# Prescribed Performance Tracking Control for the Hypersonic Vehicle with Actuator Faults

Authors: Zhenghui Yang; Wentao He; Yushan He; Yaen Xie; Jun Li; Shuo Song
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

This paper provides a solution for the trajectory tracking control of a hypersonic flight vehicle (HFV), which is encountered performance constraints, actuator faults, external disturbances, and system uncertainties. For the altitude and velocity control subsystems, the backstepping-based dynamic surface control (DSC) strategy is constructed to guarantee the predefined constraint of tracking errors. The introduction of first-order low-pass filters effectively remedies the problem of “complexity explosion” existing in high-order backstepping design. Simultaneously, radial basis function neural networks (RBFNNs) are adopted for approximating the unavailable dynamics, in which the minimum learning parameter (MLP) algorithm brilliantly alleviates the excessive occupation of the computational resource. Specially, in consideration of the unknown actuator failures, the adaptive signals are designed to enhance the reliability of the closed-loop system. Finally, according to rigorous theoretical analysis and simulation experiment, the stability of the proposed controller is verified, and its superiority is exhibited intuitively.


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## Body

## 1. Introduction

The recent few years have witnessed the burgeoning interest in hypersonic flight vehicles (HFVs) from researchers owing to its unique advantages like rapid maneuver and high efficiency. In light of these inherent characteristics, HFVs have been extensively applied in both military and civilian fields. However, there are still many obstructions in constructing controllers for HFVs, involving the harsh flight environment, the uncertainty of aerodynamics, and the strong nonlinearity and coupling. In order to surmount these challenges, numerous control frameworks have been examined, just to name a few, sliding mode control (SMC) [1–4], backstepping control [5–8], dynamic surface control (DSC) [9, 10], adaptive control [11, 12], and so forth.As a classical nonlinear control approach, SMC is celebrated for its antidisturbance capability, easy implementation, and excellent robustness [13–16]. In [1], SMC-based architecture was established to achieve the trajectory tracking control for HFV, while it comes with the undesired phenomenon of chattering. This defect will undoubtedly generate a heavy burden on actuators and shorten their service lives. Fortunately, [2] provided a continuous antichattering sliding mode controller for the flexible HFVs. Besides the SMC methods, the backstepping-based procedure design possesses a high application value in high-order nonlinear systems. The disadvantage of traditional backstepping methods lies in the issue of “complexity explosion,” which always leads to waste of computational resources and excessive hardware requirements. Therefore, the DSC-based solution was developed in [9] so that the above problem could be relaxed by resorting to the introduction of low-pass filters. In addition, complicated aerodynamics is another threat for trajectory tracking of HFVs. Inaccurate dynamics makes model-dependent control schemes uncapable for practical engineering. Inspired by this condition, a neural adaptive sliding mode control algorithm was proposed in [17], where the radial basis function neural networks (RBFNNs) were utilized to approximate the available dynamics with an exacting precision.At the same time, while accounting for the uncertainties of HFV models, great demands were placed on the convergence rate and tracking accuracy of control systems [18]. From this view of point, most of the aforementioned controllers possess satisfactory steady-state characteristics but lack specific constraints on transient performances. Under this urgent requirement, the prescribed performance control (PPC) technology is introduced to restrict certain states within a predefined region globally [19–24], such that the system transient indicators can be flexibly specified by the designers. Therefore, Shao et al. imposed time performance constraints in event-triggered robust control for quadrotors, which apparently optimized the stable rate of the closed-loop system [19]. As for the PPC strategy in vehicle control field, Bu constructed a RBFNN-based robust algorithm for air-breathing hypersonic vehicles (AHVs) with unknown dead-zone input nonlinearity, where the trajectory tracking errors are effectively suppressed within a predetermined range [21].However, the controllers mentioned above are all designed on basis of the assumption that actuators of HFVs work normally. Obviously, it is of impractical conceit for HFV servicing in ideal environments. To prevent the resulting performance degradation and instable phenomenon, the actuator faults must be taken into account while formulating the trajectory tracking control strategies with sufficient applicability and reliability [25–27]. In [26], a quasi-continuous high-order sliding mode framework was presented, in which the neural network observer is introduced to ensure the convergence of estimation errors. Further in [28], an adaptive fault-tolerant control scheme was derived in the presence of actuator saturation to improve system reliability and robustness. By the utilization of PPC, a fault-tolerant protocol was designed to achieve the accurate trajectory tracking for HFVs.Driven by the aforementioned observations, this paper aims at the prescribed performance fault-tolerant control for HFV suffering from system uncertainties and actuator failures. The backstepping-based dynamic surface design is conducted to guarantee the finite-time convergence of altitude and velocity tracking errors. Unknown multiplicative and additive faults of actuators are handled by resorting to the construction of adaptive architectures. On the other hand, unmodeled dynamics are approximated via RBFNNs; especially, the minimum learning parameter (MLP) algorithm requires less computational resource. Finally, the stability and superiority of the proposed controller will be corroborated by theoretical analysis and simulation experiments. It will be elaborated that the transient and steady-state performance constraints are always satisfied. Contributions of this article are presented as follows:(1)
Different from the traditional approximation of RBFNN [12, 17], MLP algorithm is utilized to cope with the uncertain dynamics in this paper. In this way, it is no longer necessary to estimate the entire weight matrix, but the upper bound of its norm is taken as the object of online updating. Therefore, the computational complexity is considerably degraded, and the hardware requirement is reduced significantly(2)
In [23, 24], PPC control methods were presented for MEMS Gyroscopes and Networked Uncertain Quadrotors, respectively, which cannot be directly applied for hypersonic vehicles. Considering this point, the PPC-based dynamic surface controller is synthesized for hypersonic vehicles in the presence of uncertainties, actuator faults, and disturbances. The results of this paper could be treated as an application of PPC for hypersonic vehicles(3)
Instead of restricting the time performance [19, 20], the PPC-based amplitude constraints [23, 24] of HFVs’ trajectory tracking errors are addressed in this paper. In virtue of the hyperbolic tangent function, the open-loop tracking error dynamics with certain designer-specified index constraints is transformed into an equivalent “state-constrained” system, in which both transient and steady-state responses obtain the satisfactory performanceThe remainder of this paper is organized as follows. Section2 gives a longitudinal model of HFVs and the relative preliminaries. The altitude and velocity controllers are designed in Section 3 and Section 4, respectively. In Section 5, the validity of the controllers is verified through a simulation experiment. Finally, Section 6 shows the conclusion of this work.


## 2. Problem Formulation

### 2.1. Longitudinal Model of HFV

The trajectory tracking control problem for a HFV is considered in this paper. According to [29], the longitudinal model of the HFV can be expressed as:
(1)V̇=Tcosα−Dm−μsinγr2,(2)γ̇=L+TsinαmV−μ−V2rcosγVr2,(3)ḣ=Vsinγ,(4)α̇=q−γ̇,(5)q̇=MyyIyy.The corresponding dynamics are given as:(6)L=q¯SCL,(7)D=q¯SCD,(8)T=q¯SCT,(9)Myy=q¯Sc¯CMα+CMδ+CMq,(10)r=h+re,(11)CL=0.6203α,(12)CD=0.6450α2+0.0043378α+0.003772,(13)CT=0.022576ββ<10.0224+0.00336ββ>1,(14)CMα=−0.035α2+0.036117α+5.3261×10−6,(15)CMq=c¯q−0.6796α2+0.3015α−0.22892V,(16)CMδe=ceδe−α,(17)ce=0.0292,where c¯ and ce denote constants; q¯ denotes the dynamic pressure satisfying q¯=1/2ρV2; V, h, γ, α, and q denote velocity, altitude, flight path angle, angle of attack, and pitch rate of HFV, respectively [30]; T, D, L, and Myy denote the thrust of the engine, drag force, lift force, and pitching moment, respectively [31]; and S, m, ρ, Iyy, μ, and re denote the reference area, mass, density of air, moment of inertia, the gravitational constant, and radius of the Earth, respectively [32].As a matter of fact, various unexpected failures occur to the actuators frequently in practical engineering, which will cause severe performance degradation and even collapse for whole close-loop system. Therefore, it is necessary to take the actuator faults into consideration while designing the trajectory tracking controllers. With the introduction of multiplicative and additive faults, the actual input signal is described as:(18)δe=ρiuc+dit+Δi,where ρi denotes the unknown effectiveness factor of actuator and satisfies 0<ιi≤ρi≤1, ιi≤1 represents the minimum value of ρi, dit denotes the unknown external disturbance, and Δi is the additive fault with i=1,2.


### 2.2. Fundamental of RBFNNs

RBFNN is an effective tool to obtain the approximation of unknown system dynamics. The following lemma sketches the basic principle of RBFNNs.(19)fx=W∗THx+r,0<r≤O.Lemma 1 [33].
An arbitrary continuous smooth functionfx can be described as the following form by defining a basis function Hx.Here,W∗=Wa1,Wa2,⋯,WapT is the ideal weight matrix, and x=xa1,xa2,⋯,xam represents the input vector. r and O are the additional approximation error and its upper bound; Hx=Ha1x,Ha2x,⋯,HamxT is selected as the Gaussian basis function, which can be expressed as:
(20)Haix=exp−x−ci222bi2,i=1,⋯,p,withci∈Rm and b∈Rp denoting the center vector and the width of Gaussian basis function, respectively.


### 2.3. Other Preliminaries

Lemma 2 [33].
For arbitrary scalarx∈R and μ>0, the inequation about hyperbolic tangent function is shown as:
(21)0<x−xtanhμx≤κμ,in which the constant κ=0.2785.Assumption 3.
The additive actuator fault and external disturbances are both unknown but bounded, which are supposed to satisfydit+Δi≤λii=1,2 with λi being positive constants.


## 2.1. Longitudinal Model of HFV

The trajectory tracking control problem for a HFV is considered in this paper. According to [29], the longitudinal model of the HFV can be expressed as:
(1)V̇=Tcosα−Dm−μsinγr2,(2)γ̇=L+TsinαmV−μ−V2rcosγVr2,(3)ḣ=Vsinγ,(4)α̇=q−γ̇,(5)q̇=MyyIyy.The corresponding dynamics are given as:(6)L=q¯SCL,(7)D=q¯SCD,(8)T=q¯SCT,(9)Myy=q¯Sc¯CMα+CMδ+CMq,(10)r=h+re,(11)CL=0.6203α,(12)CD=0.6450α2+0.0043378α+0.003772,(13)CT=0.022576ββ<10.0224+0.00336ββ>1,(14)CMα=−0.035α2+0.036117α+5.3261×10−6,(15)CMq=c¯q−0.6796α2+0.3015α−0.22892V,(16)CMδe=ceδe−α,(17)ce=0.0292,where c¯ and ce denote constants; q¯ denotes the dynamic pressure satisfying q¯=1/2ρV2; V, h, γ, α, and q denote velocity, altitude, flight path angle, angle of attack, and pitch rate of HFV, respectively [30]; T, D, L, and Myy denote the thrust of the engine, drag force, lift force, and pitching moment, respectively [31]; and S, m, ρ, Iyy, μ, and re denote the reference area, mass, density of air, moment of inertia, the gravitational constant, and radius of the Earth, respectively [32].As a matter of fact, various unexpected failures occur to the actuators frequently in practical engineering, which will cause severe performance degradation and even collapse for whole close-loop system. Therefore, it is necessary to take the actuator faults into consideration while designing the trajectory tracking controllers. With the introduction of multiplicative and additive faults, the actual input signal is described as:(18)δe=ρiuc+dit+Δi,where ρi denotes the unknown effectiveness factor of actuator and satisfies 0<ιi≤ρi≤1, ιi≤1 represents the minimum value of ρi, dit denotes the unknown external disturbance, and Δi is the additive fault with i=1,2.


## 2.2. Fundamental of RBFNNs

RBFNN is an effective tool to obtain the approximation of unknown system dynamics. The following lemma sketches the basic principle of RBFNNs.(19)fx=W∗THx+r,0<r≤O.Lemma 1 [33].
An arbitrary continuous smooth functionfx can be described as the following form by defining a basis function Hx.Here,W∗=Wa1,Wa2,⋯,WapT is the ideal weight matrix, and x=xa1,xa2,⋯,xam represents the input vector. r and O are the additional approximation error and its upper bound; Hx=Ha1x,Ha2x,⋯,HamxT is selected as the Gaussian basis function, which can be expressed as:
(20)Haix=exp−x−ci222bi2,i=1,⋯,p,withci∈Rm and b∈Rp denoting the center vector and the width of Gaussian basis function, respectively.


## 2.3. Other Preliminaries

Lemma 2 [33].
For arbitrary scalarx∈R and μ>0, the inequation about hyperbolic tangent function is shown as:
(21)0<x−xtanhμx≤κμ,in which the constant κ=0.2785.Assumption 3.
The additive actuator fault and external disturbances are both unknown but bounded, which are supposed to satisfydit+Δi≤λii=1,2 with λi being positive constants.


## 3. Altitude Controller Design

Aiming towards the altitude and velocity tracking missions, two subsystems are formulated on basis of the longitudinal model of HFV. In this section, a DSC-based adaptive fault-tolerant controller is constructed for the altitude tracking subsystem. Due to the high-order characteristic of the dynamics, traditional backstepping design is inevitably accompanied by the phenomenon of “complexity explosion,” which is arising from the frequent differential to virtual commands. Therefore, low-pass filters are introduced to solve this problem, while system uncertainties are handled via RBFNN. Finally, it is validated via Lyapunov-based analysis that altitude tracking errors exponentially converge to a tiny region containing the origin.In altitude tracking control subsystem, three states are selected asγ, ϑp, and q. To facilitate the subsequent design, a state vector is defined as:
(22)x=x1,x2,x3T=γ,ϑp,qT,where ϑp=α+γ. Thus, the altitude dynamics can be described as [30]:
(23)ẋ1=f1x1,V+g1Vx2,ẋ2=f2+g2x3,ẋ3=f3x1,x2,x3,V+g3Vδe=f3x1,x2,x3,V+g3Vρ1uc1+d1t+Δ1,with
(24)g1V=0.31015ρVSm,f1x1,V=TsinαmV−μ‐V2rcosγVr2,f2=0,g2=1,f3x1,x2,x3,V=0.5ρV2SCMα+CMq−ceαIyy,g3V=q¯Sc¯ceIyy.Assumption 4.
The uncertain termgiV possesses an upper bound g¯i, which satisfies 0<giV≤g¯i, i=1,3.At this moment, the control objective has been transformed into forcingxii=0,1,2,3 to track the desired trajectories, where x0=h. For this purpose, the tracking error variables are introduced as xei=xi−xid with x0d=hd, x1d=γd, x2d=ϑpd, and x3d=qd denoting reference states.To ensure the prescribed performance, the boundary of variablexei is defined as:
(25)x¯ei≤xei≤x¯ei.The error transformation is designed as:(26)xei=x¯ei−x¯ei2tanhsi+x¯ei+x¯ei2,(27)si=atanh2xei−x¯ei−x¯eix¯ei−x¯ei,in which si is the transformed tracking error. It indicates that the transformed error increases monotonically with the original variable.Note additionally that there exist limits as:(28)limsi→−∞xei=x¯ei,limsi→+∞xei=x¯ei.Necessarily, the time derivative ofsi is derived as:
(29)ṡi=miẋei+ni,in which mi=∂si/∂xei, ni=∂si/∂x¯eix¯̇ei+∂si/∂x−eiẋ−ei with i=1,2,3.Step 1.
According to Equation (29), the derivative of s0 satisfies the following equation:
(30)ṡ0=m0ẋe0+n0=m0Vsinx1+n0.Based on the similar analysis in [34], the asymptotic stability of the altitude tracking error xe0 is ensured if the virtual command is designed as:
(31)x1d=arcsin−khs0−khhs0+ḣd+n0m0V,with kh>0,khh>0. For the purpose of stabilizing xe1, the following equations are firstly presented by utilizing Equations (24) and (26)–(29):
(32)ẋe1=f1+g1x2−ẋ1d,ṡ1=m1ẋe1+n1=m1f1+g1x2−ẋ1d+n1.A first-order low-pass filter is introduced as:(33)ε2ẋ2d+x2d=x2c,x2d0=x2c0,where the variablex2c represents the filtered signal and ε2>0 is a constant.To cope with the unavailable dynamicm1f1, Lemma 1 is applied here. Then, one has:
(34)m1f1=Wa1∗THa1x+r1,0<r1≤O1,Wa1∗THa1≤Wa1∗THa1≤w1h1,where h1=Ha1 and w1≥Wa1∗T.By defining thatR1=O1, the virtual control input and the relevant adaptive laws are designed as:
(35)x2c=1g1ẋ1d−g1xe2+1m1−w^1h1tanhs1μ1−R^1tanhs1μ2−k1s1−k2tanhs1−n1,(36)w^̇1=α1s1h1tanhs1μ1−c1w^1,(37)R^̇1=β1s1tanhs1μ2−c2R^1,where w^1 and R^1 are the estimation values of w1 and R1, respectively, and k1 and k2 are the positive constants.Step 2.
As forx2, according to Equations (24) and (26)–(29), one has:
(38)ẋe2=ẋ2−ẋ2d=x3−ẋ2d,ṡ2=m2ẋe2+n2=m2x3−ẋ2d+n2.
A first-order filter is given as:(39)ε3ẋ3d+x3d=x3c,x3d0=x3c0,where ε3>0 is a positive constant.Introduce the virtual commandx3c, which is taken as:
(40)x3c=1m2−k3s2−k4tanhs2−n2+ẋ2d−xe3,where k3 and k4 are the positive constants.Step 3.
To ensure the convergence ofxe3, Equations (24) and (26)–(29) are combined, and the following equations are obtained:
(41)ẋe3=ẋ3−ẋ3d=f3+g3ρ1uc1+d1t+Δ1−ẋ3d,(42)ṡ3=m3ẋe3+n3=m3f3+g3ρ1uc1+d1t+Δ1−ẋ3d+n3.
By taking the unavailable dynamicm3f3 into consideration and according to Lemma 1, one can obtain:
(43)m3f3=Wa3∗THa3x+r3,0<r3≤O3,(44)Wa3∗THa3≤Wa3∗THa3≤w3h3,where h3=Ha3 and w3≥Wa3∗T.With the definitions ofR3=m3g¯3λ1+O3 and η1=1−ρ1m3/ρ1, the control input and corresponding adaptive laws are designed as:
(45)u1=−g3−1unom1+un1,(46)unom1=1m3w^3h3tanhs3μ3+R^3tanhs3μ4+k5s3+k6tanhs3+n3−ẋ3d,(47)un1=1m3η^1unom1tanhunom1s3μ5,(48)η^̇1=k7unom1s3tanhunom1s3μ5−k8η^1,(49)w^̇3=α3s3h3tanhs3μ3−c5w^3,(50)R^̇3=β3s3tanhs3μ4−c6R^3,where η^1, w^3, and R^3 are estimations of η1, w3, and R3, respectively, and k5, k6,k7, k8, α3, β3, c5, c6, μ3, and μ4 are all positive constants.Theorem 5.
For the HFV system (24), if the virtual commands are designed as Equations (31), (35), and (40), control signals are designed as Equations (45)–(47); it can be concluded that the tracking errors satisfy exponential convergence, and the prescribed performance constraints are guaranteed.Proof.
Lyapunov function candidate is selected as:(51)φ0=∑i=13φi,where φi are defined as:
(52)φ1=12s12+12y22+12α1w~12+12β1R~12,φ2=12s22+12y32,φ3=12s32+ρ12k7η~12+12α3w~32+12β3R~32.The differential ofyi, i=2,3, can be expressed as:
(53)ẏi=ẋid−ẋic=−yiεi+Bi⋅,Bi⋅=−ẋic.It is worthy to point that there must exist constantsοi>0, i=2,3, satisfying the inequality [32]:
(54)Bi⋅≤οi,in which o1 are unknown positive constants.Therefore, the time derivative ofφi can be developed as:
(55)φ̇1=s1m1f1+g1x2−ẋ1d+n1+y2ẏ2−1α1w~1w^̇1−1β1R~1R^̇1=s1m1f1+m1g1xe2+m1g1y2−w^1h1tanhs1μ1−R^1tanhs1μ2+m1ẋ1d−m1g1xe2−k1s1−k2tanhs1−n1−m1ẋ1d+n1+y2ẏ2−1α1w~1w^̇1−1β1R~1R^̇1≤s1w1h1+R1+m1g¯1s12+y222−w^1h1s1+μ1w^1h1κ−R^1s1+μ2R^1κ−k1s12−k2s1+k2κ−1ε2−12y22+ο222−1α1w~1α1s1h1tanhs1μ1−c1w^1−1β1R~1β1s1tanhs1μ2−c2R^1,≤−k1−m1g¯12s12−1ε2−m1g¯12−12y22+ο222+c1w~1w^1+c2R~1R^1+μ1w1h1κ+μ2R1κ+k2κ≤−k1−m1g¯12s12−1ε2−m1g¯12−12y22−c12w~21−c22R~12+c12w12+c22R12+ο222+μ1w1h1κ+μ2R1κ+k2κ=−θ1φ1+ζ1,where θ1=min2k1−m1g¯1,2/ε2−m1g¯1−1,α1c1,β1c2, ζ1=c1/2w12+c2/2R12+ο22/2+μ1w1h1κ+μ2R1κ+k2κ.With the application of Equations (41)–(44), the time derivative of φ2 can be expressed as:
(56)φ̇2=s2m2x3−ẋ2d+n2+y3ẏ3=s2m2xe3+y3+x3c−ẋ2d+n2+y3ẏ3=s2m2xe3+m2y3+m2ẋ2d−k3s2−k4tanhs2−m2xe3−n2−m2ẋ2d+n2+y3ẏ3≤s2m2y3−k3s2−k4tanhs2−y32ε3+y3ο3≤m2s2y3−k3s22−k4s2+k4κ−y32ε3+y3ο3≤−k3−m22s22−1ε3−m22−12y32+ο322+k4κ=−θ2φ2+ζ2,where θ2=min2k3−m2,2/ε3−m2−1,, ζ2=ο32/2+k4κ.With the substitution of Equations (47)–(50), the time derivative of φ3 is written as:
(57)φ̇3=s3m3f3+g3ρ1uc1+d1t+Δ1−ẋ3d+n3−ρ1k7η~1η^̇1−1α3w~3w^̇3−1β3R~3R^̇3≤s3w3h3+r3+m3s31−ρ1unom1−unom1−ρ1un1−ẋ3d+n3s3+m3g¯3λ1s3−ρ1k7η~1k7unom1s3tanhunom1s3μ5−k8η^1−1α3w~3w^̇3−1β3R~3R^̇3≤s3w~3h3+s3R~3−k5s32−k6s3+m3s31−ρ1unom1−ρ1κη^1unom1s3+κμ3w^3h3+μ4R^3+k6+μ5ρ1η1−ρ1κη~1unom1s3+ρ1k8η~1η^1−1α3w~3w^̇3−1β3R~3R^̇3.By utilizing Equations (51) and (52), the above inequality can be rewritten as:
(58)φ̇3≤s3w~3h3+s3R~3−k5s32−k6s3+m3s31−ρ1unom1−ρ1κη^1unom1s3+κμ3w^3h3+μ4R^3+k6+μ5ρ1η1−ρ1κη~1unom1s3+ρ1k8η~1η^1−1α3w~3α3s3h3tanhs3μ3−c5w^3,−1β3R~3β3s3tanhs3μ4−c6R^3≤−k5s32+ρ1k8η~1η^1+c5w~3w^3+c6R~3R^3+κμ3w3h3+μ4R3+k6+μ5ρ1η1≤−k5s32−ρ1k82η~12−c52w~32−c62R~32+ρ1k82η12+c52w32+c62R32+κμ3w3h3+μ4R3+k6+μ5ρ1η1=−θ3φ3+ζ3,where θ3=min2k5,k7k8,α3c5,β3c6, ζ3=κμ3w3h3+μ4R3+k6+μ5ρ1η1+ρ1k8/2η12+c5/2w32+c6/2R32.Consequently, the differential ofφ0 can be calculated as:
(59)φ̇0≤−k1−m1g¯12s12−1ε2−m1g¯12−12y22−c12w~21−c22R~12−k3−m22s22−1ε3−m22−12y32−k5s32−ρ1k82η~12−c52w~32−c62R~32+c12w12+c22R12+ο222+μ1w1h1κ+μ2R1κ+k2κ+ο322+k4κ+ρ1k82η12+c52w32+c62R32+κμ3w3h3+μ4R3+k6+μ5ρ1η1≤−θ0φ0+ζ0,with
(60)θ0=min2k1−m1g¯1,2ε2−m1g¯1−1,α1c1,β1c2,2k3−m2g¯2,2ε3−m2g¯2−1,2k5,k7k8,α3c5,β3c6,(61)ζ0=c12w12+c22R12+ο222+μ1w1h1κ+μ2R1κ+k2κ+ο322+k4κ+ρ1k82η12+c62R32+c52w32+κμ3w3h3+μ4R3+k6+μ5ρ1η1.To conclude,s1, s2, and s3 exponentially converge to a neighborhood around the origin, only if the designed parameters k1, k2, k3, k4, k5, k6, k7, k8, α1, α3, β1, β3, c1, c2, c3, c4, c5, c6, μ1, μ2, μ3, μ4, and μ5 are chosen to satisfy θ0>0. According to the transformation Equation (26), the stabilization of tracking errors xei is guaranteed, while their trajectories always remain within the predefined region. The proof of Theorem 5 is completed.


## 4. Velocity Controller Design

The velocity tracking control subsystem for HFVs is established as:(62)V̇=fv+ρ2uc2+d2t+Δ2.The definition of velocity tracking errorVe and its time derivative are given as:
(63)Ve=V−Vd,V̇e=V̇−V̇d=fv+ρ2uc2+d2t+Δ2−V̇d.The predesigned boundary of variableVe is denoted as:
(64)V¯e≤Ve≤V¯e.The error transformation is redesigned as:(65)Ve=V¯e+V¯e2tanhs4+V¯e+V¯e2,s4=atanh2Ve−V¯e−V¯eV¯e−V¯e,where s4 is the transformed velocity error. Obviously, it indicates that the transformed error increases monotonically with the original error.Hence, there exist limits as:(66)lims4→−∞Ve=V¯e,lims4→+∞Ve=V¯e.Additionally, the time derivative ofs4 is derived as:
(67)ṡ4=m4V̇e+n4=m4fv+ρ2uc2+d2t+Δ2−V̇d+n4.Subsequently, the uncertain dynamicm4fv is approximated by adopting RBFNN, which can be expressed as:
(68)m4fv=Wv1∗THv1x+rv,0<rv≤Ov,(69)Wv1∗THv1≤Wv1∗THv1≤wvhv,where hv=Hv1 and wv≥Wv1∗T.The control input and adaptive laws are given as:(70)uc2=−unom2+un2,(71)unom2=1m4w^vhvtanhs4μ6+R^4tanhs4μ7+k9s4+k10tanhs4+n4−v̇d,(72)un2=1m4η^2unom2tanhunom2s4μ8,(73)η^̇2=k11unom2s4tanhunom2s4μ8−k12η^2,(74)w^̇v=α4hvs4tanhs4μ6−c7w^v,(75)R^̇4=β4s4tanhs4μ7−c8R^4,where η2=1−ρ2m4/ρ2 and R4=m4λ2+Ov; η^2, w^v, and R^4 are the estimate values of η2, wv, and R4, respectively; and k9, k10, k11, k12, α4, β4, c7, c8, μ6, μ7, and μ8 are all positive parameters.Theorem 6.
Noting the velocity dynamic (61) of HFV and considering the fault tolerant controller Equations (69)–(71), exponential convergence and prescribed performance constraints will be ensured for the tracking error.Proof.
To illustrate the convergence of velocity tracking error, the Lyapunov function is selected as:(76)φ4=12s42+ρ22k11η~22+12α4w~v2+12β4R~42.With the substitution of Equations (69)–(71), the time derivative of φ4 is derived as:
(77)φ̇4=s4m4fv+ρ2uc+d2t+Δ2−v̇d+n4s4−ρ2k11η~2η^̇2−1α4w~vw^̇v−1β4R~4R^̇4≤s4wvhv+Rv+m4s41−ρ2unom2−unom2−ρ2un2−v̇d+n4s4−ρ2k11η~2η^̇2−1α4w~vw^̇v−1β4R~4R^̇4≤s4w~vhv+s4R~4−k9s42−k10s4+m4s41−ρ2unom2−ρ2unom2η^2s4+κμ6w^vhv+μ7R^4+k10+μ8ρ2η2−ρ2unom2η~2s4+ρ2k12η~2η^2−1α4w~vw^̇v−1β4R~4R^̇4.Taking Equations (71)–(73) into account, Equation (77) can be further expressed as:
(78)φ̇4≤s4w~vhv+s4R~4−k9s42−k10s4+m4s41−ρ2unom2−ρ2unom2η^2s4+κμ6w^vhv+μ7R^4+k10+μ8ρ2η2−ρ2unom2η~2s4+ρ2k12η~2η^2−1α4w~vα4hvs4tanhs4μ6−c7w^v−1β4R~4β4s4tanhs4μ7−c8R^4≤−k9s42+ρ2k12η~2η^2+c7w~vw^v+c8R~4R^4+κμ6wvhv+μ7R4+k10+μ8ρ2η2≤−k9s42−ρ2k122η~22−c72w~2v−c82R~42+ρ2k122η22+c72wv2+c82R42+κμ6wvhv+μ7R4+k10+μ8ρ2η2=−θ4φ4+ζ4,where θ4=min2k9,k11k12,α4c7,β4c8 and ζ4=ρ2k12/2η22+c7/2wv2+κμ6wvhv+μ7R4+k10+μ8ρ2η2+c8/2R42. Therefore, it can be concluded that s4 is asymptotically stable, and the velocity tracking error Ve completes convergence within the prespecified range. Theorem 6 is validated sufficiently.


## 5. Simulation

To illustrate the specific performances of the proposed controller, a numerical simulation is executed based on the longitudinal dynamics (1)–(17) of HFV. By giving the initial states and reference trajectories, the tracking control objective will be achieved with predefined performance constraints. The initial values of states are provided in Table 1, and the control parameters are shown in Table 2.Table 1
Initial values of states.
ItemValueUnitV15060ft/sh110000ftγ0radα1.6325π/180radq0rad/sTable 2
Values of control parameters.
SectionParametersPPCxei=λ¯i−λ¯iexp−ιit+λ¯i,i=0,1,2,3,4,λ¯0=105,λ¯0=3,ι0=0.2,λ¯1=0.05,λ¯1=0.04,ι1=0.3,λ¯2=0.1,λ¯2=0.035,ι2=0.1,λ¯1=0.1,λ¯1=0.05,ι1=0.1,λ¯1=105,λ¯1=5,ι1=0.1Low-pass filterε2=10,ε3=10Input of MLP-NNReferring to [35]Controllerkh=10,khh=1.5,k1=0.8,k2=0.2,α1=0.0001,c1=2,β1=0.001,c2=10,k3=2.5,k4=0.01,k5=0.05,k6=0.05,k7=0.05,k8=30,α3=0.001,c5=10,β3=0.05,c6=10,k9=3,k10=2,k11=0.005,k12=2,α4=0.05,c7=2,β4=0.01,c8=2Meanwhile, the desired trajectories, environmental disturbances, and actuator effectiveness parameters are defined as:(79)Vd=15160ft/s,hd=110100ft,d1=0.01×sin0.05t,d2=0.01×cos0.05t,ρ1=1,t<100s0.7,t≥100s,Δ1=0.001×sin0.05t,ρ2=1,t<100s0.8,t≥100s,Δ2=0.001×cos0.05t.Consequently, the main results are exhibited in Figures1–10. Firstly, the curves of tracking errors are shown in Figures 1 and 2. It follows from these two pictures that the altitude tracking error and velocity tracking error are stabilized into a tiny neighborhood around origin within 30 s and 50 s, respectively. Obviously, not only the transient performance constraint is guaranteed but also the steady-state errors are maintained within ranges of 5 m and 5 m/s, which means undoubtedly a very high accuracy for hypersonic flight. What is subsequently given in Figures 3 and 4 is the curves of control inputs. In particular, even if there is an unknown actuator failure occurring at 100 s, the control signal is capable of returning to a constant within an extremely short time. Figures 5 and 6 show the curves of adaptive estimates R^3 and R^4, which always remain bounded under the designed algorithm. The estimations about weight matrixes of RBFNNs are provided in Figures 7 and 8, and the adaptive learning parameters for actuator failure are settled in Figures 9 and 10. As is shown in figures, all the estimations finally converge to constants. Thus, the proposed controller is capable of tracking the desired trajectory in case of actuator faults.Figure 1
The altitude tracking error of HFV.Figure 2
The velocity tracking error of HFV.Figure 3
The input signal of altitude control subsystem.Figure 4
The input signal of velocity control subsystem.Figure 5
Estimation ofR3.Figure 6
Estimation ofR4.Figure 7
Estimation ofw3.Figure 8
Estimation ofwv.Figure 9
Estimation ofη1.Figure 10
Estimation ofη2.For better illustration of the merits PPC, simulation results without performance constraints are presented in Figures11 and 12. Observing the tracking errors of altitude and velocity, one can conclude that the performance constraints will be violated before the system’s stabilization. From this aspect, it deduces that the PPC control strategies are necessary for hypersonic vehicles’ tracking control.Figure 11
The tracking error of altitude without PPC.Figure 12
The tracking error of velocity without PPC.


## 6. Conclusion

This study concentrates on the robust prescribed performance tracking control for HFV in the presence of system uncertainties and unknown actuator faults. On the basis of longitudinal model of HFV, an antidisturbance neural adaptive control strategy is constructed, in which MLP algorithm is employed to approximate the unavailable dynamics with a reduced computational burden. In addition, the multiplicative and additive failures of actuator are taken into account simultaneously. Finally, the stability and effectiveness of proposed controller are elaborated by rigorous theoretical analysis. The results of simulation example further indicate that the altitude and velocity tracking errors satisfy the prescribed performance constraints.



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*Source: 8880387-2021-03-31.xml*