# Finite-Time Antisaturation Tracking Control for Hypersonic Vehicle with Uncertain Dynamics

Authors: Yuwei Cui; Yushan He; Aijun Li; Jun Li; Shuo Song
Journal: International Journal of Aerospace Engineering
 (2020)

---

## Abstract

This paper concentrates on the problem of finite-time altitude and velocity tracking control for hypersonic flight vehicles that encounter unmodeled dynamics and input saturations. An adaptive neural finite-time backstepping control strategy is constructed by designing modified virtual commands and compensation signals. The minimum learning parameter algorithm based on a radial basis function is employed to approximate the unknown dynamics with low computational burden. Furthermore, an auxiliary system is established to cope with the nonlinearity caused by actuator saturation. It is concluded by a Lyapunov-based analysis that the finite-time stability is guaranteed under the developed architecture. Finally, numeral simulation is provided to demonstrate the effectiveness of the proposed controller.


---

## Body

## 1. Introduction

In the research of near space aircrafts, generic hypersonic vehicles (GHVs) have become a hotspot in countries all over the world. It owes much to the significant advantages in flight speed and concealment, which makes GHVs possess high application value in both military and civilian fields. However, the unique integrated design and complex flight environment have brought tremendous challenges to the control of GHVs. Facing such a dynamic system with strong coupling, fast time-varying dynamics, high nonlinearity, and uncertainties, scholars have contributed strenuous attempts, including but not limited to sliding mode control [1–3], dynamic compensation control (DCC) [4–6], and adaptive control [7–9].The backstepping method is an effective strategy to cope with higher-order nonlinear systems, which is widely utilized in trajectory tracking control for GHVs. However, repeated derivation for virtual commands leads to a problem called “explosion of complexity (EOC).” The solution for this obstacle is a dynamic surface control (DSC), in which the low-pass filter is introduced to provide the differential signal of virtual command. In [10], a compound control algorithm based on dynamic compensation was proposed so that the desired trajectory is tracked under strong coupling and high nonlinearity. After that, Zhao and Li designed a robust controller for GHVs with actuator failures, in which the trajectory tracking errors are restricted with a prescribed range [11]. Unfortunately, the above results are not satisfactory enough in terms of convergence speed, which is especially important for a GHV in continual rapid motion. For this reason, an adaptive robust control strategy was constructed for finite-time tracking missions when external disturbances and model uncertainties were encountered [12]. What is more, Wu et al. established a fixed-time sliding mode backstepping framework, and the time-varying disturbances were estimated via a modified observer [3].Meanwhile, another ineluctable problem is that the nonlinearity and uncertainty in the GHV dynamics must be paid full attention. As a result, it is difficult to achieve the control objectives through traditional control methods deeply relying on the accurate model information. To relax the conservatism, fuzzy logic control (FLC) [13–15] and neural network (NN) [1, 16, 17] have become excellent instruments for approximation of unknown dynamics. In [13], two fuzzy logic systems were developed to handle the aerodynamic uncertainties of the GHV model. Meanwhile, the constraints of the attitude tracking errors are guaranteed by virtue of the nonlinear mapping method. On the other hand, aiming at the unknown measurement noises, Liu et al. exploited an adaptive fuzzy approximator, which could provide a safeguard for the performances of the proposed controller [14]. Different from the above methodologies, the advantage of the radial basis function neural network (RBFNN) is that it can approximate nonlinear smooth functions with arbitrary precision. Therefore, it is extensively applied in the tracking control of GHV. In [16], adaptive neural networks were employed to tackle unmodeled dynamics, while the unavailable external disturbances were estimated accurately via a novel disturbance observer. To proceed further, Ping et al. formulated a neuroadaptive sliding mode control strategy to eliminate the system uncertainties, where the nonlinearity arising from the actuator failures was alleviated significantly [1].Besides the capability of antiuncertainty, another important indicator is the stabilization rate. To the best of our knowledge, compared with an asymptotically stable control, the finite-time control performs better in terms of rapidity and robustness. On the basis of a finite time disturbance observer [18], constructs a trajectory tracking backstepping algorithm for a hypersonic flight vehicle with the presence of time-varying environmental disturbance. Subsequently, the finite-time tracking control problem of flexible GHV is investigated in [19], while the unavailable system parameters are taken into consideration. To proceed further, the finite-time tracking control scheme is further extended to the case of unknown actuator failures in [12], where the fixed-time observer is adopted to estimate fault information and other uncertain parameters.It is noteworthy that another caveat here is input saturation constraint. From a practical point of view, it is impossible for actuators to output infinite control torque due to the restriction of mechanical structure. In [20], an adaptive sliding mode control algorithm was proposed for flexible GHVs, in which adaptive laws were designed to cope with the actuator saturations. Subsequently, Sun et al. constructed a backstepping-based dynamic surface control technique to achieve the altitude and velocity tracking control for GHVs [21]. Not only the phenomenon of EOC is removed by a first-order filter, but also the errors induced by the filter are eliminated via designing compensation signals. Moreover, a specific auxiliary system is developed to conquer input saturation. With further exploration, an adaptive antisaturation controller for GHVs is established in [22], which can ensure that the system still has the capability of stable tracking even in the presence of actuator faults.Motivated by the above observations, this paper addresses the finite-time trajectory tracking control for GHVs exposed to unmodeled dynamics and input saturations. On the basis of the longitudinal model of GHVs, the altitude and velocity tracking control is achieved via the DCC-based algorithm design. The parameter uncertainties are approximated by RBFNN, and actuator saturations are handled via an auxiliary system. The main contributions of the research are summarized as follows:(1)
Compared with [10, 11], the DCC applied in this paper ensures the global finite-time convergence of tracking errors. In addition, the problem of EOC is relaxed, and all the tracking errors and estimation errors will converge to a tiny region containing the origin in finite time(2)
Different from [16], the minimum learning parameter (MLP) algorithm-based RBFNN is employed to approximate the unknown dynamics. In this way, only a single parameter is adaptively updated rather than the whole weight matrix; thus, the excessive occupation of the computational resource is avoided effectively(3)
In consideration of the antisaturation control in [23], the nonlinearity arising from actuator saturation is compensated for effectively, while the auxiliary variable is restricted to a bounded range. In this paper, the hyperbolic tangent function-based modified auxiliary relaxes the conservatism creativelyThe remainder of this paper is structured as follows: Section2 shows the mathematical model and preliminaries; Section 3 details the structure and design procedure of the controller; Section 4 theoretically authenticates the stability of the proposed controller; finally, a simulation example is provided in Section 5 and the conclusion is shown in Section 6.


## 2. System Description and Preliminaries

### 2.1. Longitudinal Model of GHV

According to the longitudinal model of GHVs in [24], the velocity and attitude dynamic equations are given as
(1)V̇=Tcosα−Dm−μsinγr2(2)γ̇=L+TsinαmV−μ−V2rcosγVr2,(3)ḣ=Vsinγ,(4)α̇=Q−γ̇,(5)Q̇=MyyIyy,which include five rigid-body states: the velocity V, the flight-path angle γ, the altitude h, the angle of attack α, and the pitch rate Q. Meanwhile, four forces or moments are introduced as the thrust T, the drag D, the lift moment L, and the pitching moment Myy. Other parameters m,μ, and Iyy denote the mass, the gravitational constant, and the moment of inertia, respectively. The relevant aerodynamic coefficient is shown 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=0.022576β,β<1,0.0224+0.00336β,β>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 the dynamic pressure Q¯ satisfies Q¯=1/2ρV2 with ρ denoting the density of air; S and re denote the reference area and the radius of the Earth, respectively; and c¯ and ce are both constants.It is noteworthy that most of the existing studies about GHV control lack relevant discussions on actuator saturation. In practice, there is no doubt that the actuator can only output a limited control torque due to the inherent mechanical structure. This paper will provide a solution for tracking control for GHVs in the presence of input saturation constraints and external disturbances. To this end,δe is defined as
(18)δe=satu1+d1,where d1 represents the lumped external disturbance. The saturation nonlinearity satu is expressed as
(19)satu=signuum,u≥um,u,u<um,where um denotes the upper bound of satu and u is the desired control input.


### 2.2. NN-Based Radial Basis Function

In this paper, the unmodeled dynamics of GHVs are approximated via RBFNN. The basic principle of RBFNN is expounded as the following lemma.Lemma 1 (see [25]).
By defining a basis functionHx, an arbitrary continuous smooth function fx can be written in the following form:
(20)fx=W∗THx+ο,0<ο≤O,where x=x1,x2,,⋯,xm represents the input vector and W∗=w1,w2,⋯,wpT is the ideal weight matrix; ο and O denote the approximation error and its upper bound, respectively; and hx=h1x,h2x,⋯,hmxT is selected as the Gaussian basis function, whose definition is shown as
(21)hix=exp−x−ci222bi2,i=1,⋯,p,
whereci∈Rm and b∈Rp denote the center vector and the width of Gaussian basis function, respectively.


### 2.3. Other Preliminaries

Lemma 2 (see [26]).
Forx∈R and μ>0, the hyperbolic tangent function possesses the following property:
(22)0<x−xtanhμx≤Kμ,with K=0.2785.Assumption 3.
The external disturbances in the GHV system is unknown but bounded, i.e.,0<di≤Di,i=1,2, in which Di are positive constants.Remark 4.
In contrast to the traditional aircraft, GHV mainly works in the near-space area, which possesses the characteristics of complicated environment and dramatically changing aerodynamics. The actual external disturbance torque is usually difficult to observe accurately. Therefore, it is necessary to enhance the controllers’ robustness to disturbances and uncertainties.


## 2.1. Longitudinal Model of GHV

According to the longitudinal model of GHVs in [24], the velocity and attitude dynamic equations are given as
(1)V̇=Tcosα−Dm−μsinγr2(2)γ̇=L+TsinαmV−μ−V2rcosγVr2,(3)ḣ=Vsinγ,(4)α̇=Q−γ̇,(5)Q̇=MyyIyy,which include five rigid-body states: the velocity V, the flight-path angle γ, the altitude h, the angle of attack α, and the pitch rate Q. Meanwhile, four forces or moments are introduced as the thrust T, the drag D, the lift moment L, and the pitching moment Myy. Other parameters m,μ, and Iyy denote the mass, the gravitational constant, and the moment of inertia, respectively. The relevant aerodynamic coefficient is shown 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=0.022576β,β<1,0.0224+0.00336β,β>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 the dynamic pressure Q¯ satisfies Q¯=1/2ρV2 with ρ denoting the density of air; S and re denote the reference area and the radius of the Earth, respectively; and c¯ and ce are both constants.It is noteworthy that most of the existing studies about GHV control lack relevant discussions on actuator saturation. In practice, there is no doubt that the actuator can only output a limited control torque due to the inherent mechanical structure. This paper will provide a solution for tracking control for GHVs in the presence of input saturation constraints and external disturbances. To this end,δe is defined as
(18)δe=satu1+d1,where d1 represents the lumped external disturbance. The saturation nonlinearity satu is expressed as
(19)satu=signuum,u≥um,u,u<um,where um denotes the upper bound of satu and u is the desired control input.


## 2.2. NN-Based Radial Basis Function

In this paper, the unmodeled dynamics of GHVs are approximated via RBFNN. The basic principle of RBFNN is expounded as the following lemma.Lemma 1 (see [25]).
By defining a basis functionHx, an arbitrary continuous smooth function fx can be written in the following form:
(20)fx=W∗THx+ο,0<ο≤O,where x=x1,x2,,⋯,xm represents the input vector and W∗=w1,w2,⋯,wpT is the ideal weight matrix; ο and O denote the approximation error and its upper bound, respectively; and hx=h1x,h2x,⋯,hmxT is selected as the Gaussian basis function, whose definition is shown as
(21)hix=exp−x−ci222bi2,i=1,⋯,p,
whereci∈Rm and b∈Rp denote the center vector and the width of Gaussian basis function, respectively.


## 2.3. Other Preliminaries

Lemma 2 (see [26]).
Forx∈R and μ>0, the hyperbolic tangent function possesses the following property:
(22)0<x−xtanhμx≤Kμ,with K=0.2785.Assumption 3.
The external disturbances in the GHV system is unknown but bounded, i.e.,0<di≤Di,i=1,2, in which Di are positive constants.Remark 4.
In contrast to the traditional aircraft, GHV mainly works in the near-space area, which possesses the characteristics of complicated environment and dramatically changing aerodynamics. The actual external disturbance torque is usually difficult to observe accurately. Therefore, it is necessary to enhance the controllers’ robustness to disturbances and uncertainties.


## 3. Finite-Time Controller Design

In this section, a finite-time tracking controller is developed in virtue of the MLP-based adaptive backstepping strategy. To complete the trajectory tracking control, two subsystems are formulated according to the longitudinal model of GHV in Section2. For each subsystem, the novel control commands and adaptive laws are designed to ensure the global finite-time stability of a closed-loop system. One caveat here is that the altitude tracking dynamic is a three-order system, in which the effect of EOC is too severe to be neglected. Therefore, the compensation signals are constructed on the basis of a command-filtered control. Meanwhile, a RBFNN-based MLP algorithm is employed to approximate unknown dynamics, such that the burden of computation is relaxed significantly. Input saturations are addressed by resorting to an auxiliary system. All the approximation errors are eliminated along with the unknown external disturbances through the design of the adaptive laws.


### 3.1. Altitude Control

As the reference trajectory of altitude is denoted byhd, the corresponding flight path angle γd can be written as
(23)γd=arcsin−khh~−khhtanhh~+ḣdV,where kh and khh are positive constants and γd∈−90°,90°.Considering the GHV model in Section2, there are three states in the altitude tracking control system, including γ,ϑp, and q. Hence, a vector is introduced as
(24)x=x1,x2,x3T=γ,ϑp,qT,where ϑp=α+γ. Subsequently, according to Equations (2), (3), (4) and (5), the altitude tracking subsystem of GHV can be expressed as
(25)ẋ1=f1x1,V+g1Vx2,ẋ2=x3,ẋ3=f3x1,x2,x3,V+g3Vδe=f3x1,x2,x3,V+g3Vsatu1+d1,with
(26)g1V=0.31015ρVSm,f1x1,V=TsinαmV−μ‐V2rcosγVr2,f3x1,x2,x3,V=0.5ρV2SCMα+CMq−ceαIyy,g3V=q¯Sc¯ceIyy.Assumption 5.
The uncertain termgiV possesses an upper bound g¯i, which satisfies 0<giV≤g¯i,i=1,3.Remark 6.
As pointed above, there exists high uncertainty in the GHV model, due to not only the strong coupling and nonlinearity caused by its own structure but also the fast time variability caused by the high-speed flight. Accordingly, it is difficult to obtain an accurate model for GHVs. In this paper, a RBFNN-based control strategy is proposed to facilitate controller design. Concretely speaking, the functionsfi and gi with i=1,3 possess unknown structures, which will be approximated by NN. And the resulting approximation error will be eliminated by resorting to the design of adaptive laws.With the introduction of reference trajectory of states, the tracking errors are defined as(27)ε1=x1−x1d,ε2=x2−x2d,ε3=x3−x3d−ξ1tanhχ1,where
(28)χ̇1=cosh2χ1ξ1−ζ1tanhχ1+satu1−u1,χ10=0,with the designed parameters satisfying ξ1,ζ1>0, xid being the designed trajectory.Remark 7.
In [23], the auxiliary system and the transformed error are defined as χ̇=−ζχ+satu−u and ε=x−xd−ξχ, respectively. However, this solution is effective only if the auxiliary variable χ is bounded, while the dynamics (27) and (28) employs the hyperbolic tangent function to construct a bounded compensation term ζtanhχ, so that the conservatism of above assumption is relaxed.Taking the time derivative of equations and substituting Equation (25) yields
(29)ε̇1=f1+g1x2−ẋ1d,ε̇2=x3−ẋ2d,ε̇3=f3+g3satu1+d1−ẋ3d+ζ1tanhχ1−satu1+u1.For the purpose of relaxing the issue of EOC, a first-order low-pass filter is introduced as(30)c2ẋ2d+x2d=x2c,x2d0=0,c3ẋ3d+x3d=x3c,x3d0=0,where x2c,x3c are both output signals and the filter parameters satisfy c2,c3>0. Resulting error variables are defined as
(31)y2=x2d−x2c,y3=x3d−x3c.Step 1.
Taking the unavailable dynamicf1 into account and using Lemma 2 yield
(32)f1=W1∗TH1x+ο1,0<ο1≤O1,W1∗TH1≤W1∗TH1≤w1h1,where h1=H1 and w1≥W1∗T.
The compensation signal is designed as(33)φ̇1=−ϑ1φ1−ϑ2tanhφ1−g1ε1y2+0.5y22φ1+y2,where ϑ1,ϑ2 are both positive constants.
Subsequently, the adaptive laws and the virtual command are designed as(34)w^̇1=α1ε1h1−r1w^1,(35)R^̇1=β1ε1−r2R^1,(36)x2c=1g1ẋ1d−k1ε1−k2tanhε1−w^1h1tanhε1−R^1tanhε1+l1φ1,in which R1=O1; w^1 and R^1 denote estimations of w1 and R1, respectively; and parameters α1,β1,r1,r2,k1,k2,l1 are all designed as positive constants.Remark 8.
Different from the conventional algorithm of RBFNN, MLP algorithms select the upper bound of the weight matrix norm as the online updating parameter. Hence, only a single parameter is estimated rather than the whole weight matrix, so that the excessive occupation of the computational resource is relaxed significantly.To show the stability ofε1, the Lyapunov function (LF) is chosen as
(37)F1=12ε12+12φ12.DifferentiatingF1 and using Equations (33) and (40) yield
(38)Ḟ1=ε1ε̇1+φ1φ̇1=ε1f1+g1ε2+x2d−ẋ1d+φ1−ϑ1φ1−ϑ2tanhφ1−g1ε1y2+0.5y22φ1+y2=ε1f1+g1ε1ε2+g1ε1y2+ε1−k1ε1−k2tanhε1−w^1h1tanhε1−R^1tanhε1+l1φ1−ϑ1φ12−ϑ2φ1tanhφ1−g1ε1y2−12y22+φ1y2≤ε1f1+g1ε1ε2−k1ε12−k2ε1tanhε1−w^1h1ε1tanhε1−R^1ε1tanhε1+ε12+14l12φ12−ϑ1φ12−ϑ2φ1tanhφ1+12φ12.According to Lemma2, one has
(39)−k2ε1tanhε1≤−k2ε1+k2ϖ,−w^1h1ε1tanhε1≤−w^1h1ε1+w^1h1ϖ,−R^1ε1tanhε1≤−R^1ε1+R^1ϖ,−ϑ2φ1tanhφ1≤−ϑ2φ1+ϑ2ϖ.Substituting Equation (39) into Equation (38) leads to
(40)Ḟ1≤ε1W1∗TH1x+ο1+g1ε1ε2−k1−1ε12−k2ε1+k2ϖ−w^1h1ε1+w^1h1ϖ−R^1ε1+R^1ϖ−ϑ1−14l12−1φ12−ϑ2φ1+ϑ2ϖ≤ε1w1h1+ε1R1+g1ε1ε2−k1−1ε12−k2ε1+k2ϖ−w^1h1ε1+h12w^122+ϖ22−R^1ε1+R^122+ϖ22−ϑ1−14l12−12φ12−ϑ2φ1+ϑ2ϖ=ε1w~1h1+ε1R~1−k1−1−12g¯1ε12+12g¯1ε22−k2ε1−ϑ1−14l12−12φ12−ϑ2φ1+h12w^122+R^122+ϖ2+k2ϖ+ϑ2ϖ.Step 2.
For the purpose of the stabilization ofε2, the virtual command x3d is given as
(41)x3c=−k3ε2−k4ε2tanhε2+ẋ2d+l2φ2,where k3, k4, and l2 are all positive parameters.
The filter compensation signal is expressed as(42)φ̇2=−ϑ3φ2−ϑ4tanhφ2−ε2y3+0.5y32φ2+y3,where ϑ3,ϑ4 are both positive parameters.
The second LF is selected, and its derivatives are presented as(43)F2=12ε22+12φ22,(44)Ḟ2=ε2ε̇2+φ2φ̇2=ε2ε3+x3d−ẋ2d+φ2−ϑ3φ2−ϑ4tanhφ2−12φ2y32+y3=ε2ε3+ε2y3−k3ε22−k4ε2tanhε2−g1ε1ε2+ε2l2φ2−ϑ3φ22−ϑ4φ2tanhφ2−ε2y3−12y32+φ2y3≤ε2ε3−k3ε22−k4ε2tanhε2−g1ε1ε2+ε22+14l22φ22−ϑ3φ22−ϑ4φ2tanhφ2+12φ22.
It can be derived from Lemma2 that
(45)−k4ε1tanhε2≤−k4ε2+k4ϖ,−ϑ4φ2tanhφ2≤−ϑ4φ2+ϑ4ϖ.
Substituting the result into Equation (44), it obtains
(46)Ḟ2≤ε2ε3−k3−1−12g¯1ε22+12g¯1ε12−k4ε2+k4ϖ−ϑ3−14l22−12φ22−ϑ4φ2+ϑ4ϖ.Step 3.
In this part, the uncertain dynamicf3 is approximated via RBFNN, which is shown as
(47)f3=W3∗TH3x+ο3,0<ο3≤O3,(48)W3∗TH3≤W3∗TH3≤w3h3,where h3=H3 and w3≥W3∗T.
Owing to the effect of external disturbances, it is defined according to Assumptions3 and 5 that R3=g¯3um+D1+O3+ζ1−um. Afterward, the adaptive laws and the control input are given as
(49)w^̇3=α3ε3h3−r3w^3,(50)R^̇3=β3ε3−r4R^3,(51)u1=1g3−k5ε3−k6ε3ε3+ρ1−w^3h3ε3ε3+ρ2−R^3ε3ε3+ρ3+ẋ3d,(52)ρ1=b11+k6,ρ2=b21+w^3h3,ρ3=b31+R^3,where parameters α3,β3,r3,r4,k5,k6,ρ1,ρ2,ρ3 are all positive constants.Remark 9.
Considering the controller (51), several ingredients should to be emphasized as follows: (i) The term −k5ε3 and the term −k6ε3/ε3+ρ1 guarantee the finite time convergence of ε3. Meanwhile, the term −w^3h3ε3/ε3+ρ2 with the construction (49) is utilized to estimate the weight parameter of NN, and the term −R^3ε3/ε3+ρ3 is for unknown lumped disturbances. (ii) The introduction of variables εii=1,2,3 not only leads to the attenuation of chattering phenomenon but also improves the controllability of steady-state errors. This point will be proved in subsequent derivations.Consequently, the third LF and its derivative are presented as(53)F3=12ε32,Ḟ3=ε3ε̇3=ε3f3+g3satu1+d1−ẋ3d+ζ1tanhχ1−satu1+u1=ε3f3−k5ε32−k6ε32/ε3+ρ1−w^3h3ε32/ε3+ρ2−R^3ε32/ε3+ρ3−ε2ε3+ε3g3satu1+d1+ζ1tanhχ1−satu1.Taking Equations (47) and (48) into account, it leads to
(54)Ḟ3=ε3W3∗TH3x+ο3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3g3satu1+d1+ζ1tanhχ1−satu1≤ε3w3h3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3g3satu1+d1+O3+ζ1tanhχ1−satu1≤ε3w3h3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3R3.Some of the items in Equation (54) are scaled as
(55)−k6ε32ε3+ρ1=−k6ε3+k6ρ1×ε3ε3+ρ1=−k6ε3+k6b11+k6×ε3ε3+ρ1≤−k6ε3+b1,−w^3h3ε32ε3+ρ2=−w^3h3ε3+w^3h3ρ2×ε3ε3+ρ2=−w^3h3ε3+w^3h3b21+w^3h3×ε3ε3+ρ2≤−w^3h3ε3+b2,−R^3ε32ε3+ρ3=−R^3ε3+R^3ρ3×ε3ε3+ρ3=−R^3ε3+R^3b31+R^3×ε3ε3+ρ3≤−R^3ε3+b3.With the substitution of these results, Equation (54) becomes
(56)Ḟ3≤ε3w3h3−k5ε32−k6ε3+b1−w^3h3ε3+b2−R^1ε3+b3−ε2ε3+ε3R3≤ε3w~3h3+ε3R~3−k5−12ε32+12ε22−k6ε3+b1+b2+b3.


### 3.2. Velocity Control

As shown in Section2, the closed-loop system for velocity tracking control is expressed as
(57)V̇=fv+satu2+d2,with
(58)fv=Tcosθ−γ−Dm−gsinγ−u2.According to Lemma2, the unavailable state fv can be written as
(59)fv=Wv∗THvx+οv,0<οv≤Ov,(60)Wv∗THv≤Wv∗THv≤wvhv,where hv=Hv and wv≥Wv∗T.The velocity tracking errorev is introduced as
(61)εv=V−Vd−ζ2tanhχ2,where
(62)χ̇2=cosh2χ2ξ2−ζ2tanhχ2+satu2−u2,χ20=0,with the parameters satisfying ξ2,ζ2>0.Differentiatingεv along Equation (61) and using Equation (57) yield
(63)ε̇v=fv+d2−V̇d+ζ2tanhχ2+u2.It is defined thatRv=D2+Ov+ζ1. Subsequently, the adaptive laws and the control input are given as
(64)w^̇v=αvεvhv−r5w^v,(65)R^̇v=βvεv−r6R^v,(66)u2=−k7εv−k8εvεv+ρ4−w^vhvεvεv+ρ5−R^vεvεv+ρ6−V̇d,(67)ρ4=b41+k8,ρ5=b51+w^vhv,ρ6=b61+R^v,where the parameters αv,βv,r5,r6,k7,k8,b4,b5,b6 are all designed as positive constants.To illustrate the convergence of the velocity tracking error, the LF is selected as(68)Fv=12εv2.DifferentiatingFv along (78) yields
(69)Ḟv=εvε̇v=εvfv+d2−V̇d+ζ2tanhχ2+u2=εvfv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+ζ2tanhχ2.With the substitution of Equations (59) and (60), it is further derived as
(70)Ḟv=εvWv∗THvx+οv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+ζ2tanhχ2≤εvwvhv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+Ov+ζ2tanhχ2≤εvwvhv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvRv.There holds the inequations that(71)−k8εv2εv+ρ4=−k8εv+k8ρ4εvεv+ρ4=−k8εv+k8b41+k8×εvεv+ρ4≤−k8εv+b4,−w^vhvεv2εv+ρ5=−w^vhvεv+w^vhvρ5εvεv+ρ5=−w^vhvεv+w^vhvb51+w^vhv×εvεv+ρ5≤−w^vhvεv+b5,−R^vεv2εv+ρ6=−R^vεv+R^vρ6εvεv+ρ6=−R^vεv+R^vb61+R^v×εvεv+ρ6≤−R^vεv+b6.Substituting the results into Equation (70), one obtains
(72)Ḟv≤εvwvhv−k7εv2−k8εv+b4−w^vhvεv+b5−R^vεv+b6+εvRv≤εvw~vhv+εvR~v−k7εv2−k8εv+b4+b5+b6.


## 3.1. Altitude Control

As the reference trajectory of altitude is denoted byhd, the corresponding flight path angle γd can be written as
(23)γd=arcsin−khh~−khhtanhh~+ḣdV,where kh and khh are positive constants and γd∈−90°,90°.Considering the GHV model in Section2, there are three states in the altitude tracking control system, including γ,ϑp, and q. Hence, a vector is introduced as
(24)x=x1,x2,x3T=γ,ϑp,qT,where ϑp=α+γ. Subsequently, according to Equations (2), (3), (4) and (5), the altitude tracking subsystem of GHV can be expressed as
(25)ẋ1=f1x1,V+g1Vx2,ẋ2=x3,ẋ3=f3x1,x2,x3,V+g3Vδe=f3x1,x2,x3,V+g3Vsatu1+d1,with
(26)g1V=0.31015ρVSm,f1x1,V=TsinαmV−μ‐V2rcosγVr2,f3x1,x2,x3,V=0.5ρV2SCMα+CMq−ceαIyy,g3V=q¯Sc¯ceIyy.Assumption 5.
The uncertain termgiV possesses an upper bound g¯i, which satisfies 0<giV≤g¯i,i=1,3.Remark 6.
As pointed above, there exists high uncertainty in the GHV model, due to not only the strong coupling and nonlinearity caused by its own structure but also the fast time variability caused by the high-speed flight. Accordingly, it is difficult to obtain an accurate model for GHVs. In this paper, a RBFNN-based control strategy is proposed to facilitate controller design. Concretely speaking, the functionsfi and gi with i=1,3 possess unknown structures, which will be approximated by NN. And the resulting approximation error will be eliminated by resorting to the design of adaptive laws.With the introduction of reference trajectory of states, the tracking errors are defined as(27)ε1=x1−x1d,ε2=x2−x2d,ε3=x3−x3d−ξ1tanhχ1,where
(28)χ̇1=cosh2χ1ξ1−ζ1tanhχ1+satu1−u1,χ10=0,with the designed parameters satisfying ξ1,ζ1>0, xid being the designed trajectory.Remark 7.
In [23], the auxiliary system and the transformed error are defined as χ̇=−ζχ+satu−u and ε=x−xd−ξχ, respectively. However, this solution is effective only if the auxiliary variable χ is bounded, while the dynamics (27) and (28) employs the hyperbolic tangent function to construct a bounded compensation term ζtanhχ, so that the conservatism of above assumption is relaxed.Taking the time derivative of equations and substituting Equation (25) yields
(29)ε̇1=f1+g1x2−ẋ1d,ε̇2=x3−ẋ2d,ε̇3=f3+g3satu1+d1−ẋ3d+ζ1tanhχ1−satu1+u1.For the purpose of relaxing the issue of EOC, a first-order low-pass filter is introduced as(30)c2ẋ2d+x2d=x2c,x2d0=0,c3ẋ3d+x3d=x3c,x3d0=0,where x2c,x3c are both output signals and the filter parameters satisfy c2,c3>0. Resulting error variables are defined as
(31)y2=x2d−x2c,y3=x3d−x3c.Step 1.
Taking the unavailable dynamicf1 into account and using Lemma 2 yield
(32)f1=W1∗TH1x+ο1,0<ο1≤O1,W1∗TH1≤W1∗TH1≤w1h1,where h1=H1 and w1≥W1∗T.
The compensation signal is designed as(33)φ̇1=−ϑ1φ1−ϑ2tanhφ1−g1ε1y2+0.5y22φ1+y2,where ϑ1,ϑ2 are both positive constants.
Subsequently, the adaptive laws and the virtual command are designed as(34)w^̇1=α1ε1h1−r1w^1,(35)R^̇1=β1ε1−r2R^1,(36)x2c=1g1ẋ1d−k1ε1−k2tanhε1−w^1h1tanhε1−R^1tanhε1+l1φ1,in which R1=O1; w^1 and R^1 denote estimations of w1 and R1, respectively; and parameters α1,β1,r1,r2,k1,k2,l1 are all designed as positive constants.Remark 8.
Different from the conventional algorithm of RBFNN, MLP algorithms select the upper bound of the weight matrix norm as the online updating parameter. Hence, only a single parameter is estimated rather than the whole weight matrix, so that the excessive occupation of the computational resource is relaxed significantly.To show the stability ofε1, the Lyapunov function (LF) is chosen as
(37)F1=12ε12+12φ12.DifferentiatingF1 and using Equations (33) and (40) yield
(38)Ḟ1=ε1ε̇1+φ1φ̇1=ε1f1+g1ε2+x2d−ẋ1d+φ1−ϑ1φ1−ϑ2tanhφ1−g1ε1y2+0.5y22φ1+y2=ε1f1+g1ε1ε2+g1ε1y2+ε1−k1ε1−k2tanhε1−w^1h1tanhε1−R^1tanhε1+l1φ1−ϑ1φ12−ϑ2φ1tanhφ1−g1ε1y2−12y22+φ1y2≤ε1f1+g1ε1ε2−k1ε12−k2ε1tanhε1−w^1h1ε1tanhε1−R^1ε1tanhε1+ε12+14l12φ12−ϑ1φ12−ϑ2φ1tanhφ1+12φ12.According to Lemma2, one has
(39)−k2ε1tanhε1≤−k2ε1+k2ϖ,−w^1h1ε1tanhε1≤−w^1h1ε1+w^1h1ϖ,−R^1ε1tanhε1≤−R^1ε1+R^1ϖ,−ϑ2φ1tanhφ1≤−ϑ2φ1+ϑ2ϖ.Substituting Equation (39) into Equation (38) leads to
(40)Ḟ1≤ε1W1∗TH1x+ο1+g1ε1ε2−k1−1ε12−k2ε1+k2ϖ−w^1h1ε1+w^1h1ϖ−R^1ε1+R^1ϖ−ϑ1−14l12−1φ12−ϑ2φ1+ϑ2ϖ≤ε1w1h1+ε1R1+g1ε1ε2−k1−1ε12−k2ε1+k2ϖ−w^1h1ε1+h12w^122+ϖ22−R^1ε1+R^122+ϖ22−ϑ1−14l12−12φ12−ϑ2φ1+ϑ2ϖ=ε1w~1h1+ε1R~1−k1−1−12g¯1ε12+12g¯1ε22−k2ε1−ϑ1−14l12−12φ12−ϑ2φ1+h12w^122+R^122+ϖ2+k2ϖ+ϑ2ϖ.Step 2.
For the purpose of the stabilization ofε2, the virtual command x3d is given as
(41)x3c=−k3ε2−k4ε2tanhε2+ẋ2d+l2φ2,where k3, k4, and l2 are all positive parameters.
The filter compensation signal is expressed as(42)φ̇2=−ϑ3φ2−ϑ4tanhφ2−ε2y3+0.5y32φ2+y3,where ϑ3,ϑ4 are both positive parameters.
The second LF is selected, and its derivatives are presented as(43)F2=12ε22+12φ22,(44)Ḟ2=ε2ε̇2+φ2φ̇2=ε2ε3+x3d−ẋ2d+φ2−ϑ3φ2−ϑ4tanhφ2−12φ2y32+y3=ε2ε3+ε2y3−k3ε22−k4ε2tanhε2−g1ε1ε2+ε2l2φ2−ϑ3φ22−ϑ4φ2tanhφ2−ε2y3−12y32+φ2y3≤ε2ε3−k3ε22−k4ε2tanhε2−g1ε1ε2+ε22+14l22φ22−ϑ3φ22−ϑ4φ2tanhφ2+12φ22.
It can be derived from Lemma2 that
(45)−k4ε1tanhε2≤−k4ε2+k4ϖ,−ϑ4φ2tanhφ2≤−ϑ4φ2+ϑ4ϖ.
Substituting the result into Equation (44), it obtains
(46)Ḟ2≤ε2ε3−k3−1−12g¯1ε22+12g¯1ε12−k4ε2+k4ϖ−ϑ3−14l22−12φ22−ϑ4φ2+ϑ4ϖ.Step 3.
In this part, the uncertain dynamicf3 is approximated via RBFNN, which is shown as
(47)f3=W3∗TH3x+ο3,0<ο3≤O3,(48)W3∗TH3≤W3∗TH3≤w3h3,where h3=H3 and w3≥W3∗T.
Owing to the effect of external disturbances, it is defined according to Assumptions3 and 5 that R3=g¯3um+D1+O3+ζ1−um. Afterward, the adaptive laws and the control input are given as
(49)w^̇3=α3ε3h3−r3w^3,(50)R^̇3=β3ε3−r4R^3,(51)u1=1g3−k5ε3−k6ε3ε3+ρ1−w^3h3ε3ε3+ρ2−R^3ε3ε3+ρ3+ẋ3d,(52)ρ1=b11+k6,ρ2=b21+w^3h3,ρ3=b31+R^3,where parameters α3,β3,r3,r4,k5,k6,ρ1,ρ2,ρ3 are all positive constants.Remark 9.
Considering the controller (51), several ingredients should to be emphasized as follows: (i) The term −k5ε3 and the term −k6ε3/ε3+ρ1 guarantee the finite time convergence of ε3. Meanwhile, the term −w^3h3ε3/ε3+ρ2 with the construction (49) is utilized to estimate the weight parameter of NN, and the term −R^3ε3/ε3+ρ3 is for unknown lumped disturbances. (ii) The introduction of variables εii=1,2,3 not only leads to the attenuation of chattering phenomenon but also improves the controllability of steady-state errors. This point will be proved in subsequent derivations.Consequently, the third LF and its derivative are presented as(53)F3=12ε32,Ḟ3=ε3ε̇3=ε3f3+g3satu1+d1−ẋ3d+ζ1tanhχ1−satu1+u1=ε3f3−k5ε32−k6ε32/ε3+ρ1−w^3h3ε32/ε3+ρ2−R^3ε32/ε3+ρ3−ε2ε3+ε3g3satu1+d1+ζ1tanhχ1−satu1.Taking Equations (47) and (48) into account, it leads to
(54)Ḟ3=ε3W3∗TH3x+ο3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3g3satu1+d1+ζ1tanhχ1−satu1≤ε3w3h3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3g3satu1+d1+O3+ζ1tanhχ1−satu1≤ε3w3h3−k5ε32−k6ε32ε3+ρ1−w^3h3ε32ε3+ρ2−R^3ε32ε3+ρ3−ε2ε3+ε3R3.Some of the items in Equation (54) are scaled as
(55)−k6ε32ε3+ρ1=−k6ε3+k6ρ1×ε3ε3+ρ1=−k6ε3+k6b11+k6×ε3ε3+ρ1≤−k6ε3+b1,−w^3h3ε32ε3+ρ2=−w^3h3ε3+w^3h3ρ2×ε3ε3+ρ2=−w^3h3ε3+w^3h3b21+w^3h3×ε3ε3+ρ2≤−w^3h3ε3+b2,−R^3ε32ε3+ρ3=−R^3ε3+R^3ρ3×ε3ε3+ρ3=−R^3ε3+R^3b31+R^3×ε3ε3+ρ3≤−R^3ε3+b3.With the substitution of these results, Equation (54) becomes
(56)Ḟ3≤ε3w3h3−k5ε32−k6ε3+b1−w^3h3ε3+b2−R^1ε3+b3−ε2ε3+ε3R3≤ε3w~3h3+ε3R~3−k5−12ε32+12ε22−k6ε3+b1+b2+b3.


## 3.2. Velocity Control

As shown in Section2, the closed-loop system for velocity tracking control is expressed as
(57)V̇=fv+satu2+d2,with
(58)fv=Tcosθ−γ−Dm−gsinγ−u2.According to Lemma2, the unavailable state fv can be written as
(59)fv=Wv∗THvx+οv,0<οv≤Ov,(60)Wv∗THv≤Wv∗THv≤wvhv,where hv=Hv and wv≥Wv∗T.The velocity tracking errorev is introduced as
(61)εv=V−Vd−ζ2tanhχ2,where
(62)χ̇2=cosh2χ2ξ2−ζ2tanhχ2+satu2−u2,χ20=0,with the parameters satisfying ξ2,ζ2>0.Differentiatingεv along Equation (61) and using Equation (57) yield
(63)ε̇v=fv+d2−V̇d+ζ2tanhχ2+u2.It is defined thatRv=D2+Ov+ζ1. Subsequently, the adaptive laws and the control input are given as
(64)w^̇v=αvεvhv−r5w^v,(65)R^̇v=βvεv−r6R^v,(66)u2=−k7εv−k8εvεv+ρ4−w^vhvεvεv+ρ5−R^vεvεv+ρ6−V̇d,(67)ρ4=b41+k8,ρ5=b51+w^vhv,ρ6=b61+R^v,where the parameters αv,βv,r5,r6,k7,k8,b4,b5,b6 are all designed as positive constants.To illustrate the convergence of the velocity tracking error, the LF is selected as(68)Fv=12εv2.DifferentiatingFv along (78) yields
(69)Ḟv=εvε̇v=εvfv+d2−V̇d+ζ2tanhχ2+u2=εvfv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+ζ2tanhχ2.With the substitution of Equations (59) and (60), it is further derived as
(70)Ḟv=εvWv∗THvx+οv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+ζ2tanhχ2≤εvwvhv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvd2+Ov+ζ2tanhχ2≤εvwvhv−k7εv2−k8εv2εv+ρ4−w^vhvεv2εv+ρ5−R^vεv2εv+ρ6+εvRv.There holds the inequations that(71)−k8εv2εv+ρ4=−k8εv+k8ρ4εvεv+ρ4=−k8εv+k8b41+k8×εvεv+ρ4≤−k8εv+b4,−w^vhvεv2εv+ρ5=−w^vhvεv+w^vhvρ5εvεv+ρ5=−w^vhvεv+w^vhvb51+w^vhv×εvεv+ρ5≤−w^vhvεv+b5,−R^vεv2εv+ρ6=−R^vεv+R^vρ6εvεv+ρ6=−R^vεv+R^vb61+R^v×εvεv+ρ6≤−R^vεv+b6.Substituting the results into Equation (70), one obtains
(72)Ḟv≤εvwvhv−k7εv2−k8εv+b4−w^vhvεv+b5−R^vεv+b6+εvRv≤εvw~vhv+εvR~v−k7εv2−k8εv+b4+b5+b6.


## 4. Stability Analysis

In this section, the stability of the developed controller is demonstrated via Lyapunov-based analysis. It is concluded that the control system is capable of maintaining global finite-time convergence. Furthermore, the altitude and velocity tracking errors of GHV will converge to a tiny region containing the origin in finite time.Theorem 10.
For the GHV dynamics (25) and (57) with Assumptions 3 and 5, the global finite-time convergence is guaranteed under controllers (36), (41), (51), and (66); the adaptive laws (34) and(35), (49) and (50), and (64) and (65); the compensation signals (33) and (42); and the auxiliary systems (28) and (62).Proof.
The LF is constructed as(73)F4=F1+F2+F3+Fv+12α1w~12+12α3w~32+12αvw~v2+12β1R~12+12β3R~32+12βvR~v2.
Taking the time derivation and substituting above results, it has:(74)Ḟ4=Ḟ1+Ḟ2+Ḟ3+Ḟv−1α1w~1w^̇1−1α3w~3w^̇3−1αvw~vw^̇v−1β1R~1R^̇1−1β3R~3R^̇3−1βvR~vR^̇v≤−k1−1−g¯1ε12−k3−1−g¯1ε22−k5−1ε32−k7εv2−ϑ1−14l12−12φ12−ϑ3−14l22−12φ22−k2ε1−k4ε2−k6ε3−k8εv−ϑ2φ1−ϑ4φ2+h12w^122+R^122+r1w^1w~1+r3w^3w~3+r5w^vw~v+r2R^1R~1+r4R^3R~3+r6R^vR~v+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6≤−k1−1−g¯1ε12−k3−1−g¯1ε22−k5−1ε32−k7εv2−ϑ1−14l12−12φ12−ϑ3−14l22−12φ22−w~12−w~32−w~v2−R~12−R~32−R~v2+w~12+h12w^122+r1w^1w~1+w~32+r3w^3w~3+w~v2+r5w^vw~v+R~12+R^122+r2R^1R~1+R~32+r4R^3R~3+R~v2+r6R^vR~v+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6.
Take some of the items as quadratic functions to obtain their maximum:(75)w~12+h12w^122+r1w^1w~1=w1−w∧12+h12w^122+r1w^1w1−w^1=−r1−1−h122w^12+r1−2w1w^1+w12≤r12−2h12w124r1−4−2h12,w~32+r3w^3w~3=w3−w∧32+r3w^3w3−w^3=−r3−1w^32+r3−2w3w^3+w32≤r32w324r3−1,w~v2+r5w^vw~v≤r52wv24r5−1,R~12+R^122+r2R^1R~1≤r22−2R124r2−6,R~32+r4R^3R~3≤r42R324r4−1,R~v2+r6R^vR~v≤r62Rv24r6−1.
With the substitution of these results, one has(76)Ḟ4≤−k1−1−g¯1ε12−k3−1−g¯1ε22−k5−1ε32−k7εv2−ϑ1−14l12−12φ12−ϑ3−14l22−12φ22−w~12−w~32−w~v2−R~12−R~32−R~v2+r12−2h12w124r1−4−2h12+r32w324r3−1+r52wv24r5−1+r22−2R124r2−6+r42R324r4−1+r62Rv24r6−1+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6≤−κ1F4+Δ1,where
(77)κ1=min2k1−1−g¯1,2k3−1−g¯1,2k5−1,2k7,2ϑ1−12l12−1,2ϑ3−12l22−1,2α1,2α3,2αv,2β1,2β3,2βv,(78)Δ1=r12−2h12w124r1−4−2h12+r32w324r3−1+r52wv24r5−1+r22−2R124r2−6+r42R324r4−1+r62Rv24r6−1+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6.
The result indicates that the designed controller is asymptotically stable. Therefore, as long as the initial conditions are bounded, the estimation errors must have uniform ultimate boundedness. Thus, there exist constantsw¯i,i=1,3,v, satisfying w¯i>w^i and w¯i>wi. R¯i,i=1,3,v, follow the same principle, i.e.,
(79)w¯i−w^i>wi−w^i=w~i,i=1,3,v,R¯i−R^i>Ri−R^i=R~i,i=1,3,v.
Rebuild the LF as(80)F5=F1+F2+F3+Fv+1α1w¯1−w∧12+1α3w¯3−w∧32+1αvw¯v−w∧v2+1β1R¯1−R∧12+1β3R¯3−R∧32+1βvR¯v−R∧v2.
DifferentiatingF5 along Equation (80) yields
(81)Ḟ5=Ḟ1+Ḟ2+Ḟ3+Ḟv−2α1w¯1−w^1w^̇1−2α3w¯3−w^3w^̇3−2αvw¯v−w^vw^̇v−2β1R¯1−R^1R^̇1−2β3R¯3−R^3R^̇3−2βvR¯v−R^vR^̇v≤−k1−1−g¯1ε12−k3−1−g¯1ε22−k5−1ε32−k7εv2−ϑ1−14l12−12φ12−ϑ3−14l22−12φ22−k2ε1−k4ε2−k6ε3−k8εv−ϑ2φ1−ϑ4φ2−ε1h1w¯1−w^1−ε3h3w¯3−w^3−εvhvw¯v−w^v−ε1R¯1−R^1−ε3R¯3−R^3−εvR¯v−R^v+2r1w¯1−w^1w^1+2r3w¯3−w^3w^3+2r5w¯v−w^vw^v+2r2R¯1−R^1R^1+2r4R¯3−R^3R^3+2r6R¯v−R^vR^v+h12w^122+R^122+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6.
In particular, the results hold as(82)2r1w¯1−w^1w^1+h12w^122=−2r1−h122w^12+2r1w¯1w^1≤2r12w¯124r1−h12,2r3w¯3−w^3w^3=−2r3w^32+2r3w¯3w^3≤r3w¯322,2r5w¯v−w^vw^v≤r5w¯v22,2r2R¯1−R^1R^1+R^122≤2r22R¯124r2−1,2r4R¯3−R^3R^3≤r4R¯322,2r6R¯v−R^vR^v≤r6R¯v22.
Substituting these results into Equation (81), it leads to
(83)Ḟ5≤−k2ε1−k4ε2−k6ε3−k8εv−ϑ2φ1−ϑ4φ2−ε1h1w¯1−w^1−ε3h3w¯3−w^3−εvhvw¯v−w^v−ε1R¯1−R^1−ε3R¯3−R^3−εvR¯v−R^v+2r12w¯124r1−h12+r3w¯322+r5w¯v22+2r22R¯124r2−1+r4R¯322+r6R¯v22+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6≤−κ2F5+Δ2,where
(84)κ2=min2k2,2k4,2k6,2k8,2ϑ2,2ϑ4,α1ε1h1,α3ε3h3,αvεvhv,β1ε1,β3ε3,βvεv,(85)Δ2=2r12w¯124r1−h12+r3w¯322+r5w¯v22+2r22R¯124r2−1+r4R¯322+r6R¯v22+ϖ2+k2ϖ+k4ϖ+ϑ2ϖ+ϑ4ϖ+b1+b2+b3+b4+b5+b6.
As a result, it is indicated that all the tracking errors and estimation errors converge into a tiny region in finite time. The proof of Theorem10 is completed.Remark 11.
Consequently, it is necessary to make a guiding explanation for the selection of controller parameters from a theoretical perspective. Considering Equations (83), (84) and (85), the overall feedback gain depends on ki, which indicates that the larger ki leads to the faster stabilization rate, and accordingly, the required control torque will increase. On the other hand, the design parameters bi play a significant role in steady-state error. Obviously, if the value of bi is small, the steady-state error of the closed-loop system will decrease, while the chattering phenomenon of the actuator will increase. Therefore, the best choice is to compromise the two indicators and determine the values of these parameters, so that the controller possesses a satisfactory overall performance.


## 5. Numerical Simulations

In this section, a simulation example is provided to demonstrate the stability and effectiveness of the designed controller. According to the dynamics (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16) and (17) of GHV, the initial values and desired trajectories are given to facilitate the implementation of the tracking controller, while various performances will be simultaneously recorded and exhibited. Necessarily, the initial values of system states are set as Table 1, and the designed parameters of the control algorithm are shown in Table 2.Table 1
Initial values of states.
ItemValueUnitV15060ft/sh110000ftγ0radα1.6325π/180radq0rad/sTable 2
Parameters for the designed control algorithm.
SectionParametersLow-pass filterε1=10,ε2=10Compensation signalsϑ1=10,ϑ2=0.5Auxiliary systemξ1=2,ζ1=0.01ξ2=2,ζ2=0.01Controllerkh=1.3,khh=13,k1=0.8,k2=0.2,l1=0.01,ϑ1=10,ϑ2=0.5,ϑ3=10,ϑ4=0.5,α1=50,r1=0.04,β1=10,r2=2,k3=1.5,k4=0.2,k5=2,k6=0.5,α3=0.005,r3=10,β3=0.05,r4=0.5,b1=b2=b3=0.001,ξ1=2,ζ1=0.01,ξ2=2,ζ2=0.01,αv=0.001,r5=10,βv=0.001,r6=10,k7=0.6,k8=0.1,b4=b5=b6=0.001Subsequently, to test the trajectory tracking performance and robustness against disturbance, the reference altitude and velocity signals, the lumped external disturbances, and the saturation constraint of actuator are defined as(86)Vd=15110,hd=110100,d1=0.01∗sin0.1∗t,d2=0.01∗cos0.1∗t,u1m=0.2rad,u2m=0.9.Consequently, the main results are given in Figures1–10.Figure 1
Trajectory tracking error of altitude of GHV.Figure 2
Trajectory tracking error of velocity of GHV.Figure 3
The control input of altitude subsystem.Figure 4
The control input of velocity subsystem.Figure 5
The estimation ofR^1.Figure 6
The estimation ofR^3.Figure 7
The estimation ofR^v.Figure 8
The estimation ofw^1.Figure 9
The estimation ofw^3.Figure 10
The estimation ofw^v.To start with, curves of trajectory tracking errors are given in Figures1 and 2, which indicate that the altitude and velocity tracking missions are completed within 10 s. Undoubtedly, the finite-time controller shows great superiority in contrast with ordinary asymptotic convergence. Afterwards, the feedback input signals of the closed-loop system are shown in Figures 3 and 4. Practically speaking, the altitude control torque and velocity control torque converges into a tiny region containing origin within 10 s. It is noteworthy that the saturation nonlinearity of actuators is well accommodated during the control process. What is more, Figures 5–7 give the curves of estimated values R^1, R^3, and R^v, which converge to a constant in finite time. The adaptive parameters of RBFNNs are addressed in Figures 8–10. Obviously, estimations of the norm of the weight matrix will always remain bounded under the proposed method. To conclude, performances of the proposed controller satisfy the design requirements.


## 6. Conclusion

This paper investigates the trajectory tracking control of GHVs exposed to system uncertainties and input saturation constraints. On the basis of the longitudinal model of GHVs, the command filter-based adaptive neural controller is constructed to guarantee the finite-time stability of the closed-loop system. Specially, considering the defect of EOC associated with a conservative backstepping design, the low-pass filters are introduced and compensation signals are designed to avoid the repeat derivation on virtual command. Meanwhile, the MLP algorithm-based function approximation is utilized to cope with the inaccurate model information, in which the tedious analytical computation is simplified significantly. The input nonlinearity caused by actuator saturation is addressed via a modified auxiliary system. Finally, the rigorous mathematical analysis and simulation results are provided to confirm the validity of the proposed architecture.



---

*Source: 8852339-2020-11-09.xml*