Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 374838, 14 pages
http://dx.doi.org/10.1155/2015/374838

Research Article

Adaptive Neural Back-Stepping Control with Constrains for a Flexible Air-Breathing Hypersonic Vehicle

Pengfei Wang, Jie Wang, Jianming Shi, Chang Luo, Shili Tan, and Qiyun Xu

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

Received 17 May 2015; Revised 25 October 2015; Accepted 8 November 2015

Academic Editor: Xinggang Yan

Copyright © 2015 Pengfei Wang 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.

The design of an adaptive neural back-stepping control for a flexible air-breathing hypersonic vehicle (AHV) in the presence of input constraint and aerodynamic uncertainty is discussed. Based on functional decomposition, the dynamics can be decomposed into the velocity subsystem and the altitude subsystem. To guarantee the exploited controller’s robustness with respect to parametric uncertainties, neural network (NN) is applied to approximate the lumped uncertainty of each subsystem of AHV model. The exceptional contribution is that novel auxiliary systems are introduced to compensate both the tracking errors and desired control laws, based on which the explored controller can still provide effective tracking of velocity and altitude commands when the actuators are saturated. Finally, simulation studies are made to illustrate the effectiveness of the proposed control approach in spite of the flexible effects, system uncertainties, and varying disturbances.

1. Introduction

Air-breathing hypersonic vehicles (AHV) are crucial because they may represent a more efficient way to make access to space routine or even make the space travel routine and intercontinental travel as easy as intercity travel. A key issue in making AHV feasible and efficient is the flight control design [1]. However, the flight control of AHV is still an open challenge due to its peculiarity of flight dynamics. It is worth noting that there exists strong coupling between the propulsive and the aerodynamic forces, which makes the aerodynamic characteristics of AHV very difficult to be estimated and measured [2, 3].

Recently, lots of efforts have been put into flight control for AHV. Linear control theory was widely employed for flight control design based on linearized model. As shown in [4], the robust control of AHV is studied by introducing a linear quadratic regulator (LQR) with stochastic robustness analysis. In [5], LQR controller is proposed for the linearization model based on any equilibrium points of flight envelope.

Nonlinear control scheme is also employed for AHV. Back-stepping has been proved to be a powerful tool for the tracking control of a large class of strict-feedback systems or pure-feedback ones [6–12]. Back-stepping design for nonlinear systems consists of a recursive design procedure, which breaks down the full system control problem into a sequence of designs for lower-order subsystems. A back-stepping controller is designed with multilayer online adaptive neural networks, which can provide good tracking performance [13]. In [14], a combination of novelty command filtered back-stepping technology and dynamic inversion methodology is adopted for designing a dynamic state-feedback controller that provides stable tracking of the altitude and velocity reference commands. However, it is well known that there exists a problem of “explosion of terms” in the traditional back-stepping design, which is caused by the repeated differentiations of virtual control laws. To solve this problem, dynamic surface control [15] and tracking differentiators [16] should be applied.

Since the aerodynamic characteristics of AHV are sensitive to the flight condition changes, they are difficult to be measured. Thus, the aerodynamic uncertainty needs to be dealt with at the control design as the main issues. Conventionally, the flight control is accomplished by feedback linearization with neural networks to deal with the uncertainties. Efforts in [17] approximate the unknown nonlinear functions by radial basis function networks and incorporating the dynamic surface technique into a neural network based adaptive control design framework. Reference [18] investigates the discrete time controller for the longitudinal dynamics of the hypersonic flight vehicle with throttle setting constraint. The controller is proposed by estimating the system uncertainty and unknown control gain separately with neural networks. The auxiliary error signal is designed to compensate the effect of throttle setting constraint.

Moreover, input constraint also cannot be ignored in practice since the outputs of actuators are constrained for physical limitations. If the input constraint is ignored, control systems may suffer from performance limitations or even lose stability [19–21]. Much literature theoretically focuses on the control problem with input constraint [22–26]. In practice, when input constraint occurs, aircraft body may change seriously even disintegrating. So it is necessary to research the control design problem with input constraint.

Motivated by the results of the previous studies, an adaptive neural back-stepping control approach is addressed for the longitudinal dynamical model of AHV. By viewing the flexible effects as system uncertainties, the longitudinal dynamics of AHV are decomposed into two functional subsystems, namely, the respective velocity subsystem and altitude subsystem. To ensure the controller’s robustness, NNs are applied to estimate the lumped uncertainty of each subsystem. Particularly, novel auxiliary systems are exploited to deal with the problem of control input constraint. Finally, simulation results are presented to demonstrate the efficacy of the proposed control methodology. The special advantages of the approach proposed herein include the following:

(1)The novel auxiliary systems are employed to eliminate the error between the desired control laws and actual control laws, which makes sure that the semi-globally uniformly bounded stability of closed-loop system can be still achieved even when the physical limitations are in effect.(2)The second-order reference model is designed for the precise estimation of the derivatives of virtual control laws, which predigests the design of controller.

The organization of paper is outlined as follows. Firstly, the longitudinal motion model and control-oriented model of an AHV are described in Section 2. Section 3 presents the design procedures of adaptive neural back-stepping controller. Then, simulation results are given in Section 4. Finally, brief concluding remarks end the paper in Section 5.

2. Problem Formulation

2.1. Longitudinal Dynamic Model of AHV

The model taken into consideration in this paper is developed by Bolender and Doman [27] for the longitudinal dynamics of an AHV. The flexible effects are included in these equations by modeling the vehicle as a single flexible structure with mass-normalized mode shapes.

Assuming a flat Earth and normalizing the span of an AHV to unit depth, the nonlinear motion equations are written as [27]

where the rigid body states , , , , and represent velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively; and are gravity constant and the radial distance from center of the earth; and are mass of vehicle and moment of inertia about pitch axis; , , , and represent thrust, drag, lift, and pitch moment, respectively; denotes the thrust moment arm; the flexible states denote the first three bending modes of the fuselage; and () mean the damping ratio and natural frequency of the mass-normalized generalized coordinates of the flexible structure.

The dynamic system of an AHV can be decomposed as velocity subsystem and altitude-related subsystem. Define the control inputs , which are fuel equivalence ratio and elevator angular deflection, respectively.

Remark 1. In [28], the controller design requires an auxiliary actuator as the canard. However, the presence of a canard is quite problematic for the vehicle configuration, as this control surface must withstand the expected high temperatures at hypersonic speed. Therefore, it is assumed that only the fuel-to-air ratio and elevator are the actuators available for controlling the vehicle.

Let

The thrust , drag , lift , pitch moment , and generalized forces () are defined as

where , , and stand for the respective dynamic pressure, reference area, and aerodynamic chord. , , , and are denoted as the fitting error of , , , and , respectively.

To be convenient for design of back-stepping, (1) can be rewritten as

with

The model errors are considered as

Remark 2. The flexible dynamics are taken as perturbations on the rigid body system, and their effects are evaluated in simulation. According to (1) and (4), would be asymptotic stability if , , and are bounded.

2.2. Control Objective

In practice, due to physical limitations, the outputs of the actuator are constrained. Input constraints studied in the paper include the constraint on fuel equivalence ratio, elevator deflection. The constraint on fuel equivalence ratio is imposed by the very nature of the propulsion system, which is required to maintain the conditions that sustain scramjet operation [29]. If the limit is violated, the thermal choking will occur. It could induce that engine unstarts which could jeopardize mission, vehicle, and its contents [30]. The constraints on elevator deflection and canard deflection are mainly imposed by the limits on control surface displacement. The above input constraint can be expressed as

where and denote the upper and lower bound of , respectively; and stand for the upper and lower bound of , respectively; and are the desired control inputs to be designed in the subsequent section.

A second-order reference model with amplitude, rate, and bandwidth limitation is introduced to deal with the input constraint. The structure of second-order reference model is given in Figure 1. , , and , , and are the designed actual control variables and virtual control variables. Define , , and , , and as the executable control laws and virtual control laws with the second-order reference model.

Figure 1: Second-order reference model.

2.3. Radial Basis Function NN Approximation

The radial basis function NN (RBFNN) will be introduced to approach the unknown functions and owing to its excellent performance and global approximation. It has been proved that RBFNN can approximate an arbitrary continuous function over a compact set to an arbitrary accuracy. RBFNN is formulated as with the input vector , the weight vector , the basis function vector , the node number , and the input number . The basis function is selected as the following Gaussian function:

where and stand for the center and width of Gaussian function, respectively.

Assumption 3. and are continuous real function. The unknown functions and can be designed as

where and are radial basis function vector; and denote the approximation error, and and ; and represent optimal weight vector, and are the estimation of and , respectively, and and are estimation error of and , respectively.

Lemma 4. According to

there exist

Lemma 5 (see [31]). The adaptive law of is designed as

where

then ; is positive constant.

3. Controller Design

The control objective pursued in this section is to develop an adaptive neural back-stepping controller for an AHV to provide robust tracking of velocity and altitude commands and . It is assumed that the rigid body states , , , , and are available for measurement. It is easy to note that the velocity is mainly related to and the altitude is mainly affected by since the thrust affects and has a dominant contribution to change in (1). In what follows, the respective control laws and will be designed to make and .

Assumption 6. Consider ; is positive constant. Define which stand for the estimation error of .

Lemma 7 (see [32]). If and , then ,

3.1. Controller Design for Velocity Subsystem

Differentiating the velocity track error with respect to time results in

with

To eliminate the error between and , a novel auxiliary system is introduced as

where is a design parameter.

The adaptive law of weight vector can be designed as

where , , and and are design parameters.

The adaptive law of is given as follows:

where and are design parameters.

The desired controller is introduced as

where and are positive constants to be designed.

Plugging (23) into (18) results in

The Lyapunov function can be designed as

Differentiating with respect to time results in

When and , ; when and , .

Invoking Lemma 7, we have

Substituting (27) into (26) results in

where

According to , we obtain

Invoking Lemma 4, it is deduced as

Combining (30)–(33), we have the expression of

3.2. Controller Design for Altitude Subsystem

Define the tracking errors of , , and , respectively, as

Differentiating (35)–(38) and invoking (5)–(8) yield

where

The auxiliary systems are introduced as

where stand for design parameters.

The adaptive laws of weight vector can be designed as

where and ; and are design parameters.

The adaptive laws of () are given as follows:

where and are design parameters.

The desired control laws , , , and are designed as

where ,    are positive constants to be designed.

Substituting (47)–(50) into (39)–(42) gives

The Lyapunov function can be designed as

Differentiating and invoking (49)–(52) yield

Invoking Lemma 5, we can derive

where .

The following inequalities hold:

where , , and .

Plugging (44)–(46) and (57)–(58) into (56) results in

where , , , , and , .

When and , ; when and , .

Invoking Lemma 4, we can derive

Combining (60)–(63), we have the expression of

3.3. Stability Analysis

The Lyapunov function can be designed as

Combining (34) and (60)–(63) and differentiating with respect to time result in

where .

Let

Define

where ; .

Let

If , , and , , do not belong to the compact set , , and , respectively, . , , , and are bounded. Further, also is bounded according to (66). By regulating the designed parameters, the radius of , , , and can be made arbitrarily small.

Theorem 8. Consider the system of (1) under Assumptions 3 and 6, auxiliary analytical system (20) and (44), controllers (23) and (47)–(50), adaptive laws (21) and (45), and upper bound estimation of system error (22) and (46). If (71) is satisfied, all the signals involved are bounded.

4. Simulation Results

Simulations are done to illustrate the effectiveness of the robust adaptive approximate back-stepping control scheme proposed in the previous section. The vehicle model parameters are referred to in [27]. The damping of flexibility is , , and vibrational frequencies are  rad/s,  rad/s, and  rad/s.

The aircraft is commanded to perform a cruising flight mission that the initial height is 25908 m and dynamic pressure is 95760 Pa. The initial trim conditions are designed as follows: , , , , ,  mkg0.5/m0.5,  mkg0.5/m0.5,  mkg0.5/m0.5, and  m/skg0.5/m0.5, .

The reference commands of velocity and altitude are generated via the following filters:

The control laws are limited by , , , , and . The bandwidth of , , , , and is limited by 20 rad/s, 20 rad/s, 30 rad/s, 40 rad/s, and 60 rad/s, respectively.

The design parameters of controller are proposed as follows: , , , , , and . The adaptive law of RBF is considered as follows: , , , , , , , , , and the upper bound of weight vectors , , , , , and .

4.1. Trajectory Tracking Simulation without External Disturbance

To show efficiency of the proposed control law, the following two cases are proposed.

Case 1. The track height command is a square wave signal, whose amplitude  m and period is 600 s.

Case 2. The track height command is a square wave signal, whose amplitude  m and period is 600 s.

Time responses of velocity, altitude, flight path angle, attack angle, pitch rate, fuel equivalence ratio, elevator deflection, and flexible states in two cases are given in Figures 2–11.

Figure 2: Response curves of velocity and velocity tracking error (Case 1).

Figure 3: Response curves of altitude and altitude tracking error (Case 1).

Figure 4: Time history of fuel equivalence ratio and elevator deflection (Case 1)

Figure 5: Response curves of flight path angle, attack angle, and pitch rate (Case 1).

Figure 6: The flexible states (Case 1).

Figure 7: Response curves of velocity and velocity tracking error (Case 2).

Figure 8: Response curves of altitude and altitude tracking error (Case 2).

Figure 9: Time history of fuel equivalence ratio and elevator deflection (Case 2).

Figure 10: Response curves of flight path angle, attack angle, and pitch rate (Case 2).

Figure 11: The flexible states (Case 2).

Time responses to altitude step change in Case 1 are presented in Figures 2–6. As shown in Figures 2 and 3, the velocity and altitude can track their commands timely and without any steady tracking errors. The control inputs and attitude angles are smooth and within their bounds, just as shown in Figures 4 and 5. The flexible states can reach equilibrium state, which can be observed from Figure 6.

Time responses to altitude step change in Case 2 are presented in Figures 7–11. It can be seen from Figure 9 that the saturation occurs in elevator deflection during simulation, but the controller still provides a good track characteristic according to Figures 7–10. Furthermore, the flexible states also can maintain stability, which can be observed from Figure 11.

4.2. Trajectory Tracking Simulation with External Disturbance

To illustrate the robustness to external disturbance of the control laws developed herein, two vertical wind disturbances whose amplitudes are 17 m/s and 14 m/s are added to the control system in 100 s and 300 s, respectively, and the sustaining time of wind disturbance is 10 s. The additional attack angle with wind disturbance can be expressed as

where and are horizontal additional attack angle and vertical additional attack angle, respectively; and are level wind speed and vertical wind speed, respectively.

The aircraft is commanded to perform a cruising flight mission that the initial height is 25908 m and dynamic pressure is 95760 Pa. When the simulation time , altitude follows the step command with 50 ft/s; when the simulation time , altitude follows the step command with 139 ft/s.

The simulation results are illustrated in Figures 12–15. It can be seen from Figure 14 that the saturation occurs in elevator deflection with wind disturbance in 100 s and 300 s, but the controller still provides a good track characteristic according to Figures 12–15, which indicate that the proposed adaptive neural back-stepping controller has a good performance on disposing input constraint under external disturbance.

Figure 12: Response curves of velocity and velocity tracking error.

Figure 13: Response curves of altitude and altitude tracking error.

Figure 14: Time history of fuel equivalence ratio and elevator deflection.

Figure 15: Response curves of flight path angle, attack angle, and pitch rate.

5. Conclusions

An adaptive neural back-stepping control design is presented in this paper for hypersonic vehicle with input constraint and uncertain aerodynamic parameters. To reduce the complexity of the controller design, the nonlinear model is divided into two subsystems, and the adaptive neural back-stepping controllers are developed for them, respectively. The neural network based radial basis function is designed to approximate the unknown system nonlinearity, which needs less knowledge of dynamic model. In order to eliminate the problem of “explosion of term,” a second-order reference model is designed for the precise estimation of the derivatives of virtual control laws. Expressly, the auxiliary systems are constructed to eliminate the saturation effect, where a novel auxiliary error compensation scheme is proposed. A comprehensive Lyapunov based stability analysis is provided to show that the control design achieves ultimate boundedness tracking for the hypersonic vehicle. Simulation results demonstrate that the proposed control strategy can provide satisfactory velocity and altitude tracking performance with input constraint and uncertain aerodynamic parameters.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Aeronautical Science Foundation of China (Grant no. 20120916006).

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