Constrained Adaptive Neural Control for Air-Breathing Hypersonic Vehicles without Backstepping

An adaptive neural control scheme without backstepping is proposed for the air-breathing hypersonic vehicle subject to input constraints. To estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, two radial basis function neural networks (RBFNNs) are constructed. Since the complex backstepping design steps are not needed, the proposed control structure is quite concise and the problemof “explosionof terms” is avoided.Moreover, a novel nonlinear auxiliary system is constructed to solve the problem of input constraints.The advantage of the proposed auxiliary system is that its high-order form has good performance and the parameter tuning is relatively easy. Simulation results show that the designed controllers achieve stable tracking of reference commands with good performance.


Introduction
Air-breathing hypersonic vehicle (AHV) has attracted wide concern, because it offers a promising and economic access to near space [1,2].The design of the flight control system is an important issue to ensure that the AHV can accomplish its intended task.Due to the complex flight environment, the highly nonlinear dynamic behavior, and the large uncertainty [3,4], it is a huge challenge to design the controllers with satisfactory performance.
Recently, there are already numerous publications on the design of the flight control system for AHV.A robust linear output-control strategy is investigated for AHV, achieving the stable tracking of reference trajectories [5].Since the feedback linearization (FL) remains based on the accurate model which is difficult to be obtained, the robustness of FL is weak when there is the model uncertainty.To enhance the robustness with respect to model parameters, a nonfragile control scheme is studied based on the linear parameter-varying (LPV) model [6,7].Also the sliding mode control is an effective nonlinear control method and is widely studied for AHV subject to the uncertainty.A key technique for sliding mode control is how to solve the chattering problem [8].To reduce the chattering, the quasi-continuous high-order mode controllers are exploited, effectively reducing the flutter of the AHV [9].By introducing the nonlinear disturbance observers (NDO), the total uncertainty is estimated online, and the robustness of the controller is guaranteed [10,11].Noticing that the robustness of the designed controllers is closely related to the NDO, an novel NDO based on hyperbolic sine function is investigated and the robustness is enhanced [12].Moreover, much concern has been on the intelligent controls such as the fuzzy logical system (FLS) [13][14][15] and the neural network (NN) [16,17].The minimal learning parameter (MLP) scheme [13,14,16] and the composite learning-based parameter adaptive law [15,17] are constructed to update the FLS/NN weights.By estimating the model uncertainty accurately and compensating for the controllers, the robustness of the proposed control scheme is ensured.
Backstepping control is regarded as an effective tool to solve the problem of unmatched uncertainty [18,19].Noting that the traditional backstepping control needs the tedious analytical calculation of derivatives, the problem of the "explosion of terms" is inevitable.Therefore, the dynamic surface control [20,21], command filtered approaches [22], and tracking differentiator (TD) [12] are proposed to solve the problem of "explosion of terms".Though good results Mathematical Problems in Engineering in terms of command tracking can be gained by the above method, the whole control system becomes cumbersome and it is unfavorable to the design of the controller.To simplify the design process of the controllers, the controllers without the virtual control laws are creatively designed [16].However, the expression of the final actual controller becomes complex with a large number of state variables and design parameters.Motivated by the above studies [13,14,16,17], the radial basis function neural networks (RBFNNs) with MLP algorithm are constructed to approximate the model uncertainty.Different from the previous studies, the control scheme without backstepping is proposed by introducing the high-order differentiator.Thus there is no need of complex strict-feedback form and virtual controllers.Also only two RBFNNs are needed to estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, respectively.
From a practical perspective, the control inputs are limited to a certain range.The designed controller without considering the input constraints may suffer performance degradation or even instability appearance.Therefore, it is necessary to take the input constraints into consideration.The Nussbaum function based auxiliary design system is employed for the strict-feedback systems and the nonstrictfeedback systems, preventing the outputs from violating the constraints [23,24].Also a linear auxiliary system is adopted for the backstepping scheme [25] and the disturbance observer-based feedback linearization control [26].Moreover, an novel hybrid auxiliary system is designed to solve the problem of the nonsymmetric nonlinear input constraints [27], and it has been applied to the AHV systems [28,29] and the flexible spacecraft systems [30].However, these antiwindup methods can only be applied in some specific control scheme such as the backstepping control.Regardless of the linear auxiliary system [25,26] or the hybrid auxiliary system [27][28][29][30], it is not suitable to be extended to the high-order forms.Since a high-order auxiliary system is required for the altitude subsystem under the proposed control scheme, a novel nonlinear auxiliary system is constructed.
The remainder of the paper is organized as follows: the inputs constrained AHV model and the description of RBFNN are presented in Section 2.Then, the main design process of the controllers is provided in Section 3. Next, Sections 4 and 5 show the stability analysis and simulation, respectively.Finally, the conclusions are summarized in Section 6.The main contributions of this paper are summarized as follows.
(1) Different from the adaptive neural control schemes in [16,17,28], there is no need of complex backstepping design steps in this paper.Thus the proposed control structure is quite concise and the problem of "explosion of terms" is avoided.
(2) Compared with the linear auxiliary system [25,26] and the hybrid auxiliary system [27][28][29][30], the nonlinear auxiliary system proposed in this paper can be extended to the high-order forms and applied to the altitude subsystem under the proposed control scheme.Moreover, the parameter tuning for the proposed auxiliary system is relatively easy.[31].This mode has been widely used in the design of flight control systems.Then, Parker simplified the model by ignoring the weak couplings and slow dynamics [32].The dynamic equations are formulated as

Inputs Constrained AHV Model and RBFNN Description
where , ℎ, , , and  denote the velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively.The flexible are the first two vibrational modes. is the mass,   the moment of inertia,   the damping ratio, and   the natural frequency.  = 1 + ψ /  , with ψ the constrained beam coupling constant for   .The thrust , lift , drag , pitching moment , and generalized forces   ( = 1, 2) are given as follows [17]: with where  and   are the fuel equivalence ratio and the elevator deflection, respectively;  and  denote the reference area and the aerodynamic chord, respectively;  is the dynamic pressure,  the air density, and   the thrust to moment coupling coefficient.

Input Constraints.
Two inputs of the AHV are  = [,   ]  .In engineering practice, the inputs cannot be any desired values, but must be limited within a certain range.
must be limited to a certain range to ensure the normal operation of the scramjet.Similarly, the limitation of   is determined by the physical structure of the elevator.The above input constraints can be described as where   and   are the inputs to be designed; the "max" and "min" denote the maximum and minimum value in the constrained range, respectively.

Description of RBFNN.
The RBFNN has been widely used because it can approximate any continuous function to an arbitrary precision [16,33].Assume that the number of RBFNN nodes is  ≥ 1.The mapping from the input layer X ∈ Ω ⊂ R  to the output layer  ∈  is where where   = [ 1 ,  2 , . . .,   ] T denotes the center vector and   is the width vector.Assume that (X) : R  → R is the function to be approximated.There exists W * such that where W * denotes the ideal weight vector;  is the approximation error.

Controller Design
The control target is selected as developing a controller  = [,   ] T so that the outputs  = [,ℎ] T can follow the reference commands   = [  , ℎ  ] T .Considering that  is steered by  to a large degree and ℎ is mainly affected by   , we naturally decompose the AHV model into the -subsystem and the ℎ-subsystem.In what follows, two constrained adaptive neural controllers will be separately designed for the two subsystems such that  →   and ℎ → ℎ  .
The error between  and   is expressed by Differentiating   by time , we have Inspired by the MLP method [13,14,16], the neural controller  is designed as where  1 > 0,  2 > 0 are design parameters;   = ‖W *  ‖ 2 , with the estimation λ .λ is decided by the adaptive law as follows: where   > 0 is a design parameter.Noting that  is constrained as (11), a novel auxiliary system is designed as where   > 0,   > 0,  1 > 0 are design parameters;   is the state variable.Define the compensated velocity tracking error as Differentiating    by time  and invoking ( 19) and ( 22), we have Control law Eq.( 25) Auxiliary syst.Eq. ( 22) RBFNN The desired control input   is determined by the chosen control law: The controller structure of the V-subsystem is presented in Figure 1.
Remark 1.As shown in Figure 1, if the designed control law   directly contains the variable  and   , or the RBFNN input layer X  contains the element , the "algebraic loop" problem will appear.Thus, it can be concluded from (25) that the "algebraic loop" problem will not exist as long as  ∉ X  .
Step 2 (designing control law for the ℎ-subsystem).The error between ℎ and ℎ  is defined as Choose the command of  as where  ℎ1 > 0,  ℎ2 > 0 are design parameters.
According to [33],  ℎ can converge to zero exponentially, if we choose  →   .Hence, a constrained adaptive neural controller is developed to steer  →   .
in (34) cannot be obtained directly, we get their estimations by introducing the finite-time-convergent differentiator [34].Assume that the estimations are γ , γ , γ  , γ  , and ... γ .The estimation errors are defined as According to [34], there exist positive constants   such that The desired control input   is determined by the chosen control law: where   > 0 is design parameter; λℎ is the approximation of  ℎ = ‖W * ℎ ‖ 2 with the following adaptive law: where  ℎ > 0 is design parameter.  in the chosen control law   , there is no need of complex strict-feedback form and virtual controller.(b) Similarly, the RBFNN input layer X ℎ must meet the requirement that   ∉ X ℎ .

Simulation Results
The aerodynamic parameters and physical properties of the AHV model are given in Tables 2-6.To characterize model uncertainty (including parameter uncertainties and external disturbances), we add the perturbation into the nominal aerodynamic coefficient.Define the actual coefficient as where  0 represents the nominal value and Δ denotes the amplitude of coefficient perturbation.The states initialization of the AHV are given in Table 1.
The reference commands   and ℎ  are obtained through a second-order system with a natural frequency   = 0.1 and a damping coefficient  = 0.9.The velocity steps are from 8200 f t/s to 8800 f t/s and the altitude steps from 88600 f t to 89200 f t.
Case II.All simulation conditions remain the same except that the auxiliary systems ( ( 22) and ( 32)) do not work.
For the designed controllers without the auxiliary systems, simulation results are shown in Figures 10-14.
Figures 2 and 3 depict the tracking performance that  →   and ℎ → ℎ  .It is shown that the outputs  and ℎ steadily track the reference commands with the satisfactory tracking error.The states , , , and the flexible states, as shown in Figures 4 and 5, change smoothly without the chattering and mutation during the entire simulation process.Figures 6-8 depict the inputs and the states of the auxiliary systems.It is obvious that the auxiliary systems compensate for the inputs when the actuators are saturated.Figure 9 shows that the estimations of   and  ℎ are bounded and smooth.As shown in Figures 10-12, the velocity and altitude become divergent because the control inputs hit their saturation for a long time.It can be observed from Figure 13 that the angle of attack oscillates near zero when the elevator deflection is constrained.Negative angle of attack will directly lead to engine flameout.Figure 14 reveals that the flexible states become divergent, and the aircraft will disintegrate by violent vibration.In conclusion, the designed controllers without the auxiliary systems fail to achieve the tracking of reference commands.Scenario 2. To show the robustness and effectiveness of the proposed algorithm, the cases are considered as follows.
Case I. Simulations with and without parameter uncertainties and external disturbances are included and compared.The aerodynamic coefficient perturbations are selected as Δ = 40% and Δ = 0, respectively.
Case II.To show how the proposal works under a noisy environment, the quantization errors in the state variables are described as V  = 0.6×sin(0.1),ℎ  = 0.2×sin(0.1),  = 0.1 × /180 × sin(0.1).In addition, the control input signals are added with the white noise whose mean is 0 and variance is 0.01.Also the aerodynamic coefficient perturbations are selected as Δ = 40%.Case III.The backstepping control scheme based on novel NDO in [12] is selected for comparison with the proposed controller in this article.All simulation conditions remain the same as Case II.
In order to ensure the fairness of the comparison, the input constraints are not considered.Simulation results in Case I and Case II are shown in Figures 15 and 16 and simulation results in CaseIII are presented in Figures 17-20.
From Figures 15-16, it can be seen that the proposed control scheme achieved robust tracking of the velocity and altitude under the environment of parameter uncertainties and additive noise.Figures 17 and 18 reveal that the control method proposed in this article has higher accuracy in the velocity and altitude tracking, compared with the method in [12].It can be observed from Figures 19 and 20 that the inputs of the method in [12] cannot be adjusted in time, especially in the initial time period, resulting in the flight-path angle, pitch angle, and pitch rate not meeting the requirements of the tracking task.

Conclusions
A constrained adaptive neural control without backstepping is proposed for the AHV in this article.It is shown that the proposed control structure is quite concise and the problem of "explosion of terms" is avoided.By conducting the Method in [12] Method in this article Reference command V ＞ Method in [12] Method in this article Reference command h ＞ ×10 4 comparative simulations, the effectiveness of the proposed nonlinear auxiliary systems is verified.Compared with the backstepping control scheme based on novel NDO, the proposed algorithm has stronger robustness.Our further works will concentrate on designing better learning algorithms for RBFNN.Method in [12] Method in this article coefficients for the air-breathing hypersonic vehicle (AHV).Readers can get this data in [32].

Figure 1 :
Figure 1: The controller structure of the V-subsystem.
Based on the X43A aircraft developed by NASA, M. Bolender et al. presented an AHV model 2.1.AHV Model.

Table 3 :
Lift and drag coefficient values.