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

Authors: Zhu Guoqiang; Liu Jinkun
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

An adaptive neural control scheme is proposed for a class of generic hypersonic flight vehicles. The main 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.


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## Body

## 1. 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 [1–5]. In this paper a nonlinear generic model of HFVs is adopted, which has been widely used by various researchers [6–8]. 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 [17–19]. 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 [23–25], 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 strict-feedback 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 [31–33].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 [31–33], only one parameter needs to be updated 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, which generally depends on the neural network nodes, and the computational burden are greatly reduced.(3)
With the bounded of the virtual control gaingi(·), the singularity problem by the estimation of gi(·) 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 Section2, 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.


## 2. Problem Formulation and Preliminaries

### 2.1. Longitudinal Model of HFVs

The model considered in this paper is taken from the NASA Langley Research Center [2, 3]. Cruising at a Mach number of 15 and at an altitude of 110000 ft, the longitudinal hypersonic flight model is given by(1)V˙=TV,βcos⁡α-DV,αm-μsin⁡γr2,h˙=Vsin⁡γ,γ˙=LV,α+TV,βsin⁡αmV-μ-V2rcos⁡γVr2,α˙=q-γ˙,q˙=MyyV,α,q,δEIyy,where V is the velocity, γ the flight path angle, h the altitude, α the attack angle, q the pitch rate, δE the elevator deflection, and β the throttle setting. TV,β, DV,α, LV,α, and MyyV,α,q,δE represent the thrust, drag, lift-force, and pitching moment, respectively, which can be expressed as(2)TV,β=ρV2SCT2,LV,α=ρV2SCL2,DV,α=ρV2SCD2,Myy=12ρV2Sc¯CMα+CMδE+CMq,with(3)CL=0.6203α,CD=0.6450α2+0.0043378α+0.003772,CT=0.02576βifβ<10.0224+0.00336βifβ>1,r=h+RE,CMα=-0.035α2+0.0366171+ΔCMαα+5.3261×10-6,CMq=c2Vq-6.796α2+0.3015α-0.2289,CMδE=ceδE-α.The nominal values of inertial and aerodynamic parameters are given in Table1. Besides, at trimmed cruise condition, V=15060ft/s, h=11000ft, γ=0 rad, α=0.0315, and q=0 rad/s.Table 1
Parameters of the HFV.
Symbol
Parameter
Value
c
-
Mean aerodynamic chord
80 ft
I
y
y
Moment of inertia
7
×
1
0
6 slug·ft2
S
Reference area
3603 ft2
ρ
Air density
0.24325
×
1
0
-
4 slugs·ft−3
m
Mass of aircraft
9375 slug
R
E
Radius of the earth
20903500 ft
μ
Gravitational constant
1.39
×
1
0
16 ft3/s2The engine dynamics can be modeled by a second-order system:(4)β¨=-2ξωnβ˙-ωn2β+ωn2βc.Therefore, by selecting the commanded valueβc as the new control input, the HFV is composed of five state variables X=[V,h,γ,α,q]T and two control inputs U=[βc,δE]T, while the outputs to be controlled are selected as Y=[V,h]T. The design objective is such that the outputs track the desired altitude and velocity commands Yd=[Vd,hd]T 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 βc. The altitude change is related mainly to the elevator deflection δE. Thus, it is reasonable to divide the system into two loops: the velocity loop and the altitude loop.Note that the thrust termTsin⁡α is generally much smaller than the lift L, velocity V is high, and the flight path angle γ is typically very small during the trimmed cruise condition, which justify the following approximation.Assumption 1 (see [7, 19]).
The thrust termTsin⁡α≈0, and the term Vsin⁡γ≈Vγ.
Defining thatθ denotes the pitch angle, we have θ=α+γ. Then, we define state variables as x=[x1,x2,x3,x4]T, with x1=h, x2=γ, x3=θ, and x4=q. For simplicity, let x¯i=[x1,…,xi]T, so the altitude subsystem can be written as(5)x˙1=f1x¯1+g1x¯1x2,x˙2=f2x¯2+g2x¯2x3,x˙3=f3x¯3+g3x¯3x4,x˙4=f4x¯4+g4x¯4δE,y=x1,where (6)f1x¯1=0,f3x¯3=0,f2x¯2=-ρV2S2mV0.6203x2-μ-V2rVr2cos⁡x2,f4x¯4=c¯ρV2S2IyyCMx3-x2+CMx4-0.0292x3-x2,g1x¯1=V,g2x¯2=ρV2S2mV0.6203,g3x¯3=1,g4x¯4=cec¯ρV2S2Iyy.Since the values of the inertial and the aerodynamic parameters are uncertain, the aforementionedfi(x¯i) and gi(x¯i), i=1,2,3,4, are unknown smooth functions. Moreover, it is easy to check that gi(x¯i) are always strictly positive. With these observations in mind, we have the following assumption.Assumption 2.
There exist positive constantsbi and di such that 0<bi≤gi(x¯i)≤di.Remark 3.
It is worth noting that, in the proposed scheme, bothbi and di will not appear in the control law and will be used only for analysis; they can be unknown.Assumption 4.
V
d and its first derivative are known and bounded, while hd and its first four derivatives are continuous and bounded.


### 2.2. Description of RBF NNs

In this paper, RBF NNs will be employed to approximate unknown functions. Mathematically, an RBF NN can be expressed as(7)Fξ=WTψξ,where F∈R and ξ∈Rn are the NN outputs and input, W∈Rn is the weight vector, and ψξ=ψ1ξ,…,ψNξT is the basis function vector with ψiξ commonly chosen as the Gaussian functions:(8)ψiξ=12πϕexp⁡-ξ-ξi22ϕ2,ϕ>0,i=0,…,N,where ξi∈Rn and ϕ∈R are constants called the center and width of the basis function, respectively.Lemma 5 (see [17]).
Given any continuous functionF(ξ):Ωξ→R with Ωξ⊂Rn a compact set and any constant ϵ>0, by appropriately choosing ϕ and ξi,i=1,…,N, for some sufficiently large integer N, there exists an RBF NN W∗Tψ(ξ) such that(9)Fξ=W∗Tψξ+Δξ,Δξ≤ϵ,∀ξ∈Ωξ,where W∗ is the optimal weight vector defined as(10)W∗=arg⁡minW∈Rn⁡supξ∈Ωξ⁡Ψξ-WTψξ,and Δ(ξ) denotes the approximation error.


## 2.1. Longitudinal Model of HFVs

The model considered in this paper is taken from the NASA Langley Research Center [2, 3]. Cruising at a Mach number of 15 and at an altitude of 110000 ft, the longitudinal hypersonic flight model is given by(1)V˙=TV,βcos⁡α-DV,αm-μsin⁡γr2,h˙=Vsin⁡γ,γ˙=LV,α+TV,βsin⁡αmV-μ-V2rcos⁡γVr2,α˙=q-γ˙,q˙=MyyV,α,q,δEIyy,where V is the velocity, γ the flight path angle, h the altitude, α the attack angle, q the pitch rate, δE the elevator deflection, and β the throttle setting. TV,β, DV,α, LV,α, and MyyV,α,q,δE represent the thrust, drag, lift-force, and pitching moment, respectively, which can be expressed as(2)TV,β=ρV2SCT2,LV,α=ρV2SCL2,DV,α=ρV2SCD2,Myy=12ρV2Sc¯CMα+CMδE+CMq,with(3)CL=0.6203α,CD=0.6450α2+0.0043378α+0.003772,CT=0.02576βifβ<10.0224+0.00336βifβ>1,r=h+RE,CMα=-0.035α2+0.0366171+ΔCMαα+5.3261×10-6,CMq=c2Vq-6.796α2+0.3015α-0.2289,CMδE=ceδE-α.The nominal values of inertial and aerodynamic parameters are given in Table1. Besides, at trimmed cruise condition, V=15060ft/s, h=11000ft, γ=0 rad, α=0.0315, and q=0 rad/s.Table 1
Parameters of the HFV.
Symbol
Parameter
Value
c
-
Mean aerodynamic chord
80 ft
I
y
y
Moment of inertia
7
×
1
0
6 slug·ft2
S
Reference area
3603 ft2
ρ
Air density
0.24325
×
1
0
-
4 slugs·ft−3
m
Mass of aircraft
9375 slug
R
E
Radius of the earth
20903500 ft
μ
Gravitational constant
1.39
×
1
0
16 ft3/s2The engine dynamics can be modeled by a second-order system:(4)β¨=-2ξωnβ˙-ωn2β+ωn2βc.Therefore, by selecting the commanded valueβc as the new control input, the HFV is composed of five state variables X=[V,h,γ,α,q]T and two control inputs U=[βc,δE]T, while the outputs to be controlled are selected as Y=[V,h]T. The design objective is such that the outputs track the desired altitude and velocity commands Yd=[Vd,hd]T 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 βc. The altitude change is related mainly to the elevator deflection δE. Thus, it is reasonable to divide the system into two loops: the velocity loop and the altitude loop.Note that the thrust termTsin⁡α is generally much smaller than the lift L, velocity V is high, and the flight path angle γ is typically very small during the trimmed cruise condition, which justify the following approximation.Assumption 1 (see [7, 19]).
The thrust termTsin⁡α≈0, and the term Vsin⁡γ≈Vγ.
Defining thatθ denotes the pitch angle, we have θ=α+γ. Then, we define state variables as x=[x1,x2,x3,x4]T, with x1=h, x2=γ, x3=θ, and x4=q. For simplicity, let x¯i=[x1,…,xi]T, so the altitude subsystem can be written as(5)x˙1=f1x¯1+g1x¯1x2,x˙2=f2x¯2+g2x¯2x3,x˙3=f3x¯3+g3x¯3x4,x˙4=f4x¯4+g4x¯4δE,y=x1,where (6)f1x¯1=0,f3x¯3=0,f2x¯2=-ρV2S2mV0.6203x2-μ-V2rVr2cos⁡x2,f4x¯4=c¯ρV2S2IyyCMx3-x2+CMx4-0.0292x3-x2,g1x¯1=V,g2x¯2=ρV2S2mV0.6203,g3x¯3=1,g4x¯4=cec¯ρV2S2Iyy.Since the values of the inertial and the aerodynamic parameters are uncertain, the aforementionedfi(x¯i) and gi(x¯i), i=1,2,3,4, are unknown smooth functions. Moreover, it is easy to check that gi(x¯i) are always strictly positive. With these observations in mind, we have the following assumption.Assumption 2.
There exist positive constantsbi and di such that 0<bi≤gi(x¯i)≤di.Remark 3.
It is worth noting that, in the proposed scheme, bothbi and di will not appear in the control law and will be used only for analysis; they can be unknown.Assumption 4.
V
d and its first derivative are known and bounded, while hd and its first four derivatives are continuous and bounded.


## 2.2. Description of RBF NNs

In this paper, RBF NNs will be employed to approximate unknown functions. Mathematically, an RBF NN can be expressed as(7)Fξ=WTψξ,where F∈R and ξ∈Rn are the NN outputs and input, W∈Rn is the weight vector, and ψξ=ψ1ξ,…,ψNξT is the basis function vector with ψiξ commonly chosen as the Gaussian functions:(8)ψiξ=12πϕexp⁡-ξ-ξi22ϕ2,ϕ>0,i=0,…,N,where ξi∈Rn and ϕ∈R are constants called the center and width of the basis function, respectively.Lemma 5 (see [17]).
Given any continuous functionF(ξ):Ωξ→R with Ωξ⊂Rn a compact set and any constant ϵ>0, by appropriately choosing ϕ and ξi,i=1,…,N, for some sufficiently large integer N, there exists an RBF NN W∗Tψ(ξ) such that(9)Fξ=W∗Tψξ+Δξ,Δξ≤ϵ,∀ξ∈Ωξ,where W∗ is the optimal weight vector defined as(10)W∗=arg⁡minW∈Rn⁡supξ∈Ωξ⁡Ψξ-WTψξ,and Δ(ξ) denotes the approximation error.


## 3. Adaptive Neural Controller Design

### 3.1. Performance and Error Transformation Functions

Let the tracking error be defined as(11)e=y-yd,where yd is the desired trajectory. Similar to [23, 24], the mathematical expression of the prescribed tracking performance is given by(12)-κεt<et<τεt,where κ and τ are given positive constants and the smooth function is given by(13)εt=ε0-ε∞exp⁡-lt+ε∞,in which ε0 is the initial value of ε(t),  ε∞ represents the value of ε(t) at the steady state, and l is the decreasing rate of ε(t). Then, introduce the following error transformation:(14)Sz=etεt,where z is the transformed error and Sz is a smooth, strictly increasing, and thus invertible function possessing the following properties:(15)limz→-∞⁡Sz=-κ,limz→+∞⁡Sz=τ.Note that ifz is kept bounded, we have -κ<Sz<τ, and thus (12) holds. The inverse transformation of Sz can be written as(16)z=S-1etεt≔Θetεt.In this paper, we choose(17)z=Θetεt=ln⁡κ+et/εtτ-et/εt.Differentiating (17) yields(18)z˙=ηy˙-ηυ,where η=∂Θ/∂(e/ε)1/ε and υ=y˙d+eε˙/ε. From the properties of the transformation, it is clear that η and υ are bounded and 0<η0≤η.Remark 6.
From (12) and (13), one can see that τε(0) and -κε(0) serve as the upper bound of the overshoot and the lower bound of the undershoot of e(t), respectively, the decreasing rate of ε(t) introduces a lower bound of the convergence rate of e(t), and max⁡{κε∞,τε∞} represents the maximum allowable size of the steady-state value of e(t). Note that ε(0), τ, and κ should be properly chosen such that -κε(0)<e(0)<τε(0).


### 3.2. Attitude Controller Design via Backstepping

After the error transformation (18), the altitude subsystem (5) is equivalent to(19)z˙1=η1g1x¯1x2-η1υ1,x˙2=f2x¯2+η2g2x¯2x3,x˙3=η3g3x¯3x4,x˙4=η4g4x¯4δE+f4x¯4,y=x1,where 0<η10≤η1 and ηk=ηk0=1,k=2,3,4. The stabilization of the transformed system (19) is sufficient to guarantee the prescribed tracking performance of system (5).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 Fiξi and Ωξi denote the unknown function to be estimated by RBF NNs and the corresponding compact set in the ith step, respectively. Then by using Lemma 5, we have(20)Fiξi=Wi∗Tψiξi+Δiξi,Δiξi≤ϵi,where ψiξi and ξi denote the vector valued function and the RBF NN input in the step with proper dimensions that are given below.Step  1. Let z1 given by (19) be the first error variable. Define z2=x2-u1, where u1 is the first virtual control signal. Then the derivative of z1 can be expressed as(21)z˙1=η1g1x¯1z2+u1-12z1+F1ξ1,where F1ξ1=1/2z1-η1υ1 and ξ1≔x1,hd,h˙d,ε1T⊂R4. Since F1ξ1 is unknown, we employ an RBF NN to approximate it on a compact set Ωξ1⊂R4. By properly choosing the basis function vectors we have(22)F1ξ1=W1∗Tψ1ξ1+Δ1ξ1,Δ1ξ1≤ϵ1,where ϵ1 is a positive constant. With respect to the unknown optimal weight vector in (22), define(23)ϑ1=W1∗2η10b1.Besides, letϑ^1 be the estimation of ϑ1 and ϑ~1≔ϑ^1-ϑ1. Consider the first Lyapunov function(24)L1=12z12+η10b12λ1ϑ~12.Taking the time derivation of (24) yields(25)L˙1=z1η1g1x¯1z2+z1η1g1x¯1u1-12z12+z1W1∗Tψ1ξ1+z1Δ1ξ1+η10b1λ1ϑ~1ϑ^˙1.Using Young’s inequality and (23), it can be verified that(26)z1W1∗Tψ1ξ1≤12z12W1∗2ψ1Tξ1ψ1ξ1+12≤η10b12ϑ1z12ψ1Tξ1ψ1ξ1+12,z1Δ1ξ1≤12z12+12ϵ12.Thus, (25) can be rewritten as(27)L˙1≤z1η1g1x¯1z2+z1η1g1x¯1u1+η10b12ϑ^1z12ψ1Tξ1ψ1ξ1+12+12ϵ12+η10b1λ1ϑ~1ϑ^˙1-λ12z12ψ1Tξ1ψ1ξ1,which suggests that we choose the first virtual control signal u1 as(28)u1=-c1z1-12ϑ^1z1ψ1Tξ1ψ1ξ1.Let(29)ϑ^˙1=λ12z12ψ1Tξ1ψ1ξ1-λ1σ1ϑ^1,ϑ^10≥0,where c1, λ1, and σ1 are positive design parameters. Then substituting (28) and (29) into (27), we get(30)L˙1≤-η10b1c1z12+z1η1g1x¯1z2-η10b1σ1ϑ~1ϑ^1+12+12ϵ12.Step i
i
=
2,3. Define zi+1=xi+1-ui, where ui is the ith virtual control signal. Then the time derivation of zi is(31)z˙i=ηigix¯izi+1+ηigix¯iui-ηi-1gi-1x¯i-1zi-1-12zi+Fiξi,where Fiξi=fi(x¯i)+ηi-1gi-1x¯i-1zi-1+1/2zi-u˙i-1 and ξi=x¯i,hd,…,hdi,ϑ^1,…,ϑ^i,ε1T∈Ωξi. Consider the ith Lyapunov function(32)Li=Li-1+12zi2+ηi0bi2λiϑ~i2,where λi is a positive design parameter, ϑ~i=ϑ^i-ϑi with ϑi=Wi∗2/ηi0bi. By taking the time derivation of (32), we have(33)L˙i=L˙i-1-ηi-1gi-1x¯i-1zi-1zi+ziηigix¯izi+ziηigix¯iui-12zi2+ziFiξi+ηi0biλiϑ~iϑ^˙i.Similar to (26), we have(34)L˙i≤L˙i-1-ηi-1gi-1x¯i-1zi-1zi+ziηigix¯izi+1+ziηigix¯iui+ηi0bi2ϑ^izi2ψiTξiψiξi+ηi0biλiϑ~iϑ^˙i-λi2zi2ψiTξiψiξi+12+12ϵi2.Choose theith virtual control signal(35)ui=-cizi-12ϑ^iziψiTξiψiξi,where ϑ^i is updated by(36)ϑ^˙i=λi2zi2ψiTξiψiξi-λiσiϑ^i,ϑ^i0≥0,with ci, λi, and σi being positive design parameters. Substituting (35), (36), and (30) into (34), we get(37)L˙i≤-∑k=1iηk0bkckzk2+ziηigix¯izi+1-∑k=1iηk0bkσkϑ~kϑ^k+∑k=1i121+ϵk2.Step  4. The time derivative of z4 is(38)z˙4=-η3g3x¯3z3+η4g4x¯4δE-12z4+F4ξ4,where F4ξ4=f4(x¯4)+η3g3(x¯3)z3+1/2z4-u˙3 and ξ4=x¯4,hd,…,hd4,ϑ^1,ϑ^2,ϑ^3,εhT∈Ωξ4⊂R13. Let(39)L4=L3+12z42+η40b42λ4ϑ~42,where λ4 is a positive design parameter and ϑ~4=ϑ^4-ϑ4 with ϑ4=W4∗2/η40b4. Differentiating (39) we have(40)L˙4=L˙3+η40b4λ4ϑ~4ϑ^˙4-z4η3g3x¯3z3+z4η4g4x¯4δE-12z42+z4F4ξ4.Similar to (26), (40) can be rewritten as(41)L˙4≤L˙3-z4η3g3x¯3z3+z4η4g4x¯4δE+η40b42ϑ^4z42ψ4Tξ4ψ4ξ4+12+12ϵ42+η40b4λ4ϑ~4ϑ^˙4-λ42z42ψ4Tξ4ψ4ξ4.Choose the control signal(42)δE=-c4z4-12ϑ^4z4ψ4Tξ4ψ4ξ4,where ϑ^4 is updated by(43)ϑ^˙4=λ42z42ψ4Tξ4ψ4ξ4-λ4σ4ϑ^4,ϑ^40≥0,with c4, λ4, and σ4 being positive design parameters. Substituting (42), (43), and (37) into (41), we arrive at(44)L˙4≤-∑k=14ckηk0bkzk2-∑k=14ηk0bkσkϑ~kϑ^k+∑k=1412+12ϵk2.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 signalui 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 ηigi 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.Theorem 9.
Consider system (5) under the Assumptions 2 and 4, with the error transformation (17), the virtual control signals (28) and (35), the control law (42), and the adaptive laws (29), (36), and (43). Then all closed-loop signals are uniformly bounded and the prescribed tracking performance (12) can be guaranteed.Proof.
Using the following facts:(45)-ϑ~iϑ^i≤-12ϑ~i2+12ϑ^i2,i=1,…,4,we rewrite (44) as(46)L˙4≤-∑k=14ckηk0bkzk2-∑k=1412ηk0bkσkϑ~k2+ς,where(47)ς=∑k=1412ηk0bkσkϑ^k2+∑k=1412+12ϵk2.
Let(48)ℏ=min⁡2ηi0bici,λiσi,i=1,…,4.
Then we have(49)L˙4≤-ℏL4+ς.
Solving (49) gives(50)0≤L4t≤ςℏ+L40-ςℏe-ℏt,t≥0,which implies that L4, zi, ϑ~i, and ϑ^i are bounded. Since z1 is bounded, according to the error transformation of (15), to (17) we can obtain that -κ<Sz<τ; as a result, we have -κεt<et<τεt; that is, the prescribed tracking performance is guaranteed. This completes the proof.


### 3.3. Velocity Controller Design via Dynamic Inversion

The velocity subsystem of (1) can be rewritten as follows:(51)V˙=fVx¯4,V+gVx¯4,Vβc,where fVx¯4,V and gVx¯4,V are unknown nonlinear function gV(x¯4,V)≥bV>0. Then define the velocity tracking error as eV=V-Vd. According to (17) and (18) we obtain(52)z˙V=-ηVυV+ηVfVx¯4,V+ηVgVx¯4,Vβc.We assume that0<ηV0≤ηV. The transformed system dynamics of (52) can be rewritten as(53)z˙V=ηVgVx¯4,Vβc-12zV+FVξV,where FVξV=1/2zV-ηVυV+ηVfVx¯4,V and ξV≔[x¯4,V,V˙d,εV]T∈ΩξV. Since FVξV is an unknown nonlinear function, we use an RBF NN to approximate it:(54)FVξV=WV∗TψVξV+ΔVξV,ΔVξV≤ϵV.The control law and the adaptive update law are designed as follows:(55)βc=-cVzV-12zVϑ^VψVTξVψVξV,ϑ^˙V=λV2zV2ψVTξVψVξV-λVσVϑ^V,where ϑ^V is the estimate of ϑV=WV∗2/ηV0bV, and cV, γV, and σV are positive design parameters. Consider the Lyapunov function(56)LV=12zV2+ηV0bV2λVϑ~V2.DifferentiatingLV we have(57)L˙V≤zVηV0bVβc+ηV0bV2zV2ϑ^ψVTξ1ψVξV+ηV0bVλVϑ~Vϑ^˙V-λV2zV2ψVTξ1ψVξV+12+12ϵV2.Substituting (55) into (57) then(58)L˙V≤-cVηV0bVzV2-ηV0bVσVϑ~Vϑ^V+12+12ϵV2,with inequation(59)-ϑ~Vϑ^V≤-12ϑ~V2+12ϑ^2,and then(60)L˙V≤-cVηV0bVzV2-12ηV0bVσVϑ~V2+12ηV0bVσVϑ^2+12+12ϵV2≤-κVLV+μV,where(61)κV=min⁡2cVηV0bV,λVσV,μV=12ηV0bVσVϑ^2+12+12ϵV2.Solving (60) gives(62)0≤LVt≤μVκV+LV0-μVκVe-κVt,t≥0.It is clear thatzV, ϑ^V, and ϑ~V are bounded. Owning that zV is bounded, together with the error transformation of (11) into (17), implies that the prescribed tracking performance is guaranteed.


## 3.1. Performance and Error Transformation Functions

Let the tracking error be defined as(11)e=y-yd,where yd is the desired trajectory. Similar to [23, 24], the mathematical expression of the prescribed tracking performance is given by(12)-κεt<et<τεt,where κ and τ are given positive constants and the smooth function is given by(13)εt=ε0-ε∞exp⁡-lt+ε∞,in which ε0 is the initial value of ε(t),  ε∞ represents the value of ε(t) at the steady state, and l is the decreasing rate of ε(t). Then, introduce the following error transformation:(14)Sz=etεt,where z is the transformed error and Sz is a smooth, strictly increasing, and thus invertible function possessing the following properties:(15)limz→-∞⁡Sz=-κ,limz→+∞⁡Sz=τ.Note that ifz is kept bounded, we have -κ<Sz<τ, and thus (12) holds. The inverse transformation of Sz can be written as(16)z=S-1etεt≔Θetεt.In this paper, we choose(17)z=Θetεt=ln⁡κ+et/εtτ-et/εt.Differentiating (17) yields(18)z˙=ηy˙-ηυ,where η=∂Θ/∂(e/ε)1/ε and υ=y˙d+eε˙/ε. From the properties of the transformation, it is clear that η and υ are bounded and 0<η0≤η.Remark 6.
From (12) and (13), one can see that τε(0) and -κε(0) serve as the upper bound of the overshoot and the lower bound of the undershoot of e(t), respectively, the decreasing rate of ε(t) introduces a lower bound of the convergence rate of e(t), and max⁡{κε∞,τε∞} represents the maximum allowable size of the steady-state value of e(t). Note that ε(0), τ, and κ should be properly chosen such that -κε(0)<e(0)<τε(0).


## 3.2. Attitude Controller Design via Backstepping

After the error transformation (18), the altitude subsystem (5) is equivalent to(19)z˙1=η1g1x¯1x2-η1υ1,x˙2=f2x¯2+η2g2x¯2x3,x˙3=η3g3x¯3x4,x˙4=η4g4x¯4δE+f4x¯4,y=x1,where 0<η10≤η1 and ηk=ηk0=1,k=2,3,4. The stabilization of the transformed system (19) is sufficient to guarantee the prescribed tracking performance of system (5).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 Fiξi and Ωξi denote the unknown function to be estimated by RBF NNs and the corresponding compact set in the ith step, respectively. Then by using Lemma 5, we have(20)Fiξi=Wi∗Tψiξi+Δiξi,Δiξi≤ϵi,where ψiξi and ξi denote the vector valued function and the RBF NN input in the step with proper dimensions that are given below.Step  1. Let z1 given by (19) be the first error variable. Define z2=x2-u1, where u1 is the first virtual control signal. Then the derivative of z1 can be expressed as(21)z˙1=η1g1x¯1z2+u1-12z1+F1ξ1,where F1ξ1=1/2z1-η1υ1 and ξ1≔x1,hd,h˙d,ε1T⊂R4. Since F1ξ1 is unknown, we employ an RBF NN to approximate it on a compact set Ωξ1⊂R4. By properly choosing the basis function vectors we have(22)F1ξ1=W1∗Tψ1ξ1+Δ1ξ1,Δ1ξ1≤ϵ1,where ϵ1 is a positive constant. With respect to the unknown optimal weight vector in (22), define(23)ϑ1=W1∗2η10b1.Besides, letϑ^1 be the estimation of ϑ1 and ϑ~1≔ϑ^1-ϑ1. Consider the first Lyapunov function(24)L1=12z12+η10b12λ1ϑ~12.Taking the time derivation of (24) yields(25)L˙1=z1η1g1x¯1z2+z1η1g1x¯1u1-12z12+z1W1∗Tψ1ξ1+z1Δ1ξ1+η10b1λ1ϑ~1ϑ^˙1.Using Young’s inequality and (23), it can be verified that(26)z1W1∗Tψ1ξ1≤12z12W1∗2ψ1Tξ1ψ1ξ1+12≤η10b12ϑ1z12ψ1Tξ1ψ1ξ1+12,z1Δ1ξ1≤12z12+12ϵ12.Thus, (25) can be rewritten as(27)L˙1≤z1η1g1x¯1z2+z1η1g1x¯1u1+η10b12ϑ^1z12ψ1Tξ1ψ1ξ1+12+12ϵ12+η10b1λ1ϑ~1ϑ^˙1-λ12z12ψ1Tξ1ψ1ξ1,which suggests that we choose the first virtual control signal u1 as(28)u1=-c1z1-12ϑ^1z1ψ1Tξ1ψ1ξ1.Let(29)ϑ^˙1=λ12z12ψ1Tξ1ψ1ξ1-λ1σ1ϑ^1,ϑ^10≥0,where c1, λ1, and σ1 are positive design parameters. Then substituting (28) and (29) into (27), we get(30)L˙1≤-η10b1c1z12+z1η1g1x¯1z2-η10b1σ1ϑ~1ϑ^1+12+12ϵ12.Step i
i
=
2,3. Define zi+1=xi+1-ui, where ui is the ith virtual control signal. Then the time derivation of zi is(31)z˙i=ηigix¯izi+1+ηigix¯iui-ηi-1gi-1x¯i-1zi-1-12zi+Fiξi,where Fiξi=fi(x¯i)+ηi-1gi-1x¯i-1zi-1+1/2zi-u˙i-1 and ξi=x¯i,hd,…,hdi,ϑ^1,…,ϑ^i,ε1T∈Ωξi. Consider the ith Lyapunov function(32)Li=Li-1+12zi2+ηi0bi2λiϑ~i2,where λi is a positive design parameter, ϑ~i=ϑ^i-ϑi with ϑi=Wi∗2/ηi0bi. By taking the time derivation of (32), we have(33)L˙i=L˙i-1-ηi-1gi-1x¯i-1zi-1zi+ziηigix¯izi+ziηigix¯iui-12zi2+ziFiξi+ηi0biλiϑ~iϑ^˙i.Similar to (26), we have(34)L˙i≤L˙i-1-ηi-1gi-1x¯i-1zi-1zi+ziηigix¯izi+1+ziηigix¯iui+ηi0bi2ϑ^izi2ψiTξiψiξi+ηi0biλiϑ~iϑ^˙i-λi2zi2ψiTξiψiξi+12+12ϵi2.Choose theith virtual control signal(35)ui=-cizi-12ϑ^iziψiTξiψiξi,where ϑ^i is updated by(36)ϑ^˙i=λi2zi2ψiTξiψiξi-λiσiϑ^i,ϑ^i0≥0,with ci, λi, and σi being positive design parameters. Substituting (35), (36), and (30) into (34), we get(37)L˙i≤-∑k=1iηk0bkckzk2+ziηigix¯izi+1-∑k=1iηk0bkσkϑ~kϑ^k+∑k=1i121+ϵk2.Step  4. The time derivative of z4 is(38)z˙4=-η3g3x¯3z3+η4g4x¯4δE-12z4+F4ξ4,where F4ξ4=f4(x¯4)+η3g3(x¯3)z3+1/2z4-u˙3 and ξ4=x¯4,hd,…,hd4,ϑ^1,ϑ^2,ϑ^3,εhT∈Ωξ4⊂R13. Let(39)L4=L3+12z42+η40b42λ4ϑ~42,where λ4 is a positive design parameter and ϑ~4=ϑ^4-ϑ4 with ϑ4=W4∗2/η40b4. Differentiating (39) we have(40)L˙4=L˙3+η40b4λ4ϑ~4ϑ^˙4-z4η3g3x¯3z3+z4η4g4x¯4δE-12z42+z4F4ξ4.Similar to (26), (40) can be rewritten as(41)L˙4≤L˙3-z4η3g3x¯3z3+z4η4g4x¯4δE+η40b42ϑ^4z42ψ4Tξ4ψ4ξ4+12+12ϵ42+η40b4λ4ϑ~4ϑ^˙4-λ42z42ψ4Tξ4ψ4ξ4.Choose the control signal(42)δE=-c4z4-12ϑ^4z4ψ4Tξ4ψ4ξ4,where ϑ^4 is updated by(43)ϑ^˙4=λ42z42ψ4Tξ4ψ4ξ4-λ4σ4ϑ^4,ϑ^40≥0,with c4, λ4, and σ4 being positive design parameters. Substituting (42), (43), and (37) into (41), we arrive at(44)L˙4≤-∑k=14ckηk0bkzk2-∑k=14ηk0bkσkϑ~kϑ^k+∑k=1412+12ϵk2.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 signalui 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 ηigi 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.Theorem 9.
Consider system (5) under the Assumptions 2 and 4, with the error transformation (17), the virtual control signals (28) and (35), the control law (42), and the adaptive laws (29), (36), and (43). Then all closed-loop signals are uniformly bounded and the prescribed tracking performance (12) can be guaranteed.Proof.
Using the following facts:(45)-ϑ~iϑ^i≤-12ϑ~i2+12ϑ^i2,i=1,…,4,we rewrite (44) as(46)L˙4≤-∑k=14ckηk0bkzk2-∑k=1412ηk0bkσkϑ~k2+ς,where(47)ς=∑k=1412ηk0bkσkϑ^k2+∑k=1412+12ϵk2.
Let(48)ℏ=min⁡2ηi0bici,λiσi,i=1,…,4.
Then we have(49)L˙4≤-ℏL4+ς.
Solving (49) gives(50)0≤L4t≤ςℏ+L40-ςℏe-ℏt,t≥0,which implies that L4, zi, ϑ~i, and ϑ^i are bounded. Since z1 is bounded, according to the error transformation of (15), to (17) we can obtain that -κ<Sz<τ; as a result, we have -κεt<et<τεt; that is, the prescribed tracking performance is guaranteed. This completes the proof.


## 3.3. Velocity Controller Design via Dynamic Inversion

The velocity subsystem of (1) can be rewritten as follows:(51)V˙=fVx¯4,V+gVx¯4,Vβc,where fVx¯4,V and gVx¯4,V are unknown nonlinear function gV(x¯4,V)≥bV>0. Then define the velocity tracking error as eV=V-Vd. According to (17) and (18) we obtain(52)z˙V=-ηVυV+ηVfVx¯4,V+ηVgVx¯4,Vβc.We assume that0<ηV0≤ηV. The transformed system dynamics of (52) can be rewritten as(53)z˙V=ηVgVx¯4,Vβc-12zV+FVξV,where FVξV=1/2zV-ηVυV+ηVfVx¯4,V and ξV≔[x¯4,V,V˙d,εV]T∈ΩξV. Since FVξV is an unknown nonlinear function, we use an RBF NN to approximate it:(54)FVξV=WV∗TψVξV+ΔVξV,ΔVξV≤ϵV.The control law and the adaptive update law are designed as follows:(55)βc=-cVzV-12zVϑ^VψVTξVψVξV,ϑ^˙V=λV2zV2ψVTξVψVξV-λVσVϑ^V,where ϑ^V is the estimate of ϑV=WV∗2/ηV0bV, and cV, γV, and σV are positive design parameters. Consider the Lyapunov function(56)LV=12zV2+ηV0bV2λVϑ~V2.DifferentiatingLV we have(57)L˙V≤zVηV0bVβc+ηV0bV2zV2ϑ^ψVTξ1ψVξV+ηV0bVλVϑ~Vϑ^˙V-λV2zV2ψVTξ1ψVξV+12+12ϵV2.Substituting (55) into (57) then(58)L˙V≤-cVηV0bVzV2-ηV0bVσVϑ~Vϑ^V+12+12ϵV2,with inequation(59)-ϑ~Vϑ^V≤-12ϑ~V2+12ϑ^2,and then(60)L˙V≤-cVηV0bVzV2-12ηV0bVσVϑ~V2+12ηV0bVσVϑ^2+12+12ϵV2≤-κVLV+μV,where(61)κV=min⁡2cVηV0bV,λVσV,μV=12ηV0bVσVϑ^2+12+12ϵV2.Solving (60) gives(62)0≤LVt≤μVκV+LV0-μVκVe-κVt,t≥0.It is clear thatzV, ϑ^V, and ϑ~V are bounded. Owning that zV is bounded, together with the error transformation of (11) into (17), implies that the prescribed tracking performance is guaranteed.


## 4. 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:(63)hdhc=ϖn12ϖn22p+ϖn12p2+2ζϖn2p+ϖn22,VdVc=ϖn12p2+2ζϖn1p+ϖn12,where p is Laplace operator, ϖn1=0.5, ϖn2=0.3, and ζh=0.95. hc and Vc are step commands. The inputs of the RBF NNs are ξ1,ξ2,ξ3,ξ4, and ξV and the nonlinear functions are approximated with width ϕ=1. The initial values of the adaptive laws are ϑ^10=ϑ^20=ϑ^30=ϑ^40=ϑ^V0=0. In addition we choose the design parameters c1=0.04, c2=150, c3=120, c4=100, cV=0.5, λ1=0.1, λ2=0.1, λ3=0.01, λ4=0.01, λV=10, σ1=0.1, σ2=0.1, σ3=0.1, σ4=0.1, and σV=1. The parameters of performance functions are given by εh0=30, εh∞=5, lh=0.2, εV0=20, εV∞=2, lV=0.5, and κ=τ=1. The initial states are chosen as V=15060ft/s, h=110000ft, γ=0 rad, θ=0.0315 rad, and q=0 rad/s.The simulation results are presented in Figures1−10. 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 Tsinα/(mV) is 3×10-5; it is much smaller than the term L/(mV) whose minimum value is 1×10-3 and the maximum value of flight path angle γ is less than 0.012 rad. Therefore, Assumption 1 is reasonable.Figure 1
Velocity tracking.Figure 2
The velocity tracking erroreV and the prescribed error bounds.Figure 3
Throttle settingβ.Figure 4
The altitude tracking.Figure 5
Altitude tracking erroreh and the prescribed error bounds.Figure 6
Elevator deflectionδE.Figure 7
Altitude tracking.Figure 8
Altitude tracking erroreh and the prescribed error bounds.Figure 9
Elevator deflectionδE.Figure 10
(a) Path angelγ; (b) attack angel α; (c) pitch rate q.
(a)
(b)
(c)


## 5. 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.



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*Source: 591789-2015-06-14.xml*