], optimal control [14], and the approach of reducing and avoiding coning motion [15].

Consequently, in order to design the velocity control system for a missile with the constraint of the terminal velocity, a deep understanding of the interaction among flight mechanics, optimizations, and control is necessary. The aim of this research is to propose a design methodology for the velocity control system by utilizing the coning motion based on a direct discretization and dynamic inversion method. The remaining of this paper is organized as follows. Section 2 describes the problem formulation and design scheme for velocity control. In Section 3, a direct discretization method is deduced to develop the standard trajectory optimization and the velocity control profile, and the velocity control methodology is presented in detail through the coning motion technique. Moreover, the structure and parameter design of velocity controller are explored for a coning motion missile. In Section 4, a simulation case is demonstrated to govern the terminal velocity of the missile through using the coning motion algorithm. In order to illustrate the velocity control performance, a traditional control method is also carried out for comparative study. Finally, some conclusions are presented in Section 5.

#### 2. Problem Formulation

##### 2.1. Motion Equations

For the optimal flight velocity control problem under consideration, a mathematical model based on point mass dynamics can be used for determining the trajectory and velocity, yieldingwhere , , , and denote velocity, trajectory inclination angle, path angle, and bank angle of missile, respectively; , , and represent distance along -, -, -axis; and and are mass of missile and acceleration of gravity, separately.

Here the earth is assumed to be flat, and the model for the drag (), the lift (), and the side force () can be formulated aswhere is dynamic pressure and is reference area. Here the coefficients of aerodynamics , , and are usually obtained by CFD or wind tunnel test.

##### 2.2. Cost Function

The performance index used for minimization of missile miss-distance and control iswhere denotes flight time and is angle of attack (AOA). Here subscript 0 and represent initial and terminal time, respectively.

##### 2.3. Flight Path Constraints

The constraints corresponding to the flight path optimization for a given missile are as follows.

The constraints on states and parameters control arewhere the subscript lower and upper represent lower and upper limit, respectively.

These numbers are completely based on the flight envelope characteristics, representing the possible values for the maximum and minimum state variables.

The terminal constraints of the missile are

Similarly, the nonlinear state inequality constraints are

For the purpose of the velocity control design, two subproblems are imperative to be solved. One is to obtain the optimal trajectory in the ideal condition. In consideration of this subproblem, the flight performance should be optimized to produce the trajectory and velocity profile satisfying the above requirements. The other one is to control the missile flying path along the optimal trajectory, in the same time to ensure the coherence of the missile speed and the ideal velocity profile; that is, the integration of the trajectory and velocity control should be considered.

#### 3. Velocity Control System Design

The process of the velocity control system design is conducted in Figure 1. Primarily, the optimal trajectory and velocity profile are carried out as a criterion for meeting the index requirements. Then, in order to perform this velocity control, a virtual moving target moving along the critical trajectory is applied to guide the missile. In terms of this, the velocity control schematic and model can be built, and the velocity control law and control parameters will be implemented including the integration design for velocity control and trajectory control for a good performance of the coning motion missile system.