Discrete Dynamics in Nature and Society

Volume 2016, Article ID 5098784, 9 pages

http://dx.doi.org/10.1155/2016/5098784

## A Novel Optimal Control Method for Impulsive-Correction Projectile Based on Particle Swarm Optimization

^{1}Nanjing University of Science and Technology, Nanjing 210094, China^{2}China Academy of Engineering Physics, Mianyang, Sichuan 621900, China

Received 25 May 2016; Revised 9 September 2016; Accepted 5 October 2016

Academic Editor: Seenith Sivasundaram

Copyright © 2016 Ruisheng Sun et al. 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 parametric optimization approach based on a modified particle swarm optimization (PSO) to design a class of impulsive-correction projectiles with discrete, flexible-time interval, and finite-energy control. In terms of optimal control theory, the task is described as the formulation of minimum working number of impulses and minimum control error, which involves reference model linearization, boundary conditions, and discontinuous objective function. These result in difficulties in finding the global optimum solution by directly utilizing any other optimization approaches, for example, Hp-adaptive pseudospectral method. Consequently, PSO mechanism is employed for optimal setting of impulsive control by considering the time intervals between two neighboring lateral impulses as design variables, which makes the briefness of the optimization process. A modification on basic PSO algorithm is developed to improve the convergence speed of this optimization through linearly decreasing the inertial weight. In addition, a suboptimal control and guidance law based on PSO technique are put forward for the real-time consideration of the online design in practice. Finally, a simulation case coupled with a nonlinear flight dynamic model is applied to validate the modified PSO control algorithm. The results of comparative study illustrate that the proposed optimal control algorithm has a good performance in obtaining the optimal control efficiently and accurately and provides a reference approach to handling such impulsive-correction problem.

#### 1. Introduction

Impulsive-correction projectile, as a type of precision-guided munitions, produces directive force for trajectory control, which results in quick-maneuvering and precise-correcting via small impulsive rockets equipped around the cross section of the airframe. This has also attracted significant interests of research due to its short-time response, high ratio of effectiveness-consumption, and simpleness of guidance [1–5]. Considerable work on the design of magnitude of impulse thrust and its position arrangement [4], real-time computational algorithm [6], flight stability of impulsive-correction trajectory [7], and so on has been developed. For instance, Liu and Willms [8] provided a novel approach to obtain necessary and sufficient conditions for impulsive controllability of continuous linear dynamics exercising discrete-time actions for applying into maneuvers of spacecraft. Rempala and Zabczyk [9] developed a simple and direct proof of a version of Blaquiere’s maximum principle for deterministic fixed-time impulsive control problems. Prado [10] considered the problem of optimal maneuvers to insert a satellite in a constellation in the application of two impulses to the satellite and the objective to perform this maneuver with minimum fuel consumption. Nevertheless, due to its nondifferentiable and discontinuous characteristics of impulsive control and its limit of finite working number of impulses, that is, discrete and finite-times correction, it is impossible to implement continuous control like conventional aerodynamic-fin control for the airframe.

Nowadays, numerical optimization controls are categorized into two different classes with their own advantages and characteristics, that is, direct method based on mathematical programming and parameterization of state and control histories and indirect method grounded on solution of two-point boundary value problem (TPBVP) using optimal control principle [11–13]. They are of probability to be applied for achieving an optimal performance of such problem. In general, direct method is more popular in application than indirect method due to the difficulty to obtain analytical solutions of indirect approach for nonlinear complex system. Hp-adaptive pseudospectral method, as one kind of the most popular and efficient direct methods, is combining Legendre pseudospectral method [14, 15] and Hp-adaptive method [16], which discretizes state variables and control variables on a series of Legendre-Gauss-Lobatto (LGL) points. What limits its application in impulsive system is that impulsive control variable is nondifferentiable with flexible-time intervals, which does not satisfy the Karush-Kuhn-Tucker (KKT) conditions.

To tackle such problems effectively, swarm intelligence- (SI-) based methods among those evolutionary algorithms [17–20] such as genetic algorithms (GAs), simulated annealing (SA), and ant-colony optimization are becoming more popular due to their speed and accuracy qualities. They are inspired by natural phenomena, for example, the behavior of groups of birds, ant colonies, herds of animals, and even social connections between human beings [19]. As a type of SI-based methods, the particle swarm optimization (PSO), primarily introduced in 1995 by Eberhart and Kennedy [21] and then extended by other researchers [22, 23], has been showing brilliant effect in optimizing discontinuous problems for its briefness in concept, easiness to implement, and high computational efficiency. Some mathematical approaches also make contributions to the PSO. For example, Couceiro and Sivasundaram provided a modified PSO algorithm to overcome traditional PSO algorithm’s drawbacks by considering a fraction calculus approach [24]. Pires et al. proposed a novel method for controlling the convergence rate of the PSO algorithm using fractional calculus concepts and observed the relationship between the fractional order velocity and the convergence of the algorithm [25]. Reference [26] interpreted PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE) and proposed a class of modified PSO iteration methods based on local attractors of the SODE which behaved differently for different problems. Unlike the traditional optimization techniques, the PSO does not rely on the rigid mathematical characteristics (continuity, derivability) of the optimization problem itself and constraints in the optimization process. Reference [27] developed a PSO approach with a punish function to design the static parameters such as the working number of impulses, the magnitude and axial eccentricity of each single impulse thrust, and the fin oblique angle for impulsive-correction projectile. Yang et al. [28] presented a new approach to a fuel-optimal impulsive control problem of the guided projectile by using an improved PSO technique; this method did not consider the optimal working modes and flexible-time intervals in detail. Reference [29] proposed a new method for solving an optimal control problem applied to spacecraft reentry trajectory by using a PSO method and avoiding the calculations needed in the common analytical approaches.

Therefore, the aim of this paper is to present a new method for solving an optimal impulsive control problem with discrete, flexible-time interval, and finite-times correction using a modified PSO method, where the Hp-adaptive pseudospectral method has not been fully solved. The remaining of this paper is organized as follows. Section 2 deduces the mathematic model of impulsive-correction projectile system and states the optimal impulsive control problem. In Section 3, a modification based on the basic PSO is presented in detail for the impulsive optimization control design and the structure and parameter design of the controller, which solve the optimal setting of impulsive control for impulsive-correction projectile. Moreover, a suboptimal control and guidance law based on PSO technique are developed for the real-time consideration of the online design in practice. In Section 4, a simulation case is shown to demonstrate impulsive-correction projectile by implementing the modified PSO algorithm. In order to validate the performance of the impulsive control design, a specific nonlinear flight dynamic model coupled with different conditions such as optimal and suboptimal algorithm is also carried out. Finally, some conclusions are presented in Section 5.

#### 2. Problem Statement

For the convenience of discussion, the motion of impulsive-correction projectile in longitude plane is chosen in this paper under assumption that the Earth is flat and motionless. The flight dynamic equations of impulsive-correction projectile arewhere the state variables include true airspeed of the projectile , trajectory inclination angle , pitch angle , pitch angular ratio , horizontal position , height position , and mass of the projectile . and are the magnitude and axial eccentricity of the impulsive thrust, respectively; means the pitching moment; is the moment of inertia about -axis; denotes the mass flow rate; represents the attack of angle; and the drag force and lift force are both the functions of dynamic pressure , reference area , and the drag and lift coefficients , as shown in (2).

Considering the wind disturbance during the flying period of the projectile, additional equations can be governed aswhere the subscript means the relative movement between the projectile and the air stream. (+ for downwind and − for upwind) is the horizontal velocity of wind during the flight, and Mach number is the function of and sonic speed .

Compared with the time-of-flight of the entire trajectory, the working time of the impulse is so transient that it can be treated as instantaneous mutation. Meanwhile, in consideration of low-cost design, impulsive rocket is of open loop, which cannot change the magnitude of its thrust force; that is, impulsive control only stands two statuses:where denotes the constant magnitude of the impulsive force (see Figure 1) and represents the flag of the impulse actuated with order number , to the contrary . Here is the total number of the impulses equipped.