Mathematical Problems in Engineering

Volume 2018, Article ID 6968526, 13 pages

https://doi.org/10.1155/2018/6968526

## Constrained Adaptive Neural Control for Air-Breathing Hypersonic Vehicles without Backstepping

Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China

Correspondence should be addressed to Shili Tan; moc.361@nhclsnat

Received 29 April 2018; Revised 13 July 2018; Accepted 30 July 2018; Published 6 August 2018

Academic Editor: Javier Moreno-Valenzuela

Copyright © 2018 Shili Tan 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

An adaptive neural control scheme without backstepping is proposed for the air-breathing hypersonic vehicle subject to input constraints. To estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, two radial basis function neural networks (RBFNNs) are constructed. Since the complex backstepping design steps are not needed, the proposed control structure is quite concise and the problem of “explosion of terms” is avoided. Moreover, a novel nonlinear auxiliary system is constructed to solve the problem of input constraints. The advantage of the proposed auxiliary system is that its high-order form has good performance and the parameter tuning is relatively easy. Simulation results show that the designed controllers achieve stable tracking of reference commands with good performance.

#### 1. Introduction

Air-breathing hypersonic vehicle (AHV) has attracted wide concern, because it offers a promising and economic access to near space [1, 2]. The design of the flight control system is an important issue to ensure that the AHV can accomplish its intended task. Due to the complex flight environment, the highly nonlinear dynamic behavior, and the large uncertainty [3, 4], it is a huge challenge to design the controllers with satisfactory performance.

Recently, there are already numerous publications on the design of the flight control system for AHV. A robust linear output-control strategy is investigated for AHV, achieving the stable tracking of reference trajectories [5]. Since the feedback linearization (FL) remains based on the accurate model which is difficult to be obtained, the robustness of FL is weak when there is the model uncertainty. To enhance the robustness with respect to model parameters, a nonfragile control scheme is studied based on the linear parameter-varying (LPV) model [6, 7]. Also the sliding mode control is an effective nonlinear control method and is widely studied for AHV subject to the uncertainty. A key technique for sliding mode control is how to solve the chattering problem [8]. To reduce the chattering, the quasi-continuous high-order mode controllers are exploited, effectively reducing the flutter of the AHV [9]. By introducing the nonlinear disturbance observers (NDO), the total uncertainty is estimated online, and the robustness of the controller is guaranteed [10, 11]. Noticing that the robustness of the designed controllers is closely related to the NDO, an novel NDO based on hyperbolic sine function is investigated and the robustness is enhanced [12]. Moreover, much concern has been on the intelligent controls such as the fuzzy logical system (FLS)[13–15] and the neural network (NN)[16, 17]. The minimal learning parameter (MLP) scheme [13, 14, 16] and the composite learning-based parameter adaptive law [15, 17] are constructed to update the FLS/NN weights. By estimating the model uncertainty accurately and compensating for the controllers, the robustness of the proposed control scheme is ensured.

Backstepping control is regarded as an effective tool to solve the problem of unmatched uncertainty [18, 19]. Noting that the traditional backstepping control needs the tedious analytical calculation of derivatives, the problem of the “explosion of terms” is inevitable. Therefore, the dynamic surface control [20, 21], command filtered approaches [22], and tracking differentiator (TD)[12] are proposed to solve the problem of “explosion of terms”. Though good results in terms of command tracking can be gained by the above method, the whole control system becomes cumbersome and it is unfavorable to the design of the controller. To simplify the design process of the controllers, the controllers without the virtual control laws are creatively designed [16]. However, the expression of the final actual controller becomes complex with a large number of state variables and design parameters. Motivated by the above studies [13, 14, 16, 17], the radial basis function neural networks (RBFNNs) with MLP algorithm are constructed to approximate the model uncertainty. Different from the previous studies, the control scheme without backstepping is proposed by introducing the high-order differentiator. Thus there is no need of complex strict-feedback form and virtual controllers. Also only two RBFNNs are needed to estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, respectively.

From a practical perspective, the control inputs are limited to a certain range. The designed controller without considering the input constraints may suffer performance degradation or even instability appearance. Therefore, it is necessary to take the input constraints into consideration. The Nussbaum function based auxiliary design system is employed for the strict-feedback systems and the nonstrict-feedback systems, preventing the outputs from violating the constraints [23, 24]. Also a linear auxiliary system is adopted for the backstepping scheme [25] and the disturbance observer-based feedback linearization control [26]. Moreover, an novel hybrid auxiliary system is designed to solve the problem of the nonsymmetric nonlinear input constraints [27], and it has been applied to the AHV systems [28, 29] and the flexible spacecraft systems [30]. However, these antiwindup methods can only be applied in some specific control scheme such as the backstepping control. Regardless of the linear auxiliary system [25, 26] or the hybrid auxiliary system [27–30], it is not suitable to be extended to the high-order forms. Since a high-order auxiliary system is required for the altitude subsystem under the proposed control scheme, a novel nonlinear auxiliary system is constructed.

The remainder of the paper is organized as follows: the inputs constrained AHV model and the description of RBFNN are presented in Section 2. Then, the main design process of the controllers is provided in Section 3. Next, Sections 4 and 5 show the stability analysis and simulation, respectively. Finally, the conclusions are summarized in Section 6. The main contributions of this paper are summarized as follows.

Different from the adaptive neural control schemes in [16, 17, 28], there is no need of complex backstepping design steps in this paper. Thus the proposed control structure is quite concise and the problem of “explosion of terms” is avoided.

Compared with the linear auxiliary system [25, 26] and the hybrid auxiliary system [27–30], the nonlinear auxiliary system proposed in this paper can be extended to the high-order forms and applied to the altitude subsystem under the proposed control scheme. Moreover, the parameter tuning for the proposed auxiliary system is relatively easy.

#### 2. Inputs Constrained AHV Model and RBFNN Description

##### 2.1. AHV Model

Based on the X43A aircraft developed by NASA, M. Bolender et al. presented an AHV model [31]. This mode has been widely used in the design of flight control systems. Then, Parker simplified the model by ignoring the weak couplings and slow dynamics [32]. The dynamic equations are formulated as where , , , , and denote the velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively. The flexible states are the first two vibrational modes. is the mass, the moment of inertia, the damping ratio, and the natural frequency. , with the constrained beam coupling constant for . The thrust , lift , drag , pitching moment , and generalized forces are given as follows [17]:withwhere and are the fuel equivalence ratio and the elevator deflection, respectively; and denote the reference area and the aerodynamic chord, respectively; is the dynamic pressure, the air density, and the thrust to moment coupling coefficient.

##### 2.2. Input Constraints

Two inputs of the AHV are . In engineering practice, the inputs cannot be any desired values, but must be limited within a certain range. must be limited to a certain range to ensure the normal operation of the scramjet. Similarly, the limitation of is determined by the physical structure of the elevator. The above input constraints can be described aswhere and are the inputs to be designed; the “max” and “min” denote the maximum and minimum value in the constrained range, respectively.

##### 2.3. Description of RBFNN

The RBFNN has been widely used because it can approximate any continuous function to an arbitrary precision [16, 33]. Assume that the number of RBFNN nodes is . The mapping from the input layer to the output layer is where denotes the weight vector; , is chosen aswhere denotes the center vector and is the width vector.

Assume that is the function to be approximated. There exists such thatwhere denotes the ideal weight vector; is the approximation error.

#### 3. Controller Design

The control target is selected as developing a controller so that the outputs can follow the reference commands . Considering that is steered by to a large degree and is mainly affected by , we naturally decompose the AHV model into the and the . In what follows, two constrained adaptive neural controllers will be separately designed for the two subsystems such that and .

*Step 1 (designing control law for the ). *The dynamic equation of is described aswhere contains large uncertainty and is approximated by one RBFNN:where is the ideal weight vector; ; , has same formulation to (14); is the approximation error.

The error between and is expressed byDifferentiating by time , we have Inspired by the MLP method [13, 14, 16], the neural controller is designed aswhere , are design parameters; , with the estimation . is decided by the adaptive law as follows:where is a design parameter.

Noting that is constrained as (11), a novel auxiliary system is designed aswhere , , are design parameters; is the state variable.

Define the compensated velocity tracking error asDifferentiating by time and invoking (19) and (22), we have The desired control input is determined by the chosen control law: The controller structure of the V-subsystem is presented in Figure 1.