Mathematical Problems in Engineering

Volume 2015, Article ID 293480, 11 pages

http://dx.doi.org/10.1155/2015/293480

## Robust Adaptive Attitude Control for Airbreathing Hypersonic Vehicle with Attitude Constraints and Propulsive Disturbance

^{1}College of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China^{2}College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 16 July 2015; Accepted 26 November 2015

Academic Editor: Marek Lefik

Copyright © 2015 Jian Fu 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

A robust adaptive backstepping attitude control scheme, combined with invariant-set-based sliding mode control and fast-nonlinear disturbance observer, is proposed for the airbreathing hypersonic vehicle with attitude constraints and propulsive disturbance. Based on the positive invariant set and backstepping method, an innovative sliding surface is firstly developed for the attitude constraints. And the propulsive disturbance of airbreathing hypersonic vehicle is described as a differential equation which is motivated by attitude angles in this paper. Then, an adaptive fast-nonlinear disturbance observer for the proposed sliding surface is designed to estimate this kind of disturbance. The convergence of all closed-loop signals is rigorously proved via Lyapunov analysis method under the developed robust attitude control scheme. Finally, simulation results are given to illustrate the effectiveness of the proposed attitude control scheme.

#### 1. Introduction

In recent years, a new kind of aerospace vehicle, which is called airbreathing hypersonic vehicle (AHV), has attracted the considerable attention in both military and civil communities. Comparing with conventional flight vehicles, this new aerospace vehicle can sustain flight for a long time with high flight speed and cover a large envelope [1, 2]. Due to the strong disturbance and fast changed pneumatics parameters, it is still a challenging task to design a robust AHV control system. Therefore, considerable efforts have been made to develop an efficient flight controller for this kind of aerospace vehicles. In [3, 4], adaptive function link network control and fuzzy logical system were proposed for the aerospace vehicle with uncertainties. And baseline nonlinear model predictive controller was presented in [5] for AHV with nonvanishing mismatched disturbances. In [6], an adaptive fault-tolerant control strategy with input saturation is proposed for AHV flight system with actuator faults and external disturbance. However, the robust attitude control method should be further developed for the AHV with attitude constraints and propulsive disturbance.

In order to suppress the unknown external disturbance from changeable flight environment, many papers have focused on the disturbance observer of AHV flight control system [7–11]. A nonlinear disturbance observer based control (NOBC) method is proposed in [7] for the longitudinal dynamics of a generic airbreathing hypersonic vehicle under mismatched disturbance. In [8], the NDO technique is combined with Takagi-Sugeno (T-S) fuzzy linear model to design a disturbance observer based guaranteed cost fuzzy controller (GCFC) for the AHV flight systems. To achieve satisfactory tracking performance, a composite tracking controller is proposed for AHV using NDO and backstepping technologies [9]. However, there are few disturbance observer based control results for AHV with the propulsive disturbance. This kind of disturbance for AHV is usually motivated from the coupling between the scramjet propulsion system and attitude angles [10, 11]. Thus, this kind of disturbance will lead to a challenging problem for the attitude control of the AHV. In this paper, the propulsive disturbance is described as a differential equation which is motivated by attitude constraints in this paper. And a composite controller is proposed to handle this disturbance using adaptive NDO and backstepping technologies.

Furthermore, sliding mode control (SMC) is considered as an important method due to its high precision control, relative simplicity of control design, and high robust features with respect to system internal perturbations and external disturbances. Thus, the SMC and its application of AHV have been widely studied in [12–21]. A composite control treatment based on sliding model method is presented to the nonlinear longitudinal dynamics of AHV vehicles [12]. And dynamic terminal sliding mode technique is proposed for the robust control design of AHV [13, 14]. Considering the input constraints, a second-order terminal sliding mode control method is discussed for AHV flight system in reentry phase [15]. Meanwhile, an adaptive dynamic sliding mode control method is presented for the fault-tolerant control problem of AHV in [16]. However, most of the AHV control systems combining backstepping control with SMC have not considered the attitude constraints. Thus, it is necessary to design an innovative sliding mode control for the AHV using backstepping technique and considering the attitude constraints.

This work is motivated by the robust attitude control of AHV with attitude constraints and propulsive disturbance. The control objective is that the proposed robust control scheme can track a desired trajectory in the presence of the unknown propulsive disturbance and attitude constraints. An innovative sliding surface is firstly developed in this paper. And the invariant sets and saturation function are utilized in this surface for the attitude constraints of AHV. To guarantee the control effects under propulsive disturbance, the switching information from invariant-set-based sliding surface is included in the adaptive disturbance observer to increase the convergence speed. Therefore, the organization of the paper is as follows. Section 2 details the problem formulation, while main results are given in Section 3. Simulation results are presented in Section 4 to show the effectiveness of proposed robust adaptive backstepping attitude control for AHV. And some conclusion results are shown in Section 5.

#### 2. Problem Statement

##### 2.1. Model Description

To study the robust attitude controller, a nonlinear attitude motion model of AHV flight system is described as the following nonlinear affine nonlinear system [22–24]:where is the attitude angle vector of slow-loop states, is the body-axis angular rate vector of fast-loop states, is the deflection vector of control surfaces which are the system control inputs, is the system output, and the unknown disturbance is defined aswhere is the unknown propulsive disturbance which is coupling with the attitude angles and is the rest of unknown disturbance. The detailed expression of , , , and can be found in [24].

##### 2.2. Assumptions and Control Objective

In order to reduce the coupling effects between the scramjet propulsion system and attitude angles, the attitude constraints are defined aswhere is the attitude angle vector and is the desired output signal. When the attitude angle vector satisfies constraints (3), the propulsive disturbance will be a small value which can be neglected in the control design process. However, if exceeds the boundary of constraints (3), this disturbance will be increased significantly. Then, it is supposed that the disturbances and will satisfy the following assumptions.

*Assumption 1. *For all , there exists such that , .

*Assumption 2. *The propulsive disturbance can be described as the following differential equation:where , and we can obtain if . For the element , there exists an unknown positive constant such that , . The vector is a special designed sliding surface as shown in (22), and there exists while .

*Assumption 3. *The disturbance can be described as the following differential equation:where , and there exists a known positive constant such that , .

*Assumption 4. *The generalized matrix inverses of and are always existing for the nonlinear attitude motion model of the AHV, and the elements in matrixes , , , and are always continuous for the nonlinear attitude motion model of the AHV.

*Definition 1 (see [25]). *The set is said to be positively invariant (PI) for nonlinear system (1), if, for all , there exists the solution , .

#### 3. Robust Adaptive Backstepping Attitude Control for the AHV with Attitude Constraints and Propulsive Disturbance

##### 3.1. Preliminary

In this section, a few results are given to support the design process of robust adaptive backstepping attitude controller. Considering the standard backstepping control design method, we definewhere is a designed virtual control law. Then, considering (1), the derivatives can be written aswhere are the state errors of slow-loop states and are the state errors of fast-loop states. Invoking (6), the attitude constraints (3) can be rewritten aswhere and .

Considering the attitude constraints of AHV, the unidirectional auxiliary surfaces (UAS) for slow-loop and fast-loop states are utilized to design the invariant-set-based sliding mode controller. The detailed design process for these surfaces is given as follows.

*Step 1. *Considering the slow-loop system (7), we define the switching surfaces aswhere , , , , , , and is denoted by for brevity. The conditions and are given to guarantee the stability of switching surfaces and . And the condition is used to avoid the overlap of switching surfaces.

*Step 2. *Invoking the coefficients in (10), the unidirectional auxiliary surface can be designed aswhere where are the designed parameters which satisfy , , and , and the rest of coefficients are given as follows:where and . Then, the compact form of unidirectional auxiliary surfacewhere , , , , and is a constant vector.

As shown in Figure 1, the UAS can form a convex set which satisfies , and the expression of set is written asThe compact form of convex set can be written as , where implies , .