# Novel Prescribed Performance Control Scheme for Flexible Hypersonic Flight Vehicles with Nonaffine Dynamics and Neural Approximation

Authors: Yong Liu; Gang Li; Yuchen Li; Yahui Wu
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

This study develops a novel neural-approximation-based prescribed performance controller for flexible hypersonic flight vehicles (HFVs). Firstly, a new prescribed performance mechanism is exploited, which develops new performance functions guaranteeing velocity and altitude tracking errors with small overshoots. Compared with the existing prescribed performance mechanism, it has better preselected transient and steady-state performance. Then, the nonaffine model of HFV is decomposed into a velocity subsystem and an altitude subsystem. A prescribed performance-based proportional-integral controller is designed in the velocity subsystem. In the altitude subsystem, the model is expressed as a nonaffine pure feedback form, and control inputs are derived from neural approximations. In order to reduce the amount of computation, only one neural network approximator is used to approximate the subsystem uncertainties, and an advanced regulation algorithm is applied to the devise adaptive law for neural estimation. At the same time, the complex design process of back-stepping can be avoided. Finally, numerical simulation results are presented to verify the efficiency of the design.


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## Body

## 1. Introduction

Hypersonic flight vehicles (HFVs) fly in near space and fly at speeds greater than five times the speed of sound, including aerospace vehicles with horizontal takeoff and landing, reentry vehicles, hypersonic cruise missiles, and many other types of flight vehicles [1–3]. Compared with existing spacecraft, near space hypersonic flight vehicles can reach global targets in a short time and have great potential in military and civilian applications. Designing a rational control system is not only one of the core technologies of HFVs but also a challenging subject. During the high-speed flight of HFVs, the drastic changes of aerodynamic and aerothermal characteristics will change many parameters of the vehicles, so it is very important to study the steady-state performance of the control system. At the same time, under the constraints of narrow flight corridors, improving the transient performance of the system is also one of the problems to be solved.Most of the research results of control systems for HFVs focus on the longitudinal model of HFV due to the fact that HFVs themselves are characterized by fast time variation, strong nonlinearity, strong coupling, and uncertainty, and the longitudinal model is complex enough. In order to save fuel and keep the control system stable, HFVs should try to avoid lateral maneuver [4]. Meanwhile, many advanced methods are used in control systems for HFVs, including robust control [5, 6], sliding mode control [7, 8], back-stepping control [2, 6], and intelligent control.In [9], a self-scheduled robust decoupling control law is designed for HFVs; the simulation results verify the reliability of the method. Aiming at the parameter perturbation and external disturbance existing in HFVs, a robust controller is designed in [10], which combines with nonlinear disturbance observer and back-stepping control method. However, as the order of the system increases, the differential expansion problem occurs. In [11], a dynamic surface robust controller is designed for the input constraints and parameter uncertainties of HFV. According to the longitudinal model of HFV, [12] introduces a nonlinear disturbance observer and designs a novel integrated sliding mode controller. When the system has external disturbances, chattering, and model uncertainties, it can ensure stable tracking. In [13, 14], in order to achieve quasicontinuous and continuous switching of control inputs, a high-order sliding mode control scheme is studied. In [15], a fuzzy back-stepping control law is proposed. Through the back-stepping method and the fuzzy logic system, stable tracking is achieved. Under strict assumptions, the altitude dynamics of HFVs can be rewritten as a strict-feedback system. And with the new back-stepping control scheme, only the final control law is required to be implemented in the new back-stepping control method. In [16], the longitudinal model of HFV is expressed by the Takagi–Sugeno (T-S) fuzzy system, and a H2/H∞ tracking control law is designed. The above works are mainly aimed at the stability performance of HFV system. If we limit the transient performance of the system, it is necessary to adjust the controller parameters instantaneously. However, due to the numerous and complex parameters in the controller, it is difficult to ensure that the multiple performance indices of the system can meet the prescribed requirements at the same time.Bechlioulis et al. first propose a prescribed performance control strategy [17, 18]. The main idea of the prescribed performance control scheme is to introduce the performance function to standardize corresponding indices of the system. By constructing the error conversion function, an original “constrained” system is transformed into an equivalent “unconstrained” one. In [19], considering a class of nonlinear systems with unknown state variables, a fuzzy state observer is introduced to reconstruct the unknown state variables, and a prescribed performance fuzzy back-stepping controller is designed. Similarly, [20] integrates prescribed performance with high-order extended state observer via dynamic surface control, which can estimate unknown state variables and eliminate involving errors. But in [17–20], the performance function needs to know the sign of the initial error. In [21], a prescribed performance control strategy is applied to the HFV control system; the model is simplified to a strict-feedback formulation. However, the design process of the control law is too cumbersome, and the initial value of the performance function is required to be large enough. In view of the above, a novel prescribed performance control scheme guaranteeing the tracking error with a small, even zero, overshoot is proposed. And the scheme is applied to the HFV model with nonaffine dynamics. The special contributions are summarized as follows:
(1)
A novel prescribed performance function is proposed. Compared with [21, 22], the proposed prescribed performance mechanism has better transient performance, which guarantees that a small, even zero, overshoot can be imposed on tracking errors(2)
Compared with [23], the prescribed performance mechanism proposed in this paper does not simplify the key nonaffine dynamics(3)
In this paper, the total system uncertainties are approximated by a neural network approximator, and the adaptive law is designed by the norm estimation approach [24]. Compared with [25], the computational complexity is greatly reduced.


## 2. Problem Formulation

### 2.1. HFV Model Description

Considering the longitudinal model of HFV, Bolender and Doman propose theFirst Principle model [26]. Based on the First Principle model, Parker et al. neglect some weak coupling and slow dynamics in the model and establish a control-oriented longitudinal parameter fitting model of HFV [27].
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)δ1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)δ2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyywhere(8)δ1=1+ψ~1Iyy,δ2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ.ϕf⋅ and ϕa⋅ are structural mode shapes [26]. The geometry and force map of HFV model is shown in Figure 1.Figure 1
Geometry and force map of HFV model.The longitudinal model of HFV includes rigid body states and flexible states. The rigid body states areV,h,γ,θ,Q, wherein, V and h represent the flight velocity and altitude and γ and θ are the attitude angles of HFV, representing the flight path angle and the pitch angle, respectively. Q represents the pitch rate, m is the mass of the HFV, r is the radial distance from the center of the earth, Iyy is the moment of inertia about the pitch axis, and η1 and η2 are the flexible states. T, D, L, M, N1, and N2 represent the thrust, drag force, lift force, pitching moment, first generalized force, and second generalized force, respectively. Their fitting form is shown in
(9)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=12ρ¯V2,ρ¯=ρ¯0exph0−hhs,whereρ¯ represents the average air density and q¯ represents the dynamic pressure of HFV. h0 represents nominal altitude and hs represents the altitude constant. ρ¯0 represents the air density at h0. For more detailed definitions of the model variables and coefficients, we can refer to [26, 27]. Φ and δe represent control inputs; furthermore, it is observed from the longitudinal model that the control inputs Φ and δe do not occur explicitly in (1), (2), (3), (4), (5), (6), and (7). Since there is no actual actuator to control the two flexible states, and the two flexible states are also unmeasurable, in the controller, the flexible states are treated as unknown disturbances.Remark 1.
It can be seen from Equation (9) that the parameter fitting form of the drag force D contains δe2, so from the perspective of the control inputs, the HFV model is nonaffine, if it is simply to simplify the nonaffine dynamics into affine dynamics, which will cause the loss of key dynamics, and make the simplified affine model control law partially invalid or a control failure.


### 2.2. Novel Prescribed Performance Mechanism

The primary task of the prescribed performance control mechanism is to design the prescribed performance function and then construct the error conversion function to ensure that the tracking erroret can converge to an adjustable residual set with the expected convergence time and the maximum overshoot less than a designed value. The traditional form of performance function is shown in [17, 28]
(10)ρt=ρ0−ρ∞e−lt+ρ∞.Among them,l∈R+, ρ0∈R+, and ρ∞∈R+ are parameters to be designed. According to the performance requirements, tracking error et needs to satisfy
(11)−m¯ρt<et<ρt,e0>0,−ρt<et<m¯ρt,e0<0,wherem¯ is the parameter to be designed and 0≤m¯≤1. According to the above analysis, in fact, we need to know the sign of e0 a priori, and according to the sign, we can choose the appropriate conditions in Equation (11), but the initial value of the tracking error is generally difficult to obtain. Therefore, in the process of control law design and stability analysis, a variety of cases need to be considered. The relationship between tracking error et and performance function ρt is clearly illustrated in Figure 2.Figure 2
Graphical illustration of the traditional prescribed performance definition (11): (a) e0>0; (b) e0<0.In this paper, we propose a novel formulation of performance as follows:(12)ƛLt<et<ƛUt.The novel performance functionsƛLt and ƛUt are constructed as
(13)ƛLt=signe0−ιLsinhδt+σ−ιLρ∞,ƛUt=signe0+ιUsinhδt+σ+ιUρ∞.Taking the time derivative of (13),
(14)ƛ̇Lt=−δsigne0−ιLcoshδt+σsinhδt+σ2,ƛ̇Ut=−δsigne0+ιUcoshδt+σsinhδt+σ2,whereιL∈R+, ιU∈R+, δ∈R+, σ∈R+, and ρ∞∈R+ are parameters to be designed.Lemma 2 (see [17]).
If the smoothing functionρt is a monotonically decreasing positive function, and limt→∞ρt=ρ∞, then ρt is a prescribed performance function.
According to Lemma2,
(15)ƛL0=signe0−ιLsinhσ−ιLρ∞>−ιLρ∞,ƛL∞=signet−ιLsinhδt+σ−ιLρ∞=−ιLρ∞,ƛU0=signe0+ιUsinhσ+ιUρ∞>ιUρ∞,ƛU∞=signet+ιUsinhδt+σ+ιUρ∞=ιUρ∞.The new prescribed performance functions proposed in this paper satisfies Lemma2. The performance constraints on tracking errors are shown in Figure 3.Remark 3.
Since the traditional prescribed performance function must obtain the initial error sign a priori, this disadvantage limits the operability of prescribed performance control. It can be seen from Figure3 that the new prescribed performance functions proposed in this paper do not depend on the initial error; meanwhile, ƛLt and ƛUt can change the shape of the functions according to the different signs of e0. By selecting appropriate parameters, et can converge to the prescribed performance with a small overshoot.Figure 3
Graphical illustration of the novel prescribed performance definition (12): (a) e0>0; (b) e0=0; (c) e0<0.


## 2.1. HFV Model Description

Considering the longitudinal model of HFV, Bolender and Doman propose theFirst Principle model [26]. Based on the First Principle model, Parker et al. neglect some weak coupling and slow dynamics in the model and establish a control-oriented longitudinal parameter fitting model of HFV [27].
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)δ1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)δ2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyywhere(8)δ1=1+ψ~1Iyy,δ2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ.ϕf⋅ and ϕa⋅ are structural mode shapes [26]. The geometry and force map of HFV model is shown in Figure 1.Figure 1
Geometry and force map of HFV model.The longitudinal model of HFV includes rigid body states and flexible states. The rigid body states areV,h,γ,θ,Q, wherein, V and h represent the flight velocity and altitude and γ and θ are the attitude angles of HFV, representing the flight path angle and the pitch angle, respectively. Q represents the pitch rate, m is the mass of the HFV, r is the radial distance from the center of the earth, Iyy is the moment of inertia about the pitch axis, and η1 and η2 are the flexible states. T, D, L, M, N1, and N2 represent the thrust, drag force, lift force, pitching moment, first generalized force, and second generalized force, respectively. Their fitting form is shown in
(9)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=12ρ¯V2,ρ¯=ρ¯0exph0−hhs,whereρ¯ represents the average air density and q¯ represents the dynamic pressure of HFV. h0 represents nominal altitude and hs represents the altitude constant. ρ¯0 represents the air density at h0. For more detailed definitions of the model variables and coefficients, we can refer to [26, 27]. Φ and δe represent control inputs; furthermore, it is observed from the longitudinal model that the control inputs Φ and δe do not occur explicitly in (1), (2), (3), (4), (5), (6), and (7). Since there is no actual actuator to control the two flexible states, and the two flexible states are also unmeasurable, in the controller, the flexible states are treated as unknown disturbances.Remark 1.
It can be seen from Equation (9) that the parameter fitting form of the drag force D contains δe2, so from the perspective of the control inputs, the HFV model is nonaffine, if it is simply to simplify the nonaffine dynamics into affine dynamics, which will cause the loss of key dynamics, and make the simplified affine model control law partially invalid or a control failure.


## 2.2. Novel Prescribed Performance Mechanism

The primary task of the prescribed performance control mechanism is to design the prescribed performance function and then construct the error conversion function to ensure that the tracking erroret can converge to an adjustable residual set with the expected convergence time and the maximum overshoot less than a designed value. The traditional form of performance function is shown in [17, 28]
(10)ρt=ρ0−ρ∞e−lt+ρ∞.Among them,l∈R+, ρ0∈R+, and ρ∞∈R+ are parameters to be designed. According to the performance requirements, tracking error et needs to satisfy
(11)−m¯ρt<et<ρt,e0>0,−ρt<et<m¯ρt,e0<0,wherem¯ is the parameter to be designed and 0≤m¯≤1. According to the above analysis, in fact, we need to know the sign of e0 a priori, and according to the sign, we can choose the appropriate conditions in Equation (11), but the initial value of the tracking error is generally difficult to obtain. Therefore, in the process of control law design and stability analysis, a variety of cases need to be considered. The relationship between tracking error et and performance function ρt is clearly illustrated in Figure 2.Figure 2
Graphical illustration of the traditional prescribed performance definition (11): (a) e0>0; (b) e0<0.In this paper, we propose a novel formulation of performance as follows:(12)ƛLt<et<ƛUt.The novel performance functionsƛLt and ƛUt are constructed as
(13)ƛLt=signe0−ιLsinhδt+σ−ιLρ∞,ƛUt=signe0+ιUsinhδt+σ+ιUρ∞.Taking the time derivative of (13),
(14)ƛ̇Lt=−δsigne0−ιLcoshδt+σsinhδt+σ2,ƛ̇Ut=−δsigne0+ιUcoshδt+σsinhδt+σ2,whereιL∈R+, ιU∈R+, δ∈R+, σ∈R+, and ρ∞∈R+ are parameters to be designed.Lemma 2 (see [17]).
If the smoothing functionρt is a monotonically decreasing positive function, and limt→∞ρt=ρ∞, then ρt is a prescribed performance function.
According to Lemma2,
(15)ƛL0=signe0−ιLsinhσ−ιLρ∞>−ιLρ∞,ƛL∞=signet−ιLsinhδt+σ−ιLρ∞=−ιLρ∞,ƛU0=signe0+ιUsinhσ+ιUρ∞>ιUρ∞,ƛU∞=signet+ιUsinhδt+σ+ιUρ∞=ιUρ∞.The new prescribed performance functions proposed in this paper satisfies Lemma2. The performance constraints on tracking errors are shown in Figure 3.Remark 3.
Since the traditional prescribed performance function must obtain the initial error sign a priori, this disadvantage limits the operability of prescribed performance control. It can be seen from Figure3 that the new prescribed performance functions proposed in this paper do not depend on the initial error; meanwhile, ƛLt and ƛUt can change the shape of the functions according to the different signs of e0. By selecting appropriate parameters, et can converge to the prescribed performance with a small overshoot.Figure 3
Graphical illustration of the novel prescribed performance definition (12): (a) e0>0; (b) e0=0; (c) e0<0.


## 3. Preliminaries

### 3.1. Error Transformation

Since it is cumbersome to design the control law directly for the prescribed performance functions, the functions need to be transformed. Define the transformed error as(16)μt=lnς¯t1−ς¯t,whereς¯t=et−ƛLt/ƛUt−ƛLt.
(17)eμt=ς¯t1−ς¯t.Theorem 4.
If the transformed errorμt is bounded, then et satisfies Equation (12).Proof.
The inverse of Equation (16) isAfter converting Equation (17), we can get
(18)ς¯t=eμt1+eμt.Sinceμt is bounded, there must be a bounded constant μM∈R+ such that μt≤μM. Equation (18) can be changed to
(19)0<e−μM1+e−μM<ς¯t<eμM1+eμM<1.Sinceς¯t=et−ƛLt/ƛUt−ƛLt, there will be
(20)0<et−ƛLtƛUt−ƛLt<1.That is,(21)ƛLt<et<ƛUt.Remark 5.
As can be seen from [17], if we can ensure that μt is bounded, then the tracking error et can meet the performance requirements. If the appropriate values are chosen for ƛLt and ƛUt, it is guaranteed that et has satisfactory dynamic performance and steady-state accuracy. In what follows, the controller will be explored using the transformed error μt instead of the tracking error et.


### 3.2. Neural Network Approximation

The RBF neural network (NN) has three layers: input layer, hidden layer, and output layer. The activation function of neurons in the hidden layer is the radial basis function. The array operation units that make up the hidden layer are called the hidden layer nodes. The RBF neural network can represent an input-to-output mapping, which is(22)y=WTϕX,whereX=x1,x2,⋯,xnT∈Rn is the input vector, W=w1,w2,⋯,wvT∈Rv represents the weight vector, and the output of the hidden layer is ϕX, ϕX=ϕ1X,ϕ2X,⋯,ϕvXT∈Rv. Each hidden layer contains a central vector c; ϕjX is the output of the j-th neuron of the hidden layer and can be expressed as
(23)ϕjX=exp−X−cj22bj2,j=1,2,⋯,v,wherebj represents the width of the j-th neuron Gaussian function and b=b1,b2,⋯,bvT∈Rv,
(24)c=c11⋯c1v⋮⋱⋮cn1⋯cnv.n and v are the dimensions of the input vector and the number of nodes, respectively. For any nonlinear continuous function FX, there must be an ideal weight vector W∗=w1∗,w2∗,⋯,wv∗T∈Rv, so that
(25)FX=W∗ThX+ε,ε≤εM,whereε∈R represents the approximation error of NN and εM∈R+ represents the upper bound of the approximation error. As long as we choose a large enough v, εM can be arbitrarily small [29]. However, the online computational load will increase with a large v; it should be considered in the controller design [30].


### 3.3. Nussbaum-Type Function

Lemma 6 (see [31]).
For functionℏϑ, if the following properties hold
(26)limβ0→∞sup1β0∫0β0ℏϑdζ=+∞,limβ0→∞inf1β0∫0β0ℏϑdζ=−∞.
Then, the functionℏϑ is a Nussbaum-type function.Lemma 7 (see [32]).
DefineZ0t≥0 on 0,t1 and ℏϑ, if inequality holds
(27)Z0t≤∫0tƛ0ℏϑ+1ϑ̇τdτ+H0,
whereƛ0 is nonzero and H0 is a suitable constant, then ℏϑ, ∫0tƛ0ℏϑ+1ϑ̇τdτ, and H0 are all bounded on 0,t1.


## 3.1. Error Transformation

Since it is cumbersome to design the control law directly for the prescribed performance functions, the functions need to be transformed. Define the transformed error as(16)μt=lnς¯t1−ς¯t,whereς¯t=et−ƛLt/ƛUt−ƛLt.
(17)eμt=ς¯t1−ς¯t.Theorem 4.
If the transformed errorμt is bounded, then et satisfies Equation (12).Proof.
The inverse of Equation (16) isAfter converting Equation (17), we can get
(18)ς¯t=eμt1+eμt.Sinceμt is bounded, there must be a bounded constant μM∈R+ such that μt≤μM. Equation (18) can be changed to
(19)0<e−μM1+e−μM<ς¯t<eμM1+eμM<1.Sinceς¯t=et−ƛLt/ƛUt−ƛLt, there will be
(20)0<et−ƛLtƛUt−ƛLt<1.That is,(21)ƛLt<et<ƛUt.Remark 5.
As can be seen from [17], if we can ensure that μt is bounded, then the tracking error et can meet the performance requirements. If the appropriate values are chosen for ƛLt and ƛUt, it is guaranteed that et has satisfactory dynamic performance and steady-state accuracy. In what follows, the controller will be explored using the transformed error μt instead of the tracking error et.


## 3.2. Neural Network Approximation

The RBF neural network (NN) has three layers: input layer, hidden layer, and output layer. The activation function of neurons in the hidden layer is the radial basis function. The array operation units that make up the hidden layer are called the hidden layer nodes. The RBF neural network can represent an input-to-output mapping, which is(22)y=WTϕX,whereX=x1,x2,⋯,xnT∈Rn is the input vector, W=w1,w2,⋯,wvT∈Rv represents the weight vector, and the output of the hidden layer is ϕX, ϕX=ϕ1X,ϕ2X,⋯,ϕvXT∈Rv. Each hidden layer contains a central vector c; ϕjX is the output of the j-th neuron of the hidden layer and can be expressed as
(23)ϕjX=exp−X−cj22bj2,j=1,2,⋯,v,wherebj represents the width of the j-th neuron Gaussian function and b=b1,b2,⋯,bvT∈Rv,
(24)c=c11⋯c1v⋮⋱⋮cn1⋯cnv.n and v are the dimensions of the input vector and the number of nodes, respectively. For any nonlinear continuous function FX, there must be an ideal weight vector W∗=w1∗,w2∗,⋯,wv∗T∈Rv, so that
(25)FX=W∗ThX+ε,ε≤εM,whereε∈R represents the approximation error of NN and εM∈R+ represents the upper bound of the approximation error. As long as we choose a large enough v, εM can be arbitrarily small [29]. However, the online computational load will increase with a large v; it should be considered in the controller design [30].


## 3.3. Nussbaum-Type Function

Lemma 6 (see [31]).
For functionℏϑ, if the following properties hold
(26)limβ0→∞sup1β0∫0β0ℏϑdζ=+∞,limβ0→∞inf1β0∫0β0ℏϑdζ=−∞.
Then, the functionℏϑ is a Nussbaum-type function.Lemma 7 (see [32]).
DefineZ0t≥0 on 0,t1 and ℏϑ, if inequality holds
(27)Z0t≤∫0tƛ0ℏϑ+1ϑ̇τdτ+H0,
whereƛ0 is nonzero and H0 is a suitable constant, then ℏϑ, ∫0tƛ0ℏϑ+1ϑ̇τdτ, and H0 are all bounded on 0,t1.


## 4. Controller Design

In what follows, we would present a prescribed performance control scheme with nonaffine dynamics and neural approximation, based on the transformed error (16), which leads to a simple controller, capable of guaranteeing the satisfaction of the constraint μt≤μM and the boundedness of all other closed-loop signals.


### 4.1. Velocity Subsystem

The control goal of this section is to achieve stable tracking ofV⟶Vref by designing an appropriate prescribed performance controller of velocity subsystem.Since the velocity subsystem controller is relatively simple, based on the idea of [32], we design a prescribed performance proportional-integral (PI) controller without estimation parameters.Velocity tracking error is defined as(28)V~=V−Vref.Select novel prescribed performance functionsƛLVt and ƛUVt(29)ƛLVt=signV~0−ιLVsinhδVt+σV−ιLVρ∞V,ƛUVt=signV~0+ιUVsinhδVt+σV+ιUVρ∞V,whereιLV∈R+, ιUV∈R+, δV∈R+, σV∈R+, and ρ∞V∈R+ are parameters to be designed.Define transformed errorμVt(30)μVt=lnς¯Vt1−ς¯Vt,whereς¯Vt=et−ƛUVt/ƛUVt−ƛLVt.Design the PI control law as(31)Φ=−λV1μVt−λV2∫0tμVtdτ.λV1∈R+ and λV2∈R+ are parameters to be designed. According to the PI controller, it can be seen that μVt is bounded. From the conclusion of Theorem 4, if μVt is bounded, then the velocity tracking error V~ satisfies
(32)ƛLVt<V~<ƛUVt.By selecting the appropriateƛLVt and ƛUVt, V~ can have good dynamic performance and steady-state accuracy.


### 4.2. Altitude Subsystem

In this section, we will design a prescribed performance controller for altitude subsystems (2), (3), (4), and (5) such that h tracks its reference trajectory href with the altitude tracking error satisfying the preselected transient performance.We define the altitude tracking error as(33)h~=h−href.We select the prescribed performance functions of altitude subsystem as(34)ƛLht=signh~0−ιLhsinhδht+σh−ιLhρ∞h,ƛUht=signh~0+ιUhsinhδht+σh+ιUhρ∞h.In Equation (34), ιLh∈R+, ιUh∈R+, δh∈R+, σh∈R+, and ρ∞h∈R+ are parameters to be designed. Transformed error μht is defined as
(35)μht=lnς¯ht1−ς¯ht.whereς¯ht=et−ƛUht/ƛUht−ƛLht.Through feedback transformation, the control objective of the altitude subsystem is transformed intoγ⟶γd by selecting appropriate feedback control input δe [33].The reference trajectory ofγ is chosen as
(36)γd=arcsin−λhμht+ḣref+ρ̇hth~/ρhtV,whereλh∈R+ is the parameter to be designed, ρht=sinh−1δht+σh+ρ∞h, and ρ̇t=−δhcoshδht+σh/sinhδht+σh2.Lemma 8 (see [32]).
Ifγ⟶γd, then the dynamic response of μht is
(37)λhμ̇ht+μht=0.
Then,μht must be bounded.
We further conclude that the prescribed performance ofh~ can be guaranteed according to Theorem 4. And the control goal of subsystem becomes γ⟶γd.
We define(38)χ1=γ,χ2=θ,χ3=Q.
Express the other parts of the altitude subsystem (Equations (3), (4), and (5)) as nonaffine forms as follows:
(39)χ̇1=Γ1χ1,χ2,δe,χ̇2=Q,χ̇3=Γ2χ1,χ2,χ3,Φ,δe,
whereΓ1χ1,χ2,δe and Γ2χ1,χ2,χ3,Φ,δe are completely unknown continuous functions; in the following section, we will transform the original nonaffine models (3), (4), and (5) into a norm output feedback formulation.
We defineϖ1=χ1=γ and ϖ2=ϖ̇1=Γ1χ1,χ2,δe; applying (39), the time derivative of ϖ2 is derived as
(40)ϖ̇2=∂Γ1χ1,χ2,δe∂χ1χ̇1+∂Γ1χ1,χ2,δe∂χ2χ̇2+∂Γ1χ1,χ2,δe∂δeδ̇e=∂Γ1χ1,χ2,δe∂χ1Γ1χ1,χ2,δe+∂Γ1χ1,χ2,δe∂χ2χ3+∂Γ1χ1,χ2,δe∂δeδ̇e≜ψχ1,χ2,χ3,δe.
Then, we defineϖ3=ϖ̇2=ψχ1,χ2,χ3,δe; substituting (39) and (40), the time derivative of ϖ3 is derived as
(41)ϖ̇3=∂ψχ1,χ2,χ3,δe∂χ1χ̇1+∂ψχ1,χ2,χ3,δe∂χ2χ̇2+∂ψχ1,χ2,χ3,δe∂χ3χ̇3+∂ψχ1,χ2,χ3,δe∂δeδ̇e=∂ψχ1,χ2,χ3,δe∂χ1Γ1χ1,χ2,δe+∂ψχ1,χ2,χ3,δe∂χ2χ3+∂ψχ1,χ2,χ3,δe∂χ3Γ2χ1,χ2,χ3,Φ,δe+∂ψχ1,χ2,χ3,δe∂δeδ̇e≜ϒχ,Φ,δe,with χ=χ1,χ2,χ3T∈R+.
Therefore, according to Equation (39), we can get a pure feedback nonaffine model.
(42)ϖ̇1=ϖ2,ϖ̇2=ϖ3,ϖ̇3=ϒχ,Φ,δe,
whereϒχ,Φ,δe is a continuous unknown function.In order to facilitate the next step of controller design, the implicit function theorem is derived.Lemma 9 (implicit function theorem).
If the implicit functionGϖ,σ∈Rm and σ∈Rn satisfy the following conditions:
(1)
Gϖ,σ=0 is continuous in the region D⊂Rm×n with P0ϖ0,σ0 as the interior point(2)
Gϖ0,σ0=0(3)
∂G/∂ϖϖ,σ and ∂G/∂σϖ,σ are continuous in region D(4)
∂G/∂σϖ0,σ0≠0
In the regionD of P0, if Gϖ,σ=0, it can uniquely obtain a function σ0=g0ϖ0 defined on the neighborhood H⊂Rm and get Gϖ,g0ϖ0=0.Remark 10.
If the implicit function meets the above 4 conditions,σ can be expressed as a continuous differentiable function of ϖ, i.e., σ0=g0ϖ0. And then, Gϖ,σ can also be viewed as a continuous differentiable function of ϖ.Assumption 11.
It is assumed that∂ϒχ,δe/∂δe≠0.Remark 12.
Assumption11 imposes the global controllability condition on Equation (42) and also is the satisfying condition of Lemma 9. Compared with previous studies [25, 34, 35], the proposed scheme based on the Nussbaum-type function does not need to guarantee that the control gain ∂ϒχ,δe/∂δe is strictly positive. The formulation of Equation (42) is more concise than Equation (39), and there is no need of complicated recursive design process of back-stepping and the virtual control law with a first-order filter, which greatly simplifies the complexity of the control law.
Define the tracking errorγ~ and the error function fE as shown in
(43)γ~=γ−γd=ϖ1−γd,fE=ddt+ℓ3∫0tγ~dτ=γ~¨+3ℓγ~̇+3ℓ2γ~+ℓ3∫0tγ~dτ,
whereℓ∈R+ is the parameter to be designed.
The time derivative offE is derived as
(44)ḟE=γ~⃛+3ℓγ~¨+3ℓ2γ~̇+ℓ3∫0tγ~dτ=ϒχ,δe−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~=ϒχ,δe−ϒχ,δe∗+ϒχ,δe∗−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~.
Based on Assumption11 and Lemma 9, there must be a δe∗ satisfying ϒχ,δe∗−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~=0, then Equation (44) can be changed to
(45)ḟE=ϒχ,δe−ϒχ,δe∗.
We introduce the mean value theorem below.Lemma 13 (mean value theorem).
If the functionfx,y satisfies the following conditions:
(1)
fx,y has a derivative at each point of an open set Rn×a,b(2)
fx,y is continuous at both endpoints y=a and y=b
Then, there is a pointξ∈a,b such that
(46)ḟx,ξ=fx,b−fx,ab−a.
According to Lemma13, we convert Equation (45) into
(47)ḟE=ϒχ,δe−ϒχ,δe∗=Ξχ,δe∗δe−δe∗,
whereΞ⋅=Ξχ,δe∗=∂ϒχ,ς/∂ς≠0 and ς=ϑhδe+1−ϑhδe∗,ϑh∈0,1.
Then, Equation (47) can be converted to
(48)ḟE=Ξχ,δe∗δe−Ξχ,δe∗δe∗.
We employ the NN to approachδe∗, the input vector is Xh=γ,θ,QT∈R3, and the ideal weight vector is Wh∗=wh1∗,wh2∗,⋯,whv1∗T∈Rv1. The output form of the hidden layer is the same as Equation (23). εh∈R and εhM∈R+ represent the approximation error and the upper bound of the approximation error, respectively. NN approximator can be expressed as
(49)δe∗=Wh∗ΤϕhXh+εh,εh≤εhM.
We defineφ=Wh∗2 and choose the following control law:
(50)δe=ℏϑκhfE+fEφ^ϕhTXhϕhXh2,ℏϑ=expϑ2cosπϑ2,ϑ=κhfE2+fE2φ^ϕhTXhϕhXh2,
whereκh is a parameter to be designed. φ^ is the estimation of φ, and the adaptive law is designed as
(51)φ^̇=ωh2fE2ϕhTXhϕhXh−2φ^,
whereωh∈R+ is the parameter to be designed.Remark 14.
Based on [36], we utilize the minimal learning parameter to define φ^. In Equation (51), an overlarge ωh in the adaptive law will depress the tracking performance at the transient, which should be designed appropriately. However, unlike [20, 36], the controller does not require high-order extended state observer with dynamic surface control; therefore, it reduces the amount of calculations.


### 4.3. Stability Analysis of Altitude Subsystem

Theorem 15.
We consider the closed-loop altitude subsystem of HFV, under the premise of Assumption11, consisting of nonaffine plant (42), control law (50), and adaptive law (51), and then, all the signals involved are bounded.Proof.
We define the estimation error as(52)φ~=φ^−φ.
Select the following Lyapunov function as(53)Lh=fE22Ξχ,δe∗+φ~22ωh.
Taking time derivative along (53), and substituting Equations (48), (49), and (51),
(54)L̇h=fEḟEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+φ~φ^̇ωh=Ξχ,δe∗δe−δe∗fEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+φ~ωhωh2fE2ϕhTXhϕhXh−2φ^=Ξχ,δe∗δe−WhΤϕhXh+εhMfEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh+φ~2fE2ϕhTXhϕhXh=ΘδefE−ΘWhΤϕhXhfE+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh+φ~2fE2ϕhTXhϕhXh,
whereΘ=Ξχ,δe∗/Ξχ,δe∗∈−1,1.
Substituting Equation (50) and Equation (52) into Equation (54), we can get
(55)L̇h=ΘℏϑκhfE+φ^2fEϕhTXhϕhXhfE−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+fE22φ^−φϕhTXhϕhXh−2φ^φ~ωh=ΘℏϑκhfE2+φ^2fE2ϕhTXhϕhXh−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+fE22φ^ϕhTXhϕhXh−fE22φϕhTXhϕhXh−2φ^φ~ωh.
According to Equation (55),
(56)L̇h=−κhfE2+κhfE2+fE22φ^ϕhTXhϕhXh+ΘℏϑκhfE2+φ^2fE2ϕhTXhϕhXh−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh−fE22φϕhTXhϕhXh=−κh+Ξ̇χ,δe∗2Ξχ,δe∗2fE2+1+Θℏϑϑ−ΘfEWhΤϕhXh+ΘεhMfE−2φ^φ~ωh−fE22φϕhTXhϕhXh.
Owing to(57)φ~2+2φ~φ^−φ~+φ2=φ~2+2φ~φ+φ2=φ~+φ2≥0,
we obtain(58)2φ~φ^≥φ~2−φ2,
by further noting that(59)−ΘfEWhΤϕhXh=±fEWhΤϕhXh≤fE22WhΤϕhXh2+12=fE22Wh2ϕhXh2+12=fE22φϕhTXhϕhXh+12.
According to Young’s inequality [37], the following inequality holds
(60)L̇h≤−κh+Ξ̇χ,δe∗2Ξχ,δe∗2fE2+1+Θℏϑϑ+fE22φϕhTXhϕhXh+12+fE24+εhM2−fE22φϕhTXhϕhXh−φ~2−φ2ωh=−κh+Ξ̇χ,δe∗2Ξχ,δe∗2−14fE2−φ~2ωh+φ2ωh+1+Θℏϑϑ+12+εhM2.
Let(61)κh+Ξ̇χ,δe∗2Ξχ,δe∗2−14>0.
Then, we have(62)L̇h≤−ιLh+1+Θℏϑϑ+12+εhM2+φ2ωh,with, ι=min2Ξχ,δe∗κh+Ξ̇χ,δe∗/2Ξχ,δe∗2−1/4,2.
WhenL̇h≤0 if Lh≥1+Θℏϑϑ+1/2+εhM2/ι.
Using the multiplication of Equation (62) by eιt, it leads to
(63)ddtLheιt≤−ιLh+1+Θℏϑϑeιt+12+εhM2+φ2ωheιt.
Integrating Equation (63) over 0,t, we have
(64)Lht≤e−ιt∫0t1+Θℏϑϑeιτdτ+1ι12+εhM2+φ2ωh+Lh0−1ι12+εhM2+φ2ωhe−ιt.
Based on Lemma7, we know that Lh is bounded. Meanwhile, φ~ and fE are bounded. Since the polynomial s+r3 is Hurwitz, the tracking error γ~ is also bounded. By choosing an amply large κh and sufficiently small ωh, the errors γ~, φ~, and fE can be arbitrarily small. Therefore, the closed-loop control system is locally uniformly asymptotically stable, and this is the end of the proof.Remark 16.
The proposed control scheme only requires one NN to approximate the uncertaintyδe∗. By introducing the norm estimation approach [24], only one learning parameter φ^ is included in the NN approximator. Compared with [25], we do not require adjusting the elements of the weight vector online and needing the complex recursive processes based on back-stepping control either. Therefore, the amount of calculations in this paper is greatly reduced.


## 4.1. Velocity Subsystem

The control goal of this section is to achieve stable tracking ofV⟶Vref by designing an appropriate prescribed performance controller of velocity subsystem.Since the velocity subsystem controller is relatively simple, based on the idea of [32], we design a prescribed performance proportional-integral (PI) controller without estimation parameters.Velocity tracking error is defined as(28)V~=V−Vref.Select novel prescribed performance functionsƛLVt and ƛUVt(29)ƛLVt=signV~0−ιLVsinhδVt+σV−ιLVρ∞V,ƛUVt=signV~0+ιUVsinhδVt+σV+ιUVρ∞V,whereιLV∈R+, ιUV∈R+, δV∈R+, σV∈R+, and ρ∞V∈R+ are parameters to be designed.Define transformed errorμVt(30)μVt=lnς¯Vt1−ς¯Vt,whereς¯Vt=et−ƛUVt/ƛUVt−ƛLVt.Design the PI control law as(31)Φ=−λV1μVt−λV2∫0tμVtdτ.λV1∈R+ and λV2∈R+ are parameters to be designed. According to the PI controller, it can be seen that μVt is bounded. From the conclusion of Theorem 4, if μVt is bounded, then the velocity tracking error V~ satisfies
(32)ƛLVt<V~<ƛUVt.By selecting the appropriateƛLVt and ƛUVt, V~ can have good dynamic performance and steady-state accuracy.


## 4.2. Altitude Subsystem

In this section, we will design a prescribed performance controller for altitude subsystems (2), (3), (4), and (5) such that h tracks its reference trajectory href with the altitude tracking error satisfying the preselected transient performance.We define the altitude tracking error as(33)h~=h−href.We select the prescribed performance functions of altitude subsystem as(34)ƛLht=signh~0−ιLhsinhδht+σh−ιLhρ∞h,ƛUht=signh~0+ιUhsinhδht+σh+ιUhρ∞h.In Equation (34), ιLh∈R+, ιUh∈R+, δh∈R+, σh∈R+, and ρ∞h∈R+ are parameters to be designed. Transformed error μht is defined as
(35)μht=lnς¯ht1−ς¯ht.whereς¯ht=et−ƛUht/ƛUht−ƛLht.Through feedback transformation, the control objective of the altitude subsystem is transformed intoγ⟶γd by selecting appropriate feedback control input δe [33].The reference trajectory ofγ is chosen as
(36)γd=arcsin−λhμht+ḣref+ρ̇hth~/ρhtV,whereλh∈R+ is the parameter to be designed, ρht=sinh−1δht+σh+ρ∞h, and ρ̇t=−δhcoshδht+σh/sinhδht+σh2.Lemma 8 (see [32]).
Ifγ⟶γd, then the dynamic response of μht is
(37)λhμ̇ht+μht=0.
Then,μht must be bounded.
We further conclude that the prescribed performance ofh~ can be guaranteed according to Theorem 4. And the control goal of subsystem becomes γ⟶γd.
We define(38)χ1=γ,χ2=θ,χ3=Q.
Express the other parts of the altitude subsystem (Equations (3), (4), and (5)) as nonaffine forms as follows:
(39)χ̇1=Γ1χ1,χ2,δe,χ̇2=Q,χ̇3=Γ2χ1,χ2,χ3,Φ,δe,
whereΓ1χ1,χ2,δe and Γ2χ1,χ2,χ3,Φ,δe are completely unknown continuous functions; in the following section, we will transform the original nonaffine models (3), (4), and (5) into a norm output feedback formulation.
We defineϖ1=χ1=γ and ϖ2=ϖ̇1=Γ1χ1,χ2,δe; applying (39), the time derivative of ϖ2 is derived as
(40)ϖ̇2=∂Γ1χ1,χ2,δe∂χ1χ̇1+∂Γ1χ1,χ2,δe∂χ2χ̇2+∂Γ1χ1,χ2,δe∂δeδ̇e=∂Γ1χ1,χ2,δe∂χ1Γ1χ1,χ2,δe+∂Γ1χ1,χ2,δe∂χ2χ3+∂Γ1χ1,χ2,δe∂δeδ̇e≜ψχ1,χ2,χ3,δe.
Then, we defineϖ3=ϖ̇2=ψχ1,χ2,χ3,δe; substituting (39) and (40), the time derivative of ϖ3 is derived as
(41)ϖ̇3=∂ψχ1,χ2,χ3,δe∂χ1χ̇1+∂ψχ1,χ2,χ3,δe∂χ2χ̇2+∂ψχ1,χ2,χ3,δe∂χ3χ̇3+∂ψχ1,χ2,χ3,δe∂δeδ̇e=∂ψχ1,χ2,χ3,δe∂χ1Γ1χ1,χ2,δe+∂ψχ1,χ2,χ3,δe∂χ2χ3+∂ψχ1,χ2,χ3,δe∂χ3Γ2χ1,χ2,χ3,Φ,δe+∂ψχ1,χ2,χ3,δe∂δeδ̇e≜ϒχ,Φ,δe,with χ=χ1,χ2,χ3T∈R+.
Therefore, according to Equation (39), we can get a pure feedback nonaffine model.
(42)ϖ̇1=ϖ2,ϖ̇2=ϖ3,ϖ̇3=ϒχ,Φ,δe,
whereϒχ,Φ,δe is a continuous unknown function.In order to facilitate the next step of controller design, the implicit function theorem is derived.Lemma 9 (implicit function theorem).
If the implicit functionGϖ,σ∈Rm and σ∈Rn satisfy the following conditions:
(1)
Gϖ,σ=0 is continuous in the region D⊂Rm×n with P0ϖ0,σ0 as the interior point(2)
Gϖ0,σ0=0(3)
∂G/∂ϖϖ,σ and ∂G/∂σϖ,σ are continuous in region D(4)
∂G/∂σϖ0,σ0≠0
In the regionD of P0, if Gϖ,σ=0, it can uniquely obtain a function σ0=g0ϖ0 defined on the neighborhood H⊂Rm and get Gϖ,g0ϖ0=0.Remark 10.
If the implicit function meets the above 4 conditions,σ can be expressed as a continuous differentiable function of ϖ, i.e., σ0=g0ϖ0. And then, Gϖ,σ can also be viewed as a continuous differentiable function of ϖ.Assumption 11.
It is assumed that∂ϒχ,δe/∂δe≠0.Remark 12.
Assumption11 imposes the global controllability condition on Equation (42) and also is the satisfying condition of Lemma 9. Compared with previous studies [25, 34, 35], the proposed scheme based on the Nussbaum-type function does not need to guarantee that the control gain ∂ϒχ,δe/∂δe is strictly positive. The formulation of Equation (42) is more concise than Equation (39), and there is no need of complicated recursive design process of back-stepping and the virtual control law with a first-order filter, which greatly simplifies the complexity of the control law.
Define the tracking errorγ~ and the error function fE as shown in
(43)γ~=γ−γd=ϖ1−γd,fE=ddt+ℓ3∫0tγ~dτ=γ~¨+3ℓγ~̇+3ℓ2γ~+ℓ3∫0tγ~dτ,
whereℓ∈R+ is the parameter to be designed.
The time derivative offE is derived as
(44)ḟE=γ~⃛+3ℓγ~¨+3ℓ2γ~̇+ℓ3∫0tγ~dτ=ϒχ,δe−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~=ϒχ,δe−ϒχ,δe∗+ϒχ,δe∗−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~.
Based on Assumption11 and Lemma 9, there must be a δe∗ satisfying ϒχ,δe∗−γ⃛d+3ℓγ~¨+3ℓ2γ~̇+ℓ3γ~=0, then Equation (44) can be changed to
(45)ḟE=ϒχ,δe−ϒχ,δe∗.
We introduce the mean value theorem below.Lemma 13 (mean value theorem).
If the functionfx,y satisfies the following conditions:
(1)
fx,y has a derivative at each point of an open set Rn×a,b(2)
fx,y is continuous at both endpoints y=a and y=b
Then, there is a pointξ∈a,b such that
(46)ḟx,ξ=fx,b−fx,ab−a.
According to Lemma13, we convert Equation (45) into
(47)ḟE=ϒχ,δe−ϒχ,δe∗=Ξχ,δe∗δe−δe∗,
whereΞ⋅=Ξχ,δe∗=∂ϒχ,ς/∂ς≠0 and ς=ϑhδe+1−ϑhδe∗,ϑh∈0,1.
Then, Equation (47) can be converted to
(48)ḟE=Ξχ,δe∗δe−Ξχ,δe∗δe∗.
We employ the NN to approachδe∗, the input vector is Xh=γ,θ,QT∈R3, and the ideal weight vector is Wh∗=wh1∗,wh2∗,⋯,whv1∗T∈Rv1. The output form of the hidden layer is the same as Equation (23). εh∈R and εhM∈R+ represent the approximation error and the upper bound of the approximation error, respectively. NN approximator can be expressed as
(49)δe∗=Wh∗ΤϕhXh+εh,εh≤εhM.
We defineφ=Wh∗2 and choose the following control law:
(50)δe=ℏϑκhfE+fEφ^ϕhTXhϕhXh2,ℏϑ=expϑ2cosπϑ2,ϑ=κhfE2+fE2φ^ϕhTXhϕhXh2,
whereκh is a parameter to be designed. φ^ is the estimation of φ, and the adaptive law is designed as
(51)φ^̇=ωh2fE2ϕhTXhϕhXh−2φ^,
whereωh∈R+ is the parameter to be designed.Remark 14.
Based on [36], we utilize the minimal learning parameter to define φ^. In Equation (51), an overlarge ωh in the adaptive law will depress the tracking performance at the transient, which should be designed appropriately. However, unlike [20, 36], the controller does not require high-order extended state observer with dynamic surface control; therefore, it reduces the amount of calculations.


## 4.3. Stability Analysis of Altitude Subsystem

Theorem 15.
We consider the closed-loop altitude subsystem of HFV, under the premise of Assumption11, consisting of nonaffine plant (42), control law (50), and adaptive law (51), and then, all the signals involved are bounded.Proof.
We define the estimation error as(52)φ~=φ^−φ.
Select the following Lyapunov function as(53)Lh=fE22Ξχ,δe∗+φ~22ωh.
Taking time derivative along (53), and substituting Equations (48), (49), and (51),
(54)L̇h=fEḟEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+φ~φ^̇ωh=Ξχ,δe∗δe−δe∗fEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+φ~ωhωh2fE2ϕhTXhϕhXh−2φ^=Ξχ,δe∗δe−WhΤϕhXh+εhMfEΞχ,δe∗−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh+φ~2fE2ϕhTXhϕhXh=ΘδefE−ΘWhΤϕhXhfE+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh+φ~2fE2ϕhTXhϕhXh,
whereΘ=Ξχ,δe∗/Ξχ,δe∗∈−1,1.
Substituting Equation (50) and Equation (52) into Equation (54), we can get
(55)L̇h=ΘℏϑκhfE+φ^2fEϕhTXhϕhXhfE−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+fE22φ^−φϕhTXhϕhXh−2φ^φ~ωh=ΘℏϑκhfE2+φ^2fE2ϕhTXhϕhXh−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2+fE22φ^ϕhTXhϕhXh−fE22φϕhTXhϕhXh−2φ^φ~ωh.
According to Equation (55),
(56)L̇h=−κhfE2+κhfE2+fE22φ^ϕhTXhϕhXh+ΘℏϑκhfE2+φ^2fE2ϕhTXhϕhXh−ΘfEWhΤϕhXh+ΘεhMfE−Ξ̇χ,δe∗2Ξχ,δe∗2fE2−2φ^φ~ωh−fE22φϕhTXhϕhXh=−κh+Ξ̇χ,δe∗2Ξχ,δe∗2fE2+1+Θℏϑϑ−ΘfEWhΤϕhXh+ΘεhMfE−2φ^φ~ωh−fE22φϕhTXhϕhXh.
Owing to(57)φ~2+2φ~φ^−φ~+φ2=φ~2+2φ~φ+φ2=φ~+φ2≥0,
we obtain(58)2φ~φ^≥φ~2−φ2,
by further noting that(59)−ΘfEWhΤϕhXh=±fEWhΤϕhXh≤fE22WhΤϕhXh2+12=fE22Wh2ϕhXh2+12=fE22φϕhTXhϕhXh+12.
According to Young’s inequality [37], the following inequality holds
(60)L̇h≤−κh+Ξ̇χ,δe∗2Ξχ,δe∗2fE2+1+Θℏϑϑ+fE22φϕhTXhϕhXh+12+fE24+εhM2−fE22φϕhTXhϕhXh−φ~2−φ2ωh=−κh+Ξ̇χ,δe∗2Ξχ,δe∗2−14fE2−φ~2ωh+φ2ωh+1+Θℏϑϑ+12+εhM2.
Let(61)κh+Ξ̇χ,δe∗2Ξχ,δe∗2−14>0.
Then, we have(62)L̇h≤−ιLh+1+Θℏϑϑ+12+εhM2+φ2ωh,with, ι=min2Ξχ,δe∗κh+Ξ̇χ,δe∗/2Ξχ,δe∗2−1/4,2.
WhenL̇h≤0 if Lh≥1+Θℏϑϑ+1/2+εhM2/ι.
Using the multiplication of Equation (62) by eιt, it leads to
(63)ddtLheιt≤−ιLh+1+Θℏϑϑeιt+12+εhM2+φ2ωheιt.
Integrating Equation (63) over 0,t, we have
(64)Lht≤e−ιt∫0t1+Θℏϑϑeιτdτ+1ι12+εhM2+φ2ωh+Lh0−1ι12+εhM2+φ2ωhe−ιt.
Based on Lemma7, we know that Lh is bounded. Meanwhile, φ~ and fE are bounded. Since the polynomial s+r3 is Hurwitz, the tracking error γ~ is also bounded. By choosing an amply large κh and sufficiently small ωh, the errors γ~, φ~, and fE can be arbitrarily small. Therefore, the closed-loop control system is locally uniformly asymptotically stable, and this is the end of the proof.Remark 16.
The proposed control scheme only requires one NN to approximate the uncertaintyδe∗. By introducing the norm estimation approach [24], only one learning parameter φ^ is included in the NN approximator. Compared with [25], we do not require adjusting the elements of the weight vector online and needing the complex recursive processes based on back-stepping control either. Therefore, the amount of calculations in this paper is greatly reduced.


## 5. Simulation Results

To clarify and verify the performance of the proposed approximation-based prescribed performance control scheme, we present simulation studies in this section. The control object is the longitudinal model of HFV. The fourth-order Runge-Kutta method is used in the simulation, and the simulation step is 0.01 s. The initial trim conditions of HFV are listed in Table1, and aerodynamic coefficients and model parameters are borrowed from [27]. Both the velocity and altitude reference inputs are given by a second-order reference model with a damping ratio of 0.9 and a natural frequency of 0.1 rad/s. The controller parameters are selected as λV1=0.3, λV2=0.8, ωh=0.05, λh=2, κh=−25, and ℓ=8. The performance functions are designed as
(65)ƛLVt=signV~0−0.5sinh0.07t+0.45−0.15,ƛUVt=signV~0+0.5sinh0.07t+0.45+0.15,ƛLht=signh~0−0.5sinh0.09t+1.3−0.05,ƛUht=signh~0+0.5sinh0.09t+1.3+0.05.Table 1
Initial trim conditions.
ItemValueUnitsV2500m/sh27000mγ0degθ1.5295degQ0deg/sη10.2857—η20.2857—The input vector of the neural network isXh=γ,θ,QT, with γ=−1°,1°, θ=0°,5°, and Q=−5°/s,5°/s. The center vectors c1 and c2 are evenly spaced in their bounds. The number of nodes in the neural network is 20. To show the superiority, the proposed novel prescribed performance control (NPC) scheme is compared with a simplified neural back-stepping control (SNBC) strategy developed in [34].Case 1.
The velocity tracks the step command with 100 m/s, and the altitude follows the step command with 100 m. To test the robustness of the proposed control strategy, all the aerodynamic coefficients are assumed to be uncertain. We define(66)C=C0,0s≤t<40s,C01+0.4sin0.05πt,40s≤t<80s,
whereC is the value of uncertain coefficient and C0 denotes the normal value of C.The obtained simulation results are depicted in Figures4–10. It is apparent from Figures 4 and 5 that, compared with SNBC, the proposed NPC guarantees that velocity and altitude tracking errors are limited to the constructed prescribed behavior bounds with small overshoot and provide better velocity and altitude tracking performance in the presence of parametric uncertainties. For both control methodologies, attitude angles, flexible states, and control inputs, μVt and μht, shown in Figures 6 and 9, are bounded and smooth (without high-frequency chattering). Figure 10 shows that φ^ is bounded. To sum up, the superiority of the explored NPC over SNBC is well proven by the simulation results.Figure 4
Velocity tracking performance of the case.Figure 5
Altitude tracking performance of the case.Figure 6
The attitude angles of the case.Figure 7
The flexible states of the case.Figure 8
The control inputs of the case.Figure 9
μVt and μht of the case.Figure 10
φ^ of the case.


## 6. Conclusions

In this paper, a novel neural approximation-based prescribed performance control scheme guaranteeing the tracking error with small, even zero, overshoot is proposed. New performance functions are exploited in the controller which can guarantee the velocity and altitude tracking errors with satisfactory transient and steady-state performance. And the new prescribed performance mechanism does not need to know the sign of the initial tracking error, and the shape of the functions can be changed according to the sign of the initial tracking error. The control scheme proposed in this paper eliminates the complex design process of back-stepping. In addition, this paper uses only one neural network and advanced norm estimation approach, which reduces the complexity and computational pressure of the algorithm. Finally, the effectiveness and superiority of the proposed control strategy are validated by simulation results.



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*Source: 8859681-2021-02-10.xml*