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

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.


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][7][8][9][10][11][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 2 Mathematical Problems in Engineering 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][20][21].Much literature theoretically focuses on the control problem with input constraint [22][23][24][25][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 semiglobally 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 backstepping controller.Then, simulation results are given in Section 4. Finally, brief concluding remarks end the paper in Section 5.

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] V where the rigid body states denote the first three bending modes of the fuselage; 0 <   < 1 and   > 0 ( = 1, 2, 3) 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  = [Φ,   ], 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  =  +  sin . ( The thrust , drag , lift , pitch moment , and generalized forces   ( = 1, 2, 3) 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 V =   (, , ,   ) +   (, , , Φ) Φ + 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 Φ,   , and  are bounded.

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 Φ max and Φ min denote the upper and lower bound of Φ, respectively;  max and  min stand for the upper and lower bound of   , respectively; Φ  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.Φ 0 ,  0  , and  0  ,  0  , and  0  are the designed actual control variables and virtual control variables.Define Φ,   , and   ,   , and   as the executable control laws and virtual control laws with the second-order reference model.

Radial Basis Function NN Approximation.
The radial basis function NN (RBFNN) will be introduced to approach the unknown functions   (z) and   (z) ( = , , ) owing to its excellent performance and global approximation.It has been proved that RBFNN can approximate an arbitrary continuous function over a compact set Ω z ⊆ R  to an arbitrary accuracy.RBFNN is formulated as   (z) with the input vector z ⊆ R  , the weight vector  ⊆ R  , the basis

Magnitude limiter
Rate limiter where c  = [ 1 ,  2 , . . .,   ]  and   stand for the center and width of Gaussian function, respectively.
Assumption 3.   () and   () ( = , , ) are continuous real function.The unknown functions   (z) and   (z) can be designed as where there exist Lemma 5 (see [31]).The adaptive law of θ is designed as where

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 with To eliminate the error between Φ and Φ 0 , a novel auxiliary system is introduced as where   > 0 is a design parameter.
The adaptive law of weight vector can be designed as where The adaptive law of Δ   is given as follows: where   and   are design parameters.The desired controller Φ 0 is introduced as where  ,1 and  ,2 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 Substituting ( 27) into ( 26) results in where According to 2 ≤  2 + (1/) 2 , we obtain Invoking Lemma 4, it is deduced as Differentiating ( 35)-(38) and invoking ( 5)-( 8) yield where The auxiliary systems are introduced as where   ( = ℎ, , , ) stand for design parameters.
The adaptive laws of weight vector can be designed as where where   and   ( = ℎ, , , ) are design parameters.

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   = 0.02,  = 1, 2, 3, and vibrational frequencies are  1 = 27.8944rad/s,  2 = 76.8933rad/s, and  3 = 150.7526 rad/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.
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  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.

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  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.

Figure 4 :Figure 5 :
Figure 4: Time history of fuel equivalence ratio and elevator deflection (Case 1)

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

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

𝜂, u) ,
and    and    are design parameters.
1,   = 1, and the upper bound of weight vectors 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 Δℎ  = 1524 m and period is 600 s.