Complexity

Volume 2018, Article ID 9098151, 16 pages

https://doi.org/10.1155/2018/9098151

## Improved Hybrid Fireworks Algorithm-Based Parameter Optimization in High-Order Sliding Mode Control of Hypersonic Vehicles

National Key Laboratory of Science and Technology on Multispectral Information Processing, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Lei Liu; nc.ude.tsuh@ieluil

Received 29 December 2017; Accepted 31 January 2018; Published 4 March 2018

Academic Editor: László T. Kóczy

Copyright © 2018 Xiaomeng Yin 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

With respect to the nonlinear hypersonic vehicle (HV) dynamics, achieving a satisfactory tracking control performance under uncertainties is always a challenge. The high-order sliding mode control (HOSMC) method with strong robustness has been applied to HVs. However, there are few methods for determining suitable HOSMC parameters for an efficacious control of HV, given that the uncertainties are randomly distributed. In this study, we introduce a hybrid fireworks algorithm- (FWA-) based parameter optimization into HV control design to satisfy the design requirements with high probability. First, the complex relation between design parameters and the cost function that evaluates the likelihood of system instability and violation of design requirements is modeled via stochastic robustness analysis. Subsequently, we propose an efficient hybrid FWA to solve the complex optimization problem concerning the uncertainties. The efficiency of the proposed hybrid FWA-based optimization method is demonstrated in the search of the optimal HV controller, in which the proposed method exhibits a better performance when compared with other algorithms.

#### 1. Introduction

Hypersonic vehicles (HVs) have attracted increasing interest given their characteristics of high speed and excellent cost effectiveness to access the space. HVs usually fight in near space at a high speed, in which the aerodynamic properties are difficult to predict [1]. Additionally, owing to the peculiar structure of HVs, the couplings related to aerodynamics, propulsion, and structural dynamics are strong, and this makes HV sensitive to uncertainties [2]. In this study, we focus on the efficacious control design of nonlinear HV dynamics given that uncertainties are randomly distributed.

As members of sliding mode control methods [3–5], high-order sliding mode control (HOSMC) methods [6–8] exhibit strong robustness and a reduced chattering effect while dealing with uncertainties. For example, Zhang et al. [8] proposed a quasi-continuous HOSMC for HV to effectively alleviate the chattering phenomena. In addition to the chattering effect, several design requirements also should be considered for practical HV control under the effects of uncertainties. The priority is guaranteeing the stability. Furthermore, in order to ensure a satisfactory control performance, high-accuracy tracking of trajectory commands and lower fuel consumption are desired. However, when uncertainties are involved in the nonlinear control structure of HV, it is a challenge to adjust design parameters to reach a satisfied level of tracking performance. Two problems have appeared because of introducing uncertainties into the HOSM control of HV.

The first problem is that the modeling of the relation between the design parameters and the HV tracking performance under the effect of uncertain parameters is complex. Dealing with uncertainty in a probabilistic way, stochastic robustness analysis (SRA) was first proposed by Stengel and Ray [9], and it is an effective method to evaluate the extent to which the specified design requirements are satisfied. A cost function for SRA is formulated to estimate the likelihood that the design requirements are not satisfied. Subsequently, the design parameter space is searched to minimize the cost function to obtain the optimal performance in the presence of uncertainties [10]. Cao et al. [11] optimized the HV controller parameters by using SRA and hybrid PSO algorithm. However, only the dynamic response indices of step command were concerned in the cost function for SRA [11–13]. In order to achieve a desired tracking performance despite uncertainties, it is necessary to introduce appropriate indices that characterize the command tracking process and corresponding indicator functions into the optimization problem modeling of HV.

The second important problem in the HOSM control of HV involves solving the optimization problem. Conventional optimization methods, such as the gradient search method, are no longer suitable given that the partial derivative of the cost function in SRA is difficult to obtain. For complex optimization problem involving uncertainties, a high efficiency computational intelligence optimization algorithm is required to determine the optimal controller parameters of HV to achieve a satisfied level of tracking performance under the influence of uncertainties. Nowadays, various computational intelligence techniques [14, 15], such as genetic algorithm (GA) [16], particle swarm optimization (PSO) [17], and differential evolutionary (DE), have been proposed for complex optimization problems with the development of computation technology.

Among computational algorithms, the fireworks algorithm (FWA) is a relatively new swarm intelligence-based algorithm proposed by Tan and Zhu [18]. It simulates the process of fireworks explosion, in which the “good” fireworks generate more sparks in smaller explosion areas. Numerical experiments indicated that FWA converges to a global optimum with a smaller number of function evaluations than PSO and GA [19]. Li et al. [20] proposed an adaptive fireworks algorithm (AFWA) in which the explosion amplitude of fireworks that fails to produce a better spark increases. To improve interaction of solutions, hybrid algorithm of FWA-DE was developed by Zheng et al. [21]. Zhang et al. [22] proposed an improved FWA by enhancing fireworks interaction. With respect to improvements in the FWA [20–23], it is recognized that the diversification mechanism of FWA does not utilize more information on other qualified solutions in the swarm. Therefore, with respect to the HV control under uncertainties that are randomly distributed, it is necessary to develop an improved FWA with enhanced solutions interaction to effectively solve the complex optimization problem of searching for the optimal controller.

In this study, an improved hybrid FWA-based parameter optimization method is proposed for HV control to achieve an excellent tracking performance in the presence of uncertainties. The main contributions are as follows:

The uncertainties that are randomly distributed are considered in the modeling phase via SRA. The cost function evaluating the probability of design requirements violation is formulated to model the complex relation between design parameters and tracking performance of the uncertain HV system. Appropriate indices of the command tracking response are developed.

A hybrid FWA to search for the optimal design parameters is proposed for the complex optimization problem involving uncertainties to satisfy design requirements with high probability. The introduction of the hybrid FWA into SRA effectively optimizes the tracking performance of the nonlinear HV system under uncertainties.

This study is organized as follows: In Section 2, the optimization problem in the HOSM control of HV is introduced. In Section 3, the complex relation between design parameters and HV performance under uncertainties is modeled. Section 4 proposes a new hybrid FWA to determine the optimal parameters of HV. Section 5 investigates the global convergence of the proposed hybrid FWA, and the simulation and comparison results are demonstrated. A few conclusions are made in Section 6.

#### 2. HOSM Control Structure of HV with Uncertainties

The control-oriented model of a generic hypersonic vehicle (HV) is described by [24]. An inverse-square-law gravitational model and centripetal acceleration are considered, and the dynamic differential equations for velocity , altitude , flight-path angle , angle of attack , and pitch rate of HV are as follows:withwhere is the lift, is the drag, is the thrust, and is the pitching moment. , , , , and denote the mass, radial distance, radius of the Earth, gravitational constant, and density of air, respectively. Additionally, , , and denote the reference area, mean aerodynamic chord, and the moment of inertia about -body axes, respectively. denotes the elevator deflection, and denotes the engine throttle setting.

The thrust in (2) is provided by the engine dynamics, and this is represented as follows [10]:where denotes the engine throttle setting command. It is adopted that and for proper modeling of engine dynamics.

In order to guarantee the robustness of the HV flight control system, the parametric uncertainties in (1)-(2) are considered as follows:where the uncertainties , , , , , and are bounded.

HV system (1) with engine dynamics is highly nonlinear. The relationship between input variables and the output variables is apparently expressed by the feedback linearization method [10]. We differentiate three times and differentiate four times, and we obtain the following expressions:where , , , and , .

In order to force the velocity and altitude to track the time-varying commanded output , we define the velocity sliding tracking error and the altitude sliding tracking error as and , respectively. Based on (5) and (7), we havewhere the formulations of , , , , , and are the same as those in [10].

As stated in [10], the matrix in (9) is nonsingular over the entire flight envelope of HV, so (9) is decoupled with the auxiliary control input as follows:

A previous study [6] indicates that if appropriate control parameters are designed, then the finite time stabilization of system (9) is guaranteed by the quasi-continuous HOSMC and , and this is given as follows:with

The HV control structure based on HOSM is shown in Figure 1.