# Dynamic Surface Control with High-Order LESO Compensation for Near-Space Vehicle

Authors: Ouxun Li; Li Deng; Shutong Huang; Jianqi Wang; Ju Jiang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

Aimed at the flight control problem of a near-space vehicle (NSV) with uncertainties such as parameter perturbation and external disturbance, a novel dynamic surface control (DSC) method with high-order linear extended state observer (HLESO) compensation is proposed in this paper. In the proposed method, a tracking differentiator (TD) based on an inverse hyperbolic sine function is firstly designed, and the convergence of TD is also analysed with the Lyapunov stability theory. Furthermore, the numerical simulations are detailed to demonstrate the superiority of the proposed TD compared with others. Then, a dynamic surface controller based on the traditional backstepping control technique is designed, where the TD is adopted to obtain the differential signal of the virtual control law, in order to solve the “differential expansion,” and the HLESO is adopted to accurately estimate the “lumped disturbance” so as to realize the dynamic compensation of the controller and enhance the disturbance suppression ability of the system. The simulation results demonstrate the validity of the proposed method.


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## Body

## 1. Introduction

A near-space vehicle (NSV) has broad application in both military and civil areas, which has become an important development trend in the global aerospace technology [1, 2]. However, compared with traditional aircraft, NSV has the many characteristics, such as strong nonlinearity, strong coupling, model uncertainty, and being vulnerable to external interference, which makes the design of an NSV flight controller face severe challenges [3, 4]. How to avoid the adverse effects of model uncertainty and external disturbances on the control performance is one of the key problems in the design of a flight control system. Different control methods for NSV have attracted much attention, and many efforts have been carried out to deal with the robust tracking problem against uncertainties and further improve the dynamic performance, such as feedback linearization control [5], sliding mode control [6], adaptive control [7, 8], backstepping control [9–14], dynamic surface control (DSC) [15–18], and intelligent control [19, 23, 24].For the control problem of uncertainty and external disturbance, the control method must have strong disturbance rejection ability. Based on the reasonable assumption of NSV, the mathematical model of the longitudinal channel of NSV can be described as a kind of radiation-type strict feedback nonlinear system. The backstepping control method is an important nonlinear control method to deal with the unmatched uncertainties and realize multivariable decoupling control. In [9], authors present a new method for online approximation-based backstepping control in the presence of known magnitude, rate, or bandwidth constraints on the intermediate states or actuators. In [10], a command filtered backstepping approach is presented that uses adaptive function approximation to control unmanned air vehicles. In [11], a dynamic surface tracking control method based on a neural network is proposed to solve the control performance degradation caused by matched and unmatched uncertainties. In [12], a robust backstepping control method based on a nonlinear disturbance observer is proposed, and good control effect is obtained. In [13], a novel adaptive backstepping control method is proposed for a nonlinear system with mismatched uncertainties and demonstrates the efficiency and superiority of the proposed method. In [14], an inversion fault-tolerant control method is proposed for the flight control problems with uncertain parameters, external disturbance, and actuator failure, and good control effect is obtained.As an improved backstepping control method, DSC can solve the “explosion of terms” by using a filter of the synthetic input at each step of the backstepping design procedure. In [15], a neural network-based adaptive DSC is proposed for a class of uncertain nonlinear systems. In [16], an adaptive fuzzy control strategy is used to compensate for the unknown nonlinearities and modeling errors caused by changes of flight conditions. In [17, 18], a DSC technique combined with the neural network adaptive framework is presented, to design a robust longitudinal dynamics controller for a generic nonlinear air-breathing hypersonic flight vehicle model. In [19, 23], authors present a new backstepping control approach for air-breathing hypersonic vehicle (AHV) nonaffine models via neural approximation. In [24], authors deal with the design of nearly optimal tracking controllers for hypersonic flight vehicles via concise fuzzy approximations. Although the above control methods improve the control performance, they all have high complexity, which is not conducive to online calculation and implementation.The disturbance observers have been introduced into the NSV to reduce the influences of the unknown uncertainties and disturbances and enhance the robustness of the closed-loop system in the literatures. In [25], an adaptive controller is proposed by employing an event-triggered control and an extended state observer, where a simple strategy to tune the observer parameters is provided. In [26], a novel sliding mode controller based on the high-order linear extended state observer is proposed for the uncertainty and external disturbance sensitivity of the near-space vehicles. In [27], hypersonic vehicle tracking control in the presence of uncertainties and external disturbance is investigated, and a disturbance observer based on gain adaptation high-order sliding mode control is proposed. In [28], a backstepping technique with a disturbance observer is proposed for the flexible air-breathing hypersonic vehicle.This paper mainly studies the controller design of NSV with parameter perturbation and external interference. A high-order linear extended state observer (HLESO) compensation DSC method is proposed. This method is based on the combination of traditional backstepping control and DSC technique and adopts HLESO to realize dynamic compensation for system uncertainty and external disturbance. The experimental results show that the proposed control method has the advantages of high speed and high steady-state precision compared with the traditional control method. The main contributions are as follows:(1)
The tracking differentiator (TD) based on the inverse hyperbolic sine function is designed, which can solve the “differential expansion” problem of the traditional inversion control. Furthermore, a simulation study is made to test the superiority of the proposed TD compared with another TD (based on the hyperbolic sine function), traditional TD, and sliding mode differentiator (SMD)(2)
Based on the traditional backstepping control and Lyapunov stability theory, the design method of the dynamic surface controller is proposed. The virtual control quantity expressions of the velocity channel and altitude channel are provided, respectively(3)
Based on the longitudinal model of aircraft, the HLESO is designed to accurately estimate the lumped disturbance of the system, and the convergence of the disturbance observer is proven according to the Lyapunov stability theory. The numerical simulations are detailed to demonstrate under the same composite disturbance conditions the tracking errors of the proposed DSC and HLESO compensation method which are about 20 times smaller than those of the traditional methodThe remainder of this paper is organized as follows. The theoretical bases are presented in Section2, including the NSV modeling and antihyperbolic sine function characteristic analysis. The design method of the dynamic surface controller with HLESO compensation is described in Section 3. The design method and convergence of the antihyperbolic sine function tracking differentiator, dynamic surface controller, and HLESO are studied, respectively. The aircraft case simulations and discussions are provided in Section 4, followed by the conclusion in Section 5.


## 2. Preliminaries

### 2.1. NSV Model

According to [20–22], based on the assumption of the horizontal ground, the longitudinal dynamic model of NSV in a cruise phase is denoted by the following differential equations:
(1)v̇=Tcosα−Dm−μsinγr2,γ̇=L+Tsinαmv−μ−v2rcosγvr2,q̇=MyIy,α̇=q−γ,̇ḣ=vsinγ,β¨=−2ξωnβ̇−ωn2β+ωn2βc,where v, γ, q, α, h, and β are the flight velocity, track angle, pitch angle rate, attack angle, flight altitude, and engine state, respectively; m, μ, r, and Iy are the mass, gravity acceleration, radial distance from the earth’s core, and rotational inertia, respectively; βc, ωn, and ξ are the throttle setting, engine natural frequency, and damp ratio, respectively. The parameters of lift L, drag D, thrust T, and pitching moment My are modeled by [20]
(2)L=12ρv2sCL,D=12ρv2sCD,T=12ρv2sCT,Cmδe=0.0292δe−α,My=12ρv2sc¯Cm+Cmδe,CT=0.02576β,β<1,0.0224+0.00336β,β≥1,where s, c¯, and ρ are the wing area, chord length, and air density, respectively; CL, CD, CT, and Cm are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting βc and elevator deflection angle δe.Define the state vectorx=v,γ,q,α,h,β,β̇Τ, according to the nonlinear system theory and the concept of the Lie derivative, using an exact feedback linearization method for the NSV model (1). Control inputs βc and δe appear after three times and four times derivatives for the flight velocity v and the flight altitude h, respectively. The NSV linearized model can be obtained as follows:
(3)v⃛=Fv+b11βc+b12δe,h4=Fh+b21βc+b22δe,where the detailed expression of the Fv, Fh, b11, b12, b21, and b22 can be found in [20].The NSV system model has uncertainty and parameter deviation. By taking the unknown dynamics and external disturbances of the system as equivalent composite disturbances, the system dynamics can be described as(4)v⃛h4=FvFh+Bβcδe+Bd1td2t,B=b11b12b21b22,where d1t and d2t are composite disturbances for the velocity channel and altitude channel, respectively. The NSV linearized model (4) is defined under the hypotheses as follows:Assumption 1.
The matrixB is a nonsingular matrix. The track angle of NSV is very small in cruise flight status, which satisfies γ≠±π/2. Therefore, the hypothesis is reasonable.Assumption 2.
The composite disturbancesditi=1,2 and their first-order derivatives are bounded as dit≤δ, ḋit≤δ¯, where δ and δ¯ are known positive constants.


### 2.2. Antihyperbolic Sine Function

An antihyperbolic sine function is a smooth continuous function, which can be expressed as(5)arshx=lnx+x2+1.Denotearsh∙ as Θ∙; the first-order differential can be expressed as
(6)Θ̇x=1x2+1and have
(7)limx⟶∞Θ̇x=0,limx⟶0Θ̇x=1.It can be seen from equation (7) that when x⟶∞, the value of function in equation (5) is close to saturation; when x⟶0, the change rate of function value is close to 1. The Taylor expansion at x=0 of an antihyperbolic sine function can be expressed as
(8)Θx=Θx0+Θ̇xx=x0x−x0+Θ¨xx=x02!x−x02+⋯+Rnx=∑n=0∞−1n2n!4nn!22n+1x2n+1,x<1=x−13!x3+95!x5−2257!x7+⋯.It can be seen from equation (8) that in the field of x=0, when the value of x is very small, Θx≈x, the antihyperbolic sine function is a linear function. Furthermore, the antihyperbolic sine function is smooth and continuous, which can be used in the state feedback to avoid the high-frequency chattering phenomenon of sliding mode variable structure control. The system response is fast and stable, and the steady-state error is small. Based on the above characteristics of the antihyperbolic sine function, the differential tracker has better dynamic performance and ideal differential effect [29, 30].


## 2.1. NSV Model

According to [20–22], based on the assumption of the horizontal ground, the longitudinal dynamic model of NSV in a cruise phase is denoted by the following differential equations:
(1)v̇=Tcosα−Dm−μsinγr2,γ̇=L+Tsinαmv−μ−v2rcosγvr2,q̇=MyIy,α̇=q−γ,̇ḣ=vsinγ,β¨=−2ξωnβ̇−ωn2β+ωn2βc,where v, γ, q, α, h, and β are the flight velocity, track angle, pitch angle rate, attack angle, flight altitude, and engine state, respectively; m, μ, r, and Iy are the mass, gravity acceleration, radial distance from the earth’s core, and rotational inertia, respectively; βc, ωn, and ξ are the throttle setting, engine natural frequency, and damp ratio, respectively. The parameters of lift L, drag D, thrust T, and pitching moment My are modeled by [20]
(2)L=12ρv2sCL,D=12ρv2sCD,T=12ρv2sCT,Cmδe=0.0292δe−α,My=12ρv2sc¯Cm+Cmδe,CT=0.02576β,β<1,0.0224+0.00336β,β≥1,where s, c¯, and ρ are the wing area, chord length, and air density, respectively; CL, CD, CT, and Cm are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting βc and elevator deflection angle δe.Define the state vectorx=v,γ,q,α,h,β,β̇Τ, according to the nonlinear system theory and the concept of the Lie derivative, using an exact feedback linearization method for the NSV model (1). Control inputs βc and δe appear after three times and four times derivatives for the flight velocity v and the flight altitude h, respectively. The NSV linearized model can be obtained as follows:
(3)v⃛=Fv+b11βc+b12δe,h4=Fh+b21βc+b22δe,where the detailed expression of the Fv, Fh, b11, b12, b21, and b22 can be found in [20].The NSV system model has uncertainty and parameter deviation. By taking the unknown dynamics and external disturbances of the system as equivalent composite disturbances, the system dynamics can be described as(4)v⃛h4=FvFh+Bβcδe+Bd1td2t,B=b11b12b21b22,where d1t and d2t are composite disturbances for the velocity channel and altitude channel, respectively. The NSV linearized model (4) is defined under the hypotheses as follows:Assumption 1.
The matrixB is a nonsingular matrix. The track angle of NSV is very small in cruise flight status, which satisfies γ≠±π/2. Therefore, the hypothesis is reasonable.Assumption 2.
The composite disturbancesditi=1,2 and their first-order derivatives are bounded as dit≤δ, ḋit≤δ¯, where δ and δ¯ are known positive constants.


## 2.2. Antihyperbolic Sine Function

An antihyperbolic sine function is a smooth continuous function, which can be expressed as(5)arshx=lnx+x2+1.Denotearsh∙ as Θ∙; the first-order differential can be expressed as
(6)Θ̇x=1x2+1and have
(7)limx⟶∞Θ̇x=0,limx⟶0Θ̇x=1.It can be seen from equation (7) that when x⟶∞, the value of function in equation (5) is close to saturation; when x⟶0, the change rate of function value is close to 1. The Taylor expansion at x=0 of an antihyperbolic sine function can be expressed as
(8)Θx=Θx0+Θ̇xx=x0x−x0+Θ¨xx=x02!x−x02+⋯+Rnx=∑n=0∞−1n2n!4nn!22n+1x2n+1,x<1=x−13!x3+95!x5−2257!x7+⋯.It can be seen from equation (8) that in the field of x=0, when the value of x is very small, Θx≈x, the antihyperbolic sine function is a linear function. Furthermore, the antihyperbolic sine function is smooth and continuous, which can be used in the state feedback to avoid the high-frequency chattering phenomenon of sliding mode variable structure control. The system response is fast and stable, and the steady-state error is small. Based on the above characteristics of the antihyperbolic sine function, the differential tracker has better dynamic performance and ideal differential effect [29, 30].


## 3. Design of the Dynamic Surface Controller

### 3.1. Design of the Tracking Differentiator

The second-order tracking differentiator constructed by the antihyperbolic sine function can be described as(9)ẋ1t=x2t,ẋ2t=−Γ2α1∙Θβ1x1t−rt+α2∙Θβ2x2tΓ,.wherert is the input signal and x1t and x2t are the system state variables. If the parameters are Γ>0, α1>0, α2>0, β1>0, and β2>0, then for an arbitrary bounded and integrable function rt, one gets
(10)limx⟶∞∫0Tx1t−rtdt=0,T>0.Theorem 3.
Consider the following second-order system:(11)ξ̇1t=ξ2t,ξ̇2t=Ξξ1t,ξ2t.
If the solutions of system (11) satisfy
(12)limt⟶∞ξ1t=0,limt⟶∞ξ2t=0,
then, for an arbitrary bounded and integrable functionrt, the solution of the following system
(13)ẋ1t=x2t,ẋ2t=Γ2Ξx1t−rt,x2tΓsatisfies
(14)limΓ⟶∞∫0Tx1t−rtdt=0,T>0.
Proof:
For equation (11), define
(15)Ξξ1t,ξ2t=−α1∙Θβ1ξ1t−α2∙Θβ2ξ2t.
Then, equation (11) can be described as
(16)ξ̇1t=ξ2t,ξ̇2t=−α1∙Θβ1ξ1t−α2∙Θβ2ξ2t.
SinceΘ∙ is an odd function, if α1>0, β1>0, and ξ1t>0, one has α1ξ1t∙Θβ1ξ1t>0. Then,
(17)∫0ξ1α1∙Θβ1τdτ=α1ξ1t∙Θβ1δ>0,0<δ<ξ1.
Thus, a Lyapunov function can be chosen in a sufficiently small domain near the origin0,0 as follows:
(18)Lξ1t,ξ2t=∫0ξ1α1∙Θβ1τdτ+12ξ22=α1ξ1∙Θβ1δ+12ξ22>0,0<δ<ξ1.
Taking the time derivative of (18), if α2>0, β2>0, one gets
(19)L̇ξ1t,ξ2t=ξ̇1α1∙Θβ1ξ1+ξ2ξ̇2=ξ2α1∙Θβ1ξ1+ξ2−α1∙Θβ1ξ1−α2∙Θβ2ξ2=−α2ξ2∙Θβ2ξ2≤0.
It can be seen from (19) that only when ξ2=0,L̇ξ1,ξ2=0. When L̇ξ1,ξ2=0, if ξ1≠0, then ξ̇2≠0 according to (16). That is, ξ2=0 is instable if ξ1≠0. Therefore, if and only if ξi=0i=1,2, L̇ξ1,ξ2=0. According to Lyapunov’s second theorem, the system (13) is asymptotically stable at the origin 0,0 [31]. When Γ is big enough, the solution x1t of system (9) can fully approach the input signal rt in a finite time T, which means
(20)ẋ1t=x2t=ṙt.
Furthermore, the numerical simulations are detailed to demonstrate the superiority of the proposed TD compared with others.The TD from [31] is formulated as
(21)ẋ1t=x2t,ẋ2t=−R2a1∙sinhl1x1t−vt+a2∙sinhl2x2tR,where vt is the input signal and the parameters are R>0, a1>0, a2>0, l1>0, l2>0, and sinhx=ex−e−x/2.The traditional TD is described as(22)ẋ1t=x2t,ẋ2t=−R2x1t−vt+x2tR,wherevt is the input signal and the parameter is R>0.The sliding mode differentiator is described as(23)ẋ1t=−k1x1t−vt0.5signx1t−vt+x2t,ẋ2t=−k2signx1t−vt,where vt is the input signal and the parameters are k1>0 and k2>0.In the simulation, the input signal is selected asvt=1t+nt, where 1t is the unit step signal and nt is the band limited white noise of variance 0.02 and bandwidth 10. The design parameters of the aforementioned differentiators are listed in Table 1. Figure 1 presents the tracking performances of TD (9), TD (21), TD (22), and TD (23). It can be seen that the proposed TD (9) has a higher accuracy of tracking and differentiation and a faster convergence than other TD. In the same parameter conditions, TD (9) and TD (21) have roughly the same tracking accuracy and convergence speed. The differentiation peak value of TD (9) is smaller than TD (21); that is, the tracking performances of TD (9) are faster and more stable.Table 1
Values of the TD parameters.
DifferentiatorThe values of design parametersTD (9)Γ=40, α1=4, α2=4, β1=2, β2=2TD (21)R=40, a1=4, a2=4, l1=2, l2=2TD (22)R=40TD (23)k1=10, k2=5Figure 1
The tracking performances of TD: (a) tracking signal and (b) differentiation signal.
(a)(b)


### 3.2. Design of the Dynamic Surface Controller

The equations of motion are decomposed into a velocity subsystem and altitude subsystem. The block diagram of the overall control scheme of the DSC design is shown in Figure2. The controller uses the signals from reference trajectories, system states, and the estimates of the HLESO to control the NSV.Figure 2
Block diagram of the DSC design.According to the NSV model (4), considering the velocity channel, we have
(24)v1=v,v̇1=v2,v̇2=v3,v̇3=Fv+Bvϕvy,u,y=v1,By defining the error functionev=v−vd, Ev=ev,ėv,e¨v,e⃛vT, the error state equation can be described as
(25)Ev1=ev=v−vd,Ėv1=ėv=Ev2,Ėv2=e¨v=Ev3,Ėv3=e⃛v=v⃛−v⃛d,Define velocity channel error variables:(26)zv1=Ev1,zv2=Ev2−λv1Ev1,zv3=Ev3−λv2Ev1,Ev2,where the virtual control quantity λvii=1,2 is described as
(27)λv1Ev1=−ρv1zv1,λv2Ev1,Ev2=−zv1−ρv2zv2+λ̇v1Ev1,ρv2>0,ρv2>0.Forzv1 in the error variable system (26), the Lyapunov function is defined as
(28)Lv1=12zv12>0.The derivative of function (28) can be described as
(29)L̇v1=zv1∙żv1=zv1zv2+λv1Ev1.Substituting the virtual control quantity λv1Ev1 in equation (27) into (29), we have
(30)L̇v1=zv1zv2−ρv1zv1=zv1zv2−ρv1zv12.It can be seen from equation (30) that the condition for zv1 converging to 0 is L̇v1<0. Therefore, zv2 needs to converge to 0.Forzv1, zv2 in the error variable system (26), the Lyapunov function is defined as
(31)Lv2=Lv1+12zv22>0.The derivative of function (31) can be described as
(32)L̇v2=L̇v1+zv2∙żv2=zv1zv2−ρv1zv12+zv2Ėv2−λ̇v1Ev1=zv1zv2−ρv1zv12+zv2Ev3−λ̇v1Ev1=zv1zv2−ρv1zv12+zv2zv3+λv2Ev1,Ev2−λ̇v1Ev1.Substituting the virtual control quantity λv2Ev1,Ev2 in equation (27) into (32), we have
(33)L̇v2=zv1zv2−ρv1zv12+zv2zv3−zv1−ρv2zv2+λ̇v1Ev1−λ̇v1Ev1=−ρv1zv12−ρv2zv22+zv2zv3.It can be seen from equation (33) that the condition for zv1 and zv2 converging to 0 is L̇v2<0. Therefore, zv3 needs to converge to 0.Forzv1, zv2, zv3 in the error variable system (26), the Lyapunov function is defined as
(34)Lv3=Lv2+12zv32>0.The derivative of function (31) can be described as
(35)L̇v3=L̇v2+zv3∙żv3=−ρv1zv12−ρv2zv22+zv2zv3+zv3∙żv3=−ρv1zv12−ρv2zv22+zv3zv2+żv3.Let(36)żv3=−zv2−ρv3zv3,ρv3>0.Then, substituting equation (36) into (35), we can get
(37)L̇v3=−ρv1zv12−ρv2zv22−ρv3zv32<0.It can be seen from equation (37) that zv1, zv2, and zv3 also can converge to 0, so as to ensure that the speed tracking error converges to 0.Combined with (25), (26), and (36), we have
(38)v⃛=v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Then, substituting equation (38) into (24), we can get
(39)Fv+Bvϕvy,u=v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Therefore, the velocity channel control law can be described as(40)ϕvy,u=Bv−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Similarly, define the virtual control quantity of the altitude channel as(41)λh1Eh1=−ρh1zh1,λh2Eh1,Eh2=−zh1−ρh2zh2+λ̇h1Eh1,λh2Eh1,Eh2,Eh3=−zh2−ρh3zh3+λ̇h2Eh1,Eh2.After four steps of deduction,h4 can be described as
(42)h4=hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4,ρh4>0.Therefore, the altitude channel control law can be described as(43)ϕhy,u=Bh−1−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4.Combining equations (40) and (43), the dynamic surface control law of NSV can be described as
(44)U=B−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4,where the differential signal of each virtual control quantity is obtained by the tracking differentiator system (9).


### 3.3. Design of HLESO

Assumption 4.
Then-th-order derivatives of lumped disturbance ft,x,w are bounded, and ∃θ>0 have
(45)dnft,x,wdtn≤θ.
Then, according to the altitude channel of the NSV model (4), the extended n-th-order state equation can be described as
(46)ẋh1=xh2,ẋh2=xh3,ẋh3=xh4,ẋh4=Fh+Bu+dht=f+Bu,ẋh4+1=ḟ=xh4+2⋮,ẋh4+n=fn,y=xh1,where xhii=1,2,3,4 are the system state variables, xh4+ii=1,2,⋯,n are extended state variables, and f=Fh+dht is the lumped disturbance of the system. According to ESO theory, HLESO can be described as [26]
(47)x¯̇h1=x¯h2+a1ω0y−x¯h1,x¯̇h2=x¯h3+a2ω02y−x¯h1,x¯̇h3=x¯h3+a3ω03y−x¯h1,x¯̇h4=x¯h4+1+Bu+a4ω04y−x¯h1,x¯̇h4+1=x¯h4+2+a4+1ω04+1y−x¯h1,x¯̇h4+2=x¯h4+3+a4+2ω04+2y−x¯h1,⋮x¯̇h4+n−1=x¯h4+n+a4+n−1ω04+n−1y−x¯h1,x¯̇h4+n=a4+nω04+ny−x¯h1,where x¯hii=1,2,3,4 are estimation values of the system state variables, x¯h4+1 is the estimation value of the lumped disturbance of the system, x¯h4+ii=2,3,⋯,n are derivative estimation values of the lumped disturbance, aiω0ii=1,2,⋯,n+4 are LESO parameters, ω0 is the observer bandwidth, and ai=n+4!/i!n+4−i!i=1,2,⋯,n+4.


#### 3.3.1. Convergence Analysis

Define the observation error vector as(48)et=e1t,e2t,⋯,en+4t,eit=xit−x¯it,i=1,2,⋯,n+4,ηit=eiεtεn+4−i,ε=1ω0,i=1,2,⋯,n+4.Then, the error dynamics can be described as(49)η̇1t=η2−g1η1tη̇2t=η3−g2η1tη̇3t=η4−g3η1tη̇4t=η4+1−g4η1tη̇4+1t=η4+2−g4+1η1tη̇4+2t=η4+3−g4+2η1t⋮η̇4+n-1t=η4+n−g4+n-1η1tη̇n+4t=−gn+4ω1t+εdnftdtnAssumption 5.
∀ηt=η1t,η2t,⋯,ηn+4t∈ℝn+4, there are certain constants of λii=1,2,3,4, β, and the positive definite continuous differentiable functions V and W: ℝn+4⟶ℝ, which make
(50)λ1∥η∥2≤Vη≤λ2∥η∥2,λ3∥η∥2≤Wη≤λ4∥η∥2,∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1≤−Wη,∂V∂ηn+4≤βη,where ∙ is the Euclid norm.Theorem 6.
If Assumptions4 and 5 are satisfied, then ∀ε∈0,1 and tε=t0−2λ2/λ3n+4εlnε for the system equation (47), the following conclusion is true:
(51)SUPxit−x¯it≤Οεn+5−i,∀t∈tε,∞,0<ε<1.


#### 3.3.2. Proof

The derivative of the Lyapunov functionVη along system (49) can be expressed as
(52)ddtVη=∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1+∂V∂ηn+4εdnftdtn≤−Wη+εθβη≤−λ3λ2Vη+λ1λ1εθβVη.Considering the relationship betweenVη and Vη, there is
(53)ddtVη≤−λ3λ2Vη+λ12λ1εθβ.Then, we can obtain(54)ηt≤Vηλ1≤λ1Vηt0λ1e−λ3t−t02ελ2+εθβ2λ1∫t0te−λ32λ2t−t0−τdτ.Finally, the observation errors satisfy the following relationship:(55)eit=ηitεεn+4−i≤ηtεεn+4−i≤εn+4−iλ1Vηt0λ1e−λ3t−t02ελ2+θβ2λ1∫t0/εt/εe−λ3/2λ2t−t0/ε−τdτ≤Οεn+5−i,∀t∈tε,∞,tε=maxt0−2λ2λ3n+4εlnε,t0.From Theorem6 and its proof process, it can be seen that when the parameter ε is small enough, the estimation state of the designed HLESO can fully approximate the state of the system of equation (46) in the whole time domain, and the estimation errors converge to Οεn+5−i. Similarly, the HLESO applied to the velocity subsystem can effectively estimate the total disturbance.The dynamic surface controller for NSV based on HLESO can be described as(56)U=B−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3−d^1−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4−d^2,where d^1 and d^2 are estimation values of the lumped disturbance of the velocity channel and altitude channel, respectively.


## 3.1. Design of the Tracking Differentiator

The second-order tracking differentiator constructed by the antihyperbolic sine function can be described as(9)ẋ1t=x2t,ẋ2t=−Γ2α1∙Θβ1x1t−rt+α2∙Θβ2x2tΓ,.wherert is the input signal and x1t and x2t are the system state variables. If the parameters are Γ>0, α1>0, α2>0, β1>0, and β2>0, then for an arbitrary bounded and integrable function rt, one gets
(10)limx⟶∞∫0Tx1t−rtdt=0,T>0.Theorem 3.
Consider the following second-order system:(11)ξ̇1t=ξ2t,ξ̇2t=Ξξ1t,ξ2t.
If the solutions of system (11) satisfy
(12)limt⟶∞ξ1t=0,limt⟶∞ξ2t=0,
then, for an arbitrary bounded and integrable functionrt, the solution of the following system
(13)ẋ1t=x2t,ẋ2t=Γ2Ξx1t−rt,x2tΓsatisfies
(14)limΓ⟶∞∫0Tx1t−rtdt=0,T>0.
Proof:
For equation (11), define
(15)Ξξ1t,ξ2t=−α1∙Θβ1ξ1t−α2∙Θβ2ξ2t.
Then, equation (11) can be described as
(16)ξ̇1t=ξ2t,ξ̇2t=−α1∙Θβ1ξ1t−α2∙Θβ2ξ2t.
SinceΘ∙ is an odd function, if α1>0, β1>0, and ξ1t>0, one has α1ξ1t∙Θβ1ξ1t>0. Then,
(17)∫0ξ1α1∙Θβ1τdτ=α1ξ1t∙Θβ1δ>0,0<δ<ξ1.
Thus, a Lyapunov function can be chosen in a sufficiently small domain near the origin0,0 as follows:
(18)Lξ1t,ξ2t=∫0ξ1α1∙Θβ1τdτ+12ξ22=α1ξ1∙Θβ1δ+12ξ22>0,0<δ<ξ1.
Taking the time derivative of (18), if α2>0, β2>0, one gets
(19)L̇ξ1t,ξ2t=ξ̇1α1∙Θβ1ξ1+ξ2ξ̇2=ξ2α1∙Θβ1ξ1+ξ2−α1∙Θβ1ξ1−α2∙Θβ2ξ2=−α2ξ2∙Θβ2ξ2≤0.
It can be seen from (19) that only when ξ2=0,L̇ξ1,ξ2=0. When L̇ξ1,ξ2=0, if ξ1≠0, then ξ̇2≠0 according to (16). That is, ξ2=0 is instable if ξ1≠0. Therefore, if and only if ξi=0i=1,2, L̇ξ1,ξ2=0. According to Lyapunov’s second theorem, the system (13) is asymptotically stable at the origin 0,0 [31]. When Γ is big enough, the solution x1t of system (9) can fully approach the input signal rt in a finite time T, which means
(20)ẋ1t=x2t=ṙt.
Furthermore, the numerical simulations are detailed to demonstrate the superiority of the proposed TD compared with others.The TD from [31] is formulated as
(21)ẋ1t=x2t,ẋ2t=−R2a1∙sinhl1x1t−vt+a2∙sinhl2x2tR,where vt is the input signal and the parameters are R>0, a1>0, a2>0, l1>0, l2>0, and sinhx=ex−e−x/2.The traditional TD is described as(22)ẋ1t=x2t,ẋ2t=−R2x1t−vt+x2tR,wherevt is the input signal and the parameter is R>0.The sliding mode differentiator is described as(23)ẋ1t=−k1x1t−vt0.5signx1t−vt+x2t,ẋ2t=−k2signx1t−vt,where vt is the input signal and the parameters are k1>0 and k2>0.In the simulation, the input signal is selected asvt=1t+nt, where 1t is the unit step signal and nt is the band limited white noise of variance 0.02 and bandwidth 10. The design parameters of the aforementioned differentiators are listed in Table 1. Figure 1 presents the tracking performances of TD (9), TD (21), TD (22), and TD (23). It can be seen that the proposed TD (9) has a higher accuracy of tracking and differentiation and a faster convergence than other TD. In the same parameter conditions, TD (9) and TD (21) have roughly the same tracking accuracy and convergence speed. The differentiation peak value of TD (9) is smaller than TD (21); that is, the tracking performances of TD (9) are faster and more stable.Table 1
Values of the TD parameters.
DifferentiatorThe values of design parametersTD (9)Γ=40, α1=4, α2=4, β1=2, β2=2TD (21)R=40, a1=4, a2=4, l1=2, l2=2TD (22)R=40TD (23)k1=10, k2=5Figure 1
The tracking performances of TD: (a) tracking signal and (b) differentiation signal.
(a)(b)


## 3.2. Design of the Dynamic Surface Controller

The equations of motion are decomposed into a velocity subsystem and altitude subsystem. The block diagram of the overall control scheme of the DSC design is shown in Figure2. The controller uses the signals from reference trajectories, system states, and the estimates of the HLESO to control the NSV.Figure 2
Block diagram of the DSC design.According to the NSV model (4), considering the velocity channel, we have
(24)v1=v,v̇1=v2,v̇2=v3,v̇3=Fv+Bvϕvy,u,y=v1,By defining the error functionev=v−vd, Ev=ev,ėv,e¨v,e⃛vT, the error state equation can be described as
(25)Ev1=ev=v−vd,Ėv1=ėv=Ev2,Ėv2=e¨v=Ev3,Ėv3=e⃛v=v⃛−v⃛d,Define velocity channel error variables:(26)zv1=Ev1,zv2=Ev2−λv1Ev1,zv3=Ev3−λv2Ev1,Ev2,where the virtual control quantity λvii=1,2 is described as
(27)λv1Ev1=−ρv1zv1,λv2Ev1,Ev2=−zv1−ρv2zv2+λ̇v1Ev1,ρv2>0,ρv2>0.Forzv1 in the error variable system (26), the Lyapunov function is defined as
(28)Lv1=12zv12>0.The derivative of function (28) can be described as
(29)L̇v1=zv1∙żv1=zv1zv2+λv1Ev1.Substituting the virtual control quantity λv1Ev1 in equation (27) into (29), we have
(30)L̇v1=zv1zv2−ρv1zv1=zv1zv2−ρv1zv12.It can be seen from equation (30) that the condition for zv1 converging to 0 is L̇v1<0. Therefore, zv2 needs to converge to 0.Forzv1, zv2 in the error variable system (26), the Lyapunov function is defined as
(31)Lv2=Lv1+12zv22>0.The derivative of function (31) can be described as
(32)L̇v2=L̇v1+zv2∙żv2=zv1zv2−ρv1zv12+zv2Ėv2−λ̇v1Ev1=zv1zv2−ρv1zv12+zv2Ev3−λ̇v1Ev1=zv1zv2−ρv1zv12+zv2zv3+λv2Ev1,Ev2−λ̇v1Ev1.Substituting the virtual control quantity λv2Ev1,Ev2 in equation (27) into (32), we have
(33)L̇v2=zv1zv2−ρv1zv12+zv2zv3−zv1−ρv2zv2+λ̇v1Ev1−λ̇v1Ev1=−ρv1zv12−ρv2zv22+zv2zv3.It can be seen from equation (33) that the condition for zv1 and zv2 converging to 0 is L̇v2<0. Therefore, zv3 needs to converge to 0.Forzv1, zv2, zv3 in the error variable system (26), the Lyapunov function is defined as
(34)Lv3=Lv2+12zv32>0.The derivative of function (31) can be described as
(35)L̇v3=L̇v2+zv3∙żv3=−ρv1zv12−ρv2zv22+zv2zv3+zv3∙żv3=−ρv1zv12−ρv2zv22+zv3zv2+żv3.Let(36)żv3=−zv2−ρv3zv3,ρv3>0.Then, substituting equation (36) into (35), we can get
(37)L̇v3=−ρv1zv12−ρv2zv22−ρv3zv32<0.It can be seen from equation (37) that zv1, zv2, and zv3 also can converge to 0, so as to ensure that the speed tracking error converges to 0.Combined with (25), (26), and (36), we have
(38)v⃛=v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Then, substituting equation (38) into (24), we can get
(39)Fv+Bvϕvy,u=v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Therefore, the velocity channel control law can be described as(40)ϕvy,u=Bv−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3.Similarly, define the virtual control quantity of the altitude channel as(41)λh1Eh1=−ρh1zh1,λh2Eh1,Eh2=−zh1−ρh2zh2+λ̇h1Eh1,λh2Eh1,Eh2,Eh3=−zh2−ρh3zh3+λ̇h2Eh1,Eh2.After four steps of deduction,h4 can be described as
(42)h4=hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4,ρh4>0.Therefore, the altitude channel control law can be described as(43)ϕhy,u=Bh−1−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4.Combining equations (40) and (43), the dynamic surface control law of NSV can be described as
(44)U=B−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4,where the differential signal of each virtual control quantity is obtained by the tracking differentiator system (9).


## 3.3. Design of HLESO

Assumption 4.
Then-th-order derivatives of lumped disturbance ft,x,w are bounded, and ∃θ>0 have
(45)dnft,x,wdtn≤θ.
Then, according to the altitude channel of the NSV model (4), the extended n-th-order state equation can be described as
(46)ẋh1=xh2,ẋh2=xh3,ẋh3=xh4,ẋh4=Fh+Bu+dht=f+Bu,ẋh4+1=ḟ=xh4+2⋮,ẋh4+n=fn,y=xh1,where xhii=1,2,3,4 are the system state variables, xh4+ii=1,2,⋯,n are extended state variables, and f=Fh+dht is the lumped disturbance of the system. According to ESO theory, HLESO can be described as [26]
(47)x¯̇h1=x¯h2+a1ω0y−x¯h1,x¯̇h2=x¯h3+a2ω02y−x¯h1,x¯̇h3=x¯h3+a3ω03y−x¯h1,x¯̇h4=x¯h4+1+Bu+a4ω04y−x¯h1,x¯̇h4+1=x¯h4+2+a4+1ω04+1y−x¯h1,x¯̇h4+2=x¯h4+3+a4+2ω04+2y−x¯h1,⋮x¯̇h4+n−1=x¯h4+n+a4+n−1ω04+n−1y−x¯h1,x¯̇h4+n=a4+nω04+ny−x¯h1,where x¯hii=1,2,3,4 are estimation values of the system state variables, x¯h4+1 is the estimation value of the lumped disturbance of the system, x¯h4+ii=2,3,⋯,n are derivative estimation values of the lumped disturbance, aiω0ii=1,2,⋯,n+4 are LESO parameters, ω0 is the observer bandwidth, and ai=n+4!/i!n+4−i!i=1,2,⋯,n+4.


### 3.3.1. Convergence Analysis

Define the observation error vector as(48)et=e1t,e2t,⋯,en+4t,eit=xit−x¯it,i=1,2,⋯,n+4,ηit=eiεtεn+4−i,ε=1ω0,i=1,2,⋯,n+4.Then, the error dynamics can be described as(49)η̇1t=η2−g1η1tη̇2t=η3−g2η1tη̇3t=η4−g3η1tη̇4t=η4+1−g4η1tη̇4+1t=η4+2−g4+1η1tη̇4+2t=η4+3−g4+2η1t⋮η̇4+n-1t=η4+n−g4+n-1η1tη̇n+4t=−gn+4ω1t+εdnftdtnAssumption 5.
∀ηt=η1t,η2t,⋯,ηn+4t∈ℝn+4, there are certain constants of λii=1,2,3,4, β, and the positive definite continuous differentiable functions V and W: ℝn+4⟶ℝ, which make
(50)λ1∥η∥2≤Vη≤λ2∥η∥2,λ3∥η∥2≤Wη≤λ4∥η∥2,∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1≤−Wη,∂V∂ηn+4≤βη,where ∙ is the Euclid norm.Theorem 6.
If Assumptions4 and 5 are satisfied, then ∀ε∈0,1 and tε=t0−2λ2/λ3n+4εlnε for the system equation (47), the following conclusion is true:
(51)SUPxit−x¯it≤Οεn+5−i,∀t∈tε,∞,0<ε<1.


### 3.3.2. Proof

The derivative of the Lyapunov functionVη along system (49) can be expressed as
(52)ddtVη=∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1+∂V∂ηn+4εdnftdtn≤−Wη+εθβη≤−λ3λ2Vη+λ1λ1εθβVη.Considering the relationship betweenVη and Vη, there is
(53)ddtVη≤−λ3λ2Vη+λ12λ1εθβ.Then, we can obtain(54)ηt≤Vηλ1≤λ1Vηt0λ1e−λ3t−t02ελ2+εθβ2λ1∫t0te−λ32λ2t−t0−τdτ.Finally, the observation errors satisfy the following relationship:(55)eit=ηitεεn+4−i≤ηtεεn+4−i≤εn+4−iλ1Vηt0λ1e−λ3t−t02ελ2+θβ2λ1∫t0/εt/εe−λ3/2λ2t−t0/ε−τdτ≤Οεn+5−i,∀t∈tε,∞,tε=maxt0−2λ2λ3n+4εlnε,t0.From Theorem6 and its proof process, it can be seen that when the parameter ε is small enough, the estimation state of the designed HLESO can fully approximate the state of the system of equation (46) in the whole time domain, and the estimation errors converge to Οεn+5−i. Similarly, the HLESO applied to the velocity subsystem can effectively estimate the total disturbance.The dynamic surface controller for NSV based on HLESO can be described as(56)U=B−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3−d^1−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4−d^2,where d^1 and d^2 are estimation values of the lumped disturbance of the velocity channel and altitude channel, respectively.


## 3.3.1. Convergence Analysis

Define the observation error vector as(48)et=e1t,e2t,⋯,en+4t,eit=xit−x¯it,i=1,2,⋯,n+4,ηit=eiεtεn+4−i,ε=1ω0,i=1,2,⋯,n+4.Then, the error dynamics can be described as(49)η̇1t=η2−g1η1tη̇2t=η3−g2η1tη̇3t=η4−g3η1tη̇4t=η4+1−g4η1tη̇4+1t=η4+2−g4+1η1tη̇4+2t=η4+3−g4+2η1t⋮η̇4+n-1t=η4+n−g4+n-1η1tη̇n+4t=−gn+4ω1t+εdnftdtnAssumption 5.
∀ηt=η1t,η2t,⋯,ηn+4t∈ℝn+4, there are certain constants of λii=1,2,3,4, β, and the positive definite continuous differentiable functions V and W: ℝn+4⟶ℝ, which make
(50)λ1∥η∥2≤Vη≤λ2∥η∥2,λ3∥η∥2≤Wη≤λ4∥η∥2,∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1≤−Wη,∂V∂ηn+4≤βη,where ∙ is the Euclid norm.Theorem 6.
If Assumptions4 and 5 are satisfied, then ∀ε∈0,1 and tε=t0−2λ2/λ3n+4εlnε for the system equation (47), the following conclusion is true:
(51)SUPxit−x¯it≤Οεn+5−i,∀t∈tε,∞,0<ε<1.


## 3.3.2. Proof

The derivative of the Lyapunov functionVη along system (49) can be expressed as
(52)ddtVη=∑i=1n+4∂V∂ηiηi+1−giη1−∂V∂ηn+4gn+4η1+∂V∂ηn+4εdnftdtn≤−Wη+εθβη≤−λ3λ2Vη+λ1λ1εθβVη.Considering the relationship betweenVη and Vη, there is
(53)ddtVη≤−λ3λ2Vη+λ12λ1εθβ.Then, we can obtain(54)ηt≤Vηλ1≤λ1Vηt0λ1e−λ3t−t02ελ2+εθβ2λ1∫t0te−λ32λ2t−t0−τdτ.Finally, the observation errors satisfy the following relationship:(55)eit=ηitεεn+4−i≤ηtεεn+4−i≤εn+4−iλ1Vηt0λ1e−λ3t−t02ελ2+θβ2λ1∫t0/εt/εe−λ3/2λ2t−t0/ε−τdτ≤Οεn+5−i,∀t∈tε,∞,tε=maxt0−2λ2λ3n+4εlnε,t0.From Theorem6 and its proof process, it can be seen that when the parameter ε is small enough, the estimation state of the designed HLESO can fully approximate the state of the system of equation (46) in the whole time domain, and the estimation errors converge to Οεn+5−i. Similarly, the HLESO applied to the velocity subsystem can effectively estimate the total disturbance.The dynamic surface controller for NSV based on HLESO can be described as(56)U=B−1−Fv+v⃛d+λ̇v2Ev1,Ev2−zv2−ρv3zv3−d^1−Fh+hd4+λ̇h2Eh1,Eh2,Eh3−zh3−ρh4zh4−d^2,where d^1 and d^2 are estimation values of the lumped disturbance of the velocity channel and altitude channel, respectively.


## 4. Simulations and Discussion

In this section, the numerical simulations are detailed to demonstrate the effectiveness of the proposed dynamic surface control method, the mathematical model of the NSV is built on the MATLAB/Simulink platform, and the designed controller is also simulated and verified with MATLAB.In the simulation, the NSV is flying in the cruise phase with the altitude of 33 km andMa=15. The initial velocity and altitude are h0=33528m and v0=4590m/s, respectively. The damping coefficient and natural frequency of the engine model are ξ=0.7 and ω=5, respectively. Considering the limitations of the actuator, βc∈0,1 and δe∈−30∘,30∘. The step signal for the velocity channel and altitude channel is given as 50m/s and 50m, respectively. The external disturbances d1t=0.02sin0.4t+0.04cos0.2t and d2t=0.04sin0.2t+0.02cos0.4t are introduced in the 30th second. The system parameters and uncertainty boundary are given in Table 2. The design parameters of DSC and HLESO are given in Tables 3 and 4, respectively.Table 2
The parameters and uncertainty boundary of NSV.
CoefficientValueUncertainty boundarym50200kg∆m≤0.03Iy8466900kg∙m2∆Iy≤0.02s369m2∆s≤0.01c28m∆c≤0.03ce0.0292∆ce≤0.02ρ1.2266×e−h/7315.2∆ρ≤0.03Table 3
The design parameters of DSC.
SubsystemVirtual controllerValueVelocityλv1Γv1=0.005, ρv1=0.5, α1=α2=3, β1=β2=1λv2Γv2=50, ρv2=1.5, α1=α2=3, β1=β2=1λv3ρv3=2.5Altitudeλh1Γh1=0.005, ρh1=0.5, α1=α2=3, β1=β2=1λh2Γh2=0.5, ρh2=1.5, α1=α2=3, β1=β2=1λh3Γh3=50, ρh3=2, α1=α2=3, β1=β2=1λh4ρh4=2.5Table 4
The design parameters of HLESO.
HLESOValueFirst orderω0=10, nv=1, nh=1Second orderω0=5, nv=2, nh=2Third orderω0=5, nv=3, nh=3To demonstrate the effectiveness of the proposed method, different controllers are used for simulation under the same conditions. The simulation results are given in Figures3–7. Figure 3 presents the tracking performance of velocity and altitude. It can be seen that the proposed dynamic surface control can accurately track the command signal; both of the transient response time of velocity and altitude are less than 10seconds. Compared with the sliding mode control and backstepping control, the proposed control method has faster response speed and smaller transient response fluctuation. In particular, when the composite disturbance is introduced, the proposed control method can restore the system stability in a shorter time and has smaller error and higher accuracy.Figure 3
Velocity and altitude response curves: (a) velocity channel and (b) altitude channel.
(a)(b)Figure 4
The virtual control signals: (a) velocity channelλvi and (b) altitude channel λhi.
(a)(b)Figure 5
Disturbance estimation errors: (a) velocity channel and (b) altitude channel.
(a)(b)Figure 6
Actuator response curves: (a) throttle setting and (b) elevator deflection.
(a)(b)Figure 7
Parametric response curve: (a) attack angle and (b) track angle.
(a)(b)Figure4 presents the velocity channel virtual control signals λvii=1,2 and altitude channel virtual control signals λhii=1,2,3. It can be seen that all virtual control signals converge to zero in 10 seconds. When the composite disturbance is introduced, the virtual control signal has small fluctuation. The fluctuation amplitude of the proposed DSC and HLESO compensation method is smaller than that of DSC and LESO. Figure 5 presents the disturbance estimation errors of the velocity channel and altitude channel. From Figure 5, it can be known that the proposed DSC and HLESO compensation method can make the estimation errors of velocity and altitude converge to the zero field, while the tracking errors, caused by the composite disturbances, in the velocity channel and the altitude channel cannot be totally eliminated by DSC and LESO and tracking errors are larger than those in the proposed DSC and HLESO compensation method. The tracking errors are given in Table 5. From Table 5, it can be known that under the same composite disturbance conditions, the tracking errors of the proposed compensation method is about 20 times smaller than those of the traditional method. Figures 6 and 7 are the control inputs of the throttle setting and elevator deflection and the responses of attack angle and flight path angle, respectively. It is shown that the responses of flight states are with smaller fluctuations under the DSC and HLESO compensation method. Therefore, the flight control performance and the disturbance rejection ability of the DSC and HLESO compensation method are better than those of the traditional DSC method.Table 5
The tracking errors of composite disturbance.
Tracking methodError rangeDSC & LESO [30], TD [31]ev≤0.007, eh≤0.0065DSC & HLESO (second order), proposedev≤0.003, eh≤0.002DSC & HLESO (third order), proposedev≤0.0002, eh≤0.0002


## 5. Conclusions

In this paper, the uncertainty and the parameter deviation are considered in the NSV controller design. In order to obtain better flight control performance and disturbance rejection ability, a novel dynamic surface control method with HLESO compensation is proposed. The inverse hyperbolic sine function is used to track the differentiator and obtain the virtual control signal so as to avoid the problem of differential expansion and reduce the complexity of the controller design. The numerical simulations are detailed to demonstrate higher accuracy of tracking and differentiation and faster convergence than other TD. The system simulation results show that the proposed control method can track the command signal accurately and has faster response speed and smaller transient response fluctuation; the flight control performance is better than that of the traditional DSC method. It can effectively compensate for the unknown dynamics and external disturbances of the system. The fluctuation amplitude of the virtual control signals and estimation errors is smaller than that of the traditional dynamic surface control. In particular, under the same composite disturbance conditions, the tracking errors of the proposed DSC and HLESO compensation method is about 20 times smaller than those of the traditional method. The proposed control method improves the disturbance rejection ability and the robustness of the system.In the future work, we will keep on improving the tracking control performance of the NSV flight controller, and other forms of control method will be also taken into consideration, such as intelligent control, adaptive control, and finite time control.



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*Source: 4153418-2022-03-28.xml*