Novel auxiliary error compensation design for the adaptive neural control of a constrained flexible air-breathing hypersonic vehicle
Introduction
Owing to many potential applications in both civilian and military fields, the research on air-breathing hypersonic vehicles (AHVs) has received considerable attention [1], [2], as witnessed by the success of NASA’s scramjet-powered X-43A and X-51A. However, the flight control design, as the key issue of making AHV feasible and efficient, still presents numerous challenges due to the complexity of the vehicle dynamics, the extreme level of couplings between the propulsion system and the airframe dynamics, as well as the presence of noticeable flexible effects [3], [4].
Because of the extreme complexity of the vehicle dynamics, only the longitudinal dynamic model of AHV has been extensively studied [5], [6], On the basis of which, various control schemes such as neural control [7], [8], predictive control [9], fault-tolerant control [10], [11], fuzzy control [12], active robust control [13] and sliding mode control [14] have been reported. In [15], the dynamic model of a flexible AHV (FAHV) is converted into a linearly parameterized formulation, for which the dynamic inversion based back-stepping control is addressed. Though back-stepping design possesses excellent superiority in handling unmatched system uncertainty, it requires the tedious analytic computations of time derivatives of virtual controllers. To deal with such problem, advanced technologies such as sliding mode differentiator [16] and dynamic surface control (DSC) [17], [18], [19], [20] are applied. Besides, it is well known that the robustness of back-stepping control needs to be further enhanced via special approaches such as parameter projection [15], [21] and nonlinear disturbance observer (NDO) [22], [23].
On the other hand, neural approximation has been proved to be a powerful tool for improving the controller׳s uncertainty attenuation ability [24], [25], [26], [27]. On the condition that the unknown function and control gain approached by neural networks (NNs) are strictly positive, two different NNs are introduced to estimate such uncertainties [28]. In [29], the vehicle dynamics are decomposed into two functional systems namely velocity subsystem and altitude subsystem. Moreover, for each subsystem, only one NN is needed [29]. Differently, the NN is employed to estimate the previously developed back-stepping control law instead of the model nonlinearity where a nonsingular direct neural controller is constructed for AHV [3]. It is worth mentioning that the computational ability of computer and hardware is limited. Thereby the minimal-learning-parameter (MLP) method is utilized to regulate the maximum norm of ideal weight vectors rather than their elements while less online parameters are required to be tuned [30].
From a practical perspective, one open problem is that the signal provided by the designed control law cannot be implemented because of actuator constraint [30], [31], [32]. Noting that the closed-loop system may encounter performance limitations or even lose stability theoretically if the actuator is saturated, the additional system proposed in [33] is introduced to compensate the desired control law [30], [31]. Though the auxiliary error compensation strategy of that study is little complicated, the boundedness of velocity and altitude tracking errors can still be achieved when the physical limitations on actuators are in effect [30], [31]. In [21], [28], a simple auxiliary system established in [34] is adopted to modify the initial tracking error and the desired controller. However, when the control inputs constraints are in effect, the boundedness of velocity and altitude tracking errors cannot be achieved theoretically in that studies.
Motivated by the above discussions, in this paper, a novel auxiliary error compensation strategy is proposed for the adaptive neural control of a constrained FAHV to provide robust tracking of velocity and altitude reference trajectories. By viewing the flexible effects as system uncertainties, the longitudinal dynamics of FAHV are decomposed into two functional subsystems namely the respective velocity subsystem and altitude subsystem. To ensure the controller׳s robustness, NN is applied to estimate the lumped uncertainty of each subsystem. The MLP scheme is utilized to adjust the norm of NN’s weight vector. In this way, less adaptive parameters are required for neural approximation. Especially, novel auxiliary systems are exploited to cope with the problem of control inputs constraints. Finally, simulation results are presented to demonstrate the efficacy of the proposed control methodology. The special advantages of the approach proposed herein include:
- 1.
With the novel design of auxiliary system, the effect of actuator saturation is eliminated, and the boundedness of velocity and altitude tracking errors can still be guaranteed even when the physical limitations on actuators are in effect.
- 2.
The computational cost is quite low since less NNs and adaptive parameters are needed to be regulated online in this paper.
The remainder of this paper is organized as follows. Section 2 presents the FAHV model. Adaptive neural controllers are addressed in Section 3. Numerical simulation studies are made in Section 4 and the conclusions are proposed in Section 5.
Notation: The notations used in this paper are fairly standard. Throughout this paper, represents n-dimensional Euclidean space; denotes the set of all real numbers; means the norm of a vector; stands for the absolute value of a scalar.
Section snippets
Model description
The FAHV model adopted in this study is developed by Bolender and Doman [5]. Fig. 1 shows the longitudinal sketch and force map for a FAHV. It is assumed that the FAHV presented has unity depth into the paper and that air behaves as if it is a perfect gas (i.e., the ratio of specific heats is a constant) [15]. The aerodynamic forces acting on the vehicle are synthesized as a lift component, denoted by L, and a drag component, represented by D. L and D are respectively vertical to and converse
Adaptive neural controller design
It is easy to note that V is mainly related to Φ and h is mainly affected by δe since T affects V and δe has a dominant contribution to h change [3]. Therefore, the dynamic model of FAHV is reasonable to be divided into the velocity subsystem (i.e., Eq. (1)) and the altitude subsystem (i.e., Eqs. (2), (3), (4), (5)) [28], [29], [30], [31], [32]. In what follows, all the rigid-body states V, h, γ, θ and Q are assumed to be measured and that they will be used for controller design.
Simulation results
The proposed controllers (19), (26), (31), (38) with adaptive laws (20), (27), (39), first-order filters (28), (32), and additional systems (16), (35) are tested in MATLAB/Simulink® environment. The aerodynamic coefficients and model parameters of the utilized dynamic model are referred from [6]. Let =4223.44, =4223.55, ω1=16.021 rad/s, ω2=19.582 rad/s, ζ1=ζ2=0.02, k1=k2=1.0084. The reference trajectories of velocity and altitude are smoothened via the following filters:
Conclusions
An adaptive neural controller is exploited for the longitudinal dynamics of FAHV subject to actuator saturation and parametric uncertainties. Firstly, the vehicle dynamics are decomposed into the respective velocity subsystem and altitude subsystem that are controlled separately. Then a dynamic inversion based velocity controller and a back-stepping based altitude controller are addressed. The first-order filter is applied to solve the problem of “explosion of terms”. By employing the RBFNN to
Acknowledgments
The authors would like to express their sincere thanks to anonymous reviewers for their helpful suggestions for improving the technical note. This work was supported by the Aeronautical Science Foundation of China (Grant no. 20130196004).
Xiangwei Bu was born in 1987. He received his B.S. and M.S. degrees from Air Force Engineering University in 2010 and 2012, respectively. He is currently a Ph.D. candidate in Air and Missile Defense College, Air Force Engineering University. His research interests mainly lie in hypersonic vehicle modeling and nonlinear control, back-stepping control, neural control, prescribed performance control, as well as tracking differentiator and nonlinear disturbance observer design.
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Xiangwei Bu was born in 1987. He received his B.S. and M.S. degrees from Air Force Engineering University in 2010 and 2012, respectively. He is currently a Ph.D. candidate in Air and Missile Defense College, Air Force Engineering University. His research interests mainly lie in hypersonic vehicle modeling and nonlinear control, back-stepping control, neural control, prescribed performance control, as well as tracking differentiator and nonlinear disturbance observer design.
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