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

Authors: Guoqiang Zhu; Lingfang Sun; Xiuyu Zhang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A neural network robust control is proposed for a class of generic hypersonic flight vehicles with uncertain dynamics and stochastic disturbance. Compared with the present schemes of dealing with 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.


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## Body

## 1. 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 [6–8]. 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 [9–11].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, 15–17]. To eliminate this problem, dynamic surface control (DSC) was applied to longitudinal dynamics of HFVs in [18–23], 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 [24–26]. 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 [27–30], 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 Section2, 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.


## 2. 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(1)V˙=Tcos⁡α-Dm-gsin⁡γ+ω1,h˙=Vsin⁡γ,γ˙=L+Tsin⁡αmV-gcos⁡γV,α˙=q-γ˙+ω,q˙=MyyIyy+ω5,where V is the velocity, γ is the flight path angle, h is the altitude, α is the attack angle, q is the pitch rate, and ωi,i=1,5, are the uncertain external disturbance. 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)T=CTα3α3+CTα2α2+CTαα+CT0,D=q¯SCDα,δe,L=q¯SCLα,δe,Myy=zTT+q¯Sc¯CM,αα+CM,δeδe,with CL=CLαα+CL0, CD=CDα2α2+CDαα+CD0, CM,α(α)=CM,αα2α2+CM,ααα+CM,α0, CTα3=β1Φ+β2, CTα2=β3Φ+β4, CTα=β5Φ+β6, CT0=β7Φ+β8, q¯=ρV2/2, and ρ=ρ0exp⁡-h-h0/hs, where δe is the elevator deflection and Φ is the throttle setting. Letting θ denote the pitch angle, we have θ=α+γ. Then, we define state variables as x=x1,x2,x3,x4,x5T, with x1=V, x2=h, x3=γ, x4=θ, and x5=q. Note that the flight path angle γ is typically very small during the trimmed cruise condition, which justify the approximation sin⁡(γ)≈γ, so the system (1) can be rewritten as(3)x˙=fx+gxu+ω,y=x1,x2,where f(x)=f1x,f2x,f3x,f4x,f5xT, g(x)=g1x,g2x,g3x,g4x,g5xT, f1(x)=βaφ0cos⁡α-D/m-gsin⁡γ, f2(x)=0, f3=q¯SCL0-CLαx2/mV+Tsin⁡α/mV-gcos⁡x2/V, f4(x)=0, f5=zTT+q¯Sc¯CM,α(α)/Iyy, g1(x)=βbφ0cos⁡α/m, g2(x)=V, g3(x)=q¯SCLα/mV, g4(x)=1, g5(x)=q¯Sc¯ce/Iyy, βa=β2,β4,β6,β8, βb=β1,β3,β5,β7, and φ0=α3,α2,α,1T. Since the values of the inertial and the aerodynamic parameters are uncertain, the aforementioned fi(x) and gi(x),i=1,…,5, are unknown functions. Moreover, as stated in [18], from the model of the HFV, it is easy to check that g1(x), g2(x), and g3(x) are always strictly positive and g5(x) is strictly negative since ce is negative. With these observations in mind, we have the following assumption.Assumption 1.
Notice that there exist positive constantsbim and biM such that 0<bim≤gix≤biM,i=1,…,5.Assumption 2.
V
d and its first derivative are known and bounded, while hd and its first two derivatives are known and bounded.Remark 3.
In the following proposed scheme, bothbim and biM 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.


### 2.1. 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(4)Fξ=WTψξ,where F∈R and ξ∈Rn are the NN output and input, W∈Rn is an N-dimensional weight vector, and ψξ=ψ1ξ,…,ψNξT is the Gaussian function with the following form:(5)ψiξ=12πϕexp⁡-ξ-ξi22ϕ2,ϕ>0,i=0,…,N,where ξi∈Rn is a constant called the center of the Gaussian function and ϕ∈R is the width, respectively.Lemma 5.
According to the approximation property of RBF NNs, 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 optimal weight W∗T such that W∗Tψ(ξ) can approximate the given function F(ξ) with approximation error Δ(ξ) bounded by ϵ [26].(6)Fξ=W∗Tψξ+Δξ,Δξ≤ϵ,∀ξ∈Ωξ,since W∗ is unknown; we need to estimate it online.


## 2.1. 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(4)Fξ=WTψξ,where F∈R and ξ∈Rn are the NN output and input, W∈Rn is an N-dimensional weight vector, and ψξ=ψ1ξ,…,ψNξT is the Gaussian function with the following form:(5)ψiξ=12πϕexp⁡-ξ-ξi22ϕ2,ϕ>0,i=0,…,N,where ξi∈Rn is a constant called the center of the Gaussian function and ϕ∈R is the width, respectively.Lemma 5.
According to the approximation property of RBF NNs, 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 optimal weight W∗T such that W∗Tψ(ξ) can approximate the given function F(ξ) with approximation error Δ(ξ) bounded by ϵ [26].(6)Fξ=W∗Tψξ+Δξ,Δξ≤ϵ,∀ξ∈Ωξ,since W∗ is unknown; we need to estimate it online.


## 3. 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(7)-mipit<eit<nipit,where ei=yi-yid,i=1,2, are the tracking error, mi and ni are given positive constants, and the performance function pi is defined as smooth and decreasing positive function. From (7), one can see that -mipi(0) and nipi(0) are the lower and upper bound of the undershoot of ei(t), respectively. mipi(∞) and nipi(∞) represent the maximum allowable size of the steady-state value of ei(t).To transform (7) into an equivalent unconstrained one, define the error transformation function Υi(Si)=ei(t)/pi(t), where Si is the transformed error and Υi(Si) is a smooth, strictly increasing function having the following properties:(8)limS→-∞ΥiSi=-mi,limS→+∞ΥiSi=ni.From (8), if Si is bounded, we have -mi<ΥiSi<ni, and thus (7) holds. Hence, to achieve the prespecified tracking performance, one only needs to keep Si bounded. The inverse transformation of ΥiSi can be expressed as(9)Si=Υi-1eitpit≔Θieitpit,and differentiating (9) yields(10)S˙i=ηiy˙i-ηiυi.From the properties of the transformation, it is clear that ηi and υi are bounded and 0<ηi0≤ηi.


### 3.2. 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:(11)x˙1=f1x+g1xΦ+ω1,y1=x1,where f1(x) and g1(x) are unknown nonlinear function and g1≥b1m>0. Then define the velocity tracking error as e1=x1-x1d. According to (10) and (11) we obtain(12)S˙1=-η1υ1+η1f1x+η1g1xΦ+ω1.Let F1ξ1=1/2S1-η1υ1+η1f1(x)+ω1 and ξ1≔x,x˙1d,p1,ω1T∈Ωξ1⊆R8. The transformed system dynamics of (12) can be rewritten as(13)S˙1=η1g1xΦ-12S1+F1ξ1.SinceF1ξ1 is an unknown nonlinear function, we use an RBF NN to approximate it. Then by using Lemma 5, we have(14)F1ξ1=W1∗Tψ1ξ1+Δ1ξ1,Δ1ξ1≤ϵ1.Choose the control signal and the adaptive update law as follows:(15)Φ=-c1S1-12S1ϑ^1ψ1Tξ1ψ1ξ1,ϑ^˙1=λ12S12ψ1Tξ1ψ1ξ1-λ1σ1ϑ^1,where ϑ^1 is the estimate of ϑ1=W1∗2/η10b1m and c1, λ1, and σ1 are designed positive parameters. Consider the Lyapunov function(16)Γ1=12S12+η10b1m2λ1ϑ~12.The differential of Lyapunov functionΓ1 can be found as follows.(17)Γ˙1≤S1ηV0b1mΦ+η10b1m2S12ϑ^ψ1Tξ1ψ1ξ1+η10b1mλ1ϑ~1ϑ^˙1-λ12S12ψ1Tξ1ψ1ξ1+12+12ϵ12.Substituting (15) into (17) yields(18)Γ˙1≤-c1η10b1mS12-η10b1mσ1ϑ~1ϑ^1+12+12ϵ12.According to Assumption1 and the inequality -ϑ~1ϑ^1≤-1/2ϑ~12+1/2ϑ2, we have(19)Γ˙1≤-c1η10b1mS12-12η10b1mσ1ϑ~12+12η10b1mσ1ϑ2+12+12ϵ12,Γ˙1≤-2κ1Γ1+C1,where(20)κ1=minc1η10b1m,12λ1σ1,μ1=12η10b1mσ1ϑ2+12+12ϵ12.Solving (19) gives(21)0≤Γ1t≤C12κ1+Γ10-C12κ1e-2κ1t,t≥0,from which it is clear that, by properly choosing the design parameters c1,λ1, and σ1, S1, ϑ^1, and ϑ~1 in the closed-loop system are uniformly ultimately bounded, and the prescribed tracking performance is guaranteed.


### 3.3. 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 δe is constructed at Step 4. After the error transformation (7)–(10), the altitude subsystem (1) is equivalent to(22)S˙2=η2g2xx3-η2υ2,x˙3=g3xx4+f3x,x˙4=g4xx5+f4x,x˙5=g5xδe+f5x+ω5,y2=x2.The stabilization of the transformed system (22) is sufficient to guarantee the prescribed tracking performance of altitude subsystem.Step 1.
LetS2 given by (22) be the first error variable. Define u3 as the first virtual control signal. Then the derivative of S2 can be expressed as(23)S˙2=η2g2xx3-12S2+F2ξ2,where F2ξ2=1/2S2-η2υ2. Since F2ξ2 is unknown, we employ an RBF NN to approximate it on a compact set Ωξ2. By properly choosing the basis function vectors we have(24)F2ξ2=W2∗Tψ2ξ2+Δ2ξ2,Δ2ξ2≤ϵ2,where ξ2:=x2,x2d,x˙2d,p2T∈Ωξ2⊆R4 and ϵ2 is a positive constant. With respect to the unknown optimal weight vector in (24), define(25)ϑ2=W2∗2η20b2m,where b2m are given by Assumption 1. Since ϑ2 is unknown, let ϑ^2 be the estimation of ϑ2 and ϑ~2≔ϑ^2-ϑ2. Consider the first Lyapunov function(26)Γ2=12S22+η20b2m2λ2ϑ~22.
The derivation of (26) can be found as follows:(27)Γ˙2=S2η2g2xx3-u3+S2η2g2xu3-12S22+S2W2∗Tψ2ξ2+S2Δ2ξ2+η20b2mλ2ϑ~2ϑ^˙2.
Using Young’s inequality, it can be verified that(28)S2W2∗Tψ2ξ2≤12S22W2∗2ψ2Tξ2ψ2ξ2+12≤η20b2m2ϑ2S22ψ2Tξ2ψ2ξ2+12,S2Δ2ξ2≤12S22+12ϵ22.
Thus, (27) can be rewritten as(29)Γ˙2≤S2η2g2x3-u3+S2η2g2u3+η20b2m2ϑ^2S22ψ2Tξ2ψ2ξ2+12+12ϵ22+η20b2mλ2ϑ~2ϑ^˙2-λ22S22ψ2Tξ2ψ2ξ2,which suggests that we choose the virtual control signal u3 as(30)u3=-c2S2-12ϑ^2S2ψ2Tξ2ψ2ξ2,and the adaptation law(31)ϑ^˙2=λ22S22ψ2Tξ2ψ2ξ2-λ2σ2ϑ^2,ϑ^20≥0,where c2, λ2, and σ2 are positive design parameters. Then substituting (30) and (31) into (29), we get(32)Γ˙2≤-η20b2mc2S22+S2η2g2x3-u3-η20b2mσ2ϑ~2ϑ^2+12+12ϵ22.
Introduce a new state variablex3d, which can be obtained by the following first-order filter:(33)τ3x˙3d+x3d=u3,x3d0=u30.Step j (j=2,3).
Define theith surface error Si=xi-xid,i=j+1, where ui+1 is the ith virtual control signal. Then the time derivation of Si is(34)S˙i=gixxi+1-ui+1+gixui+1-12Si+Fiξi,where Fiξi=fi(x)+1/2Si-x˙id is unknown; we use RBF NN to approximate it on a compact set Ωξi,(35)Fiξi=Wi∗Tψiξi+Δiξi,Δiξi≤ϵi,with ξi≔x,xid,x˙idT∈Ωξi⊆R6. Consider the ith Lyapunov function(36)Γi=12Si2+bim2λiϑ~i2,where λi is a positive design parameter; ϑ~i=ϑ^i-ϑi with ϑi=Wi∗2/bim. The derivation of (36) is(37)Γ˙i=gixSixi+1-ui+1+gixSiui+1-12Si2+SiFiξi+bimλiϑ~iϑ^˙i.
Similar to (27)–(29), we have(38)Γ˙i≤gixSixi+1-ui+1+gixSiui+1+bim2ϑ^iSi2ψiTξiψiξi+bimλiϑ~iϑ^˙i-λi2Si2ψiTξiψiξi+12+12ϵi2.
Choose theith virtual control signal(39)ui+1=-ciSi-12ϑ^iSiψiTξiψiξi,where ϑ^i is updated by(40)ϑ^˙i=λi2Si2ψiTξiψiξi-λiσiϑ^i,ϑ^i0≥0,with ci, λi, and σi positive design parameters. Substituting (39), (40), into (38), we get(41)Γ˙i≤-cibiSi2+gixSixi+1-ui+1-bimσiϑ^iϑ~i+12+12ϵi2.
Letui+1 pass through the following first-order filter with time constant τi+1 to obtain a new state variable xi+1d,(42)τi+1x˙i+1d+xi+1d=ui+1,xi+1d0=ui+10,Step 4.
Finally, the time derivative ofS5=x5-x5d is(43)S˙5=g5xδe-12S4+F5ξ5, where F5ξ5=f5(x)+1/2S5-x˙5d+ω5 is unknown; we use RBF NN to approximate it on a compact set Ωξ5,(44)F5ξ5=W5∗Tψ5ξ5+Δ5ξ5,Δ5ξ5≤ϵ5,and ξ5=x,x5d,x˙5d,ω5T∈Ωξ5⊂R8. Let(45)Γ5=12S52+b5m2λ5ϑ~52,where λ5 is a positive design parameter and ϑ~5=ϑ^5-ϑ5 with ϑ5=W5∗2/b5m. Differentiating (45) we have(46)Γ˙5=S5g5xδe-12S52+S5F5ξ5+b5mλ5ϑ~5ϑ^˙5.
Similar to (28), (46) can be rewritten as(47)Γ˙5≤S5g5xδe+b5m2ϑ^5S52ψ5Tξ5ψ5ξ5+12+12ϵ52+b5mλ5ϑ~5ϑ^˙5-λ52S52ψ5Tξ5ψ5ξ5.
Notingg5(x)<0, we choose the actual control signal(48)δe=c5S5+12ϑ^5S5ψ5Tξ5ψ5ξ5,where ϑ^5 is updated by(49)ϑ^˙5=λ52S52ψ5Tξ5ψ5ξ5-λ5σ5ϑ^5,ϑ^50≥0,with c5, λ5, and σ5 positive design parameters. Substituting (48), (49), into (47), we arrive at(50)Γ˙5≤-c5b5mS52-b5mσ5ϑ^5ϑ~5+12+12ϵ52.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.


### 3.4. Stability and Tracking Performance Analysis

First of all, define the filter error(51)yi+1=xi+1d-ui+1,i=2,3,4.Then, it follows that(52)xi+1-ui+1=Si+1+yi+1.From (33) and (42),(53)x˙i+1d=-yi+1τi+1.Taking (30), (40), (51), and (53) into consideration, the time derivative of y3 satisfies(54)y˙3=-y3τ3+c2S˙2+12ϑ^˙2S2ψ2Tξ2ψ2ξ2+12ϑ^2S˙2ψ2Tξ2ψ2ξ2+ϑ^2S2ψ˙2Tξ2ψ2ξ2,from which we have(55)y˙3=-y3τ3+B3S2,S3,y3,ϑ^2,x2d,x˙2d,x¨2d,p2,by the same token, we have(56)y˙i+1=-yi+1τi+1+Bi+1•,where Bi+1•,i=2,3,4, are continuous functions. From (55) and (56), the following inequalities hold:(57)yi+1y˙i+1≤-yi+12τi+1+Bi+1yi+1.Theorem 7.
Consider system (22) under Assumptions 1 and 2, with the error transformation (9), the virtual control signals (30) and (39), the control law (48), and the adaptive laws (31), (40), and (49). Then all closed-loop signals are uniformly bounded and the prescribed tracking performance (7) can be guaranteed.Proof.
Define the following Lyapunov function:(58)Γ≤∑i=25Γi+∑i=2412yi+12,where Γi,(i=2,…,5) are defined by (26), (36), and (45), respectively. The differential of the Lyapunov function Γ is(59)Γ˙≤∑i=25Γ˙i+∑i=24yi+1y˙i+1.
From (32), (41), (50), and (57) and using the following inequalities,(60)η2g2S2S3+y3≤η22b2M2S22+12S32+12y32-ϑ^iϑ~i≤-12ϑ~i2-ϑi2,i=2,…,5,SigiSi+1+yi+1≤biM2Si2+12Si+12+12yi+12,i=3,4we have(61)Γ˙≤-η2b2mc2S22+η22b2M2S22+12S32+12y32-η20b2mσ212ϑ~22-ϑ22-∑i=35cibimSi2-∑i=35bimσi2ϑ~i2-ϑi2+∑i=34biM2Si2+12Si+12+12yi+12+∑i=2512+12ϵi2+∑i=24yi+1y˙i+1.
Define the compact sets(62)Π1≔x2d,x˙2d,x¨2d:x2d+x˙2d+x¨2d≤B0∈R3,Π2≔∑i=25Γi+∑i=2412yi+12≤ϱ∈R14,where Π1×Π2 is also a compact set in R14. Then, the continuous functions Bi+1,i=2,3,4 have maximums on Π1×Π2, say, Mi+1. Thus,(63)yi+1y˙i+1≤-yi+12τi+1+Mi+12yi+122μ+μ2which together with (61) implies that(64)Γ˙≤-η2b2mc2-η22b2M2S22-c3b3m-12-b3M2S32-c4b4m-12-b4M2S42-c5b5m-12S52-1τ3-M322μ-12y32-1τ4-M422μ-12y42-1τ5-M522μ-12y52-12η20b2mσ2ϑ~22-∑i=35bimσi2ϑ~i2+∑i=2512+12ϵi2+3μ2+12η20b2mσ2ϑ22+∑i=35bimσi2ϑi2.
Let(65)κ1=min⁡η2b2mc2-η22b2M2,c3b3m-12-b3M2,c4b4m-12-b4M2,c5b5m-12κ2=min1τ3-M322μ-12,1τ4-M422μ-12,1τ5-M522μ-12,κ3=min12λ2η2σ2,λiσi2,i=3,4,5,C=∑i=2512+12ϵi2+3μ2+η2b2mσ2ϑ22+∑i=35bimσi2ϑi2.
Then, we have(66)Γ˙≤-2κΓ+C,where κ=min⁡κ1,κ2,κ3. Letting κ>C/2ϱ, we have Γ˙≤0 on Γ=ϱ, which implies that if Γ0≤ϱ, then Γt≤ϱ,∀t≥0, and Γt≤ϱ is an invariant set. Moreover, solving (66) yields(67)0≤Γt≤C2κ+Γ0-C2κe-2κt;hence, all the signals in the closed-loop system are bounded. Particularly, it follows from the boundedness of S2 that the prescribed tracking performance (7) is guaranteed. This completes the proof.


## 3.1. Performance Functions and Error Transformation

Similar to [31, 32], the mathematical expression of the prescribed tracking performance is given by(7)-mipit<eit<nipit,where ei=yi-yid,i=1,2, are the tracking error, mi and ni are given positive constants, and the performance function pi is defined as smooth and decreasing positive function. From (7), one can see that -mipi(0) and nipi(0) are the lower and upper bound of the undershoot of ei(t), respectively. mipi(∞) and nipi(∞) represent the maximum allowable size of the steady-state value of ei(t).To transform (7) into an equivalent unconstrained one, define the error transformation function Υi(Si)=ei(t)/pi(t), where Si is the transformed error and Υi(Si) is a smooth, strictly increasing function having the following properties:(8)limS→-∞ΥiSi=-mi,limS→+∞ΥiSi=ni.From (8), if Si is bounded, we have -mi<ΥiSi<ni, and thus (7) holds. Hence, to achieve the prespecified tracking performance, one only needs to keep Si bounded. The inverse transformation of ΥiSi can be expressed as(9)Si=Υi-1eitpit≔Θieitpit,and differentiating (9) yields(10)S˙i=ηiy˙i-ηiυi.From the properties of the transformation, it is clear that ηi and υi are bounded and 0<ηi0≤ηi.


## 3.2. 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:(11)x˙1=f1x+g1xΦ+ω1,y1=x1,where f1(x) and g1(x) are unknown nonlinear function and g1≥b1m>0. Then define the velocity tracking error as e1=x1-x1d. According to (10) and (11) we obtain(12)S˙1=-η1υ1+η1f1x+η1g1xΦ+ω1.Let F1ξ1=1/2S1-η1υ1+η1f1(x)+ω1 and ξ1≔x,x˙1d,p1,ω1T∈Ωξ1⊆R8. The transformed system dynamics of (12) can be rewritten as(13)S˙1=η1g1xΦ-12S1+F1ξ1.SinceF1ξ1 is an unknown nonlinear function, we use an RBF NN to approximate it. Then by using Lemma 5, we have(14)F1ξ1=W1∗Tψ1ξ1+Δ1ξ1,Δ1ξ1≤ϵ1.Choose the control signal and the adaptive update law as follows:(15)Φ=-c1S1-12S1ϑ^1ψ1Tξ1ψ1ξ1,ϑ^˙1=λ12S12ψ1Tξ1ψ1ξ1-λ1σ1ϑ^1,where ϑ^1 is the estimate of ϑ1=W1∗2/η10b1m and c1, λ1, and σ1 are designed positive parameters. Consider the Lyapunov function(16)Γ1=12S12+η10b1m2λ1ϑ~12.The differential of Lyapunov functionΓ1 can be found as follows.(17)Γ˙1≤S1ηV0b1mΦ+η10b1m2S12ϑ^ψ1Tξ1ψ1ξ1+η10b1mλ1ϑ~1ϑ^˙1-λ12S12ψ1Tξ1ψ1ξ1+12+12ϵ12.Substituting (15) into (17) yields(18)Γ˙1≤-c1η10b1mS12-η10b1mσ1ϑ~1ϑ^1+12+12ϵ12.According to Assumption1 and the inequality -ϑ~1ϑ^1≤-1/2ϑ~12+1/2ϑ2, we have(19)Γ˙1≤-c1η10b1mS12-12η10b1mσ1ϑ~12+12η10b1mσ1ϑ2+12+12ϵ12,Γ˙1≤-2κ1Γ1+C1,where(20)κ1=minc1η10b1m,12λ1σ1,μ1=12η10b1mσ1ϑ2+12+12ϵ12.Solving (19) gives(21)0≤Γ1t≤C12κ1+Γ10-C12κ1e-2κ1t,t≥0,from which it is clear that, by properly choosing the design parameters c1,λ1, and σ1, S1, ϑ^1, and ϑ~1 in the closed-loop system are uniformly ultimately bounded, and the prescribed tracking performance is guaranteed.


## 3.3. 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 δe is constructed at Step 4. After the error transformation (7)–(10), the altitude subsystem (1) is equivalent to(22)S˙2=η2g2xx3-η2υ2,x˙3=g3xx4+f3x,x˙4=g4xx5+f4x,x˙5=g5xδe+f5x+ω5,y2=x2.The stabilization of the transformed system (22) is sufficient to guarantee the prescribed tracking performance of altitude subsystem.Step 1.
LetS2 given by (22) be the first error variable. Define u3 as the first virtual control signal. Then the derivative of S2 can be expressed as(23)S˙2=η2g2xx3-12S2+F2ξ2,where F2ξ2=1/2S2-η2υ2. Since F2ξ2 is unknown, we employ an RBF NN to approximate it on a compact set Ωξ2. By properly choosing the basis function vectors we have(24)F2ξ2=W2∗Tψ2ξ2+Δ2ξ2,Δ2ξ2≤ϵ2,where ξ2:=x2,x2d,x˙2d,p2T∈Ωξ2⊆R4 and ϵ2 is a positive constant. With respect to the unknown optimal weight vector in (24), define(25)ϑ2=W2∗2η20b2m,where b2m are given by Assumption 1. Since ϑ2 is unknown, let ϑ^2 be the estimation of ϑ2 and ϑ~2≔ϑ^2-ϑ2. Consider the first Lyapunov function(26)Γ2=12S22+η20b2m2λ2ϑ~22.
The derivation of (26) can be found as follows:(27)Γ˙2=S2η2g2xx3-u3+S2η2g2xu3-12S22+S2W2∗Tψ2ξ2+S2Δ2ξ2+η20b2mλ2ϑ~2ϑ^˙2.
Using Young’s inequality, it can be verified that(28)S2W2∗Tψ2ξ2≤12S22W2∗2ψ2Tξ2ψ2ξ2+12≤η20b2m2ϑ2S22ψ2Tξ2ψ2ξ2+12,S2Δ2ξ2≤12S22+12ϵ22.
Thus, (27) can be rewritten as(29)Γ˙2≤S2η2g2x3-u3+S2η2g2u3+η20b2m2ϑ^2S22ψ2Tξ2ψ2ξ2+12+12ϵ22+η20b2mλ2ϑ~2ϑ^˙2-λ22S22ψ2Tξ2ψ2ξ2,which suggests that we choose the virtual control signal u3 as(30)u3=-c2S2-12ϑ^2S2ψ2Tξ2ψ2ξ2,and the adaptation law(31)ϑ^˙2=λ22S22ψ2Tξ2ψ2ξ2-λ2σ2ϑ^2,ϑ^20≥0,where c2, λ2, and σ2 are positive design parameters. Then substituting (30) and (31) into (29), we get(32)Γ˙2≤-η20b2mc2S22+S2η2g2x3-u3-η20b2mσ2ϑ~2ϑ^2+12+12ϵ22.
Introduce a new state variablex3d, which can be obtained by the following first-order filter:(33)τ3x˙3d+x3d=u3,x3d0=u30.Step j (j=2,3).
Define theith surface error Si=xi-xid,i=j+1, where ui+1 is the ith virtual control signal. Then the time derivation of Si is(34)S˙i=gixxi+1-ui+1+gixui+1-12Si+Fiξi,where Fiξi=fi(x)+1/2Si-x˙id is unknown; we use RBF NN to approximate it on a compact set Ωξi,(35)Fiξi=Wi∗Tψiξi+Δiξi,Δiξi≤ϵi,with ξi≔x,xid,x˙idT∈Ωξi⊆R6. Consider the ith Lyapunov function(36)Γi=12Si2+bim2λiϑ~i2,where λi is a positive design parameter; ϑ~i=ϑ^i-ϑi with ϑi=Wi∗2/bim. The derivation of (36) is(37)Γ˙i=gixSixi+1-ui+1+gixSiui+1-12Si2+SiFiξi+bimλiϑ~iϑ^˙i.
Similar to (27)–(29), we have(38)Γ˙i≤gixSixi+1-ui+1+gixSiui+1+bim2ϑ^iSi2ψiTξiψiξi+bimλiϑ~iϑ^˙i-λi2Si2ψiTξiψiξi+12+12ϵi2.
Choose theith virtual control signal(39)ui+1=-ciSi-12ϑ^iSiψiTξiψiξi,where ϑ^i is updated by(40)ϑ^˙i=λi2Si2ψiTξiψiξi-λiσiϑ^i,ϑ^i0≥0,with ci, λi, and σi positive design parameters. Substituting (39), (40), into (38), we get(41)Γ˙i≤-cibiSi2+gixSixi+1-ui+1-bimσiϑ^iϑ~i+12+12ϵi2.
Letui+1 pass through the following first-order filter with time constant τi+1 to obtain a new state variable xi+1d,(42)τi+1x˙i+1d+xi+1d=ui+1,xi+1d0=ui+10,Step 4.
Finally, the time derivative ofS5=x5-x5d is(43)S˙5=g5xδe-12S4+F5ξ5, where F5ξ5=f5(x)+1/2S5-x˙5d+ω5 is unknown; we use RBF NN to approximate it on a compact set Ωξ5,(44)F5ξ5=W5∗Tψ5ξ5+Δ5ξ5,Δ5ξ5≤ϵ5,and ξ5=x,x5d,x˙5d,ω5T∈Ωξ5⊂R8. Let(45)Γ5=12S52+b5m2λ5ϑ~52,where λ5 is a positive design parameter and ϑ~5=ϑ^5-ϑ5 with ϑ5=W5∗2/b5m. Differentiating (45) we have(46)Γ˙5=S5g5xδe-12S52+S5F5ξ5+b5mλ5ϑ~5ϑ^˙5.
Similar to (28), (46) can be rewritten as(47)Γ˙5≤S5g5xδe+b5m2ϑ^5S52ψ5Tξ5ψ5ξ5+12+12ϵ52+b5mλ5ϑ~5ϑ^˙5-λ52S52ψ5Tξ5ψ5ξ5.
Notingg5(x)<0, we choose the actual control signal(48)δe=c5S5+12ϑ^5S5ψ5Tξ5ψ5ξ5,where ϑ^5 is updated by(49)ϑ^˙5=λ52S52ψ5Tξ5ψ5ξ5-λ5σ5ϑ^5,ϑ^50≥0,with c5, λ5, and σ5 positive design parameters. Substituting (48), (49), into (47), we arrive at(50)Γ˙5≤-c5b5mS52-b5mσ5ϑ^5ϑ~5+12+12ϵ52.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.


## 3.4. Stability and Tracking Performance Analysis

First of all, define the filter error(51)yi+1=xi+1d-ui+1,i=2,3,4.Then, it follows that(52)xi+1-ui+1=Si+1+yi+1.From (33) and (42),(53)x˙i+1d=-yi+1τi+1.Taking (30), (40), (51), and (53) into consideration, the time derivative of y3 satisfies(54)y˙3=-y3τ3+c2S˙2+12ϑ^˙2S2ψ2Tξ2ψ2ξ2+12ϑ^2S˙2ψ2Tξ2ψ2ξ2+ϑ^2S2ψ˙2Tξ2ψ2ξ2,from which we have(55)y˙3=-y3τ3+B3S2,S3,y3,ϑ^2,x2d,x˙2d,x¨2d,p2,by the same token, we have(56)y˙i+1=-yi+1τi+1+Bi+1•,where Bi+1•,i=2,3,4, are continuous functions. From (55) and (56), the following inequalities hold:(57)yi+1y˙i+1≤-yi+12τi+1+Bi+1yi+1.Theorem 7.
Consider system (22) under Assumptions 1 and 2, with the error transformation (9), the virtual control signals (30) and (39), the control law (48), and the adaptive laws (31), (40), and (49). Then all closed-loop signals are uniformly bounded and the prescribed tracking performance (7) can be guaranteed.Proof.
Define the following Lyapunov function:(58)Γ≤∑i=25Γi+∑i=2412yi+12,where Γi,(i=2,…,5) are defined by (26), (36), and (45), respectively. The differential of the Lyapunov function Γ is(59)Γ˙≤∑i=25Γ˙i+∑i=24yi+1y˙i+1.
From (32), (41), (50), and (57) and using the following inequalities,(60)η2g2S2S3+y3≤η22b2M2S22+12S32+12y32-ϑ^iϑ~i≤-12ϑ~i2-ϑi2,i=2,…,5,SigiSi+1+yi+1≤biM2Si2+12Si+12+12yi+12,i=3,4we have(61)Γ˙≤-η2b2mc2S22+η22b2M2S22+12S32+12y32-η20b2mσ212ϑ~22-ϑ22-∑i=35cibimSi2-∑i=35bimσi2ϑ~i2-ϑi2+∑i=34biM2Si2+12Si+12+12yi+12+∑i=2512+12ϵi2+∑i=24yi+1y˙i+1.
Define the compact sets(62)Π1≔x2d,x˙2d,x¨2d:x2d+x˙2d+x¨2d≤B0∈R3,Π2≔∑i=25Γi+∑i=2412yi+12≤ϱ∈R14,where Π1×Π2 is also a compact set in R14. Then, the continuous functions Bi+1,i=2,3,4 have maximums on Π1×Π2, say, Mi+1. Thus,(63)yi+1y˙i+1≤-yi+12τi+1+Mi+12yi+122μ+μ2which together with (61) implies that(64)Γ˙≤-η2b2mc2-η22b2M2S22-c3b3m-12-b3M2S32-c4b4m-12-b4M2S42-c5b5m-12S52-1τ3-M322μ-12y32-1τ4-M422μ-12y42-1τ5-M522μ-12y52-12η20b2mσ2ϑ~22-∑i=35bimσi2ϑ~i2+∑i=2512+12ϵi2+3μ2+12η20b2mσ2ϑ22+∑i=35bimσi2ϑi2.
Let(65)κ1=min⁡η2b2mc2-η22b2M2,c3b3m-12-b3M2,c4b4m-12-b4M2,c5b5m-12κ2=min1τ3-M322μ-12,1τ4-M422μ-12,1τ5-M522μ-12,κ3=min12λ2η2σ2,λiσi2,i=3,4,5,C=∑i=2512+12ϵi2+3μ2+η2b2mσ2ϑ22+∑i=35bimσi2ϑi2.
Then, we have(66)Γ˙≤-2κΓ+C,where κ=min⁡κ1,κ2,κ3. Letting κ>C/2ϱ, we have Γ˙≤0 on Γ=ϱ, which implies that if Γ0≤ϱ, then Γt≤ϱ,∀t≥0, and Γt≤ϱ is an invariant set. Moreover, solving (66) yields(67)0≤Γt≤C2κ+Γ0-C2κe-2κt;hence, all the signals in the closed-loop system are bounded. Particularly, it follows from the boundedness of S2 that the prescribed tracking performance (7) is guaranteed. This completes the proof.


## 4. 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 ϖn1=0.5, ϖn2=0.3, and ζh=0.95. The reference signals of velocity and altitude are 400 ft/s and 1000 ft, respectively.(68)x1dx1c=ϖn12s2+2ζϖn1s+ϖn12,x2dx2c=ϖn12ϖn22s+ϖn12s2+2ζϖn2s+ϖn22.In the simulation, we chooseϑ^10=ϑ^20=ϑ^30=ϑ^40=ϑ^50=0. The design parameters are chosen as c1=0.5, c2=0.4, c3=120, c4=60, c5=30, λ1=10, λ2=0.1, λ3=0.1, λ4=0.01, λ5=0.01, σ1=1, σ2=0.1, σ3=0.1, σ4=0.1, and σ5=0.1. The initial conditions of the system (1) are V0=7850 ft/s, h0=85000 ft, γ0=0 rad, α0=0.0264 rad, and q0=0 rad/s. The performance functions are selected as pi(t)=(pi0-pi∞)e-lt+pi∞,i=1,2, with parameters p10=15, p1∞=0.7, l1=0.8, p20=30, p2∞=1, l2=0.4, and mi=ni=1. The tracking performance is shown in Figures 1–5, which indicate that the tracking performance is achieved by the proposed control scheme.Figure 1
Velocity tracking performance.Figure 2
Altitude tracking performance.Figure 3
Control inputsΦ and δe.Figure 4
Flight path angle, pitch angle, and pitch rate.Figure 5
Estimation of parameters.


## 5. 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.



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*Source: 7864375-2017-09-18.xml*