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.
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 [
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 [
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 [
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
To study the robust attitude controller, a nonlinear attitude motion model of AHV flight system is described as the following nonlinear affine nonlinear system [
In order to reduce the coupling effects between the scramjet propulsion system and attitude angles, the attitude constraints are defined as
For all
The propulsive disturbance
The disturbance
The generalized matrix inverses of
The set
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 define
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.
Considering the slow-loop system (
Invoking the coefficients
As shown in Figure
The convex set
Similarly, we can define the following switching surfaces for the fast-loop system (
Invoking the coefficients
For the points
See Appendix
For all
See Appendix
In the attitude control design, we combine the backstepping method, adaptive invariant-set-based sliding mode control, and the disturbance observer technique to design a robust adaptive backstepping attitude controller for the AHV system (
Invoking (
It is clear that
In this paper, the nonlinear disturbance observer which is employed to estimate
For the estimation error
To suppress the estimation error
Defining
Defining
The discussion for the stability of sliding mode surface
Substituting (
Choose the Lyapunov function candidate as
Substituting (
It is apparent that the first term on the right-hand side of (
Invoking (
Considering the error systems (
Invoking (
The discussion for the reaching phase for sliding surface is given in this part. Choose the Lyapunov function candidate as
Considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as
Considering the attitude motion dynamics (
According to (
Considering the stability of sliding surface as shown in Theorem
In this section, simulation results are given to illustrate the effectiveness of the proposed adaptive UAS-SMC schemes for AHV with attitude constraints. Suppose that the AHV vehicle lies in the cruise flight phase with the velocity 3000 m/s and flight altitude 30 km. The initial attitude and attitude angular velocity conditions are chosen as
According to the design steps in Section
The attitude responses of AHV system under SMC and UAS-SMC methods.
The state error responses under SMC + NDO and adaptive UAS-SMC + NDO.
The disturbance estimations under NDO [
Adaptive UAS-SMC + NDO attitude control input.
The attitude and state error responses are shown in Figures
In this paper, the adaptive UAS-SMC controller with nonlinear disturbance observer has been proposed for the AHV with the unknown propulsive disturbance and the attitude constraints. To handle the propulsive disturbance, a developed fast-nonlinear disturbance observer is proposed to estimate the propulsive disturbance using adaptive method. And an innovative sliding surface is firstly proposed for the attitude constraints with invariant set theory. Rigorous analysis has been given for the convergence of all closed-loop signals under the proposed control schemes. Simulation results show the effectiveness of the robust adaptive UAS-SMC scheme for the AHV. In the following study, the robust attitude control scheme can be further developed for the AHV with time-varying attitude constraints.
Take the point
From the definition of
Invoking (
Invoking (
Similarly, we have
Take
Invoking (
For every point
From the above discussion, function
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by National Natural Science Foundation (NNSF) of China under Grants 61374212, 61174102, 61304099, and 11402117; Postdoctoral Fund of Jiangsu Province 1401023; Pre-Research Foundation of General Equipment Department 9140C300305140C30140.