Neural Networks Approximator Based Robust Adaptive Controller Design of Hypersonic Flight Vehicles Systems Coupled with Stochastic Disturbance and Dynamic Uncertainties

A neural network robust control is proposed for a class of generic hypersonic flight vehicles with uncertain dynamics and stochastic disturbance. Comparedwith the present schemes of dealingwith dynamic uncertainties and stochastic disturbance, the outstanding feature of the proposed scheme is that only one parameter needs to be estimated at each design step, so that the computational burden can be greatly reduced and the designed controller is much simpler. Moreover, by introducing a performance function in controller design, the prespecified transient and performance of tracking error can be guaranteed. It is proved that all signals of closed-loop system are uniformly ultimately bounded. The simulation results are carried out to illustrate effectiveness of the proposed control algorithm.


Introduction
It is well known that stochastic disturbances appear often in many practical systems [1,2].Their existence is a source of instability of the control systems; thus, the investigations on stochastic systems have received considerable attention in recent years.Several classes of nonlinear systems with stochastic disturbances and dynamic uncertainties were stabilized by using adaptive neural networks based control.In [3] a backstepping control scheme was proposed for the stochastic nonlinear strict-feedback system.
A framework based on the stochastic Liouville Equation (SLE) was provided for both three-state and six-state Vinh's equations for hypersonic entry in Mars atmosphere [4].Wu et al. [5] applied the stochastic small-gain theorem and backstepping design technique in the stochastic nonlinear systems with uncertain nonlinear functions and unmodeled dynamics.
In recent years, hypersonic flight vehicles (HFVs) have received a great deal of attention around the world, which offer a promising technology for cost-efficient and reliable access to space and are especially suitable for prompt global response [6][7][8].However, the design of control systems for HFVs is a challenging work due to the longitudinal dynamics of HFVs being highly nonlinear and strong couplings between the propulsive and aerodynamic forces [9][10][11].
The modeling inaccuracy, parameters uncertainties, and external disturbances can result in strong adverse effects on the performance of HFVs control systems.As a result, the onboard flight control systems design of HFVs presents numerous challenges.
Based on modern control techniques, various flight control systems have been designed to the longitudinal dynamics of HFVs.In [12], the adaptive backstepping method was used to design controllers for HFVs, while fuzzy logic systems (FLSs) and neural networks (NNs) were used to approximate the unknown system dynamics in [13,14].However, backstepping design suffers from the problem of "explosion of complexity" caused by the repeated differentiations of nonlinear functions [5,[15][16][17].To eliminate this problem, dynamic surface control (DSC) was applied to longitudinal dynamics of HFVs in [18][19][20][21][22][23], which uses a low-pass filter at each design step to avoid the derivatives of nonlinear functions.In the above control schemes, the uncertain nonlinear functions in the HFVs model were approximated by NNs or FLSs using their universal approximation capability [24][25][26].However, the drawback of these control schemes is that the number of adaptation laws depends on the number of the NNs nodes or the number of the fuzzy rules.With an increase of NNs nodes or fuzzy rules, the number of estimated parameters will increase significantly.To solve this problem, in [27][28][29][30], the norm of the ideal weighting vector in NNs or FLSs was considered as the estimation parameter instead of the elements of weighting vector.Thus, the number of adaptation laws is reduced considerably.
Besides, an interesting question raised by DSC or backstepping control schemes is the output error transient performance.Recently, to guarantee a prespecified tracking performance, a backstepping design based on NNs was proposed for a class of uncertain nonlinear systems, and it is shown that the tracking errors can converge to predefined arbitrarily small residue sets with prescribed convergence rate and maximum overshoot [31,32].However, to our best knowledge, limited attention has been paid to this problem for the controller design of HFVs.
Another feature of the proposed scheme is that the radial basis function (RBF) NNs are employed to compensate for the uncertain nonlinear functions.By using the minimal learning technique, only one parameter needs to be update online at each design step regardless of the NNs input-output dimension and the number of NNs nodes.As a result, the number of adaptation laws and the computational burden are greatly reduced.
Inspired by the aforementioned discussions, in this paper, we divide the control oriented model (COM) of HFVs into two parts: velocity subsystem and altitude subsystem.Dynamic inversion method is employed to design the controller for velocity subsystem, while DSC strategy is used for altitude subsystem.Besides, a performance function is introduced to obtain a virtual error constraint variable, which can ensure the prescribed transient performance.By this transformation, the original tracking error can converge to predefined arbitrarily small residue sets with prescribed convergence rate and maximum overshoot.Simulation results are presented to demonstrate the efficiency of the proposed scheme.
The rest of this paper is organized as follows.In Section 2, the nonlinear longitudinal dynamics 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 conclusion of this paper in Section 5.

Problem Formulation and Preliminaries
The control oriented model (COM) of the longitudinal dynamics of HFV considered in this study is taken form [8,18].The equations of the COM model are expressed as where  is the velocity,  is the flight path angle, ℎ is the altitude,  is the attack angle,  is the pitch rate, and with , and  =  0 exp(−(ℎ − ℎ 0 )/ℎ  ), where   is the elevator deflection and Φ is the throttle setting.Letting  denote the pitch angle, we have  = +.Then, we define state variables as  = [ 1 ,  2 ,  3 ,  4 ,  5 ]  , with  1 = ,  2 = ℎ,  3 = ,  4 = , and  5 = .Note that the flight path angle  is typically very small during the trimmed cruise condition, which justify the approximation sin() ≈ , so the system (1) can be rewritten as where Since the values of the inertial and the aerodynamic parameters are uncertain, the aforementioned   () and   (),  = 1, . . ., 5, are unknown functions.Moreover, as stated in [18], from the model of the HFV, it is easy to check that  1 (),  2 (), and  3 () are always strictly positive and  5 () is strictly negative since   is negative.With these observations in mind, we have the following assumption.
Assumption 2.   and its first derivative are known and bounded, while ℎ  and its first two derivatives are known and bounded.Remark 3. In the following proposed scheme, both   and   do not appear in the final control law and are used only for stability analysis, so they can be unknown.
Remark 4. The COM model of HFV is an MIMO system that has cross-channel coupling.In order to retain simplicity, we divide the model into velocity and altitude subsystems, the coupling effect is treated as a part of unknown nonlinear functions, and then RBF NNs are used to approximate them.

Radial Basis Function Neural Network Approximation.
In this study, the radial basis function neural networks (RBF NNs) are used to approximate the continuous unknown functions on a given compact set.Mathematically, an RBF NN form [26] can be expressed as where  ∈ R and  ∈ R  are the NN output and input,  ∈ R  is an -dimensional weight vector, and () = [ 1 (), . . .,   ()]  is the Gaussian function with the following form: where   ∈ R  is a constant called the center of the Gaussian function and  ∈ R is the width, respectively.

Lemma 5. According to the approximation property of RBF NNs, given any continuous function 𝐹(𝜉)
: 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 optimal weight  *  such that  *  () can approximate the given function () with approximation error Δ() bounded by  [26].
since  * is unknown; we need to estimate it online.

Adaptive Controller Design and Stability Analysis
3.1.Performance Functions and Error Transformation.Similar to [31,32], the mathematical expression of the prescribed tracking performance is given by where   =   −   ,  = 1, 2, are the tracking error,   and   are given positive constants, and the performance function   is defined as smooth and decreasing positive function.From (7), one can see that −    (0) and     (0) are the lower and upper bound of the undershoot of   (), respectively.    (∞) and     (∞) represent the maximum allowable size of the steady-state value of   ().
To transform (7) into an equivalent unconstrained one, define the error transformation function Υ  (  ) =   ()/  (), where   is the transformed error and Υ  (  ) is a smooth, strictly increasing function having the following properties: From ( 8), if   is bounded, we have −  < Υ  (  ) <   , and thus (7) holds.Hence, to achieve the prespecified tracking performance, one only needs to keep   bounded.The inverse transformation of Υ  (  ) can be expressed as and differentiating (9) yields From the properties of the transformation, it is clear that   and   are bounded and 0 <   0 ≤   .

Velocity Controller Design via Dynamic Inversion.
In this paper, by functional decomposition, the dynamics of HFVs is decoupled into velocity and altitude subsystem.Velocity subsystem of HFV (1) can be rewritten as follows: where  1 () and  1 () are unknown nonlinear function and  1 ≥  1 > 0. Then define the velocity tracking error as  1 =  1 −  1 .According to (10) and ( 11) we obtain Let  1 ( The transformed system dynamics of ( 12) can be rewritten as Since  1 ( 1 ) is an unknown nonlinear function, we use an RBF NN to approximate it.Then by using Lemma 5, we have Choose the control signal and the adaptive update law as follows: where θ1 is the estimate of  1 = ‖ * 1 ‖ 2 / 10  1 and  1 ,  1 , and  1 are designed positive parameters.Consider the Lyapunov function The differential of Lyapunov function Γ 1 can be found as follows.

Attitude Controller Design via DSC.
In this section, the DSC technique will be introduced to the system described by (1); the recursive design procedure contains 4 steps.In Steps 1-3, the virtual control law is designed at each step; finally an overall control law   is constructed at Step 4. After the error transformation ( 7)-( 10), the altitude subsystem ( 1) is equivalent to The stabilization of the transformed system ( 22) is sufficient to guarantee the prescribed tracking performance of altitude subsystem.
Step 1.Let  2 given by ( 22) be the first error variable.Define  3 as the first virtual control signal.Then the derivative of  2 can be expressed as where ) is unknown, we employ an RBF NN to approximate it on a compact set Ω  2 .By properly choosing the basis function vectors we have where and  2 is a positive constant.With respect to the unknown optimal weight vector in (24), define where  2 are given by Assumption 1.Since  2 is unknown, let θ2 be the estimation of  2 and θ2 fl θ2 −  2 .Consider the first Lyapunov function The derivation of ( 26) can be found as follows: Using Young's inequality, it can be verified that Mathematical Problems in Engineering 5 Thus, ( 27) can be rewritten as which suggests that we choose the virtual control signal  3 as and the adaptation law where  2 ,  2 , and  2 are positive design parameters.Then substituting ( 30) and ( 31) into ( 29), we get Introduce a new state variable  3 , which can be obtained by the following first-order filter: Step j ( = 2, 3 where   is a positive design parameter; θ = θ −   with   = ‖ *  ‖ 2 /  .The derivation of (36) is Similar to ( 27)-( 29), we have Choose the th virtual control signal where θ is updated by with   ,   , and   positive design parameters.Substituting (39), (40), into (38), we get Let  +1 pass through the following first-order filter with time constant  +1 to obtain a new state variable  (+1) , Step 4. Finally, the time derivative of  5 =  5 −  5 is where  5 ( 5 ) =  5 () + (1/2) 5 − ẋ 5 +  5 is unknown; we use RBF NN to approximate it on a compact set Ω  5 , and where  5 is a positive design parameter and θ5 = θ5 −  5 with  5 = ‖ * 5 ‖ 2 / 5 .Differentiating (45) we have Similar to ( 28), (46) can be rewritten as Noting  5 () < 0, we choose the actual control signal where θ5 is updated by with  5 ,  5 , and  5 positive design parameters.Substituting (48), (49), into (47), we arrive at Remark 6.Compared with the exiting control schemes for hypersonic flight vehicle (1), the above proposed shows that, by combining DSC with the adaptive tracking controller, the design procedure can be greatly simplified and the computational burden, since only one parameter needs to be updated online in each step, can be greatly reduced.

Simulation Results
In this section, the simulation results are used to verify the effectiveness of the proposed dynamic surface control schemes.The simulation model of HFVs is taken from [18,21].The reference signals have been generated by filtering step reference signals through a prefilter (68), with natural frequency  1 = 0.5,  2 = 0.3, and  ℎ = 0.95.The reference signals of velocity and altitude are 400 ft/s and 1000 ft, respectively.which indicate that the tracking performance is achieved by the proposed control scheme.

Conclusion
This paper develops a neural network based dynamic surface controller for the HFV system in the presence of external disturbance and dynamic uncertainties.By introducing the performance and the error transformation function in controller design, the performance of tracking error can be guaranteed.Moreover, using the norm estimation approach, only one parameter needs to be updated online in each design step regardless of the plant order and input-output dimension and hence, the computational burden problem is circumvented.Numerical simulations on the COM of HFV demonstrated the effectiveness of the proposed control algorithm.