Neural Network-Based Adaptive Backstepping Control for Hypersonic Flight Vehicles with Prescribed Tracking Performance

An adaptive neural control scheme is proposed for a class of generic hypersonic flight vehicles.Themain advantages of the proposed scheme include the following: (1) a new constraint variable is defined to generate the virtual control that forces the tracking error to fall within prescribed boundaries; (2) RBF NNs are employed to compensate for complex and uncertain terms to solve the problem of controller complexity; (3) only one parameter needs to be updated online at each design step, which significantly reduces the computational burden. It is proved that all signals of the closed-loop system are uniformly ultimately bounded. Simulation results are presented to illustrate the effectiveness of the proposed scheme.


Introduction
During the past decades, hypersonic flight vehicles (HFVs) have received a great deal of attention.They may represent more cost-efficient and reliable access to space routine and are especially suitable for prompt global response, as well as offering worldwide air superiority because of the high speed and endurance [1][2][3][4][5].In this paper a nonlinear generic model of HFVs is adopted, which has been widely used by various researchers [6][7][8].The dynamics of HFVs are highly nonlinear with strong couplings between the propulsive and aerodynamic effects.The requirements of flight stability and high speed response make the onboard flight control of HFVs quite difficult [9,10].Besides, modeling inaccuracy can result in strong adverse effects on the performance of HFVs control systems.Thus, the controller design for HFVs is challenging and must guarantee closed-loop stability and desired performance [11].
Recently, feedback control strategy based on nonlinear control theory has been used for HFVs, such as sliding mode control [3], minimax linear quadratic regulator control [12,13], genetic algorithm [14], and sequential loop closure controller design [15].In [16], the adaptive backstepping method was used to design controller for the HFVs model, while fuzzy logic and neural networks were used to approximate the unknown system dynamics in [17][18][19].Adaptive dynamic surface control schemes were proposed by [20,21] to avoid the derivatives of nonlinear functions.The nonlinear dynamic inversion method was used to design a robust controller.In [3,14], feedback linearization techniques were applied to design nonlinear controllers for the longitudinal motion of a hypersonic aircraft containing aerodynamic uncertain parameters.This approach leads to a complicated high-order Lie derivatives and is hard to perform a robustness analysis when considering uncertainties.In [22], a neural network controller for a nonlinear flight dynamic system was designed by using the adaptation mechanism to deal with the effects of aerodynamic modeling errors.
In the control design for HFVs, an important issue is tracking performance.Traditionally, the controller for HFVs guarantees the tracking error convergence to a residual set.Moreover, the transient behavior such as overshoot, undershoot, and convergence rate are difficult to be established analytically.In [23][24][25], a prescribed performance scheme is proposed for one-class nonlinear systems; this approach is to construct a prescribed performance function that converts the tracking error into a new variable.Therefore the tracking performance can be characterized by a prescribed constraint function.Besides, the prescribed performance approach with new definition is applied in a class of uncertain strictfeedback systems [26], strict-feedback time-delay systems [27], and MIMO systems [28], respectively.
A drawback of adaptive NNs [22] or FLSs [29,30] schemes is that the number of adaptation laws generally depends on the neural network nodes or the fuzzy rules.That is, with an increase of the nodes or the rules, the parameters to be estimated may be greatly increased.To solve this problem, we propose a new method by estimating the norm of the NNs weights rather than estimating every item of the weight vector [31][32][33].
In this paper, we separate the longitudinal model of HFVs into two parts: the velocity subsystem and the altitude subsystem.Velocity and altitude controllers are designed separately.For the velocity subsystem, a dynamic inversion controller with radial basis function neural networks (RBF NNs) is proposed to track a desired velocity trajectory.The altitude subsystem is transformed into a strict-feedback form.Then an adaptive backstepping controller is designed to track a desired altitude trajectory.The main contribution of this paper is described as follows: (1) We introduce a performance function, and a new error constraint variable is used as a virtual tracking error variable to ensure the prescribed transient performance.By extending the prescribed tracking performance technique proposed in [23,24] to HFVs, it is shown that the tracking errors can converge to predefined arbitrarily small residue sets with prescribed convergence rate and maximum overshoot.
(2) RBF NNs are employed to compensate for complex and uncertain terms to solve the problem of controller complexity.By using the minimal learning technique [31][32][33], only one parameter needs to be updated online at each design step regardless of the NNs inputoutput dimension and the number of NNs nodes.As a result, the number of adaptation laws, which generally depends on the neural network nodes, and the computational burden are greatly reduced.
(3) With the bounded of the virtual control gain   (⋅), the singularity problem by the estimation of   (⋅) is avoided without any effort, and both low and up bounded will not appear in the control law and will be used only for analysis; they can be unknown.
The rest of this paper is organized as follows.In Section 2, the nonlinear longitudinal dynamic model of HFVs is presented.The controllers design and the stability analysis are given in Section 3. The simulation results are illustrated in Section 4, followed by conclusions of this paper in Section 5.
The engine dynamics can be modeled by a second-order system: Therefore, by selecting the commanded value   as the new control input, the HFV is composed of five state variables  = [, ℎ, , , ]  and two control inputs  = [  ,   ]  , while the outputs to be controlled are selected as  = [, ℎ]  .The design objective is such that the outputs track the desired altitude and velocity commands   = [  , ℎ  ]  with prescribed tracking performance.
From (1), it can be inferred that the main contribution in the change of flight vehicle velocity is from the throttle setting   .The altitude change is related mainly to the elevator deflection   .Thus, it is reasonable to divide the system into two loops: the velocity loop and the altitude loop.
Note that the thrust term  sin  is generally much smaller than the lift , velocity  is high, and the flight path angle  is typically very small during the trimmed cruise condition, which justify the following approximation.
Assumption 2. There exist positive constants   and   such that 0 Remark 3. It is worth noting that, in the proposed scheme, both   and   will not appear in the control law and will be used only for analysis; they can be unknown.Assumption 4.   and its first derivative are known and bounded, while ℎ  and its first four derivatives are continuous and bounded.

Description of RBF NNs.
In this paper, RBF NNs will be employed to approximate unknown functions.Mathematically, an RBF NN can be expressed as where  ∈ R and  ∈ R  are the NN outputs and input,  ∈ R  is the weight vector, and () = [ 1 (), . . .,   ()]  is the basis function vector with   () commonly chosen as the Gaussian functions: where   ∈ R  and  ∈ R are constants called the center and width of the basis function, respectively.
Lemma 5 (see [17]).Given any continuous function () : Ω  → R with Ω  ⊂ R  a compact set and any constant  > 0, by appropriately choosing  and   ,  = 1, . . ., , for some sufficiently large integer , there exists an RBF NN  *  () such that where  * is the optimal weight vector defined as and Δ() denotes the approximation error.

Adaptive Neural Controller Design
3.1.Performance and Error Transformation Functions.Let the tracking error be defined as where   is the desired trajectory.Similar to [23,24], the mathematical expression of the prescribed tracking performance is given by where  and  are given positive constants and the smooth function is given by in which  0 is the initial value of (),  ∞ represents the value of () at the steady state, and  is the decreasing rate of ().
Then, introduce the following error transformation: where  is the transformed error and () is a smooth, strictly increasing, and thus invertible function possessing the following properties: Note that if  is kept bounded, we have − < () < , and thus (12) holds.The inverse transformation of () can be written as In this paper, we choose Differentiating ( 17) yields where  = (Θ/(/))(1/) and  = ẏ  +  ε /.From the properties of the transformation, it is clear that  and  are bounded and 0 <  0 ≤ .Remark 6.From ( 12) and ( 13), one can see that (0) and −(0) serve as the upper bound of the overshoot and the lower bound of the undershoot of (), respectively, the decreasing rate of () introduces a lower bound of the convergence rate of (), and max{ ∞ ,  ∞ } represents the maximum allowable size of the steady-state value of ().Note that (0), , and  should be properly chosen such that −(0) < (0) < (0).
Based on the backstepping approach, a trajectory tracking controller is designed for the dynamics model given in (19).The design procedure contains 4 steps, and the actual control law will be deduced at the last step.For convenience, let   (  ) and Ω   denote the unknown function to be estimated by RBF NNs and the corresponding compact set in the th step, respectively.Then by using Lemma 5, we have where   (  ) and   denote the vector valued function and the RBF NN input in the step with proper dimensions that are given below.
Step 1.Let  1 given by ( 19) be the first error variable.Define  2 =  2 −  1 , where  1 is the first virtual control signal.Then the derivative of  1 can be expressed as where ) is unknown, we employ an RBF NN to approximate it on a compact set Ω  1 ⊂ R 4 .By properly choosing the basis function vectors we have where  1 is a positive constant.With respect to the unknown optimal weight vector in (22), define Besides, let θ1 be the estimation of  1 and θ1 := θ1 −  1 .Consider the first Lyapunov function Taking the time derivation of (24) yields Using Young's inequality and ( 23), it can be verified that Thus, (25) can be rewritten as which suggests that we choose the first virtual control signal  1 as Let where  1 ,  1 , and  1 are positive design parameters.Then substituting ( 28) and ( 29) into (27), we get Step i ( = 2, 3).Define  +1 =  +1 −   , where   is the th virtual control signal.Then the time derivation of   is where where   is a positive design parameter, θ = θ −   with   = ‖ *  ‖ 2 / 0   .By taking the time derivation of (32), we have Similar to (26), we have Choose the th virtual control signal where θ is updated by with   ,   , and   being positive design parameters.Substituting ( 35), (36), and ( 30) into (34), we get Step 4. The time derivative of  4 is where  4 ( 4 ) =  4 ( 4 ) +  3  3 ( 3 ) (41) Choose the control signal where θ4 is updated by with  4 ,  4 , and  4 being positive design parameters.Substituting (42), (43), and (37) into (41), we arrive at Remark 7. The RBF NNs are used to compensate for the complex and uncertain terms to solve the problem of controller complexity, and the repeated derivation of virtual control signal   can be avoided.Compared with the neural based control [16,21], in each design step, by using the estimation of the norm of the NNs weights, only one parameter needs to be updated online; therefore the design procedure can be greatly simplified and the computational burden is greatly reduced.Moreover, the lower bound of the virtual control coefficient     is used to avoid the singularity problem without any additional effort.
Remark 8. Since the approximation ability of RBF NNs is on a compact set, we can only guarantee the semiglobal stability of the control scheme.
Proof.Using the following facts: we rewrite (44) as where Let Then we have Solving (49) gives which implies that  4 ,   , θ , and θ are bounded.Since  1 is bounded, according to the error transformation of (15), to (17) we can obtain that − < () < ; as a result, we have −() < () < (); that is, the prescribed tracking performance is guaranteed.This completes the proof.

Velocity Controller Design via Dynamic Inversion.
The velocity subsystem of (1) can be rewritten as follows: where   ( 4 , ) and   ( 4 , ) are unknown nonlinear function   ( 4 , ) ≥   > 0. Then define the velocity tracking error as   =  −   .According to (17) and ( 18) we obtain We assume that 0 <  0 ≤   .The transformed system dynamics of (52) can be rewritten as where and then where Solving (60) gives It is clear that   , θ , and θ are bounded.Owning that   is bounded, together with the error transformation of ( 11) into (17), implies that the prescribed tracking performance is guaranteed.

Simulation Results
In this section, the numerical simulation results are presented to show the performance of the control scheme.Simulation of the HFV model is conducted for trimmed cruise conditions of 110000 ft and Mach 15.The parameters of simulation model are taken from [16,21].The control objective is to track the step change of 100 ft/s in airspeed and 2000 ft in altitude.Linear command filters are used to generate the differentiable commands: where  is Laplace operator,  1 = 0.5,  2 = 0.The simulation results are presented in Figures 1−10.The responses to 100-ft/s step-velocity and 2000-ft step-aliunde command in trimmed condition are depicted in Figures 1-3 and Figures 4−6, respectively.From Figures 2 and 5, we see that the tracking errors performance are guaranteed.Figures 7-10 show the simulation results of altitude tracking with square wave trajectory.From the results of simulations, the maximum value of terms sin/() is 3 × 10 −5 ; it is much smaller than the term /() whose minimum value is 1 × 10 −3 and the maximum value of flight path angle  is less than 0.012 rad.Therefore, Assumption 1 is reasonable.

Conclusion
An adaptive neural control scheme has been proposed for a class of longitudinal dynamics of a generic hypersonic  flight vehicle.We have shown that, by using a new constraint variable, the prescribed tracking performance can be achieved.The unknown nonlinear functions associated with each recursive step of backstepping control were approximated by using RBF NNs.For each design step, only one parameter needs to be updated online.Thus the explosion of the complex problem in backstepping control scheme and the computational burden can be greatly reduced.Numerical simulations revealed that the tracking error clearly satisfies the prescribed performance specification and verified the proposed design scheme.Currently, we assume that all of the system states are available and the controller is based on state feedback.However, some states cannot be obtained in some circumstances, especially when the sensor fault occurs.As a result, future work will be focused on output feedback control law design.

Figure 2 :
Figure 2: The velocity tracking error   and the prescribed error bounds.

Table 1 :
Parameters of the HFV.